Multidisciplinary Approach to Quantum Field Theory, Volume 1: An introduction 0750332255, 9780750332255

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction

Online at: https://doi.org/10.1088/978-0-7503-3227-9

A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie Department of Physics, Washington University, St. Louis, MO, USA

IOP Publishing, Bristol, UK

ª IOP Publishing Ltd 2022 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organizations. Permission to make use of IOP Publishing content other than as set out above may be sought at [email protected]. Michael Ogilvie has asserted his right to be identified as the author of this work in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. ISBN ISBN ISBN ISBN

978-0-7503-3227-9 978-0-7503-3225-5 978-0-7503-3228-6 978-0-7503-3226-2

(ebook) (print) (myPrint) (mobi)

DOI 10.1088/978-0-7503-3227-9 Version: 20221001 IOP ebooks British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Published by IOP Publishing, wholly owned by The Institute of Physics, London IOP Publishing, No.2 The Distillery, Glassfields, Avon Street, Bristol, BS2 0GR, UK US Office: IOP Publishing, Inc., 190 North Independence Mall West, Suite 601, Philadelphia, PA 19106, USA

To Judy, for everything

Contents Preface

xi

Acknowledgement

xiv

Author biography

xv

1

Introduction to quantum field theory

1.1 1.2 1.3 1.4 1.5

Natural units The simple harmonic oscillator in classical mechanics The harmonic oscillator in quantum mechanics Photons Paths to quantum field theory Reference

2

Quantum mechanics and path integrals

2.1

Classical mechanics and fields 2.1.1 The Lagrangian formalism 2.1.2 Functional differentiation 2.1.3 Symmetry in classical mechanics 2.1.4 The Hamiltonian formalism Quantum mechanics 2.2.1 Time evolution in the Schrödinger picture 2.2.2 The propagator for a free nonrelativistic particle* 2.2.3 The Heisenberg representation 2.2.4 Interactions The Feynman path integral for one degree of freedom 2.3.1 Defining the path integral 2.3.2 Matrix elements and time ordering 2.3.3 Generating functions 2.3.4 The simple harmonic oscillator 2.3.5 Wick’s theorem 2.3.6 Perturbation theory and Feynman diagrams Problems Bibliography

2.2

2.3

vii

1-1 1-1 1-2 1-3 1-6 1-9 1-10 2-1 2-1 2-2 2-3 2-5 2-7 2-8 2-9 2-9 2-11 2-12 2-13 2-13 2-16 2-19 2-20 2-23 2-24 2-30 2-32

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

3

Classical fields

3.1 3.2

Wave equations in classical mechanics and quantum mechanics Special relativity 3.2.1 Geometry of spacetime 3.2.2 Lorentz transformations of fields The Lagrangian formalism for fields 3.3.1 The Klein–Gordon equation 3.3.2 Maxwell’s equations 3.3.3 The Schrödinger equation Continuous symmetries in classical field theory 3.4.1 Example: translation in space and time The Hamiltonian formalism Causality Problems

3.3

3.4 3.5 3.6

4

3-1

Free quantum fields

3-1 3-2 3-4 3-5 3-6 3-7 3-9 3-10 3-11 3-13 3-15 3-15 3-19 4-1

4.1 4.2 4.3 4.4 4.5

The Feynman path integral for field theories Free scalar fields Another approach to the functional integral Interpretation of Z[0] for free fields Vacuum energy examples 4.5.1 Casimir effect 4.5.2 Energy of field interacting with a static source 4.6 Fock space 4.7 Relativistic invariance and Fock space 4.8 Free quantum fields in Fock space 4.9 The canonical commutation relations and causality 4.10 Equivalence to the functional integral formalism 4.11 Continuous symmetries in quantum field theories Problems Further reading

4-1 4-3 4-5 4-5 4-6 4-7 4-8 4-9 4-13 4-15 4-16 4-18 4-18 4-21 4-21

5

Interacting quantum fields

5-1

5.1 5.2 5.3

Perturbation theory and Feynman diagrams Feynman diagrams in position space Feynman diagrams in momentum space

5-2 5-4 5-6

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5.4 5.5

5.6 5.7

Scattering theory A toy model of nucleons and pions ¯ ¯ scattering amplitude 5.5.1 The NN and NN 5.5.2 The NN¯ scattering amplitude 5.5.3 Mandelstam variables and crossing symmetry 5.5.4 Four more processes: Nϕ → Nϕ, N¯ ϕ → N¯ ϕ, NN¯ → ϕϕ and ϕϕ → NN¯ The CPT theorem Cross-sections and decay rates 5.7.1 Decay rates 5.7.2 Cross-sections Problems Reference

6

Renormalization

6.1 6.2 6.3 6.4 6.5 6.6

Mass renormalization Coupling constant renormalization Field renormalization Renormalization: a systematic process Renormalizability Matrix elements and the LSZ reduction formula Problems Bibliography

7

Symmetries and symmetry breaking

7.1

Internal symmetries 7.1.1 Introduction to spontaneous symmetry breaking Spontaneous symmetry breaking and perturbation theory Broken continuous symmetries and Goldstone bosons 7.3.1 Examples of Goldstone bosons Renormalization of models with spontaneous symmetry breaking Problems

7.2 7.3 7.4

5-8 5-10 5-11 5-13 5-13 5-16 5-17 5-18 5-20 5-22 5-24 5-25 6-1 6-2 6-6 6-13 6-14 6-16 6-19 6-20 6-21 7-1 7-1 7-3 7-7 7-9 7-10 7-12 7-14

8

Fermions

8-1

8.1 8.2

Introduction to the Dirac equation Representations of the Lorentz group

8-1 8-3

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8.2.1 The rotation group and its representations 8.2.2 The Lorentz group 8.2.3 Representations of the Lorentz group 8.3 The Dirac equation 8.4 Solutions of the Dirac equation 8.5 The free Dirac field 8.6 Dirac bilinears 8.7 Chiral symmetry and helicity 8.8 Charge conjugation and coupling to the electromagnetic field 8.9 Functional integration for fermions 8.10 Feynman rules and scattering for a Yukawa field theory 8.10.1 Nucleon–nucleon scattering at order g 2 8.10.2 Loop diagrams with fermions 8.11 Interpreting the boson and fermion functional determinants 8.12 The linear sigma model of mesons and nucleons Problems

x

8-3 8-4 8-6 8-10 8-12 8-15 8-18 8-20 8-21 8-22 8-28 8-29 8-33 8-34 8-37 8-38

Preface There is an aphorism ‘The prerequisite for field theory is field theory’. It seems to be easily and universally understood by those who teach the subject, and it also appears to resonate with those who are starting to learn it. Roughly, the meaning is this: quantum field theory is a vast subject, with connections to most areas of modern physics and also modern mathematics. It contains many useful ideas, techniques, and methods. Some of these are sophisticated elaborations of what is taught in classes in mechanics, electromagnetism and quantum mechanics. Others, such as Feynman diagrams and the Feynman path integral, may be new to most students beginning to study field theory. When learning field theory for the first time, one must learn a great deal of technical material which may obscure the larger picture. A good grasp of what to do often seems to be a prerequisite to understanding why to do it. This book is a modern introduction to quantum field theory. It is based on a series of premises about the relation of quantum field theory to modern physics research.

Quantum field theory is ubiquitous in modern physics Traditionally, quantum field theory has been taught as the underlying basis for particle physics. It began in the post-World War II era with the successes of quantum electrodynamics (QED). These successes lay not only in the agreement of theory with experiment, but also in the successful handling of difficult issues of mathematical physics, such as the ultraviolet divergences which appear in field theory calculations. To this day, QED is the most successful fundamental physical theory we have. Of course, QED is both the progenitor and a part of the Standard Model of Particle Physics, which successfully encapsulates almost everything we currently know about elementary particles at this time. Moreover, quantum field theory finds its deepest connection to the Universe as we apprehend it in its connection to particle physics. However, quantum field theory is also the basis for much of modern theoretical physics, especially in condensed matter physics, nuclear physics and astroparticle physics. The application of the renormalization group to problems in statistical and condensed matter physics by Kadanoff, Fisher and Wilson has brought field theory to centrality within most of theoretical physics. The overlap and exchange of ideas between condensed matter physics and particle physics can scarcely be exaggerated. Nuclear physics is increasingly focused on its basis in quantum chromodynamics (QCD), one of the major components of the Standard Model. Astroparticle physics, itself a relatively new field, is based on quantum field theory combined with general relativity.

What should be taught and who learns it is changing The immediate goal of graduate education in physics is to prepare students for research in modern physics; the larger goal is to prepare them for the fruitful application of ideas from physics to whatever endeavors they turn their minds and xi

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

hands toward. The attendees in my quantum field theory courses at Washington University typically represent a mix of research interests. Although the typical attendee is a second- or third-year graduate student, their primary research interests are not necessarily in particle physics, but may lie in condensed matter physics, nuclear physics, astroparticle physics, or general relativity. Although the theoretically inclined make up a majority, there are always some sharp experimentalists as well. Classes often include one or more bright undergraduates who are looking for a challenge, and occasionally another faculty member will sit in as well. In order to serve their interests, a modern field theory course needs to be as universally applicable as possible, and introduce modern tools of quantum field theory in a way which can be readily applied in many ways. It is important that we have sufficient space in the field theory curriculum to develop these topics, even if this means deprecating other material. Three such tools stand out as particularly important: • Functional differentiation and integration; • The renormalization group; • Topology. It is the use of functional methods that marks the biggest change from traditional approaches to teaching field theory, which are based on canonical quantization and the interaction picture. While there is much value still in this approach, the majority of field theory research today uses functional techniques. It is therefore essential that these are introduced as early as possible and used throughout. The renormalization group is arguably the most important, successful and widely applicable technique to be developed in the modern era of theoretical physics. The language of renormalization group flows and fixed points allows the same kind of trans-disciplinary communication that Feynman diagrams provide. It is hard to over-emphasize the need for the development of physical intuition hand-in-hand with formalism in the study of this topic. Topology is used in different ways in different areas of theoretical physics, but there are a few simple core ideas. The topology of space and time and the topological properties of the space of fields can have crucial and surprising consequences. Topologically stable field configurations can indicate a variety of non-perturbative effects which are not directly accessible using traditional Feynman diagram methods.

How to use this book This book is intended as a gateway to modern field theory, and includes about one semesterʼs worth of material. It can be used in a traditional lecture course, a reading course, or for self-study. Although written for second- or third-year graduate students entering research, it could be used in a course for advanced undergraduates. It may be useful for self-study by researchers in any field of physics interested in modern tools of quantum field theory. Chapter 1 might or might not explicitly be covered in a lecture course, depending on the background of the participants, but

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probably should be assigned reading in any case. Chapter 2 introduces the path integral in the familiar context of quantum mechanics, and is essential to the rest of the book. The focus of the book is on teaching quantum field theory, and simple examples are drawn from several areas of physics, and it is not equivalent to a graduate text on particle physics. Many universities teach a one-semester graduate-level particle physics course, Most such courses today are centered on the Standard Model, and often teach just enough field theory in a heuristic fashion to understand basic calculations using Feynman diagrams. The audience for such a course overlaps the audience for quantum field theory, but they are not identical. There is no one book that contains all the quantum field theory a given researcher will need. Quantum field theory has proven to be a flexible tool, and a subject which is continually growing. There are many excellent quantum field theory texts. Some of them are particularly appropriate once the prerequisite of a prior field theory course is satisfied.

xiii

Acknowledgement There are many different approaches to quantum field theory. In the course of my graduate studies, I had the privilege of attending lectures on quantum field theory by Bernard Feldman, my thesis advisor Gerald Guralnik, Sidney Coleman and Steven Weinberg. I learned a great deal from their different approaches to the subject and I am grateful to all of them. I am also grateful to my colleagues at Washington University, Carl Bender, Claude Bernard, Bhupal Dev, Francesc Ferrer, Zohar Nussinov and Alexander Seidel, with whom I have had many useful discussions about field theory and how to teach it. Finally, I must thank all my students in Physics 551-552, the two-semester quantum field theory sequence at Washington University, especially those in the last few years during the COVID-19 pandemic. You have made me a better teacher, and I am proud of you all.

xiv

Author biography Michael Ogilvie Michael C Ogilvie is Professor of Physics at Washington University in St. Louis. He received his Ph.D. at Brown University, working under Gerald Guralnik. Before joining the faculty at Washington Univerity, he held postdoctoral appointments at the University of Maryland and Brookhaven National Laboratory. His research interests include lattice gauge theory, the theory of phase transitions, and the phase structure of QCD at nonzero temperature and density.

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IOP Publishing

A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 1 Introduction to quantum field theory

In this chapter, we discuss some basic concepts of quantum field theory. We introduce the natural units of nuclear and particle physics, which will be used throughout the book, and review the classical and quantum physics of the simple harmonic oscillator. We then sketch the quantization of the electromagnetic field as an example of early work on quantum field theory. This leads us to the three paths we will use in developing quantum field theory as a preeminent tool of modern theoretical physics.

1.1 Natural units It will be extremely convenient to use the natural units of nuclear and particle physics, in which both c and ℏ are taken to be one. For example, the speed of light in light-years per year is exactly one by definition. A common way to think about natural units is to think of spatial coordinates x ⃗ and time t as having the same units, [L ] = [T ]. Both can be measured in either units of length or units of time. From this point of view, our familiar MKS units are legacy units, requiring a conversion factor because 3 × 108 m ≃ 1 s. Similarly, setting ℏ = 1 is interpreted as requiring [E ][T ] = [p ][L ] = 1. Combined with c = 1, this implies a series of relations

[E ] = [p ] = [m ] = [T ]−1 = [L ]−1. Note also that frequency and energy have the same units, as do momentum and wavenumber. Example: suppose a particle lifetime τ is related to its mass m by a formula τ = a /m, The constant a is dimensionless in natural units. In order to find the relation between τ and m in our standard units, we have to find the combination of ℏ and c which converts between time and mass. We know that mc 2 has dimensions of energy, and ℏ has units of energy multiplied by time, so we arrive at τ = ℏa /mc 2 . doi:10.1088/978-0-7503-3227-9ch1

1-1

ª IOP Publishing Ltd 2022

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

1.2 The simple harmonic oscillator in classical mechanics The simple harmonic oscillator is ubiquitous in physics, and its solution provides us with most of the tools we need to begin the study of quantum field theory. The classical equation of motion for the harmonic oscillator is

m

d 2x + kx = 0 dt 2

where m is the mass and k is the spring constant of the oscillator. We define the angular frequency ω as

ω=

k m

so the equation of motion can also be written as

d 2x + ω 2x = 0, dt 2 with general solution

x(t ) = A sin(ωt ) + B cos(ωt ). The equation of motion does not depend on m and k separately, but only on ω. The Lagrangian L which leads to the correct classical equation of motion is

L(x , x )̇ =

1 ⎛ dx ⎞2 1 2 m⎜ ⎟ − kx . 2 ⎝ dt ⎠ 2

The Euler–Lagrange equation

d ∂L ∂L =0 − dt ∂x ̇ ∂x gives the equation of motion. We can obtain the same equation of motion from the Lagrangian

L=

1 ⎛ dx ⎞2 1 2 2 ⎜ ⎟ − ωx . 2 ⎝ dt ⎠ 2

This form can be obtained from the previous one by a rescaling of

x → m−1/2x. This is the natural normalization of the kinetic term to use in quantum field theory. The Hamiltonian H of the harmonic oscillator is obtained by defining the momentum p as ∂L p= ∂x ̇ and then

H (p , q ) = px ̇ − L(x , x ). ̇

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

For the harmonic oscillator Lagrangian p = x ̇ with our new convention, and the Hamiltonian is H (p, x ) = 12 p2 + 12 ω 2x 2. Exercise: Check that the inverse rescalings x → m1/2x and p → m−1/2p lead to the familiar expression of classical mechanics.

1.3 The harmonic oscillator in quantum mechanics The quantum mechanical commutation relation

[p , x ] = −i ℏ is the key to the complete solution of the harmonic oscillator. In natural units, we set ℏ = 1 and write this commutation relation as

[p , x ] = −i . Defining raising and lowering operators a + and a via

1 (a + a + ) 2ω ω + (a − a ) p=i 2

x=

or equivalently

a= a+ =

1 (ωx + ip) 2ω 1 (ωx − ip) 2ω

we find the simple commutation relation

[a , a + ] = 1 and the equally simple form for the Hamiltonian

H = ω(a +a + 1/2). From this form, we can see that the energy spectrum is bounded from below by ω/2, because for any normalized state ψ , ψ a+ a ψ ⩾ 0 so

ψ H ψ = ψ ω(a +a + 1/2) ψ ⩾ ω /2. The operator a lowers the energy of any energy eigenstate E commutator

[a , H ] = ωa implies that

Ha E = (E − ω)a E .

1-3

because the

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Repeated application of a gives rise to a chain of states of descending energy E , E − ω, E − 2ω ,... This chain of descending energy eigenstates must stop because our lower bound on H tells us that the energies must be positive. The chain therefore must stop at some point where

a 0 = 0. If this were not the case, we could use a to find a state with negative energy, and that cannot occur for the harmonic oscillator. Thus there must be at least one state satisfying a 0 = 0 and in fact there is only one such state. One way to prove this is to use the coordinate or momentum basis to show that a 0 = 0 has a unique solution, which is a Gaussian. We call this state the ground state and write is as 0 . Exercise: prove that the expectation value of H can be written as

ψHψ =

⎡

∫ dx⎢⎣ 12 ∇ψ

2

+

⎤ 1 2 2 ω x ψ 2⎥ ⎦ 2

which is always positive unless ψ = 0. Show that this implies that the expectation value is always positive for the harmonic oscillator, and therefore all eigenvalues of H are positive. Now write ψ (x ) = R(x )e iϕ(x ) to prove that the ground state can be chosen to be everywhere non-negative and is therefore unique. Given the ground state, we can construct all other energy eigenstates m from it using

[H , a + ] = ωa + so that

Ha + E = (E + ω)a + E . Starting from the ground state, this leads to a set of eigenenergies of the form

En = (n + 1/2)ω for n = 0, 1, 2 ,... We denote corresponding normalized energy eigenstate for each n as n . They are given by

n =

(a + )n 0 . n!

Each normalized energy eigenstate satisfies

H n = En n = (n + 1/2)ω n . Note that

n =

a+ n−1 n

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or equivalently

a+ n =

n+1 n+1 .

For this reason, a + is often called a raising operator. We can find the action of a by induction. We know that a 0 = 0. If we assume that for some value of n

an =

n n−1

then it follows that for that value of n that

a +a n = n n . We can then calculate that

a +a + 1 aa + n n = n+1 n+1 a +a + 1 n = n+1

an+1 =

= n+1 n and the proof by induction is complete. The operator a is called a lowering operator because it lowers by one the quantum number n of an energy eigenstate. The operator a +a is the number operator N which acts on energy eigenstates as

N n = a +a n =nn . It follows that

⎛ 1⎞ H n = ω⎜a +a + ⎟ n ⎝ 2⎠ ⎛ 1⎞ = ⎜n + ⎟ω n ⎝ 2⎠

(

from which we again recover the familiar result En = n +

1 2

)ω.

Exercise: Use induction to prove that all of these energy eigenstates are normalized to 1 if the ground states is normalized to 1. We can make these abstract results more concrete by working in the coordinate basis, where p → −id /dx . The ground state condition a 0 = 0 becomes

(ωx + )ψ (x) = 0. The properly normalized solution is a Gaussian: d dx

0

ψ0(x ) =

⎛ ω ⎞1/4 ⎛ ω ⎞ ⎜ ⎟ exp⎜ − x 2⎟ . ⎝ 2 ⎠ ⎝π ⎠

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All of the other eigenfunctions can be obtained using

ψn(x ) = x =

(a + ) n 0 ( n!)

⎛ ω ⎞ 1 ⎛⎜ ω ⎞⎟1/4 Hn( ω x ) exp ⎜ − x 2⎟ n ⎝ 2 ⎠ ⎝ ⎠ 2 n! π

where Hn(z ) is the nth Hermite polynomial.

1.4 Photons Historically, quantum field theory arose as an attempt to understand the quantum interactions of electrons and photons. Electrons are the quintessential quantum particles of non-relativistic quantum mechanics, and electromagnetic interactions are the most important and most common interaction experimentally. However, the work of Planck which began quantum physics tells us that electromagnetic fields are made up of large numbers of photons, the quanta of light. The interaction of light with electrons and atoms thus requires a many-body theory of photons, in which individual photons can be absorbed or emitted by atoms. Here we give an overview of how this was first done; a more modern treatment, including more detail, will be given later. The first step in this direction is a theory of quantized electromagnetic radiation without interactions with matter, i.e., a theory of free photons. The classical Hamiltonian for electromagnetic fields is given in terms of the electric fields E ⃗ and B ⃗ , which are vector functions of space x ⃗ and time t. In natural units, the Lagrangian is

L=

∫ d 3x 12 (E ⃗ 2 − B ⃗2)

H=

∫ d 3x 12 (E ⃗ 2 + B ⃗2).

and the Hamiltonian H is

However, the underlying dynamics is best understood in terms of the scalar potential ϕ and the vector potential A⃗ , which are related to E ⃗ and B ⃗ via

E ⃗ = − ∇ϕ −

∂A ⃗ ∂t

B ⃗ = ∇ × A⃗ . The scalar and vector potentials may be chosen in many ways to give the same E ⃗ and B ⃗ fields, because the equations relating them are invariant under the gauge transformations

∂Λ ∂t ⃗ ⃗ A → A − ∇Λ ϕ→ϕ +

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where Λ(t , x ⃗ ) is an arbitrary function of space and time. It is advantageous here to work in Coulomb gauge where the scalar potential ϕ is set to zero, and the vector potential obeys the further condition ∇ · A⃗ = 0. The classical equation of the motion for the vector potential is the wave equation:

∂ 2A ⃗ − ∇2 A ⃗ = 0. ∂t 2 The solutions of the wave equation are plane waves of the form

Ak⃗ ⃗ = ϵˆ exp (ik ⃗ · x ⃗ − iωkt ) where ωk = k = k⃗ is the angular frequency. Although these solutions are complex, we can use them to form a real solution using superposition. The polarization vector ϵ is conventionally normalized to one: ϵ⃗ 2 = 1. In Coulomb gauge, the condition ∇ · A⃗ = 0 implies k ⃗ · ϵˆ = 0. We conventionally introduce a basis for the polarization vectors ϵˆλ(k ⃗ ) which are unit vectors indexed by λ = 1, 2 which point in two directions transverse to k ⃗ such that

k ⃗ · eˆλ(k ⃗ ) = 0 and

ϵˆλ(k ⃗ ) · ϵˆλ ′(k ⃗ ) = δλλ ′. Thus we have two polarized modes for each component of the electromagnetic field; both are transverse to the direction of propagation, as we know from optics. It is convenient to consider the electromagnetic field in a large box, with Periodic boundary conditions are simplest, with 0 ⩽ x, y, z ⩽ L . ⃗ ⃗ A (0, y, z, t ) = A (L, y , z, t ) and similarly for the y and z directions. The allowed set of momenta k ⃗ are determined by the boundary conditions, so

k⃗ =

2π n⃗ L

where n ⃗ ∈ Z 3, which is to say that all three components of n ⃗ are integers. The set of functions

(

Ak⃗ λ⃗ (x ⃗ , t ) = ϵˆλ(k ⃗ ) exp ik ⃗ · x ⃗ − iωkt

)

form a complete orthonormal basis for the vector potential fields:

1 L3

∫L

3

d 3xAk⃗ λ⃗ (x ⃗ , t ) · Ak⃗ ⃗′ λ ′(x ⃗′ , t ) =

1 L3

∫L

3

⃗ ⃗ d 3xe i (k ′−k )·x e⃗ ˆλ(k ⃗ ) · ϵˆλ ′(k ′⃗ ) = δ k ⃗′ kδ⃗ λλ ′.

This holds for any value of t. This is sometimes referred to as box normalization. In quantum mechanics, space and time are treated differently in the sense that the position x ⃗ of a particle is an operator, but time t is a parameter. 1-7

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If we think of the vector field A⃗ (x ⃗, t ) as a wave function for a photon, then the wave equation

∂ 2A ⃗ − ∇2 A ⃗ = 0 ∂t 2 is the analog of the standard Schrödinger equation for a non-relativistic free particle. 2 It encodes the relativistic energy-momentum relation Eop = pop ⃗ 2 for a massless relativistic particle. Thus it seems like we have an adequate single-particle quantum mechanics for free massless particles like the photon. Two of the main virtues of box normalization are that eigenvalues of energy and momentum are discrete and that the corresponding eigenvectors are easily normalized. We can produce multi-particle states using products of such wave functions. A wave function for an n-particle state can be written as a product of single-particle wave functions, although the notation quickly becomes cumbersome. On the other hand, we can go back to the classical wave equation, and consider solutions of the form

(

)

Ak⃗ ⃗ (x ⃗ , t ) = ϵˆλ(k ⃗ )f k ⃗ (t ) exp ik ⃗ · x ⃗ . In a large box, the momentum space modes are discrete, and each mode satisfies the classical equation

d 2f k ⃗ (t ) + k⃗ dt 2

2

f k ⃗ (t ) = 0

which is nothing but the equation of a classical harmonic oscillator, which we know how to quantize. The eigenstates of a given mode will have energies

⎛ 1⎞ E k ⃗,λ = ⎜n k ⃗,λ + ⎟ωk ⎝ 2⎠ where ωk = ∣k ∣⃗ and each n k λ⃗ = 0, 1, 2 ,... The n k λ⃗ ’s are known as occupation numbers because they describe the number of photons in a given state. The energy of the system E for a given set of occupation numbers is given by summing over all the modes:

E=

⎛

∑E k ⃗,λ = ∑⎜⎝n k ⃗,λ + k ,⃗ λ

k ,⃗ λ

1⎞ ⎟ωk . 2⎠

Oddly, we see that the energy E0 of the ground state

E0 =

1

∑ 2 ωk k,⃗ λ

is an infinite sum of harmonic oscillator zero point energies. This is a first hint that quantum field theories will require new concepts and techniques beyond what quantum mechanics provides us.

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Exercise: Suppose we have an excited state specified by two non-zero occupation numbers n k1⃗ and n k2⃗ . Is the energy E of this state finite or infinite? What is the value of E − E0 ? We now have two different approaches to the quantum theory of light. The first is based on the idea that the photon field is akin to the wave function of single-particle quantum mechanics, and builds many-particle states from products of such wave functions. The second is based on treating the modes of the electromagnetic field as independent harmonic oscillators. Many-particle states are described by an infinite set of occupation numbers. Which of these approaches is correct? As with the simple harmonic oscillator, there are several equivalent approaches. In this case, these different approaches represent equivalent steps towards the quantum theory of light, which we call quantum electrodynamics or QED. It is a first-principles approach to the interaction of light with matter, and it is the greatest success story of science. The best example of the accuracy of QED is the discrepancy between the theoretical prediction for the anomalous magnetic moment of the electron and its experimental value: they differ by less than one part in 1012. The creation of the theoretical tools necessary to handle QED was the great accomplishment of post-World War II theoretical physics.

1.5 Paths to quantum field theory We will explore three different but equivalent approaches to quantum field theory. Sidney Coleman, who was an enormously influential teacher of quantum field theory, described the first two as ‘the method of the missing box’ in his well-known lectures on quantum field theory [1]. If we start from the classical mechanics of a few degrees of freedom with coordinates xj and momenta pj, where j = 1 ... N , we can pass from classical mechanics to quantum mechanics by imposing the canonical commutation relations

[pj , xk ] = −iδjk . If we wish to represent an infinite number of degrees of freedom, we must take the limit

N → ∞. This, Coleman says, is one way to define a quantum field theory. On the other hand, we can pass from the classical mechanics to classical field theory by taking the limit of infinite degrees of freedom. For example, a vibrating classical string can be conceptualized as a collection of N point masses along a line of length L. If we take the appropriate limit where N → ∞, where the masses become smaller as does the distance between them, we end up with the continuum classical field theory of a one-dimensional elastic solid. We can then ask how to quantize this classical field theory. As Coleman points out, this is also a way to define a quantum field theory, but the end result is the same one as before. It does not matter whether you take the limit N → ∞ of a classical theory and then

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Figure 1.1. Coleman’s four boxes and their connections.

quantize, or instead quantize a theory with a finite number of degrees of freedom and then take the limit. There is, however, a third major approach to quantum field theory: the Feynman path integral, often known simply as the path integral. In this approach, a quantum mechanical particle traveling between two points is conceptualized as taking all possible classical paths, which are summed over with an appropriate weighting. As shown in figure 1.1, it represents an alternative to canonical quantization for both particles and fields. The path integral approach has proven to be a powerful and flexible tool; many techniques used for quantum mechanical systems have direct translations into quantum field theory. It is the dominant approach used in modern particle physics, and is in widespread use in other branches of physics. We will develop all three approaches, but emphasize the path integral. The first two are almost essential in making a firm connection with experiment, while the path integral is the most powerful approach. Thus we follow the advice of Yogi Berra, who said ‘When you come to a fork in the road, take it’. This is always good advice in the study of quantum physics.

Reference [1] Coleman S 2018 Lectures of Sidney Coleman on Quantum Field Theory (Hackensack, NJ: World Scientific)

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 2 Quantum mechanics and path integrals

In this chapter, we introduce the path integral and important associated concepts in the context of quantum physics with a finite number of degrees of freedom. We review classical mechanics in the Lagrangian and Hamiltonian formalisms. We introduce functional differentiation as an alternate to the calculus of variations and discuss the role of symmetry. We briefly discuss the Schrödinger and Heisenberg pictures before turning to the Feynman path integral formulation of quantum mechanics. We develop generating functions, time ordering, Wick’s theorem and Feynman diagrams within the path integral framework.

2.1 Classical mechanics and fields For a classical system with one degree of freedom q(t ), the time evolution is governed by Newton’s force law, which we write as

m

d 2q = F (t ) dt 2

where m is the mass and F is the force, and often refer to as equations of motion. When the force is obtained from the derivative of a potential V (q ) with no explicit time dependence, the force law takes the form

m

d 2q dV . =− 2 dt dq

We say that the system is conservative because multiplication of the equation of motion by q ̇ implies

⎤ d ⎡ 1 ⎛ dq ⎞2 ⎢ m⎜ ⎟ + V (q )⎥ = 0 dt ⎣ 2 ⎝ dt ⎠ ⎦

doi:10.1088/978-0-7503-3227-9ch2

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which is the statement of the conservation of the energy E given by

1 ⎛ dq ⎞2 m⎜ ⎟ + V (q ). 2 ⎝ dt ⎠

E=

There are two equivalent formalisms commonly used for describing systems like this: the Hamiltonian formalism and the Lagrangian formalism. This equivalence, and the utility of both approaches, are present in classical field theory, quantum mechanics and quantum field theory. 2.1.1 The Lagrangian formalism In the Lagrangian formalism, we focus on the Lagrangian L and the action S. The Lagrangian L is a function of a set of N generalized coordinates qj (t ) as well as the time derivatives qj̇ ≡ dqj /dt . It is conventional to consider qj and qj̇ as independent quantities throughout most of the Lagrangian formalism, and write L(q, q )̇ where we use q as a notational proxy for the N-component object (q1, … , qN ). It should be stressed that this collection of coordinates need not be related by any symmetry, and need not even have the same units. The action S is defined as the time integral of L over some fixed period of time from ti to t f . The action S ⎡⎣qj , qj̇ ⎤⎦ is defined as

S [q ] =

∫t

tf

dt L(q(t ), q(̇ t ))

i

where we take q(t ) to be an arbitrary function of time t, usually referred to as a classical path. Thus the value of S depends not on the value of q at some time t, but on the functional form of q over the entire range (ti , t f ). An object such as S is referred to as a functional, and we will use square brackets, as in S [q ], to distinguish a functional from a function such as L(q(t ), q(̇ t )). The principle of least action states that, given q(ti ) and q(t f ), the dynamics determined by L will extremize the value of S over all possible functions q satisfying the initial and final conditions. The condition for an extremum is usually written as

δS [q ] = δ

∫t

tf

dt L(q(t ), q(̇ t )) = 0.

i

In standard examples from classical mechanics, the action is actually minimized, hence the term least action, but there are examples where the action is maximized. A given path q(t ) will be a local extremum of S if the action is stationary to first order in any small variation q(t ) → q(t ) + δq(t ) of the path. Note that the initial and final conditions require that any such variation satisfy δq(ti ) = δq(t f ) = 0. We have

δS =

∫t

tf

i

N ⎡ ∂L ⎤⎥ ∂L dt∑⎢ δqj + δq ̇ ∂qj̇ j ⎥⎦ ⎢ ∂q j = 1⎣ j

Using the Einstein summation convention, where pairs of repeated indices are summed over, we may write more compactly

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δS =

∫t

tf

i

⎡ ∂L ∂L ⎤⎥ dt⎢ δqj + δq ̇ ∂qj̇ j ⎥⎦ ⎢⎣ ∂qj

with an implied summation over j. We can use integration by parts to write δS as

δS =

∫t

tf

i

⎡ ∂L d ∂L ⎤⎥ dt⎢ − δq dt ∂qj̇ ⎥⎦ j ⎢⎣ ∂qj

because the boundary terms vanish. Because the variation δq(t ) is arbitrary for ti < t < t f , we must have d ∂L ∂L =0 − dt ∂qj̇ ∂qj everywhere along the extremizing path. This equation, the Euler–Lagrange equation, is a fundamental result in Lagrangian mechanics. The mathematical techniques used here belong to the calculus of variations. Exercise: For a single degree of freedom, take

L=

1 2 mq ̇ − V (q ) 2

and recover the standard equation of motion from the Euler–Lagrange equations. 2.1.2 Functional differentiation Functional differentiation is closely related to the calculus of variations, but builds on it to give us a more powerful framework for quantum field theory. It is particularly powerful when combined with the Feynman path integral, which is also known as the functional integral. Let’s suppose we have a set of N discrete variables qj which have a natural linear order. For example, we can consider the displacements qj of identical coupled masses along a line. These are independent variables, so

∂qj ∂qk

= δjk .

We can imagine taking a limit in which N → ∞ but the total length of the system is constant. Furthermore, we can arrange this limit so that that the mass of each particle goes to zero, while the mass per unit length is constant. This is the limit in which a line of coupled masses becomes a vibrating string. In this limit, the discrete variable qj becomes a continuous function of x, the distance along the line. The generalization of the rule for independent discrete variables is

δq(x ) = δ(x − x′). δq(x′) This rule can actually take you very far when combined with the chain rule. For example, consider the integral 2-3

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I [q ] =

∫ dyq n(y ).

I [q ] as a functional of q(x ), because the value of I depends on the complete functional form of q(x ) and not its value at any one point. Then

δ δ I [q ] = δq(x ) δq(x )

∫ dy q n(y ) = ∫ dy nq n−1(y )δ(x − y ) = nq n−1(x).

A more formal definition of the functional derivative is as follows: Consider a variation of the function q(x ) given by

q(x ) → q(x ) + ϵgn(x ) where the set of smooth functions gn(x ) converge to a Dirac delta function: lim g (x ) = δ (x − x′). We define the functional derivative by n→∞ n

δF [q ] F [q + δq ] − F [q ] = lim lim . δq(x′) n → ∞ ϵ → 0 ϵ The ϵ makes the variation an infinitesimal one, while the sequence of functions localizes the variation to a region near x′. As n → ∞, gn(x ) goes to infinity at x′ so ϵgn(x ) must be treated carefully. The order of limits is important, because we want to remove terms of order ϵ before taking the n → ∞ limit because squares of delta functions are not well-defined. Exercise: Apply this formal definition of the functional derivative to the calculation of δI /δq(x ) and check that this gives the same result as the chain rule. What happens if the order of limits is reversed? We can rather generally replace calculations involving the calculus of variations with the use of functional derivatives, often combined with an integration by parts. For example, consider the action S for a single degree of freedom q(t ). The action is defined in classical mechanics as the integral over time ti ⩽ t ⩽ t f of the Lagrangian:

S [q ] =

∫ dtL(q(t ))

where we show S as a functional of q but L as a function of q(t ). We replace the principle of least action δS = 0 with δS [q ] = 0. δq(t′) If we take

L (q ) =

1 ⎛ dq ⎞2 m⎜ ⎟ − V (q ) 2 ⎝ dt ⎠

with ti < t′ < t f then

δS [q ] = δq(t′)

⎡

⎤

∫ dt ⎢⎣m dqdt(t ) dtd δ(t − t′) − ∂V∂q(q(t()t )) δ(t − t′)⎥⎦. 2-4

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In order to interpret the derivative of a delta function, we can integrate by parts to obtain

d 2q ∂V (q ) δS [q ] = 0. = −m 2 − dt ∂q δq(t ) Part of the power of the functional derivatives arises from the utility of higher derivatives, which are calculated in the same manner as first derivatives. For example, in the simple case above we can calculate

⎡ d2 ∂ 2V (q ) ⎤ δS [q ] = ⎢−m 2 − ⎥ δ(t − t′) = 0. dt ∂q 2 ⎦q(t ) δq(t )δq(t′) ⎣ We can also construct functional Taylor series, which take the form ∞

F [q ] =

1 n! n=0

∑

⎡

n

⎤

δF ⎥ ∫ dt1 ... dtn⎢⎣ δq(t1)... δ q( t n ) ⎦

q(t1)... q(tn)

q(t )=0

which is a generalization of a multivariable Taylor series. 2.1.3 Symmetry in classical mechanics If L has no explicit t dependence, depending on t only through q(t ) and q(̇ t ) then there is a conserved quantity we call the energy. Of course it is easy to prove this in the familiar onedimensional case: multiplying the equations of motion by dx/dt, we observe that

⎤ dV ⎤ d ⎡ 1 ⎛ dx ⎞2 dx ⎡ d 2x ⎥ = ⎢ m⎜ ⎟ + V (x )⎥ = 0. ⎢m 2 + dx ⎦ dt ⎣ 2 ⎝ dt ⎠ dt ⎣ dt ⎦ This works in the more general case: if L has no explicit t dependence, then

dL ∂L ∂L ∂L dqj̇ ∂L dqj q̈. qj̇ + = + = dt ∂qj̇ j ∂qj ∂qj̇ dt ∂qj dt We define the energy E as

E ≡ qj̇

∂L −L ∂qj̇

and find

dL dL d ⎛ ∂L ⎞ dE ∂L ∂L ∂L qj̇ − qj̈ + qj̈ + ⎜⎜ ⎟⎟qj̇ − =0 = = dt dt dt ⎝ ∂qj̇ ⎠ dt ∂qj ∂qj̇ ∂qj̇ where we have used the equations of motion in the last step. This is general: continuous symmetries lead to conservation laws, and timetranslation invariance leads to energy conservation as a particular case. Invariance under spatial translations leads to conservation of momentum and invariance under rotations leads to conservation of angular momentum. In the cases of classical and 2-5

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quantum mechanics, we consider a general transformation q → q + δq and ask how L changes. One possibility is that L → L , and this is an easy indicator of a symmetry. However, the more general condition for a symmetry is that δL , the change in L induced by δq , can be written as

δL =

d δF dt

for some F without the use of the equations of motion. This restriction is necessary because any change δq will not change S at leading order if q is a solution of the equations of motion. If δL can be written as δF ̇ , then the action changes by

S → S + F (tf ) − F (ti ). The change in δL can also be written in terms of the changes in qj and qj̇ , so we have

d ∂L ∂L δF = δq + δq ̇ . dt ∂qj j ∂qj̇ j The right-hand side can be written using the equations of motion as

d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ d ∂L δF = ⎜⎜ ⎟⎟δqj + δqj̇ = ⎜⎜ δqj ⎟⎟ . dt ⎝ ∂qj̇ ⎠ dt ⎝ ∂qj̇ ⎠ dt ∂qj̇ This in turn implies that there is a conserved charge Q associated with the symmetry, which can be defined as ∂L Q≡ δq − δF. ∂qj̇ j For ease of calculation, it is a common practice to define δq with an explicit infinitesimal factor such as ϵ and drop that factor from the definition of Q. Example: Under time translation t → t + ϵ , the Lagrangian changes as

L→L+ϵ

dL . dt

This is completely general. The corresponding change in qj is qj → qj + ϵqj̇ . Suppose the Lagrangian is invariant under time translation, which is to say that it has no explicit time dependence, depending on t only through q(t ) and q(t) ̇ . Then the conserved charge after dropping the factor of ϵ is ∂L q̇ − L E= ∂qj̇ j which is the energy. For example, in the common case where

L=

1

∑ 2 mj q j̇2 − V (q), j

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we find

E=

1

∑ 2 mj q j̇2 + V (q) j

Example: Suppose we have a system with N vector coordinates rj⃗ where is invariant under spatial translation rj⃗ → rj⃗ + ϵa ⃗ in the sense that L does not change. This occurs, for example, when the potential depends only on differences of the r ⃗ ’s like r1⃗ − r2⃗ . In this case, where δL = 0, F is identically zero. The conserved charge is simply

Q=

∑ j

∂L · ϵa.⃗ ∂rj⃗ ̇

Not only can we remove ϵ, we also see that a ⃗ is an arbitrary spatial vector, so there are three conserved charges

P⃗ =

∑ j

∂L ∂rj⃗ ̇

which are the three components of the total momentum vector P ⃗ . 2.1.4 The Hamiltonian formalism Within the Lagrangian formalism, we can define a canonical momentum pj conjugate to the generalized coordinate qj by

pj =

∂L . ∂qj̇

A special case arises if the Lagrangian L does not depend on qj̇ so the conjugate momentum pj vanishes identically. This indicates that qj is a redundant variable, in the sense that the equation

∂L / ∂qj = 0 is a constraint on the dynamical evolution of the system; this equation is called an equation of constraint, and the variable a constraint variable. Such constraints must be handled with a little care, so for simplicity we will assume in this section that there are no constraint variables. The charge Q associated with a continuous symmetry transformation δqj can be written as

Q = pj δqj − δF. We assume that the equation for pj can be inverted to give qj̇ as a function of pj , qj , and t and define the Hamiltonian H (q, p, t ) as

H (q , p , t ) ≡

∂L q ̇ − L(q , q ,̇ t ) = pj qj̇ − L(q , q ,̇ t ). ∂qj̇ j

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Of course, when L has no explicit time dependence, H is the conserved energy of the system. Mathematically, H is a Legendre transform of L. Legendre transformations are a mathematical technology familiar in statistical mechanics, where they are similarly used to exchange independent and dependent conjugate variables in moving from one statistical ensemble to another. The action S can be rewritten as a function of q and p:

S [q , p ] =

∫ dt L = ∫ dt⎡⎣pj qj̇ − H (q, p, t )⎤⎦.

From δS = 0, we obtain Hamilton’s equations

qj̇ =

∂H ∂pj

and

pj = −

∂H ∂qj

which are the equations governing time evolution in the Hamiltonian formalism. In classical mechanics, the time evolution of any observable O is given by the chain rule:

dO ∂O ∂O ∂H ∂O ∂H ∂O ∂O dpj ∂O dqj ∂O + {O , H } = − + = + + = dt ∂t ∂pj ∂qj ∂qj ∂pj ∂t ∂pj dt ∂qj dt ∂t where we define the Poisson bracket of O with H as

{O , H } =

∂O ∂H ∂O ∂H − . ∂pj ∂qj ∂qj ∂pj

It is easy to see that taking O to be a canonical variable qk or pk reproduces the classical equations of motion. A simple case is obtained taking O = H. If H has no explicit time dependence, ∂H /∂t = 0, then {H , H } = 0 implies dH /dt = 0, and H is a constant of the motion. Poisson brackets are particularly useful in systems with many conservation laws, especially in integrable systems, where the number of independent conserved quantities equals the number of independent generalized coordinates.

2.2 Quantum mechanics Quantum mechanics inherits both ideas and structure from classical mechanics, especially from the Hamiltonian formalism. Symmetries, in particular, are generally taken over whole from classical mechanics. Occasional care must be taken, because in quantum mechanics, px is not the same as xp. These operator ordering ambiguities are often resolved by elementary arguments, but not always. The identification of observables with Hermitian operators in a Hilbert space is the origin of much of the extra mathematical technology in quantum mechanics.

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2.2.1 Time evolution in the Schrödinger picture In introductory quantum mechanics, we usually consider states as evolving according to the Schrödinger equation

i

∂ ∣ψ (t ) = H ∣ψ (t ) ∂t

using the Schrödinger picture. Observables like momentum and position are represented by operators on wave functions so that the expectation value of the position of a particle in a state ψ at time t is given by

x( t ) = ψ ( t ) ∣x∣ψ ( t ) . We can write a formal solution for ∣ψ (t ) in terms of the wave function at some other time t′ as

∣ψ (t ) = US(t , t′)∣ψ (t′) where US (t , t′) is a unitary operator satisfying

i

∂ US(t , t′) = HUS(t , t′) ∂t

and

US(t , t ) = I . We usually think of t as later than t′, but that is not required. If the Hamiltonian has no explicit time dependence this equation has the formal solution

US(t , t ) = e−i (t−t′)H . The operator US (t , t′) is often referred to as a propagator because it propagates the wave function forward in time. We say that H is the generator of time translations. In the coordinate basis, US (t , t′) is represented by an integral kernel given by

= K (x , x′ ; t , t′) = x∣US(t , t′)∣x′ . The form of this integral kernel will be a crucial ingredient in our construction of the path integral. 2.2.2 The propagator for a free nonrelativistic particle For the free nonrelativistic particle, H = p2 /2m. We choose to normalize our x-space basis elements as

x∣x′ = δ(x − x′) which in turn implies the normalization of the completeness relation

∫ dx∣x

x∣ = 1.

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Exercise: check the completeness relation by calculating

∫ dx

x1∣x x∣x2 .

The coordinate basis representation of momentum eigenstates ∣p must be proportional to exp (ipx ). If we choose the normalization x∣p = exp (ipx ) then it follows that

p∣p′ = 2πδ(p − p′) dp ∣p p∣ = 1. 2π

∫

Exercise: show that if ψ (x ) = x∣ψ , then we have

p∣ ψ =

∫

dp −ipx e ψ (x ) 2π

which is the Fourier transform ψ˜ (p ) of ψ (x ). We now can write the kernel of free particle propagator K 0(x , x′; t , t′) = x∣US (t , t′)∣x′ as

K 0(x′ , t′ ; x , t ) = x′∣e−i (t′−t )H ∣x 2 dp = 〈x′∣e−i (t′−t )p /2m∣p 〉〈p∣x〉 2π dp ip(x′−x )−i (t′−t )p 2 /2m = e 2π

∫ ∫

This is a Gaussian integral, having the form

I (a , b ) =

⎛

⎞

∫ dx exp⎝− a2 x 2 + bx⎠ ⎜

⎟

where we must have Re(a ) > 0 to insure convergence. Such integrals are ubiquitous in modern field theory. With the replacement

x → x + a −1b, we see that

I (a , b) = exp( +b 2 /2a )I (a , 0). Furthermore,

I (a , 0)2 =

∫

dxdy exp ( −ax 2 /2 − ay 2 /2)

= r exp ( −ar 2 /2) = 2π / a so we find

I (a , b ) =

2π exp( +b 2 /2a ). a 2-10

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Returning to the problem at hand, we add a small imaginary part to Δt ≡ t′ − t so Δt → Δt − iϵ in order to make our integral over p converge. We will take the limit ϵ → 0 limit after any ambiguities in our final expression are resolved. We now have that

K 0(x , x′ ; t , t′) = I (i (Δt − iϵ )/ m , i (Δx ))/2π , where Δt = t − t′ and Δx = x − x′. Now

K 0(x , x′ ; t , t′) =

⎡ m(Δx )2 ⎤ m exp ⎢i ⎥. 2πi (Δt − iϵ ) ⎣ 2(Δt − iϵ ) ⎦

We will use this result soon in the derivation of the path integral. We have been using natural units, but it is easy and instructive to temporarily restore ℏ in this expression. We see from our original expression that we need only make the replacement Δt → Δt/ℏ so we have

K 0(x , x′ ; t , t′) =

⎡ i ⎛ m(Δx )2 ⎞⎤ mℏ exp ⎢ ⎜ ⎟⎥ . 2πi (Δt − iϵ ) ⎣ ℏ ⎝ 2(Δt − iϵ ) ⎠⎦

As pointed out by Dirac, the exponential factor is exactly exp iS0 /ℏ, where S0 is the action for straight line motion from x to x′ in time t − t′, an observation which played a key role in the discovery of the path integral. 2.2.3 The Heisenberg representation In the Schrödinger picture, states evolve in time and operators are time-independent. In the Heisenberg picture, states are time-independent, conventionally placed at t = 0, but all operators now evolve in time. The difference between the two pictures lies in the grouping of the expression for expectation values. For example, using the Schrödinger picture expression for the time evolution of a state, we can write

x(t ) = ψ (0)∣e+itH xe−itH ∣ψ (0) where for simplicity we have assumed that H is time-independent. If we define Heisenberg picture states and operators by

∣ψ

≡ ∣ψ (0)

H

and

xH (t ) ≡ e+itH xe−itH = e+itH xH (0)e−itH then we calculate expectation values in the Heisenberg picture by

x( t ) =

H

ψ ∣xH (t )∣ψ

H.

This is obviously the same formula, only written differently. If OS is timeindependent, the Heisenberg operator OH (t ) evolves as

OH (t ) ≡ e+itH OSe−itH = e+itH OH (0)e−itH

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and satisfies the operator equation d OH (t ) = i [H , OH (t )] dt reminiscent of Poisson brackets in classical mechanics. There is no physical difference between the two pictures, but the Heisenberg picture is generally more useful for quantum fields for two related reasons. First, classical fields are functions of space and time so it makes sense that quantum fields will be too. Second, Einstein’s theory of special relativity unifies space and time in a way that makes it desirable to treat time and space on the same footing. Example: In the case of the simple harmonic oscillator, the raising and lowering operators

a=

1 (ωx + ip) 2ω

a +=

1 (ωx − ip) 2ω

have simple behavior in the Heisenberg picture. We have

d aH (t ) = i [H , aH (t )] = −iωaH (t ) dt so

aH (t ) = e−iωt a where we write aH (0) = a for operators at t = 0. Similarly, we have a H* (t ) = e +iωta H* (0). These two results taken together imply that H is independent of t as expected. The operator x is time-dependent, as is p: 1 (ae−iωt + a +e+iωt ) xH (t ) = 2ω

pH (t ) = i

ω + +iωt (a e − ae−iωt ). 2

2.2.4 Interactions The simple harmonic oscillator is exactly solvable but most quantum mechanics problems are not. A commonly considered case is a one-dimensional potential V (x ) with a discrete symmetry V (x ) = V ( −x ) and a single minimum at x = 0. If we assume that V (x ) can be represented as a Taylor series, then we may write ∞

V (x ) =

V 2n(0) 2n x (2n )! n=0

∑

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where

V 2n(0) =

d 2nV dx 2n

. x=0

The constant V 0(0) = V (0) represents an overall constant appearing in all the energy eigenvalues, while we can identify V 2(0) as k, the spring constant of a simple harmonic oscillator. If we have rescaled the mass to one, then we interpret V 2(0) as ω 2 , where ω is the frequency. If we stop at this second-order truncation of the potential, we have an exactly solvable problem. However, when we include the next term in the Taylor series, we have

H=

1 2 1 λ p + ω xx 2 + x 4 2 2 4!

where we have written V 4(0) as a parameter λ, which we refer to as a coupling constant. The Hamiltonian H is known as the Hamiltonian of the one-dimensional anharmonic oscillator; it has no exact solution. One of the primary approximation methods we have for models such as the anharmonic oscillator is perturbation theory, in which quantities of interest, such as energy eigenstates and eigenvalues, are calculated as power series in parameters such as λ. At a basic level, the apparatus of Feynman diagrams is a graphical method for computing quantum mechanical matrix elements as a power series in such coupling constants. In the case of QED, the coupling constant which appears in Lagrangians and Hamiltonians is the absolute value of the charge of the electron, e, but physical results can be written in terms of the fine structure constant α, which is e 2 /4πϵ0ℏc in MKS units, but simply e 2 /4π in natural units. The fine structure constant is dimensionless, and has a small value, α ≈ 1/137. This is generally taken as an indication that the power series expansions of perturbation theory are useful, even though those series are known to be divergent, as a consequence of the exponential growth in the number of Feynman diagrams as the order of perturbation theory increases.

2.3 The Feynman path integral for one degree of freedom The Feynman path integral provides a unified approach to both quantum mechanics and quantum field theory [1, 2]. At the same time, it incorporates classical physics in a simple and elegant way, and has applications and connections to many areas of physics. Here we develop the path integral for the quantum mechanics of a single degree of freedom. The extension to multiple degrees of freedom is straightforward, and quantum field theory will be obtained later as the extension to an infinite number of degrees of freedom. 2.3.1 Defining the path integral The Feynman path integral is the key to the modern approach to quantum field theory. One direct derivation of the path integral is based on the Trotter product formula, which says that 2-13

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⎡ ⎛ t ⎞ ⎛ t ⎞⎤n ⎜ − i H ⎟ exp⎜ − i V ⎟ lim exp = exp[ −it(H0 + V )]. 0 n → ∞⎢ ⎝ n ⎠⎥⎦ ⎣ ⎝ n ⎠ To see why this formula holds, note that for n large,

⎛ t ⎞ ⎛ t ⎞ ⎛ t ⎞ t ⎞⎛ exp ⎜ −i H0⎟ exp ⎜ −i V ⎟ ≈ ⎜1 − i H0⎟⎜1 − i V ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝ ⎠ ⎝ n ⎠ n ⎡ t ⎤ ≈ exp ⎢ −i (H0 + V ) + O(1/ n 2 )⎥ . ⎣ n ⎦ In the limit n → ∞, the extra terms are suppressed. We apply this formula to the matrix element

xf ∣exp[ −it(H0 + V )]∣xi for a quantum mechanical system with one degree of freedom x and a Hamiltonian H = H0 + V , where H0 = p2 /2m is the kinetic term and V (x ) is the potential. Taking x0 = xi and xn = xf we divide the time t into n equal intervals with t j − t j −1 = t /n. Introducing a complete set of states

K (x′ , t′ ; x , t ) = nlim →∞

∫

⎛ t ⎞ ⎛ t ⎞ n dxn−1…dx1 ∏ j = 1 xj exp⎜ −i H0⎟ exp⎜ −i V ⎟ xj −1 ⎝ n ⎠ ⎝ n ⎠

Each factor in the product can be evaluated as

⎛ t ⎞ ⎛ t ⎞ xj exp⎜ −i H0⎟ exp⎜ −i V ⎟ xj −1 ⎝ n ⎠ ⎝ n ⎠ ⎛ t ⎞ ⎛ t ⎞ = xj exp⎜ −i H0⎟ xj −1 exp ⎜ −i V (xj −1)⎟ ⎝ n ⎠ ⎝ n ⎠ =

⎡ m(xj − xj −1)2 ⎤ ⎡ t ⎤ m exp ⎢i ⎥ exp ⎢ −i V (xj −1)⎥ ⎣ n ⎦ 2πi (t − iϵ )/ n ⎣ 2(t − iϵ )/ n ⎦

so that

K (x′ , t′ ; x , t ) = nlim →∞

∫

⎛ ⎞n/2 m dxn−1…dx1⎜ ⎟ ⎝ 2πi (t − iϵ )/ n ⎠

⎧ n ⎡ ⎤⎫ ⎪ ⎪ m(xj − xj −1)2 t i i V (xj −1)⎥⎬ . × exp ⎨ − ⎢ ∑ ⎪ ⎪ n ⎦⎭ ⎩ j = 1⎣ 2(t − iϵ )/ n In the limit n → ∞, we have n ⎡ ⎤ m(xj − xj −1)2 t lim − i V (xj −1)⎥ = i ⎢ ∑ n→∞ n ⎣ 2(t − iϵ )/ n ⎦ j=1

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∫0

⎡ 1 ⎛ dx ⎞2 ⎤ dt⎢ m⎜ ⎟ − V (x )⎥ ⎣ 2 ⎝ dt ⎠ ⎦

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which is the classical action S. Conceptually, we think of this as a prescription for computing the path integral expression for the propagator, and write x(t )=xf

K (xf , t f ; xi , ti ) =

∫x(t′)=x

[dx ] exp iS [x ]. i

Note that we have slightly changed our notation for K to emphasize the natural grouping of arguments. The square brackets indicate that the integration [dx ] is not an integration over a variable, but over the set of all paths satisfying x(ti ) = xi and x(t f ) = xf , just as S [x ] is a functional of the entire path x(t ). Note that the measure of integration is not simply dxn−1 ... dx1 but includes an extra factor

⎛ ⎞n/2 m dxn−1…dx1⎜ ⎟ . ⎝ 2πi (t − iϵ )/ n ⎠ The path integral encodes a fundamental property of time evolution in an interesting way. Suppose we choose an intermediate time tm such that ti < tm < t f . Then US (t f , tm )US (tm, ti ) = U (t f , ti ) implies

∫ dxmK (xf , tf ; xm, tm)K (xm, tm; xi , ti ) = K (xf , tf ; xi , ti ). In the path integral formalism, this result is obtained from

K (xf , t f ; xi , ti ) =

∫

∫x(t )=x

x(t f )=xf

=

⎛i ⎞ [dx ] exp ⎜ S [x ]⎟ ⎝ℏ ⎠ m m ⎛i ⎞ [dx ] exp ⎜ S [x ]⎟ ⎝ℏ ⎠ x(t f )=xf

dxm

∫x(t )=x i

i

x(tm )=x m

∫x(t )=x i

i

⎛i ⎞ [dx ] exp ⎜ S [x ]⎟ ⎝ℏ ⎠

which is the statement that you obtain all path from xi to xf by composing all paths from xi to xm with all paths from xm to xf. As in the case of the free nonrelativistic particle, we can restore ℏ from its natural value of one, to obtain x(t f )=xf

K (xf , t f ; xi , ti ) =

∫x(t )=x i

i

⎛i ⎞ [dx ] exp ⎜ S [x ]⎟ . ⎝ℏ ⎠

If this were an ordinary integral over x xf

∫x

i

⎛i ⎞ dx exp ⎜ S (x )⎟ ⎝ℏ ⎠

rather than a path integral over all paths x(t ), we could estimate the value of the integral by the method of stationary phase, in which the behavior of the integral is dominated by the behavior near critical points where dS (x )/dx = 0. In general the phase of the integrand fluctuates wildly with x and there are large cancelations. However, in the vicinity of critical points, the phase changes much more slowly and these regions dominate. As ℏ becomes smaller, the method improves. In the functional case, the method is applied to critical paths, which satisfy

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δS = 0. δ x( t ) The critical paths are the classical paths, and the limit ℏ → 0 is the classical limit. Thus the path integral suggests a reduction to classical physics in that classical limit. We can also see from the path integral that the classical equations of motion hold as statements about expected values. Assuming that contributions from endpoints vanish, we can apply functional integration by parts in the form

∫ [dx ] δxδ(t ) eiS[x]/ℏ = 0 which implies

∫ [dx ] δδSx([xt )] eiS[x]/ℏ = 0. This in turn leads to

δS δ x( t )

= 0.

When derived in the Heisenberg representation, this is known as Ehrenfest’s theorem. 2.3.2 Matrix elements and time ordering The fundamental information about a system is contained in matrix elements of operators. Suppose we know the system is in the state ∣xi at time −T /2 in the distant past, and in a state ∣xf at time T /2 in the distant future, and want to calculate the expected value of x at time t, where −T /2 < t < +T /2. This is given by

xf ∣e−i (T /2−t )H x e−i (t+T /2)H ∣xi which represents time evolution of the initial state followed by a measurement of x at time t, and then further time evolution. By inserting a complete set of x states at the time t, it is easy to see that this matrix element is represented by

xf ∣e−i (T /2−t )H x e−i (t+T /2)H ∣xi =

x(T /2)=xf

∫x(−T /2)=x [dx ]x(t ) exp iS[x ] i

provided we interpret the right-hand side of the equation using the Trotter product formula. We can rewrite this matrix element as

xf ∣e−i (T /2−t )H x e−i (t+T /2)H ∣xi = xf ∣e−i (T /2)H e+itH xe−itH e−i (T /2)H xi . We can immediately identify e +itH xe−itH as xH (t ), the x operator in the Heisenberg representation. What can we say about e−i (T /2)H ∣xi and xf ∣e−i (T /2)H , especially in the limit T → ∞? We can insert a complete set of energy eigenstates ∣En in which case

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e−i (T /2)H ∣xi =

∑e−i(T /2)H ∣En

En∣xi =

∑e−i(T /2)E ∣En

Enxi .

n

n

n

This is a wildly oscillating sum, but it is dominated by the ground state contribution. Let us add a factor of (1 − iϵ ) to T so we have

∑e−i(1−iϵ)(T /2)E ∣En

En∣xi

n

n

where the limit ϵ → 0 will eventually be taken. In the limit where T is very large, the sum reduces to

e−i (T /2)E0∣E 0 E 0∣xi and similarly for the factor associated with xf. We must have a nonzero overlap in the matrix element E0∣xi , but this is always true in quantum mechanics where the ground state is nodeless. In field theory, the ground state is usually referred to as the vacuum, and written as ∣0 . We will follow this convention, so we now have for large T x(T /2)=xf

∫x(−T /2)=x [dx ]x(t ) exp iS[x ] = e−iTE

xf ∣0 0∣xi 0∣xH (t )∣0 .

0

i

The same arguments lead to x(T /2)=xf

∫x(−T /2)=x [dx ] exp iS[x ] = e−iTE

0

xf ∣0 0∣xi

i

where we have assumed that the vacuum is normalized to one. This leads to a formula for the vacuum expectation value of xH (t ): x(T /2)=xf

0∣xH (t )∣0 = Tlim

→∞

∫x(−T /2)=x [dx ]x(t ) exp iS[x ] i

x(T /2)=xf

.

∫x(−T /2)=x [dx ] exp iS[x ] i

The left-hand side is independent the choice of xi and xf, and the standard practice is to write simply

0∣xH (t )∣0 =

∫ [dx ]x(t )eiS[x] ∫ [dx ]eiS[x]

with the limit T → ∞ understood. This formula is reassuringly similar to standard formulas for expectation values in quantum mechanics where states must be normalized to one. If we now consider the functional integral x(T /2)=xf

∫x(−T /2)=x [dx ]x(t1)x(t2) exp iS[x ], i

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also interpreted using the Trotter product formula, we see that the order of t1 and t2 matters. If t1 > t2 , the path integral gives

e−iTE0 xf ∣0 0∣xi 0∣xH (t1)xH (t2 )∣0 and

e−iTE0 xf ∣0 0∣xi 0∣xH (t2 )xH (t1)∣0 for t2 > t1. This motivates the introduction of the time-ordered product, defined for two operators as

⎧ x(t1)x(t2 ) t1 > t2 T [x(t1)x(t2 )] = ⎨ ⎩ x(t2 )x(t1) t2 > t1 Note that no commutation of operators is implied. An equivalent compact form is

T [x(t1)x(t2 )] = θ (t1 − t2 )x(t1)x(t2 ) + θ (t2 − t1)x(t2 )x(t1). We now have the path integral expression

0∣T [x(t1)x(t2 )]∣0

∫ [dx ]x(t1)x(t2)eiS[x] = ∫ [dx ]eiS[x]

for the matrix element of the time-ordered product of two xH operators in vacuum. This has an obvious extension to the product of more than two operators

0∣T [x(t1) ... x(tn)]∣0 =

∫ [dx ]x(t1) ... x(tn)eiS[x] . iS [x ] [ dx ] e ∫

The mnemonic latest on the left may be helpful in remembering how time ordering works in practice. We now have a formula for evaluating the vacuum expectation values of timeordered products of operators in the path integral formalism. It may seem that these are somewhat exotic, but they are of fundamental importance in quantum field theory. These vacuum expectation values, sometimes abbreviated as v.e.v. and pronounced ‘vev’, are known by various other names, including n-point functions, n-point correlation functions, or n-point Green functions, and written as

G (n)(t1, … , tn) = 0∣T [x(t1)…x(tn)]∣0 . There is a fundamental result in quantum field theory, often referred to as the Wightman reconstruction theorem, which tells us that the entire Hilbert space structure of a quantum system may be obtained from the knowledge of all the npoint functions [3].

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2.3.3 Generating functions The functional Z [J ] usefully encapsulates all information about the n-point functions G (n). We define the generating function Z [J ] as a functional of the cnumber source J (t ) by the functional integral.

Z [J ] =

∫ [dx ] exp ⎡⎢⎣iS + i ∫ dtJ (t )x(t )⎤⎥⎦.

Note that

Z [0] =

∫ [dx ]eiS[x]

is the denominator of our expression for n-point functions. Functional differentiation with respect to J (t ) shows that

1 δZ [J ] = i δJ (t1)

∫

⎡ [dx ]x(t1) exp ⎣⎢iS + i

⎛ 1 ⎞2 δ 2Z [J ] ⎜ ⎟ = ⎝ i ⎠ δJ (t1)δJ (t2 )

∫

⎡ [dx ]x(t1)x(t2 ) exp ⎣⎢iS + i

∫

⎤ dtJ (t )x(t )⎦⎥

∫

⎤ dtJ (t )x(t )⎦⎥

and so on. From this and our previous results for n-point functions, we have

⎛ 1 ⎞n 1 δ nZ [J ] ⎜ ⎟ ⎝ i ⎠ Z [0] δJ (t1)…δJ (tn)

= G (n)(t1, … , tn). J =0

We say that Z [J ] is the generating functional of the n-point functions G (n). An equivalent way to state this result is to expand Z [J ] in a functional Taylor series as

Z [J ] =

∫

⎡ [dx ]e iS⎢1 + ⎢⎣

∞

in ∑n ! n=1

∫

⎤ dt1 ... dtnJ (t1)x(t1) ... J (tn)x(tn)⎥ ⎥⎦

or equivalently

⎡ Z [J ] = Z [0]⎢1 + ⎢⎣

∞

in

∑n ∫ ! n=1

⎤ dt1 ... dtnJ (t1) ... J (tn)G (n)(t1, … , tn)⎥ ⎥⎦

The generating function Z [J ] generates all the n-point functions, and is closely related to the partition function of statistical mechanics and the characteristic functions of probability distributions. It is often convenient to work with the associated generating functional W [J ] defined by

Z [J ] = e iW [J ] or equivalently

W [J ] = −i log Z [J ].

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It is straightforward to show that

δW [J ] δJ (t1) ⎛ 1 ⎞ δ 2W [J ] ⎜ ⎟ ⎝ i ⎠ δJ (t1)δJ (t2 )

= 0∣x(t1)∣0 J =0

= 0∣T [x(t1)x(t2 )]∣0 − 0∣x(t1)∣0 0∣x(t2 )∣0 J =0

and in general we can define the connected n-point functions Gc(n)(t1, … , tn ) by

⎛ 1 ⎞n−1 δ 2W [J ] Gc(n)(t1, … , tn) = ⎜ ⎟ ⎝ i ⎠ δJ (t1)…δJ (tn)

. J =0

The generating function W [J ] can be expanded in a functional Taylor series in the same manner as Z [J ], with n-point functions replaced by connected n-point functions. As we will see later, the connected n-point functions are associated with connected Feynman diagrams and scattering processes. 2.3.4 The simple harmonic oscillator The simple harmonic oscillator is the fundamental building block from which free quantum field theories are constructed. We can write the Lagrangian of the harmonic oscillator, in a form that closely parallels scalar field theories, as

L=

1 ⎛ dϕ ⎞2 1 2 2 ⎜ ⎟ − ωϕ 2 ⎝ dt ⎠ 2

by the substitution x(t ) → m−1/2ϕ(t ). The corresponding Hamiltonian is

H=

1 2 1 2 2 π + ωϕ 2 2

where the canonical momentum conjugate to ϕ is π ≡ ϕ.̇ In order to calculate the path integral for the simple harmonic oscillator, we will need to generalize our result for a single Gaussian integral to n dimensions and take the limit n → ∞. Consider an n-dimensional integral of the form 1 I (A, b ) = ∫ d nx exp⎡⎣ − 2 xtAx + b · x ⎤⎦ where b is an n-dimensional vector and A is an n × n symmetric matrix whose eigenvalues are positive. Assuming A is invertible, the substitution x → x + A−1b implies

I (A , b ) =

⎡

⎤

∫ d nx exp⎢⎣− 12 xtAx + 12 btA−1b⎥⎦

Writing A = RDRt where R is a rotation matrix and D is a diagonal matrix with Djk = djδjk , we see that the change of variable x = Ry leads to

I (A , b ) =

⎡

⎤

∫ d ny exp⎢⎣− 12 yt Dy + 12 btA−1b⎥⎦. 2-20

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The n-dimensional Gaussian integral has been reduced to n one-dimensional integrals, leading to

I (A , b ) =

⎡ 1 ⎤ 2π exp⎢ + bt A−1b⎥ . ⎣ 2 ⎦ dj

∏ j

Using det A = det D = ∏j dj , we arrived at our desired result

I (A , b ) =

⎡ 1 ⎤ (2π )n/2 exp⎢ + bt A−1b⎥ . ⎣ 2 ⎦ det A

This result remains valid in the case where all the eigenvalues have positive real parts. In order to apply this result to the path integral, we begin with the harmonic oscillator action, including a source term:

S [ϕ , J ] =

⎡

⎤

∫ dt⎢⎣ 12 (∂tϕ)2 − 12 (ω2 − iϵ)ϕ2 + Jϕ⎥⎦

where the iϵ has been added to insure convergence of the functional integral. We can expand S in a functional Taylor series in ϕ:

S=0+

∫ dt δϕδS(t )

ϕ(t ) + ϕ=0

1 2

2

S ∫ dtdt′ ϕ(t ) δϕ(tδ)δϕ (t′)

ϕ(t′) + 0 ϕ=0

where

δS δϕ(t ) δ 2S δϕ(t )δϕ(t′)

= J (t ) ϕ=0

= [ − ∂t2 − (ω 2 + iϵ )]δ(t − t′). ϕ=0

Because the action is quadratic in ϕ to start with, there are no terms of order ϕ3 or higher. Defining D(t − t′) = [−∂ t2 − (ω 2 − iϵ )]δ (t − t′), we can write the path integral as

Z [J ] =

∫ [dϕ] exp ⎡⎢⎣i ∫ dtdt′ϕ(t )D(t − t′)ϕ(t′) + i ∫ dtϕ(t )J (t )⎤⎥⎦.

This has precisely the form of an n-dimensional Gaussian integral in the limit n → ∞ is we replace an integer index running from 1 to n by a continuous index t. Thus we make the associations

Ajk → − iD(t − t′) bj → iJ (t ) suggesting that the path integral is given by

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⎡1 Z [J ] = A(det[ −iD ])−1/2 exp ⎢ ⎣2

⎤

∫ dtd ′ iJ (t )iD−1(t − t′)iJ (t′)⎥⎦

where the constant A is not determined. To make sense of this expression, we must define what we mean by D−1 and det[-iD]. The inverse of D is defined by the relation

∫ dt′D(t − t′)D−1(t′ − t″) = δ(t − t″) which is the analog of Ajk Akl−1 = δjl for a matrix inverse. From the explicit form of D in this case, this reduces to

[ −∂t2 − (ω 2 − iϵ )]D −1(t − t″) = δ(t − t″) so D−1 is a Green’s function for D. In this case, Fourier transformation easily yields the representation

iD −1(t − t′) =

∫

dE i e−iE (t−t′). 2π E 2 − ω 2 + iϵ

This integral can be explicitly carried out by contour integration. The denominator E 2 − ω 2 + iϵ has poles at E = +ω − iϵ and E = −ω + iϵ . These poles are near the real axis, but not on it. If t − t′ > 0, we can close the contour in the lower half-plane, but for t − t′ < 0 we must close the contour in the lower half-plane. The result is that

⎧ 1 −iω(t−t′) t − t′ > 0 ⎪ ⎪ ωe iD −1(t − t′) = ⎨ 2 ⎪ 1 e+iω(t−t′) t − t′ < 0 ⎪ ⎩ 2ω as can be checked directly by application of the differential operator −∂ t2 − (ω 2 − iϵ ) to

iD −1(t − t′) = θ (t − t′)

1 1 −iω(t−t′) e + θ (t′ − t ) e+iω(t−t′) 2ω 2ω

This result can also be checked using the operator

xH (t ) =

1 (ae−iωt + a +e+iωt ) 2ω

in the Heisenberg representation. The function iD−1(t − t′) is the Feynman propagator for a one-dimensional harmonic oscillator, and will be denoted hereafter as i ΔF (t − t′). We see that up to a constant, the generating function iW [J ] is given by

iW [J ] =

1 2

∫ dtd ′ iJ (t )iΔF (t − t′)iJ (t′)

from which it follows that

Gc(2)(t1, t2 ) = G (2)(t1, t2 ) = 0∣T [x(t1)x(t2 )]∣0 = i ΔF (t1 − t2 ) because 0∣x(t1)∣0 = 0 when J = 0.

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We have shown in section 2.3.3 that Z[0] should have the form

Z [0] = ae−iE0T . where a = xf ∣0 0∣xi for large T. This is represented for the harmonic oscillator as

Z [0] = A(det[ −iD ])−1/2 where A is not determined by our arguments. The determinant is formally defined from the eigenvalues of D, which satisfy

∫ dt′ D(t − t′)ψn(t′) = λnψn(t ). For the harmonic oscillator, this reduces to

[ −∂t2 − (ω 2 − iϵ )]ψn(t ) = λnψn(t ). Then the determinant of D is formally the product of the eigenvalues

det D =

∏ λn n

which is generally divergent. Later we will show how E0 can be correctly obtained from the functional determinant. The n-point functions do not depend on Z[0], which is an overall factor independent of J. It is consistent and convenient to choose the energy of the vacuum to be set at zero, and take Z[0] = 1. This choice of normalization of Z leads to the simple result

⎡1 Z [J ] = exp ⎢ ⎣2

⎤

∫ dtd ′ iJ (t )iΔF (t − t′)iJ (t′)⎥⎦.

2.3.5 Wick’s theorem The evaluation of matrix elements of the form 0∣T [x(t1) ... x(tn )]∣0 for the simple harmonic oscillator is greatly simplified by a simple algorithm due to Wick. Originally derived in field theory as an operator identity, it is revealed by the path integral to be a property of the moments of Gaussian integrals. Suppose we expand our path integral for the generating function of the simple harmonic oscillator as a functional power series in J:

Z [J ] =

∫

⎡ [dϕ ] exp ⎢⎣i

∫

dt L + i

=

∫

⎡ [dϕ ] exp ⎣⎢i

∫

1⎡ ⎤ dt L⎥⎦ ∑ ⎣⎢ n!

∞

1 n ! n=0

=∑

∫

∫

⎤ dt , Jϕ⎥⎦

∞

∫

n=0

⎤n dt iJ (t )ϕ(t )⎦⎥

dt1 ... dtniJ (t1) ... iJ (tn) 0∣T ⎡⎣ϕH (t1) ... ϕH (tn)⎤⎦∣0

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where we have chosen to normalize the path integral to Z[0] = 1. This is the functional Taylor series expansion of Z [J ]. However, we know from our results in the previous section that only the even terms in J contribute, so we can write this as ∞

Z [J ] =

1 n )!

∑ (2 n=0

∫ dt1 ... dt2niJ (t1) ... iJ (t2n) 0∣T ⎡⎣ϕH (t1) ... ϕH (t2n)⎤⎦∣0

Z [J ] =

∫ [dϕ] exp ⎡⎢⎣i ∫ dt L + i ∫ dt, Jϕ⎤⎥⎦

On the other hand, if we expand the exponential in our result

⎡1 Z [J ] = exp ⎢ ⎣2

⎤

∫ dtd ′ iJ (t )iΔF (t − t′)iJ (t′)⎥⎦

we immediately obtain

Z [J ] =

∑

1 ⎡1 ⎢ n! ⎣ 2

⎤n

∫ dtd ′ iJ (t )iΔF (t − t′)iJ (t′)⎥⎦ .

There is a combinatorial argument connecting these two results. Suppose we pair all of the fields in 0∣T [ϕH (t1) ... ϕH (t2n )]∣0 and associate with each pair a factor 0∣T ⎡⎣ϕH (t j )ϕH (tk )⎤⎦∣0 = i ΔF (t j − tk ). This association is known as a Wick contraction. With 2n fields, there are (2n )! /2nn! ways to do the pairings, because the order of the pairings and the order within a pairing are irrelevant. The process of Wick contraction is an algorithm which turns one form of Z [J ] into another. Looking at the O(J 4 ) term in Z [J ], we see that

0∣T ⎡⎣ϕH (t1)ϕH (t2 )ϕH (t3)ϕH (t 4)⎤⎦∣0 = i ΔF (t1 − t2 )i ΔF (t3 − t 4) + i ΔF (t1 − t3)i ΔF (t2 − t 4) + i ΔF (t1 − t 4)i ΔF (t2 − t3). The general result for 0∣T [ϕH (t1) ... ϕH (t2n )]∣0 is known as Wick’s theorem. 2.3.6 Perturbation theory and Feynman diagrams Let us return to the anharmonic oscillator in order to develop perturbation theory using the path integral. The Lagrangian is

L=

1 ⎛ dϕ ⎞2 1 2 2 λ ⎜ ⎟ − ω ϕ − ϕ4 . 2 ⎝ dt ⎠ 2 4!

In perturbation theory, we divide L into parts L0 and LI, where L0 is the Lagrangian of a system we can solve, and LI is an additional interaction which we cannot treat exactly. Usually, L0 represents non-interacting particles and is called the free Lagrangian, while LI is called the interaction Lagrangian. In this case,

L0 =

1 ⎛ dϕ ⎞2 1 2 2 ⎜ ⎟ − ωϕ 2 ⎝ dt ⎠ 2

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and

LI = −

λ 4 ϕ. 4!

The generating function Z [J ] is given by

Z [J ] =

∫ [dx ]ei∫ dt[L +L +Jϕ] 0

I

which can be written as

Z [J ] =

=

∫ ∫

⎡ 1+ ⎢⎣ ⎡ [dx ]e i∫ dt[L 0+Jϕ]⎢1 + ⎢⎣ [dx ]e i∫

dt [L 0+Jϕ]⎢

∞

in ∑n ! n=1

(∫

n⎤ dtLI ⎥ ⎥⎦

)

∞

in ⎛ λ ⎜− ⎝ 4! ! n=1

∫

∑n

⎞n⎤ dtϕ4(t )⎟ ⎥ . ⎠ ⎥⎦

It must be stressed that each of the t integrations is independent of the others; less compactly, we can write ∞

in

(

∑n ∫ ! n=1

∞

n

dtLI

)

=

in ⎛ λ ⎜− ! ⎝ 4!

∑n n=1

⎞ ⎛

⎞

∫ dt1ϕ4(t1)⎟⎠ ...⎜⎝− 4λ! ∫ dtnϕ4(tn)⎟⎠.

Expanding Z [J ] in this way we can calculate the generating function as a double power series in λ and J, with terms of order λ nJ p:

⎤n

⎡

∞

Z [J ] =

∞

∫ [dx ]ei∫ dt L ∑ n1! ⎢⎣−i 4λ! ∫ dt ϕ4(t )⎥⎦ · ∑ p1! ⎡⎣⎢∫ dt iJ (t )ϕ(t )⎤⎦⎥ 0

n=0

p

p=0

Note that this expansion is a normal Taylor series in λ, but a functional Taylor series in J. We can write this as ∞

Z [J ] =

∑ Z (n)[J ] n=0

where the nth term in the series is proportional to λ n , with each term Z (n)[J ] in the sum a functional Taylor series in J. Inspection tells us that

Z (0)[J ] =

∫ [dx ]ei∫ dt[L +Jϕ] 0

which is the generating function for a free theory, which we will also denote as Z0[J ]. The next coefficient is

Z (1)[J ] =

⎛

⎞

∫ [dx ]ei∫ dt[L +Jϕ]⎜⎝ −4i!λ ∫ dtϕ4(t )⎟⎠ 0

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and so on. We can evaluate Z (1)[J ] using functional differentiation. Observing that

⎛ 1 δ ⎞n ⎜ ⎟ ⎝ i δJ ( t ) ⎠

∫ [dx ]ei∫ dt[L +Jϕ] = ∫ [dx ]ei∫ dt[L +Jϕ]ϕn(t ), 0

0

we see that Z (1)[J ] can be written as

⎛ −iλ ⎛ 1 δ ⎞4 ⎞ Z (1)[J ] = ⎜⎜ dt⎜ [dx ]e i∫ ⎟⎟ ⎝ i δJ (t ) ⎠ ⎟⎠ ⎝ 4! ⎛ ⎛ 1 δ ⎞⎞ = ⎜i dt LI ⎜ ⎟⎟Z0[J ]. ⎝ i δJ (t ) ⎠⎠ ⎝

∫

∫

dt [L 0+Jϕ]

∫

A bit of thought shows that this trick can be applied to the entire series, and ⎡ ⎛ 1 δ ⎞⎤ Z [J ] = exp ⎢ dt LI ⎜ ⎟⎥Z0[J ] ⎝ i δJ (t ) ⎠⎦ ⎣

∫

which gives all of perturbation theory. The process of finding Z [J ] to a given order in perturbation theory has been reduced to a set of functional derivatives. For example, the lowest order correction to Z [J ] gives

⎛ −iλ Z (1)[J ] = ⎜⎜ ⎝ 4!

∫

⎛ 1 δ ⎞4 ⎞ ⎡1 dt⎜ ⎟ ⎟⎟ exp ⎢ ⎣2 ⎝ i δJ ( t ) ⎠ ⎠

⎤

∫ dt1dt2 iJ (t1)iD−1(t1 − t2)iJ (t2)⎥⎦

where we have set Z0[0] = 1. Taking the functional derivatives sequentially, we find after repeated functional differentiation that

iλ 4! iλ − 4! iλ − 4!

Z (1)[J ] = −

(∫

)

∫

dt

∫

dt 6 · i ΔF (0)

∫

dt 3 · (i ΔF (0))2 Z0[J ].

4

dt1i ΔF (t1 − t )iJ (t1) Z0[J ]

(∫

2

)

dt1i ΔF (t1 − t )iJ (t1) Z0[J ]

It should be stressed again that repeated integrals like 2

(∫ dt1iΔF (t1 − t)iJ (t1))

are over dummy variables. In this case, it would be more clear to write

(∫ dt iΔ (t − t)iJ (t ))(∫ dt iΔ (t − t)iJ (t )) 1

F

1

1

2

F

2

2

but this does make formulas very long and harder to comprehend. Machines can be trained to do these sorts of calculations, but an easier technology for humans to understand is the graphical method of Feynman diagrams. Let us apply Wick’s theorem to

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Z [J ] =

∫

[dx ]e i∫

1⎡ λ ∑ ⎢⎣−i 4 n! ! n=0 ∞

dt L 0

∞ 1 1 ⎛ −iλ ⎞n ⎜ ⎟ ∑ n! p = 0 p! ⎝ 4! ⎠ n=0

∫

⎤n dt ϕ4(t )⎥ · ⎦

∞

1⎡ ⎢ p! ⎣ p=0

∑

∫

⎤p dt iJ (t )ϕ(t )⎥⎦

∞

=∑

∫

dt1⋯dtndt1′ ... dt p′ iJ (t1′) ... iJ (t p′ )

× 0∣T ⎡⎣ϕ4(t1)⋯ϕ4(tn)ϕ(t1′)ϕ(t p′ )⎤⎦ 0 where we have set the factor Z0[0] = 1 according to our convention. We have also dropped the subscript H from the fields because we are not evaluating expectation values of the Heisenberg fields of the interacting theory with respect to the vacuum of the interacting theory. Each term in our expression is the expectation value of harmonic oscillator fields with respect to the harmonic oscillator vacuum, which when summed gives us the result for expectation values of the Heisenberg fields of the interacting theory with respect to the vacuum of the interacting theory. We can now apply Wick’s theorem to

0∣T ⎡⎣ϕ4(t1)⋯ϕ4(tn)ϕ(t1′)ϕ(t p′ )⎤⎦∣0 which tells us to construct all possible pairings between the 4n + p fields. This is facilitated by a graphical language in which every Wick contraction 0∣T ⎡⎣ϕH (t j )ϕH (t )⎤⎦∣0 = i ΔF (t j − tk ) is represented by a line connecting the points tj and tk which are drawn as points on the two-dimensional plane to facilitate comprehensibility of the graph; because we will integrate over all tj’s, no particular ordering is needed or desirable. The graphical notation for i ΔF (t1 − t2 ) is shown in figure 2.1. Each of the n interactions is represented as a factor of −iλ at a point tj with a vertex which connects four propagators at that point. The graphical notation for −iλ is shown in figure 2.2 Each of the p external sources is represented by a source term iJ (t j′) at a time tj which connects to one propagator at those points. These are the Feynman rules for our ϕ4 model, and the diagrams produced are Feynman diagrams. It is customary to write the Feynman rules for sources and vertices with the appropriate number of propagators appearing, but those are not part of the

Figure 2.1. The graphical representation of i ΔF (t1 − t2 ).

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Figure 2.2. The graphical representation of the ϕ4 scattering vertex. By convention, the factor of 1/4! is usually not included as part of the vertex, but is taken into account when considering the overall symmetry of the graph.

correspondence between mathematical expressions and graphs. It is not hard to write down graphs at low orders of λ and J. We show the three graphs contributing to Z (1)[J ] if figure 2.3. For each graph, there is a combinatorial term, to which we now turn. After carrying out a number of functional differentiations, we find

λ 4! λ −i 6 4! λ −i 3 4!

Z (1)[J ] = − i

∫

dt

(∫

)

4

dt1i ΔF (t1 − t )iJ (t1) Z0[J ]

(∫

2

∫

dt i ΔF (0)

∫

dt (i ΔF (0))2 Z0[J ].

)

dt1i ΔF (t1 − t )iJ (t1) Z0[J ]

Figure 2.3. The three graphs contributing to Z (1)[J ] in a ϕ4 model.

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If we look at this result for Z (1)[J ] after all functional derivatives are carried out and ignore the Z0[J ], we see that there is a factor of 1 for the diagram with four external sources, a factor of 6 for the diagram with two sources, and a factor of 3 for the diagram with no sources. This can be understood from the combinatorics of pairing the original ϕ4(t ) fields with each other, and with fields at other times coupled to J. When we couple each of the four fields at t to other fields coupled to J, there are no extra factors. However, when two of the four fields at t are joined by a propagator, there are 4 ways to choose the first field, 3 to choose the second, but the order of choice does not matter, so there are 4 · 3/2 = 6 ways to make this graph. For the graph where all four fields are joined together by a pair of propagators, there are 3 ways to do so. There is another, equivalent way to look at these combinatorial factors. The factor of 4! multiplying λ in the Lagrangian and Hamiltonian is part of the definition of λ, but it is a very convenient choice. We can rearrange the coefficients of Z (1)[J ] as

λ 4! λ −i 4 λ −i 8 −i

∫

dt

(∫

)

4

dt1i ΔF (t1 − t )iJ (t1) Z0[J ]

(∫

2

)

∫

dt i ΔF (0)

∫

dt (i ΔF (0))2 Z0[J ].

dt1i ΔF (t1 − t )iJ (t1) Z0[J ]

We now interpret the coefficients 1/4!, 1/4, and 1/8 as the symmetry factors of the graphs. When we couple each of the four fields at t to other fields coupled to J, there is a symmetry factor of 4! factorial because we can permute the legs of the graph attached to sources in 4! ways. When two of the four fields at t are joined by a propagator, there are 2 ways to permute the legs of the graph, and 2 ways to permute the internal line connecting the vertex at t to itself; this is basically a reflection of that line. This gives a symmetry factor of 4. For the graph where all four fields are joined together by a pair of propagators, there are 2 factors of 2 for each internal line, and another factor of 2 associated with exchanging the two internal lines; this gives an overall symmetry factor of 8. If we set J = 0, we see that

Z (1)[0] = −iλT ·

1 · (i ΔF (0))2 8

where

1 · (i ΔF (0))2 8 is associated with a vacuum graph, a graph which survives in the graphical expansion of Z [J ] when J = 0. To order λ, we have −iλT ·

⎡ ⎤ 1 Z [0] = ⎢1 − iλT · · (i ΔF (0))2 ⎥ . ⎣ ⎦ 8

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By convention, we have set Z0[0] = 1, which implies that the vacuum of the harmonic oscillator has energy E0(0) = 0, but the vacuum energy of the interacting theory is then not necessarily zero. Our result for Z[0] is not immediately consistent with the functional form Z [0] = ae−iE0T . However, the ground state energy of the perturbed system, E0, has an expansion in perturbation theory as

E 0 = E 0(0) + λE 0(1) + λ2E 0(2) + ... which suggests that Z (1)[0] represents a term in the expansion of an exponential, and that the first-order shift in the ground state energy is given by

λE 0(1) = λ ·

1 · (i ΔF (0))2 . 8

This is correct, as the answer can be compared with the familiar first-order formula for the ground state energy shift

λE 0(1) =0

0

λ 4 x 0 4!

, 0

which is the expectation value of the perturbing Hamiltonian in the unperturbed ground state. It is easy to see that the exponentiation of this factor does occur in Z[0]. At order λ n in the expansion of Z[0], there will be a graph consisting of n disconnected subgraphs, of the same form we have in Z (1)[0]. These subgraphs will give a contribution

⎤n 1 λn ⎡ 2 ( (0)) − iT · · i Δ F ⎢ ⎥⎦ 8 n! ⎣ to Z (n)[0]. There will be new graphs appearing at every order in λ, representing new contributions to λ nE0(n) to E0. Our argument reveals an important result: the ground state energy is the sum of connected vacuum graphs. It is a useful exercise to draw some of these and other higher-order contributions. There is a discrete symmetry which simplifies this. We are free to make the change of variable ϕ(t ) → −ϕ(t ) inside the functional integral, so that Z [−J ] = Z [J ]. In a functional Taylor expansion of Z around J = 0, there will be a term of every even order of J. The zeroth-order terms contribute to the vacuum energy, the terms of order J 2 contribute to the two-point function, the terms of order J 4 contribute to the four-point function, et cetera. From the n-point functions we can go on to calculate other quantities of interest in quantum mechanics. For example, the two-point function, obtained from Z [J ] by functional differentiation twice with respect to the source J. will gives us access to the energy of the first excited state. However, many of these kinds of results are best viewed in the context of interacting quantum field theories, so we defer that discussion to chapter 4.

Problems 1. Consider the anharmonic oscillator action

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

S=

⎡

⎤

∫ dt⎢⎣ 12 m(q1̇ 2 + q2̇ 2 ) − 12 mω02(q12 + q22 ) − 4λ (q12 + q22 )2⎥⎦

(a) Show that this action is invariant under the infinitesimal symmetry q1 → q1 − ϵq2 q2 → q2 + ϵq1 (b) Construct the conserved quantity Q associated with the symmetry and give a physical interpretation. 2. Suppose we define the generating function z (j ) for a zero-dimensional quantum field theory by

z (j ) =

∫ dxeis(x)−ijx

where s(x ) is −(m2 − iϵ )x 2 /2 − λx 4 /4!. (a) For the case λ = 0, calculate the generating function z0(j ). (b) Given that

⎡ λ ⎛ 1 d ⎞4 ⎤ z(j ) = exp⎢ −i ⎜ ⎟ ⎥z0(j ) ⎢⎣ 4! ⎝ i dj ⎠ ⎥⎦ calculate z(0) to order λ2 . (c) For each term in (b), draw the appropriate Feynman diagram(s). 3. Consider a multidimensional Gaussian probability distribution of the form 1

ρ(x ⃗ ) = Ne− 2 xjAjk xk where x ⃗ is an N-component real vector and A is an N × N real symmetric matrix. We assume that A satisfies the conditions needed for the integral over x ⃗ to exist, and that A−1 exists. The constant N is chosen so that the probability distribution ρ is normalized to one:

∫ dx ⃗ ρ(x ⃗) = 1. (a) Suppose we look at the characteristic function

Z (b ⃗ ) =

1

∫ dx ⃗ Ne− 2 x A x +ib x . j jk k

j j

Show that expectation values with respect to ρ may be obtained as

x j1 ... x jn =

1 ⎛1 ∂ ⎞ ⎛1 ∂ ⎞ ⎜ ⎟ ...⎜ ⎟Z (b ⃗ ) Z (0) ⎝ i ∂j1 ⎠ ⎝ i ∂jn ⎠

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b⃗ = 0 .

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Explain why this formula holds even if N is incorrectly chosen, e.g., set equal to one. (b) Show that

xj xk = (A−1) jk where (A−1) jk is the jk entry of the matrix inverse of A. Then show that

xj xkxl xm = (A−1) jk (A−1)lm + (A−1) jl (A−1)km + (A−1) jm (A−1)kl . (c) On the basis of (b), conjecture what the general form of x j1 ... x j2n must be. This general result is known in field theory as Wick’s theorem.

Bibliography Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (International Series in Pure and Applied Physics) (New York: McGraw-Hill) Shulman L S 2005 Techniques and Applications of Path Integration (Mineola, NY: Dover) Wightman A S and Streater R F 1964 PCT, Spin Statistics, And All That (New York, Amsterdam: Benjamin)

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IOP Publishing

A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 3 Classical fields

In this brief chapter, we apply the Lagrangian formalism to scalar and vector classical fields. We examine the role of special relativity in wave equations and in classical Lagrangian field theory. We relate continuous symmetries to conserved currents via Noether’s theorem and consider the implication. We construct the stress-energy tensor and discus conservation of energy-momentum. The causal propagator for classical theories is derived. Examples studied in this chapter include the Klein–Gordon equation, Maxwell’s equations, and the Schrödinger equation.

3.1 Wave equations in classical mechanics and quantum mechanics Wave equations are ubiquitous in physics. The most familiar is so common it is often simply known as the wave equation:

∇2 ϕ(t , x ⃗ ) −

1 ∂ 2ϕ(t , x ⃗ ) =0 ∂t 2 v2

which describes many different kinds of waves. Here the field ϕ(t , x ⃗ ) is a function of time and space and v is the phase velocity of the wave. Solutions of the wave equation can be written as a Fourier transform

ϕ(t , x ⃗ ) =

∫

d 3k ⃗ ˜ ⃗ ik·⃗ x ⃗−iωkt ϕ k e (2π )3

()

where ωk = v∣k ∣⃗ is the dispersion relation relating angular frequency to wavenumber. In the case of electromagnetic fields, v is c, the speed of light. which in natural units is simply one.

doi:10.1088/978-0-7503-3227-9ch3

3-1

ª IOP Publishing Ltd 2022

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

The Schrödinger equation is also a wave equation, which for a free particle reads

i

1 2 ∂ψ (t , x ⃗ ) =− ∇ ψ (t , x ⃗ ). 2m ∂t

Solutions of the free Schrödinger equation can also be written as a Fourier transform, but the dispersion relation is now 2

k⃗ . ωk = 2m Both of these equations are linear equations in the field, ϕ or ψ, and make finding solutions easy by the principle of superposition. In contrast, more realistic field equations are nonlinear, and typically much more difficult to solve. In quantum mechanics, we associate time and space derivatives with the energy and momentum operators via

∂ ∂t pop ⃗ = − i∇

Eop = i

so the Schrödinger equation becomes a statement about conservation of energy:

Eopψ (t , x ⃗ ) =

pop ⃗2 2m

ψ (t , x ⃗ )

in the case of a free nonrelativistic particle. If we apply this prescription to electromagnetic waves, we find 2 Eop ϕ(t , x ⃗ ) = pop ⃗ 2 ϕ(t , x ⃗ ).

This represents the relation E = ∣p∣⃗ between energy and momentum for a massless relativistic particle. In fact, the wave equation is not invariant under the Galilean transformations of Newtonian mechanics, but is rather invariant under the Lorentz transformations of special relativity. A complete theory of the quantum properties of light is necessarily a relativistic one.

3.2 Special relativity In special relativity, the laws of physics are invariant under a class of linear transformations of spacetime coordinates called Lorentz transformations which leave the invariant interval between two spacetime events invariant:

s122 = c 2(t1 − t2 )2 − (x1⃗ − x2⃗ )2 . Spacetime events are local observations of processes like the creation of a particle or the measurement of its momentum. In natural units, the invariant interval between two points can be written as

s 2 = (Δt )2 − (Δx ⃗ )2

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

where Δt = t1 − t2 and Δx ⃗ = x1⃗ − x2⃗ . An event is often idealized as occurring at a point in four-dimensional spacetime. In the case of special relativity, we refer to this space as Minkowski space. We collect both the time t and the spatial location x ⃗ of a spacetime event in a four-vector x. We write a four-vector in a variety of ways, including

x = (t , x ⃗ ) = (x 0, x1, x 2 , x 3) where x 0 = t and x1,2,3 are the spatial components. Note the use of upper indices, which denotes a contravariant four-vector. Sometimes the notation x μ is used to refer to the four-vector x, and sometimes to a particular component x μ of x. We rely on context to disambiguate the two cases. We define the inner, or dot, product of two four-vectors as

x · y ≡ x 0y 0 − x ⃗ · y ⃗ = x 0y 0 − x1y1 − x 2y 2 − x 3y 3 . The fundamental property of Minkowski space is that this inner product is the same when measured in any inertial frame. We introduce the metric tensor gμν, defined as

0 0 ⎞ ⎛+ 1 0 0 ⎟ gμν = ⎜ 0 − 1 0 − 0 0 1 0 ⎟ ⎜ 0 0 − 1⎠ ⎝ 0 so that 3

x·y=

3

∑∑ gμνx μy ν . μ = 0 ν= 0

This is written more compactly using the Einstein summation convention, in which twice-repeated indices are summed over:

x · y = gμνx μy ν . We can now define the covariant four-vector given by

xμ = gμνx ν so that xμ = (t , −x ⃗ ) = (x 0, −x1, −x 2, −x 3) ; the inner product can also be written as

x · y = xμy μ = x μyμ . Just as gμν allows us to convert contravariant vectors into covariant vectors, we can use g μν for the reverse operation

x μ = g μνxν where

g

μν

0 0 ⎞ ⎛+ 1 0 − 0 1 0 0 ⎟. =⎜ − 0 0 1 0 ⎜ ⎟ 0 0 − 1⎠ ⎝ 0

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Note that the mixed metric tensor is simply the 4 × 4 identity matrix

gνμ

0 0 ⎞ ⎛+ 1 0 0 + 1 0 0 ⎟ . = g μρgρν = ⎜ 0 0 + 1 0 ⎜ ⎟ 0 0 + 1⎠ ⎝ 0

The numerical equality between g μν and gμν is true only in special relativity using natural units. It is not true if c ≠ 1 and it is not true in general relativity. A Lorentz transformation Λ is a linear transformation x′ = Λx that preserves the invariant interval between events: x′ · y′ = (Λx ) · (Λy ). Such a transformation Λ can be written as x ′μ = Λ.μνx ν ; the ‘.’ is a placeholder often used to remind us of the ordering of indices. It is easy to prove that Λ leaves the metric tensor invariant, in the sense that Λtg Λ = g , where g is a matrix with entries given by gμν . This should be compared with the comparable relation for rotations, RtR = I . The introduction of covariant vectors may seem artificial at first. However, covariant vectors are essential to understanding the transformation properties of derivatives. Of particular importance to us is the four-dimensional generalization of ∇, given by

∂μ ≡

∂ ∂ ∂ ∂ ∂ = ⎛ 0 , 1, , 3 ⎞. μ 2 ∂x ⎝ ∂x ∂x ∂x ∂x ⎠

The derivative with respect to contravariant components yields a covariant object. Exercise: Check that ∂μ behaves as a covariant four-vector under an infinitesimal boost in the x-direction. 3.2.1 Geometry of spacetime A four-vector v can be • timelike: v μvμ > 0; vμ can be written as (v0 , 0) in some frame. • spacelike v μvμ < 0; vμ can be written as (0, v⃗ ) in some frame. • lightlike v μvμ < 0; vμ satisfies v0 = ±∣v⃗∣. We use this classification scheme to understand the geometry of spacetime. As shown in figure 3.1, spacetime can be viewed from the viewpoint of ‘now and here’, which we take to be the origin x = (0, 0)⃗ , as three separate regions: • Absolute future: x 2 > 0 and x 0 > 0. Information from the origin can reach these spacetime points. • Absolute past: x 2 > 0 and x 0 < 0. Information from these spacetime points can reach the origin. • Relative present: x 2 < 0. Regardless of the sign of x 0, there is an inertial frame in which x ′0 = 0. Absolute past and future are separated from the relative present by the lightcone, the set of spacetime points with x 2 = 0. Causal behavior in special relativity means that 3-4

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Figure 3.1. A two-dimensional section of Minkowski space, showing the absolute future and past, the relative present and the lightcone.

the origin only receives information from the absolute past and only transmits information to the absolute future. As a classical particle moves through spacetime, it traces out a worldline x μ(σ ) as a function of some parameter σ. The proper time between two timelike-separated events is given by the invariant interval. Similarly, we can calculate an invariant proper time along a worldline using

s12 =

∫1

2

ds =

∫1

2

dx μ dx ν ⎤1/2 . dσ⎡gμν ⎣ dσ dσ ⎦

Taking σ to be the time t in some inertial frame, we find

s12 =

∫ dt[1 − v⃗(t )2]1/2 .

Recalling the classical principle of least time, this is a good starting point for relativistic Lagrangian mechanics. The action S for a relativistic free particle is given by

S = −m

∫ ds = −m ∫ dt

1 − v⃗ 2 .

For small velocities, this expands to be

S=

∫ dt⎡⎣−m + 12 mv⃗2 + ⋯⎤⎦

leading to the correct nonrelativistic behavior. 3.2.2 Lorentz transformations of fields A field is a scalar under Lorentz transformations if the value of the field at a given spacetime point is the same for all inertial observers. If a spacetime point has 3-5

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coordinates x in one frame, then a Lorentz transformation Λ will map those coordinates to new values x′ in another frame via x′ = Λx . If an observer in the first frame measures a scalar field ϕ to have the value ϕ(x ) at x, then an observer in the second frame will measure the same value, so ϕ′(x′) = ϕ(x ), which we can write as ϕ′(x′) = ϕ(Λ−1x′). This transformation law is often written as

ϕ(x ) → ϕ′(x ) = ϕ(Λ−1x ) because x is a dummy index, or even more briefly as

ϕ(x ) → ϕ(Λ−1x ). This can be confusing, but eventually becomes familiar. We can understand the transformation properties of four-vector fields from the behavior of three-dimensional vector fields; after all, rotations are a proper subset of Lorentz transformations. Under a rotation x ⃗ → Rx,⃗ any vector such as a velocity v⃗ will transform in the same way: v⃗ → Rv⃗ . For a three-vector field, such as the velocity field V ⃗ (x ⃗ ) of a fluid, we must also take into account the change in coordinates, so

V ⃗ (x ⃗ ) → RV ⃗ (R−1x ⃗ ). Similarly, a four-vector field must transform as

V (x ) → ΛV (Λ−1x ) or in components

V μ(x ) → Λ.μνV ν(Λ−1x ). Tensor fields transform in an analogous way. If F μν(x ) is a tensor field of rank two, it transforms as

F μν(x ) → Λ.μρΛ.νσF ρσ(Λ−1x ).

3.3 The Lagrangian formalism for fields The generalization of the Lagrangian for fields is the Lagrangian density L which is a function of one or more fields ϕa(t , x ⃗ ) and its derivatives. The Lagrangian density is a local function, depending only on the fields and their derivatives at (t , x ⃗ ) and the Lagrangian L is the integral over space of the Lagrangian density:

L=

∫ d 3x L(ϕa(t, x ⃗)).

The action S is the integral over time of L and thus the integral of L over space and time:

S=

∫ dt L = ∫ dtd 3x ⃗ L.

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This integration is consistent with Lorentz invariance: if L is a local Lorentzinvariant scalar field, then S will be the same in all inertial frames. However, the Lagrangian formalism does not require any particular spacetime symmetry such as Lorentz invariance. The classical equations of motion are

δS = 0. δϕ (t , x ⃗ ) a

If L is a local function only of ϕa and derivatives ∂μϕa , then these equations reduce to the Euler–Lagrange equations for fields

∂L ∂L ⎛ ⎞ − ∂μ⎜ =0 a ∂ϕ (t , x ⃗ ) ∂ ∂ ϕ (t , x ⃗ )) ⎟ ⎠ ⎝ ( μ a

which generalize the Euler–Lagrange equations for particles

∂L ∂L d − ⎛⎜ a ⎞⎟ = 0. a ∂q (t ) dt ⎝ ∂q ̇ (t ) ⎠ If the Lagrangian density contains second-order derivatives and higher, it is usually easiest to work out the equations of motion directly from the functional derivative. 3.3.1 The Klein–Gordon equation To create a relativistically invariant model, we want the Lagrangian density L to be a scalar field. We will attempt to construct a classical field theory model in analogy with the simple harmonic oscillator using a single scalar field ϕ(x ) and its first derivatives. Because ϕ is a Lorentz scalar, so is ϕ2 , but the transformation properties of the derivatives ∂μϕ deserve to be checked. We have

∂μϕ(x ) → ∂μϕ(Λ−1x ) = (Λ−1).νμ∂νϕ(Λ−1x ). We can write the product of two derivatives of ϕ as

(∂μϕ(x))(∂ μϕ(x)) = g μν(∂μϕ(x))(∂νϕ(x)) which should be a scalar. The transformation is

g μν(∂μϕ(x ))(∂νϕ(x )) → g μν(Λ−1).ρμ∂ρϕ(Λ−1x )(Λ−1).σν ∂σϕ(Λ−1x ). This will be invariant if

(Λ−1).ρμ(Λ−1).σν g μν = g ρσ which is just the statement that the metric tensor is the same in all inertial frames. We now consider the Lagrangian density

L=

1 1 (∂μϕ(x))(∂ μϕ(x)) − 2 m2ϕ2(x) 2

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which is Lorentz-invariant:

L(x ) → L(Λ−1x ). In order to find the equations of motion, it is often easier to start from

L=

1 1 μν g (∂μϕ(x ))(∂νϕ(x )) − m 2ϕ 2(x ). 2 2

This makes clear that the factor of 1/2 in the kinetic terms is canceled, so the Euler– Lagrange equation for ϕ is

−m 2ϕ(x ) − ∂μ(g μν∂νϕ(x )) = 0 or

∂μ∂ μϕ + m 2ϕ = 0, the Klein–Gordon equation. The operator ∂μ∂ μ, known as the d’Alembert operator, is the four-dimensional generalization of the Laplacian operator ∇2 , given by

∂μ∂ μ = ∂t2 − ∇2 . There are several short notations for the d’Alembertian, with □ and ∂ 2 being two common choices. Similarly, the Lagrangian density is often written as

L=

1 1 ∂μϕ∂ μϕ − m 2ϕ 2(x ) 2 2

L=

1 1 (∂ϕ)2 − m 2ϕ 2(x ). 2 2

or

The quantum-mechanical interpretation of the Klein–Gordon equation is simple. With the usual associations Eop = i ∂t and pop ⃗ = −i ∇, the Klein–Gordon equation is

(p⃗

2 op

)

2 + m 2 ϕ = Eop ϕ

analogous to the Schrödinger equation, but with the energy-momentum relation of a free relativistic particle. Linear superposition allows us to construct general solutions from plane waves of the form

ϕ p ⃗(x ) = A p ⃗ e i (p ⃗ ·x ⃗−Et ) where E 2 = p ⃗ 2 + m2 . Using the four-vector p μ = (E , p ⃗ ) , we can write this as

ϕ p ⃗(x ) = A p ⃗ e−ip·x using the inner product for four-vectors. Note that both positive and negative values of E are allowed by the Klein–Gordon equation, an interpretational issue not found in the Schrödinger equation for a free particle where E is always positive.

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3.3.2 Maxwell’s equations Maxwell’s equations are field equations for the electric field E ⃗(t , x ⃗ ) and the magnetic field B ⃗(t , x ⃗ ). We will use the Heaviside–Lorentz rationalized units. The pair of equations for the divergences of E ⃗ and B ⃗ do not involve time derivatives of the fields:

∇ · E⃗ = ρ ∇ · B⃗ = 0 where ρ(t , x ⃗ ) is the local charge density. The pair of equations involving the curl of E ⃗ and B ⃗ also involve time derivatives and link the behavior of the two fields

∂B ⃗ =0 ∂t ∂E⃗ = j⃗ ∇ × B⃗ − ∂t

∇ × E⃗ +

where j ⃗ is the charge current density. The charge density and charge current form a conserved four-vector field j μ (x ) = (ρ(x ), j ⃗ (x )):

∂ρ + ∇ · j ⃗ = ∂μj μ = 0. ∂t The Lorentz transformation properties of E ⃗ and B ⃗ are not immediately obvious, as they are not spatial parts of a four-vector field. The E ⃗ and B ⃗ field may be written in terms of the scalar potential ϕ(x ) and the vector potential A⃗ (x ) as

∂A ⃗ − ∇ϕ ∂t B ⃗ = ∇ × A⃗ . E⃗ = −

The scalar and vector potentials form a four-vector potential Aμ(x ) = (ϕ(x ), A⃗ (x )), and the E ⃗ and B ⃗ fields are the off-diagonal components of the antisymmetric two-tensor

F μν = ∂ μAν − ∂ νAμ = −F νμ with

E j = − F 0j ϵ jkl B l = − F jk where ϵ jkl is the totally antisymmetric three-tensor. The covariant form of the two of Maxwell’s equations involving j u is

∂μF μν = j ν and the other two equations are

ϵ μνρσ ∂νFρσ = 0

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which follows from the definition of Fμν :

ϵ μνρσ ∂ν(∂ρAσ − ∂σAρ ) = 0. The Lagrangian for the electromagnetic field including the interaction with jμ is

1 L = − FμνF μν − Aμ jμ 4 which can be written as

L=−

1 (∂μAν ∂ μAν − ∂μAν ∂ νAμ ) + Aμ jμ . 2

From this Lagrangian, we obtain

∂μ

∂L ∂L = − ∂μ(∂ μAν − ∂ νAμ ) + j ν − ∂(Aν ) ∂(∂μAν ) = − ∂μF μν + j ν =0

Exercise: Show that the non-interacting term in the Lagrangian can be written in terms of E ⃗ and B ⃗ as

−

1 1 2 2 FμνF μν = E ⃗ − B ⃗ 4 2

(

)

3.3.3 The Schrödinger equation The Schrödinger equation is

i ∂tψ = −

1 2 ∇ ψ + Vψ 2m

where the field ψ is complex. Complex fields can always be decomposed as a real and imaginary part ψ = ψR + iψI . We can then look for L(ψR, ψI ) which will give us two Euler–Lagrange equations that can be combined to give the Schrödinger equation. Fortunately, there is a simpler procedure. We note that

1 ψR = (ψ + ψ *) 2 1 ψI = (ψ − ψ *) 2i so that

1 δψR = (δψ + δψ *) 2 1 δψI = (δψ − δψ *) 2i

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which in turns implies that any variation of ψR and ψI can be understood as a variation of ψ and ψ *. The field equations

δS =0 δψR(x ) δS =0 δψI (x ) are thus equivalent to

δS =0 δψ (x ) δS =0 δψ *(x ) when ψ and ψ * are treated as if they were independent variables. We now look for an action S which gives us

δS 1 2 = i ∂tψ + ∇ ψ − Vψ = 0. * δψ (x ) 2m The action S is simply

S=

∫

=

∫

1 2 dtd 3x ψ *⎛i ∂tψ + ∇ ψ − Vψ ⎞ 2m ⎝ ⎠ 1 (∇ψ *)(∇ψ ) − ψ *Vψ ⎤ dtd 3x ⎡ψ *i ∂tψ − 2m ⎣ ⎦

where the overall sign is chosen to give the standard Hamiltonian. Exercise: The action of the nonlinear Schrödinger equation is given by

S=

∫ dtd 3x ψ *⎛⎝i ∂tψ + 21m ∇2 ψ − 2κ (ψ *ψ )2⎞⎠.

Determine the equation of motion for ψ.

3.4 Continuous symmetries in classical field theory Continuous symmetries are more powerful for field theories than for point particles. For particles, continuous symmetries lead to conserved global charges. Time translation invariance leads to conservation of energy; translation symmetry leads to conservation of total momentum. For field theories, continuous symmetries lead to local conservation laws, which in turn lead to global conserved charges. The formal treatment in the two cases is similar, however. We consider a Lagrangian L which depends on a collection of fields ϕa . Under a general transformation ϕa → ϕa + δϕa and we ask how L changes. One possibility is that L → L, but the general case for a symmetry is that δ L, the change in L induced by δϕa , can be written as 3-11

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δ L(x ) = ∂μδF μ(x ) for some F μ without the use of the equations of motion. We do not assume that this model is relativistically invariant, and F μ need not be a four-vector This restriction on the use of the equations of motion is necessary because any change δϕa will not change S if ϕa is a solution of the equations of motion. If δ L can be put into this form, then the action S does not change under reasonable boundary conditions. By the chain rule, we have

δS =

∫ d 4x⎡⎢ ∂∂ϕLa δϕa + ∂(∂∂μLϕa ) δ(∂μϕa )⎤⎥. ⎦

⎣

If ϕa is a solution of the field equations, we have

∂L ∂L ⎛ ⎞ − ∂μ⎜ =0 ∂ϕ (t , x ⃗ ) ∂(∂μϕ a (t , x ⃗ )) ⎟ ⎠ ⎝ a

so that

δS =

∫ d 4x⎡⎢∂μ⎛⎜ ∂(∂ ϕ∂aL(t, x ⃗)) ⎞⎟δϕa + ∂(∂∂Lϕa ) δ(∂μϕa )⎤⎥ = ⎣ ⎝

μ

μ

⎠

⎦

∫ d 4x⎡⎢∂μ⎛⎜ ∂(∂ ϕ∂aL(t, x ⃗)) δϕa⎞⎟⎤⎥. ⎣ ⎝

μ

⎠⎦

It is customary to define a four-vector

π aμ =

∂L ∂(∂μϕ a (t , x ⃗ ))

so that

δS =

∫ d 4x⎡⎣∂μ(π aμδϕa )⎤⎦.

The change in the action is also equal to

δS =

∫ d 4x ∂μδF μ(x)

so

∫ d 4x ∂μ(π aμδϕa − δF μ) = 0 for any region of space and time we choose to integrate over. It follows that

∂μ(π aμδϕ a − δF μ) = 0 everywhere. This is Noether’s theorem. We define the Noether current associated with a symmetry to be 3-12

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

J μ = π aμδϕ a − δF μ and say it is a conserved current. If we write such a conserved current J μ as (ρ , J ⃗ ) then ρ is the charge density and J ⃗ is the charge current density. The local current conservation equation ∂μJ μ = 0 becomes

∂ρ + ∇ · J ⃗ = 0. ∂t Suppose we have a spatial volume V with a closed surface S. There is an amount of the conserved quantity inside V which is

∫V d 3x J 0(t, x ⃗). The time rate of change of this quantity is

d dt

∫V d 3x J 0(t, x ⃗) = ∫V d 3x ∂0J 0(t, x ⃗) = −∫V d 3x ∇ · J (⃗ t, x ⃗).

By the divergence theorem of Gauss. this is

d dt

∫V d 3x J 0(t, x ⃗) = −∫S dσ ⃗ · ∇ · J (⃗ t, x ⃗)

where the surface element dσ⃗ points outward from S. This result tells us that any decrease in the amount of charge in V arises because of a current outward from S. Furthermore, we can integrate over all space at some fixed time, or more generally a timelike surface in special relativity, to define a global charge

Q (t ) =

∫ d 3x J 0(t, x ⃗).

We now have

d Q (t ) = − dt

∫ dσ ⃗ · J (⃗ t, x ⃗) = 0

assuming there is zero current at spatial infinity. In the case of a finite volume with zero current at the boundary, this also gives a conserved charge inside the volume. 3.4.1 Example: translation in space and time We consider translations of the form x μ → x μ + ϵa μ where a μ is some fixed fourvector and for simplicity consider a single relativistic scalar field ϕ. The infinitesimal change in ϕ is δϕ = ϵa μ∂μϕ, while the change in L is

δ L = ϵa μ∂μL . This leads to a conserved current

J μ = π μa ν∂νϕ − a μL

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for every vector a μ. However, we can write this as

J μ = π μaν∂ νϕ − g μνaνL = (π μ∂ νϕ − g μνL)aν so it is actually the second-rank tensor

T μν = π μ∂ νϕ − g μνL = ∂ μϕ∂ νϕ − g μνL which is conserved

∂μT μν = 0. This tensor is known as the stress-energy tensor. Note that T μν = T νμ, so ∂νT μν = 0 as well. The globally conserved charges are

Pν =

∫ d 3x T 0ν,

with P 0 the total energy and the three components P j with j = 1, 2, 3 the total momentum in the three spatial directions. Example: In quantum field theories, an internal symmetry of a quantum field theory is associated with transformations of the fields without acting on space and time. Consider the Lagrangian of a complex field ϕ*(x ) with Lagrangian density

L = (∂μϕ*)(∂ μϕ) − m 2ϕ*ϕ − λ(ϕ*ϕ)2 . The Lagrangian is invariant under the internal symmetry transformation where ϕ → e iθϕ and ϕ*→e−iθϕ*. This is called a U(1) symmetry because the set of complex numbers of the form e iθ form a group under multiplication: e iθ1e iθ2 = e i(θ1+θ2). We can also write this Lagrangian in terms of the real and imaginary parts of ϕ, defined by

ϕ=

1 (ϕR + iϕI ) 2

where the factor of 1/ 2 is conventional. Substituting into the Lagrangian, we find

L=

1 1 1 1 λ 2 ∂μϕ1∂ μϕ1 − m 2ϕ12(x ) + ∂μϕ2∂ μϕ2 − m 2ϕ22(x ) − (ϕ12 + ϕ22 ) 2 2 4 2 2

which is invariant under the symmetry group of two-dimensional rotations SO(2), acting as

ϕ1 → cos θϕ1 − sin θϕ2 ϕ2 → sin θϕ1 + cos θϕ2 . As this example shows, the groups U (1) and SO(2) are actually two names for the same symmetry group. Using our procedure for variations of complex fields, we find the field equation for ϕ is

∂ 2ϕ + m 2ϕ + 2λ(ϕ*ϕ)ϕ = 0 and the corresponding equation for ϕ* is the complex conjugate of the equation for ϕ. Under an infinitesimal transformation ϕ → e iϵϕ, we have δϕ = iϵϕ and δϕ* = −iϵϕ*; L is invariant so δ L = 0. The fields π μ and π *μ are defined by 3-14

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

πμ =

∂L = ∂ μϕ* ∂(∂μϕ)

π *μ =

∂L = ∂ μϕ . ∂(∂μϕ*)

and

The conserved current is

J μ = π μδϕ + π *μδϕ* = iϕ∂ μϕ* − iϕ*∂ μϕ . We can check that this current is conserved by taking its divergence: μ * μ 2 * 2 ∂μJ μ = i⎡ ⎣∂μϕ∂ ϕ − ∂μϕ∂ ϕ + ϕ∂ ϕ − iϕ∂ ϕ⎤ ⎦=0

where the first two terms cancel, and the second pair of terms cancels after the use of the equations of motion.

3.5 The Hamiltonian formalism The construction of the Hamiltonian formalism proceeds by analogy with the Hamiltonian formalism for particles. For a scalar field ϕ(x ), we defined a conjugate momentum field π (x ) by

π (x ) =

∂L(x ) ∂(∂ 0ϕ(x ))

which is also the field π 0(x ) used in our discussion of symmetry. Be aware that π (x ) plays a role in quantum field theory similar to the p operator in quantum mechanics, but it is not in and of itself a physical momentum or momentum density in field theory. The Hamiltonian density is defined by

H(x ) = π (x )∂ 0ϕ(x ) − L(x ) with the understanding that we solve for ∂ 0ϕ in terms of π and ϕ. In the case of the Klein–Gordon equation, this leads to

H(x ) =

1 1 2 1 π + ( ∇ϕ ) 2 + m 2 ϕ 2 . 2 2 2

The Hamiltonian H is given by

H=

∫ d 3x ⃗ H = ∫ d 3x ⎡⃗⎣ 12 π 2 + 12 (∇ϕ)2 + 12 m2ϕ2⎤⎦.

It is easy to see that H is in fact T 00 .

3.6 Causality Suppose we want to find the behavior of a classical observable such as x(t ) or ϕ(x ) by solving a differential equation of the form 3-15

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Dx(t ) = J (t ) or

Dϕ(x ) = J (x ) where D is a differential operator and J is a known function. In the first case, for example, we might have 2

d D = ⎛ 2 + ω 2⎞ ⎝ dt ⎠ ⎜

⎟

and J (t ) is a known function of time. In the second case, we might have

D = ∂ 20 − ∇2 + m 2 . If we find a solution, it is a particular solution, but we can always a solution x0(t ) or solutions ϕ0(x ) of the homogeneous equation, Dx0(t ) = 0 or Dϕ0 = 0, to form a new particular solution via x(t ) → x(t ) + λx0(t ) or ϕ(t ) → ϕ(x ) + λϕ0(x )/ This means we must provide additional information to unambiguously determine the solution we want. We note that Dx = J and Dϕ = J look a lot like a matrix equation. That’s essentially what they are, with a formal solution x = D−1J . If D were a matrix, we would determine D−1 by requiring DD−1 to be the identity matrix. Generalizing that idea to the problem at hand, we seek in the first case a function G (t , t′) such that

D(t )G (t , t′) = δ(t − t′). If we can determine G (t , t′) then a particular solution xp(t ) is given by

xp(t ) =

∫ dt′G(t, t′)J (t′)

because

∫ =∫

D(t )xp(t ) =

dt′D(t )G (t , t′)J (t′) dt′δ(t − t′)J (t′)

J (t ). The function G (t − t′) is a Green function for the operator D(t ). Typically there are multiple Green functions for a given operator. We illustrate this with the following example. Consider the case 2

d D(t ) = ⎛ 2 + ω 2⎞ ⎝ dt ⎠ ⎜

⎟

associated with the simple harmonic oscillator. Time translation invariance allows us to write G as a function of t − t′, and we solve for G using Fourier transform methods.

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Our conventions for Fourier transforms, which we use throughout, are that frequency and wavenumber integrals have factors of 1/2π , but integrals over time and space do not. Thus

x( t ) =

dE x˜(E )e−iEt 2π

∫

and

x˜(E ) =

∫ dt x(t )e+iEt ,

consistent with the representation of the Dirac δ-function as

δ (t ) =

dω −iωt e . 2π

∫

The Fourier transform of 2

⎛ d + ω 2⎞G˜ (t − t′) = δ(t − t′) 2 ⎝ dt ⎠ ⎜

⎟

is easily found to be

( −E 2 + ω 2 )G˜ (E ) = 1 suggesting that

G˜ (ω) =

−1 . E 2 − ω2

In order to unambiguously define G (t ) as

G (t ) =

∫

−1 dω e−iωt 2 2π E − ω 2

we must specify how to deal with the poles on the real axis at E = ±ω. Because we are solving for the behavior of a classical system, we require causal behavior: xp(t ) depends on J (t′) only if t′ < t . This implies that G (t − t′) = 0 if t < t′. Equivalently, we want G (t ) = 0 if t < 0. This behavior helps define the retarded Green function GR(t ) for the simple harmonic oscillator. A standard approach to an integral of this form is contour integration, where the contour is closed in the complex plane by an arc at ∣ω∣ = ∞, either in the upper-half plane or the lower-half plane. If t > 0, the exponential factor exp( −iEt ) blows up as Im(E ) → +∞, so the contour must be closed in the lower-half plane. Similarly, if t < 0, the exponential factor exp( −iEt ) blows up as Im(E ) → +∞, so the contour must be closed in the upper-half plane. In either case, the integral will be given by Cauchy’s theorem in terms of the residues at the poles. Thus the causality requirement G (t ) = 0 for t < 0 becomes a requirement that there be no poles inside the contour around the upper-half plane. The poles at E = ±ω are then inside the contour around the lower-half plane. There are two equivalent ways to think about this. The first is to deform the contour along the real

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

axis such that a small half-circle of radius ϵ jumps over the two poles. Equivalently, we add a small imaginary part −iϵ to the poles at ±ω so that the poles now lie at ±ω − iϵ , just below the real axis. In either case, the limit ϵ → 0 limit is taken after the effects of the choice of contour are taken into account. The calculation of the retarded Green function GR(t ) for t > 0 is now easily performed as

e−iEt ⎤ GR(t ) = ( − 2πi )∑Residue⎡ − ⎢ ⎣ (E − ω + iϵ )(E + ω + iϵ ) ⎥ ⎦ −iωt +iωt e e ⎤ = i⎡ − ⎢ 2ω ⎥ 2ω ⎦ ⎣ sin (ωt ) . = ω A formula valid for all values of t can be written using the Heaviside step function θ (t ) :

GR(t ) =

sin (ωt ) θ (t ). ω

This should be compared with the Feynman propagator from chapter 2:

⎧ 1 e−iωt t > 0 ⎪ i ΔF (t ) = 2ω ⎨ 1 +iωt t < 0. ⎪ 2ω e ⎩ Obviously, the two propagators are closely related but not the same, differing in how the poles in E-space are treated. We can repeat this treatment for higher-dimensional equations like the wave equation. The prescription is the same: in order to have causal behavior, the poles in energy, or frequency, must all lie in the lower-half plane. All of this goes through essentially unchanged in the case of ϕ, where the classical equation of motion in an external field is

(∂ 2 + m 2 )ϕ(x ) = J (x ). The inhomogeneous part of the solution can be written as

ϕ(x ) =

∫ d 4y′GR(x − y )J (y ).

where the retarded Green function is

GR(x ) =

∫

d 4p −1 e−ip·x 2 4 0 (2π ) (p + iϵ ) − p ⃗ 2 + m 2

This result can also be used to obtain the causal propagator for electromagnetic waves, where m = 0.

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It is obvious that the Feynman propagator of chapter 2 and the causal propagator are different, because they treat the poles differently. We will come back to the interpretation of the Feynman propagator in chapters 4 and 5.

Problems 1. The Lagrangian for the Schrödinger equation is invariant under the internal symmetry ψ → e iθψ . Find the conserved current jμ and the associated charge Q in terms of ψ. What is its physical significance? 2. Show that the action of a massless field theory with Lagrangian

L=

1 (∂μϕ)(∂ μϕ) 2

is invariant under dilatation D: ϕ(x ) → (1 + ϵ )ϕ((1 + ϵ )x). (a) What is the corresponding conserved current jDμ? (b) Suppose we add a mass term to the Lagrangian: L → L − m2ϕ2 /2 . What is the divergence of the current jDμ now? 3. The Lagrangian density for a free, massive spin-1 particle is given in terms of a four-vector field Aμ(x ) with Lagrangian

1 1 L = − FμνF μν + m 2Aμ Aμ 4 2 where Fμν = ∂μAν − ∂νAμ. (a) Find the Euler–Lagrange equations. (b) Show that the Euler–Lagrange equations imply that the four-divergence of Aμ is zero: ∂ · A = 0. This constraint implies that there are three independent degrees of freedom, appropriate for a spin-1 excitation. (c) Using your results from (b), recover the Klein–Gordon equation for Aμ .

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IOP Publishing

A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 4 Free quantum fields

In this chapter we introduce the simplest quantum fields, corresponding to noninteracting particles. In quantum mechanics, time t and particle position x ⃗ are very different: time t is a parameter, while x ⃗ is an observable, implemented in quantum mechanics as an Hermitian operator in a Hilbert space. This asymmetry is removed in quantum field theory, where the spacetime coordinate x is a parameter, and the role played in quantum mechanics by the position operator x(t ) in the Heisenberg representation is replaced by the quantum field ϕ(t , x ⃗ ) or simply ϕ(x ) where x is a four-vector. Like x(t ), ϕ(x ) is a Heisenberg-picture operator in a Hilbert space. We will start from the functional formalism, leaning heavily on our results for the simple harmonic oscillator. We will then discuss how equivalent results are obtained in the canonical formalism and its connection to the path integral formalism. Throughout most of this chapter, the scalar field theory will be the Klein–Gordon model, the simplest relativistic quantum field theory.

4.1 The Feynman path integral for field theories The functional integral which turns a classical field theory into a quantum field theory is not very different conceptually from the method we used to go from classical mechanics to quantum mechanics. For simplicity we initially discuss the case of a single scalar field ϕ(x ) which is a function of spacetime. The classical dynamics are obtained from a Lagrangian density L(ϕ(x )). Henceforth, we will refer to such densities simply as Lagrangians. The initial value of the field at some time −T /2 in the distant past is specified by ϕ(t = −T /2, x ⃗ ) = ϕi (x ⃗ ); note that ϕi (x ⃗ ) is defined at a fixed time. Similarly, we define the final value of the field at some time in the distant future +T /2 to be ϕ(t = +T /2, x ⃗ ) = ϕf (x ⃗ ). The functional integral gives the matrix element for the time evolution

doi:10.1088/978-0-7503-3227-9ch4

4-1

ª IOP Publishing Ltd 2022

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

ϕf exp[ −iTH ] ϕi =

ϕ(T /2, x ⃗ )=ϕf (x ⃗ )

∫ϕ(−T /2, x⃗)=ϕ (x⃗) [dϕ]eiS[ϕ] i

where H is the Hamiltonian associated with L and the action S is given by S [ϕ ] = ∫ d 4x L. Instead of integrating over paths x(t ) connecting xi and xf, we are now integrating over all field configurations connecting ϕi (x ⃗ ) and ϕf (x ). Obviously, this singles out a particular inertial frame of reference in order to define H. If we insert a field ϕ(x ) at a time −T /2 < x 0 < +T /2, we argue as before that this must be interpreted as ϕ(T /2, x ⃗ )=ϕf (x ⃗ )

∫ϕ(−T /2, x⃗)=ϕ (x⃗) [dϕ]eiS[ϕ] =

ϕf e−i (T /2−x

0 )H

0

ϕ(x ⃗ ) e−i (x +T /2)H ϕi

i

where ϕ(x ⃗ ) is a Schrödinger representation quantum field. Such objects are not commonly used in quantum field theory, but we will pass immediately to the Heisenberg representation field given by

ϕH (x ) = e+itH ϕ(x ⃗ )e−itH . As in the quantum mechanical case, we identify the large T limit of e−i (T /2)H ϕi as

e−i (T /2)H ϕi → e−i (T /2)E0 0 0 ϕ

0

where 0 is the vacuum state and E0 is the vacuum energy. One change from quantum mechanics is that a non-zero constant vacuum energy density will lead to an infinite vacuum energy E0. It is common to imagine the system in a large box of volume V in order to make E0 more tractable when this is desirable. Note however that, as in quantum mechanics, we do not need the absolute value of the vacuum energy for perturbation theory. Our general expression for an n-point function is

G (n)(x1, … , xn) = 0 T [ϕ(x1) ... ϕ(xn)] 0 =

∫

[dϕ ] ϕ(x1) ... ϕ(xn)e iS[ϕ]

∫

. [dϕ ] e iS[ϕ]

If we assume the vacuum is Lorentz invariant, then the frame dependence induced by the choices ϕi and ϕf for the initial and final values of the field is removed, and we have a Lorentz-invariant formalism if L is Lorentz invariant. The generating functional Z [J ] for a quantum field theory is given by the functional integral

Z [J ] =

∫ [dϕ] exp ⎡⎣⎢iS[ϕ] + i ∫ d 4x J (x)ϕ(x)⎤⎦⎥.

We obtain the n-point functions from functional differentiation as before

⎛ 1 ⎞n 1 δ nZ [J ] G (n)(x1, … , x n) = ⎜ ⎟ ⎝ i ⎠ Z [0] δJ (x1)…δJ (xn)

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The n-point functions are the coefficients of the expansion of Z [J ] in a functional Taylor series

⎡

Z [J ] =

∞

⎤

n

∫ [dϕ]eiS[ϕ]⎢⎢⎣1 + ∑ ni ! ∫ dx1 ... dxnJ (x1)ϕ(x1) ... J (xn)ϕ(xn)⎥⎥⎦ n=1

so

⎡ Z [J ] = Z [0]⎢1 + ⎢⎣

∞

in ∑n ! n=1

∫

⎤ dx1 ... dxnJ (x1) ... J (xn)G (n)(x1, … , xn)⎥ ⎥⎦

We also have the generator of connected diagram W [J ] defined by

Z [J ] = e iW [J ] or equivalently

W [J ] = −i log Z [J ]. It is straightforward to show that

δW [J ] δJ (x1)

J =0

⎛ 1 ⎞ δ 2W [J ] ⎜ ⎟ ⎝ i ⎠ δJ (x1)δJ (x2 )

J =0

= 0 ϕ(x1) 0 = 0 T [ϕ(x1)ϕ(x2 )] 0 − 0 ϕ(x1) 0 0 ϕ(x2 ) 0

and the connected n-point functions Gc(n)(t1, … , tn ) are given by

⎛ 1 ⎞n−1 δ 2W [J ] Gc(n)(x1, … , xn) = ⎜ ⎟ ⎝ i ⎠ δJ (x1)…δJ (xn)

. J =0

4.2 Free scalar fields The Lagrangian density for the a free, real relativistic scalar field is

L=

1 1 ∂μϕ(x ))(∂ μϕ(x )) − m 2ϕ 2(x ) ( 2 2

and the action is

S [ϕ , J ] =

⎡

⎤

∫ d 4x L = ∫ d 4x ⎢⎣ 12 ∂μϕ∂ μϕ − 12 m2ϕ2 + Jϕ⎥⎦

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when an external field is included. Integrating by parts, we write this as

S [ϕ , J ] =

∫

=

∫

⎡1 ⎤ 1 d 4x ⎢ ϕ( −∂ 2ϕ) − m 2ϕ 2 + Jϕ⎥ ⎣2 ⎦ 2 1 d 4x d 4y ϕ(x )( −∂ 2x − m 2 + iϵ )δ 4(x − y )ϕ(y ) 2

∫

∫

+

d 4x J (x )ϕ(x )

in order to emphasize the analogy with finite-dimensional Gaussian integrals. We have again used the Feynman prescription and included an iϵ to ensure convergence. The operator ( −∂ 2x − m2 + iϵ )δ 4(x − y ) has an inverse ΔF (y − z ) satisfying

∫ d 4y (−∂ 2x − m2 + iϵ)δ 4(x − y )ΔF (y − z) = δ 4(x − z). This reduces to the partial differential equation 2 ( −∂ x − m 2 + iϵ )ΔF (x − z ) = δ 4(x − z )

which has the solution

ΔF (x − z ) =

∫

d 4k 1 e−ik·(x−z ). (2π )4 k 2 − m 2 + iϵ

The generating function Z0[J ] for a free field is given by

Z0[J ] =

∫ [dϕ] eiS[ϕ, J ].

If we make the change of variable

ϕ(x ) → ϕ(x ) −

∫ d 4y ΔF (x − y )J (y ),

inside the functional integral, the action becomes

S [ϕ , J ] → S [ϕ , 0] −

∫ d 4x ∫ d 4y 12 J (x)ΔF (x − y )J (y ).

This allows us to write

⎡ ⎤ 1 [dϕ ] exp ⎢iS [ϕ , 0] − i d 4x d 4y J (x )ΔF (x − y )J (y )⎥ ⎣ ⎦ 2 ⎡ ⎤ 1 = Z0[0] exp ⎢ d 4x d 4y iJ (x )i ΔF (x − y )iJ (y )⎥ . ⎣ ⎦ 2

Z0[J ] =

∫

∫

∫

∫

∫

As we have seen previously, the value of Z0[0] does not enter into the perturbative formula for the time-ordered Green functions. For the free theory, we have

G0(2)(x − y ) = 0 T [ϕ(x )ϕ(y )] 0 = i ΔF (x − y ). In this regard, note that our result also tells us that the generator of connected diagrams for the free field is

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iW0[J ] = iW0[0] +

∫ d 4x ∫ d 4y 12 iJ (x)iΔF (x − y )iJ (y )

so all the connected Green functions of order higher than two are zero: Gc(n)(x1 ... xn ) = 0 for n ⩾ 3.

4.3 Another approach to the functional integral As we have seen above, we can fix the J dependence of the generating function without a complete calculation of the functional integral. There is a widely used alternative approach to Gaussian functional integrals which is often useful. Suppose we place our free scalar field in a large spacetime box of size VT, leading to discrete eigenmodes of the d’Alembertian ∂ 2. We write the field as a sum over Fourier modes

ϕ(x ) =

1 ∑ϕ˜ (k )e−ik·x VT k

It is straightforward to show that the action can be written as

⎡1 ⎤ 1 d 4x ⎢ ϕ( −∂ 2ϕ) − m 2ϕ 2 + Jϕ⎥ ⎣2 ⎦ 2 ⎡1 ⎤ = ∑⎢ ϕ˜ ( −k )(k 2 − m 2 + iϵ )ϕ˜ (k ) + J˜ ( −k )ϕ(k )⎥ . ⎣2 ⎦

S [ϕ , J ] =

∫

k

It may seem from this expression that we have doubled the number of degrees of freedom because ϕ˜ (k ) is generally complex quantity, whereas ϕ(x ) is real. However, the reality of ϕ places a restriction on the behavior of ϕ˜ : ϕ˜ (k )* = ϕ( −k ) for the Fourier transform of a real function. Each pair of modes, k and −k may be integrated independently of the other pairs, giving ⎡ 1 ⎤ ⎡ ⎤1/2 2πi i ˜ (k )⎥ ⎢∑ iJ˜ ( −k ) exp iJ Z [J ] = ∏ ⎢ 2 ⎥ k 2 − m 2 + iϵ ⎣ (k − m 2 + iϵ ) ⎦ ⎢⎣ k 2 ⎥⎦ k Of course, this is essentially equivalent to our previous result for Z [J ], distinguished only by the discrete momentum states.

4.4 Interpretation of Z[0] for free fields Up to constant factors, the m-dependent part of Z[0] is given by

Z [0] =

⎡ 1

⎤

∏ (k 2 − m2 + iϵ)−1/2 = exp⎢⎢− 2 ∑ log (k 2 − m2 + iϵ)⎥⎥. ⎣

k

k

⎦

In the limit where the volume of spacetime VT is large, we can replace the sum over allowed values of k by an integral over the density of states:

∑ → VT ∫ k

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

and thus

⎡ 1 Z [0] = exp⎢ − VT ⎣ 2

⎤ d 4k log (k 2 − m 2 + iϵ )⎥ . 4 (2π ) ⎦

∫

This integral is divergent as k becomes large, but it nevertheless has physical import. Let us define

I=

∫

d 4p log (k 2 − m 2 + iϵ ) (2π )4

and note that

∂I = ∂m 2

∫

d 4p −1 . 4 2 (2π ) k − m 2 + iϵ

2 2 This integral has poles at k 0 = k ⃗ + m2 − iϵ and at k 0 = − k ⃗ + m2 + iϵ . We can close the contour in either the upper half-plane or the lower half-plane. Performing the contour integral over k 0 in a now-familiar way, we find

∂I = ∂m 2

∫

d 3k i (2π )3 2ωk

which can be integrated to give

I=i

d 3p 1 ωp (2π )3 2

∫

up to a constant independent of m. We now have

⎡ Z [0] ∝ e−iE0T = exp⎢ −iVT ⎣

∫

d 3k 1 ⎤ ωk ⎥ (2π )3 2 ⎦

where we identify E0 as the (infinite) vacuum energy given by

E0 = V

∫

d 3k 1 ωk . (2π )3 2

This has a clear physical interpretation: the vacuum state of a free field consists of an infinite number of quantum mechanical harmonic oscillators, all in their ground state. The energy of the vacuum is the sum of the zero-point energies for all those modes. In fact, even the vacuum energy density ϵ0 given by

ϵ0 =

E0 = V

∫

d 3p 1 ωp (2π )3 2

is infinite.

4.5 Vacuum energy examples The energy of the ground state is a fundamental quantity in quantum mechanical systems, and the energy density of the vacuum is similarly fundamental in quantum 4-6

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field theories. Although there are divergences in the energy density of a free field, differences of that quantity under different conditions are meaningful. 4.5.1 Casimir effect Although the vacuum energy in a relativistic field theory may be divergent, differences in vacuum energy densities can be finite. The Casimir effect is the change in the vacuum energy caused by the change in the boundary conditions of field due to the presence of objects made of conducting metals or dielectric materials. The most straightforward case is that of photons in the presence of two large parallel conducting plates. The change in the ground state energy as the distance between the plates varies leads to an attractive force between the plates. This effect was first predicted by Hendrik Casimir in 1947, but first quantitatively measured in experiment by Lamoreaux in 1997. The presence of the conducting plates alters the dispersion relation of the photons between the two plates. Taking the plates to be perpendicular to the z-axis and separated by a distance a, we must alter the dispersion relation from ω = k ⃗ to

k x2 + ky +

ω(a ) =

n 2π 2 a2

because the electric field perpendicular to the surface must be zero at the surface of the conducting plates. The energy of the volume between the plates is given by ∞

E 0(a ) = A∑2

∫

n=1

dkxdky 1 ω(a ) (2π )2 2

where A is the area of each plate, and an overall factor of two is included because photons have two polarization states. Note that the energy density E0(a )/aA is equal to the vacuum energy density E0 /V in the limit a → ∞. While the expression for E0(a ) is divergent, there are several approaches to extracting its finite part. Perhaps the most physical is to compute the difference E0(a ) − E0(∞); this derivation is carried out in the field theory book of Itzykson and Zuber [1]. We will instead compute the difference E0(a ) − E0(a¯ ) between the vacuum energies for two different separations a and a¯ . We will combine this with a technique called zeta function regularization. In quantum field theory, regularization techniques allow us to manipulate divergent quantities. We define an integral ∞

I (s , a ) =

∑∫ n=1

d 2k ω(a )1−s 4π 2

which reduces to the desired integral when s = 0, giving the energy per unit area. Performing the integration over the kx − ky plane using polar coordinates, we find

1 2 ⎛ 2 n 2π 2 ⎞ I (s , a ) = ⎜k + 2 ⎟ ∑ 2π n = 1 3 − s ⎝ a ⎠ ∞

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k 2=∞

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Although this integral is convergent for Re[s ] > 3, it is not convergent as s → 0. Keeping s arbitrary, we obtain after some algebra the result I (s, a ) − I (s, a ) = −

∞ ⎛ 1 1 ⎞ 1 1 π 2 −s ⎛⎜ 1 π 2 −s 1 1 ⎞ − 3 −s ⎟ = ζ (s − 3)⎜ 3 −s − 3 −s ⎟ ∑ 3 − s s − 3 ⎝a 2 ⎝a 2 3−s a¯ ⎠ 3 − s n = 1 n a¯ ⎠

where we have used the standard representation of the Riemann zeta function ∞

ζ (s ) =

1 . ns n=1

∑

Although this representation is convergent only for Re[s ] > 1, the Riemann zeta function itself can be analytically continued if s ≠ 1. The values for negative integers were first computed by Euler:

ζ( −n ) = ( −1)n

Bn+1 n+1

where Bn is the nth Euler number. Hence ζ (3) = 1/120 and

I (0, a ) − I (0, a¯ ) = −

π2 ⎛ 1 1⎞ ⎜ 3 − 3 ⎟. 720 ⎝ a a¯ ⎠

The force per unit area, which is a pressure, is given by

p=

F d π2 . = − [I (0, a ) − I (0, a¯ )] = − A da 240a 4

The force is negative, indicating that the forces between the plates are attractive. Note that the result has units of 1/[L ]4 = [E /L3], which is correct for a pressure or energy density. The factor which relates energy and inverse length in standard units is ℏc , so in standard units we have

p=−

ℏcπ 2 . 240a 4

While the Casimir effect is extremely small, it demonstrates that the zero-point energy of the vacuum leads to physical consequences. 4.5.2 Energy of field interacting with a static source Suppose we specialize our result for Z[J ] to the case of a static source, which we take to have the form

J (x ) = gρ(x ⃗ )f (x 0) where g is a coupling constant and ρ(x ⃗ ) is a static density of some unspecified stuff which interacts with the ϕ field. The function f (x 0 ) is there to adiabatically turn on and off the interaction: it is 1 for a long time period from −T /2 to T /2, and goes to zero outside of this interval smoothly and quickly. The generator iW0[J ] is given by

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1 2 1 = iW0[0] + 2

iW0[J ] = iW0[0] +

∫

d 4xd 4y(igρ(x ⃗ )f (x 0 ))i ΔF (x − y )(igρ(y ⃗ )f (y 0 ))

∫

d 4xd 4y(igρ(x ⃗ )f (x 0 ))

= iW0[0] +

− ig 2 2

∫

= iW0[0] +

− ig 2 2

∫

d 4p

i e−ik·(x−y )(igρ(y ⃗ )f (y 0 )) (2π ) 4 p 2 − m 2 + iϵ

∫

d 4k 1 (2π ) 4 p 2 − m 2 + iϵ d 4p

∫

d 4xρ(x ⃗ )f (x 0 )e−ip·x

∫

d 4y(ρ(y ⃗ )f (y 0 ))e ip·y

1 ρ˜ k ⃗ f˜ (p 0 )ρ˜* k ⃗ f˜ * (k 0 ). (2π ) 4 p 2 − m 2 + iϵ

()

()

As T → ∞, the function f converges at any point x 0 to 1, implying that f˜ (p0 ) → δ (p0 ), which fixes p0 at p0 = 0. However,

∫

dp0 ˜ 0 f (p ) 2π

2

∫ dx 0

=

f (x 0 )

2

→T

so we now have

iW0[J ] = iW0[0] − i

g2 2

∫

−1 d 3p ρ˜ (k ⃗ )ρ˜* (k ⃗ ) 3 2 (2π ) p ⃗ + m 2

which we write as iW0[J ] = iW0[0] − iEJ (J )T , identifying EJ as the extra energy of the ground state due to the static source J. Rewriting the expression for EJ in coordinate space, we have 1 EJ = d 3xd 3y ρ(x ⃗ )V (x ⃗ − y ⃗ )ρ(y ⃗ ) 2 where

∫

V (x ⃗ − y ⃗ ) =

−g 2 e−m x ⃗−y ⃗ 4π x ⃗ − y ⃗

is an attractive Yukawa potential. There are a number of important things we learn from this calculation. Suppose ρ is non-zero only in two non-overlapping regions, A and B, so that ρ can be decomposed as ρ = ρA + ρB . Exchange of a ϕ particle between the two regions leads to an interaction between the two regions, given by the potential V. The interaction due to ϕ particle exchange is attractive, as indicated by the minus sign in V. The overall strength of the interaction is controlled by the coupling g, and the range is determined by the inverse mass m−1 of the ϕ particle. These relations, between coupling constants and interaction strengths and between interaction range and inverse masses, are quite general in quantum field theory.

4.6 Fock space An alternate route to free quantum fields proceeds by constructing a Hilbert space for states with more than one particle. The familiar basis for the states of a single free particle are specified by the action of the momentum and energy operators on basis states:

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P⃗ p⃗ = p⃗ p⃗ H p⃗ = ω p⃗ p⃗ along with the normalization

p1⃗ p2⃗ = δ 3(p1⃗ − p2⃗ ) . This normalization of plane wave states is different from our standard normalization, which includes a factor of (2π )3 in the normalization of states, but this will facilitate the use of box normalization later. For a relativistic free particle, we choose ω p ⃗ = p ⃗ 2 + m2 , but for a non-relativistic system we would choose ω p ⃗ = p ⃗ 2 /2m and other choices are possible. We can construct two-particle states as products of one-particle states. This is correct if the particles are distinguishable, like an electron and a muon, but not for two particles of the same type. One of the deep results rigorously proved by the axiomatic field theory program is the connection between spin and statistics: in a local, relativistic quantum field theory, particles with integer spin, J = 0, 1, 2 ,... are bosons and have symmetric wave functions; particles with half-integer spin, J = 1/2, 3/2 ,..., are fermions and have antisymmetric wave functions. When systems of relativistic particles behave non-relativistically, these properties persist, but the spin-statistics theorem is a theorem for local relativistic quantum field theories. For indistinguishable spin-0 particles, our two-particle states must have a symmetric wave function

p1⃗ , p2⃗ = p2⃗ , p2⃗ in accordance with Bose statistics. The action of the momentum and energy operators on these states is

P ⃗ p1⃗ , p2⃗ = (p1⃗ , +p2⃗ ) p1⃗ , p2⃗ H p1⃗ , p2⃗ = (ω p1⃗ + ω p⃗ 2) p1⃗ , p2⃗ along with the normalization

p1⃗ , p2⃗ p1⃗ ′ , p2⃗ ′ = δ 3(p1⃗ − p1⃗ ′ )δ 3(p2⃗ − p2⃗ ′ ) + δ 3(p1⃗ − p2⃗ ′ )δ 3(p2⃗ − p1⃗ ′ ) . This has a clear extension to states with any number of particles. There must also be a no-particle state, the vacuum 0 which satisfies

P⃗ 0 = 0 H 0 =0 with the normalization

0 0 = 1.

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It may seem that there should be a factor of 1/ 2 in the normalization of twoparticle states and so on, but this becomes very awkward and is best handled in the completeness relation, which we write as

1= 0 0 +

∫ d 3p

p⃗ p⃗ +

1 2!

∫ d 3p1d 3p2

p1⃗ , p2⃗

p1⃗ , p2⃗ + ...

From this point of view, there is a factor of 1/2! for two-particle states, and 1/n! for nparticle states, because p1⃗ , p2⃗ and p2⃗ , p2⃗ are the same state and we are correcting over-counting. This multiparticle Hilbert space is known as Fock space. Because the particles are non-interacting, the energy and momentum of n particles in a given momentum state p ⃗ is nω p ⃗ and np ⃗ , respectively. The similarity with the even spacing of energy levels in the harmonic oscillator motivates the introduction of creation and annihilation operators on Fock space. If we use box normalization, the allowed wavenumbers p ⃗ are discrete. In that case, the creation and annihilation operators satisfy

⎡⎣a p ⃗ , a + ⎤⎦ = δ p ⃗ p ⃗ ′ p⃗ ′ [a p ⃗ , a p ⃗ ′] = 0 ⎡⎣a +, a + ⎤⎦ = 0. p⃗

p⃗ ′

We can write a typical multiparticle state using the occupation number formalism, where n p ⃗ is the number of particles in the state with momentum p ⃗ . In this notation, a state n p1⃗ , n p2⃗ has n p1⃗ particles in p1⃗ and n p2⃗ in p2⃗ . The action of a p ⃗ and a p+⃗ on such states is

a p ⃗ ..., n p ⃗ ,... = n p ⃗ ..., n p ⃗ − 1,... a p+⃗ ..., n p ⃗ ,... = n p ⃗ + 1 ..., n p ⃗ + 1,... . Note that this implies that a p ⃗ 0 = 0 for the case where all occupation numbers are zero. The momentum and energy operators are given by

P ⃗ = ∑p ⃗ a p+⃗ a p ⃗ p⃗

H = ∑ω p a⃗ p+⃗ a p ⃗ p⃗

For a scalar field we assume that these multiparticle states form a basis for the space. It is easy to see that this in turn implies that any operator O which commutes with all the creation and annihilation operators must be a multiple of the identity. It is common in many areas of physics to specify the Hamiltonian of a many-body system by giving its representation in terms of creation and annihilation operators. A typical example of such a Hamiltonian for scalar bosons is

H=

∑ω p a⃗ p+⃗ a p ⃗ + ∑V (p3⃗ , p4⃗ ; p1⃗ , p2⃗ )a p+⃗ a p+⃗ a p ⃗ a p ⃗ 3

p⃗

pj⃗

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where the interaction term represents the scattering of two particles of momentum p1⃗ and p2⃗ into particles of momentum p3⃗ and p4⃗ . The momentum space two-body interaction V (p3⃗ , p4⃗ ; p1⃗ , p2⃗ ) usually is momentum-conserving so p1⃗ + p2⃗ = p3⃗ + p4⃗ . In relativistic theories, we generally avoid box normalization unless necessary, and the commutation relations become

⎡⎣a p ⃗ , a + ⎤⎦ = δ 3(p ⃗ − p ⃗ ′) p⃗ ′ [a p ⃗ , a p ⃗ ′] = 0 ⎡⎣a +, a + ⎤⎦ = 0 p⃗

p⃗ ′

and the energy and momentum operators are given by

∫ H =∫

d 3p p ⃗ a p+⃗ a p ⃗

P⃗ =

d 3p ω p a⃗ p+⃗ a p ⃗

As an interesting application of this formalism, consider a one-dimensional lattice of lattice spacing a. A spinless boson can be located at any site n of the lattice. The Hamiltonian we take to be

H=

∑ϵ0an+an + ∑t(an++1an + an+an+1) . n

n

The parameter ϵ0 represents the energy of a boson occupying a single, isolated cite. The parameter t, often referred to as a hopping or tunneling parameter, controls the tunneling between two adjacent sites. We now define a new bosonic operator via

bθ =

∑ane inθ n

where −π < θ < π . This implies

b θ+ =

∑an+e−inθ . n

The non-zero commutator for these operators is + [bθ , b θ′ ] = δP(θ − θ′)

where δP (θ − θ′) is the periodic δ-function

δP(θ − θ′) =

∑e in(θ−θ′) = ∑2πδ(θ − θ′ + 2πm). n

m

dθ ϵ0b θ+bθ = 2π

∑ϵ0an+an.

It is easy to show that

∫

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Using the inverse Fourier transforms

an =

∫

a n+ =

∫

dθ −inθ e bθ 2π dθ −inθ + e bθ 2π

we find

∑t(an++1an + an+an+1) = ∫ n

dθ iθ (e + e−iθ )b θ+bθ . 2π

Thus the Hamiltonian can be written as

H=

∫

dθ [ϵ0 + 2t cos θ ]b θ+bθ 2π

so the single-particle eigenenergies are given by

ϵ(θ ) = ϵ0 + 2t cos θ which lie in a continuous band of width 4t centered around ϵ0 . This is nothing but band theory for bosons within the tight-binding approximation.

4.7 Relativistic invariance and Fock space Plane wave states are normalized to

p1⃗ p2⃗ = δ 3(p1⃗ − p2⃗ ) and so

∫ d 3p p ⃗

p ⃗ = 1.

These states transform in a simple way under rotations, implemented as unitary operators on the single-particle Hilbert space:

U (R ) p ⃗ = Rp ⃗ . However, these states do not transform simply under boosts. In order to maintain manifest Lorentz invariance, we would like to have single-particle states p which satisfy U (Λ) p = Λp . Such a state must clearly be proportional to p ⃗ . We can determine the relative normalization by observing that the measure

d 4p 2πδ(p 2 − m 2 )θ (p0 ) (2π )4 is the product of three Lorentz-invariant quantities. Together they restrict the value of p0 to + p ⃗ 2 + m2 , and we have upon integration over p0

d 4p d 3p 2πδ(p 2 − m 2 )θ (p0 ) → . 4 (2π ) (2π )32ω p ⃗

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Our completeness statement for p must be, up to a constant,

∫

d 3p p p = 1. (2π )32ω p ⃗

Comparing with completeness for the p ⃗ states, we see that we must have

p =

(2π )32ω p ⃗ p ⃗

and

p1 p2 = (2π )32ω p1⃗ δ 3(p1⃗ − p2⃗ ) . These are the relativistic single-particle states satisfying U (Λ) p = Λp The operators which create such states are proportional to the creation operators we have already defined:

α +(p ) = (2π )3/2 2ω p1⃗ a p+⃗ and the annihilation operators are similarly

α(p ) = (2π )3/2 2ω p1⃗ a p ⃗ so

α +(p ) 0 = p . The non-zero commutation of these operators is

[α(p ), α +(p′)] = (2π )32ω p δ⃗ 3(p ⃗ − p ⃗ ′) Under spacetime transformations, we have

α +(Λp ) 0 = Λp = U (Λ) p = U (Λ)α +(p ) 0 = U (Λ)α +(p )U +(Λ)U (Λ) 0 = U (Λ)α +(p )U +(Λ) 0 where we have used U +(Λ)U (Λ) = I and U (Λ) 0 = 0 . Thus we conclude that α+(Λp ) = U (Λ)α+(p )U +(Λ) when acting on the vacuum. This argument extends easily to all other basis states in Fock space, so α +(Λp ) = U (Λ)α +(p )U +(Λ). A similar argument shows that

e ix·Pα +(p )e−ix·P = e ix·pα +(p ) where P is the momentum operator. The adjoint of this equation is

e ix·Pα(p )e−ix·P = e−ix·pα(p ).

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

4.8 Free quantum fields in Fock space We will construct a free scalar field ϕ(x ) by imposing a set of requirements. • We require that ϕ(x ) be an observable, and hence it must be a Hermitian operator:

ϕ(x ) = ϕ+(x ). • It must be a solution of the Klein–Gordon equation:

(∂ 2 + m 2 )ϕ(x ) = 0 • It must transform properly under Lorentz transformations as well as translations:

U (Λ)ϕ(x )U +(Λ) = ϕ(Λ−1x ) e ia·Pϕ(x )e−ia·P = ϕ(x + a )

This final assumption tells us that the Fourier transform

ϕ˜ (p ) =

∫ d 4x ϕ(x)e+ip·x

of the field ϕ(x ) is a linear function of α (p ) and α+(p ). The Klein–Gordon equation tells us that (p2 − m2 )ϕ˜ (p ) = 0, so ϕ˜ (p ) has support only on the mass hyperboloid p2 = m2 . This motivates our final assumption: • The field ϕ(x ) is a linear function of the creation and annihilation operators. This is a simplifying assumption, but a very reasonable one. We expect that ϕ(x ) acts on the vacuum to create non-vacuum states, and the simplest such states are single-particle states. On the other hand, states with two or more particles cannot lie on the mass hyperboloid, because such states would have p2 ⩾ 4m2 . We write our assumed form as

ϕ(x ) =

∫

d 3p [f (x , p )α(p ) + f * (x , p )α +(p )] (2π )32ω p ⃗

where f (x , p ) is a function, not an operator, to be determined. We have used the Lorentz-invariant measure on the mass hyperboloid as well as the Hermiticity of the field. We want e ia·Pϕ(x )e−ia·P = ϕ(x + a ) so we compute ϕ(x ) = e ix·Pϕ(0)e−ix·P = =

∫

∫

d 3p × [f (0, p )e ix·Pα(p )e−ix·P + f * (0, p )e ix·Pα+(p )e−ix·P ] (2π )32ω p

d 3p [f (0, p )e−ip·xα(p ) + f * (0, p )e+ip·xα+(p )] (2π )32ω p

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which tells us that

f (x , p ) = f (0, p )e−ip·x . Using α+(Λp ) = U (Λ)α+(p )U +(Λ) we find U (Λ)ϕ(x )U +(Λ) =

∫

d 3p [f (0, p )e−ip·xα(Λp ) + f * (0, p )e+ip·xα+(Λp )] (2π )32ω p ⃗

=

∫

d 3p × [f (0, p )e−i Λp·Λxα(Λp ) + f * (0, p )e+i Λp·Λxα+(Λp )] (2π )32ω p ⃗

=

∫

d 3p′ × ⎡⎣f (0, Λ−1p′)e−ip ′·Λxα(p′) + f * ( 0, Λ−1p′)e−ip ′·Λxα+(p′)⎤⎦ (2π )32ω p ⃗ ′

where we have defined p′ = Λp and x′ = Λx , used p · x = Λp · Λx , and used the Lorentz invariance of the measure. If we compare this to

ϕ(Λx ) =

∫

d 3p′ [f (0, p′)e−ip ′·Λxα(p′) + f * (0, p′)e−ip ′·Λxα +(p′)] (2π )32ω p ⃗ ′

we see we must have f (0, Λ−1p′) = f (0, p′) so f (0, p ) must be a constant function of p. This constant is an arbitrary complex number f (0, p ) = re iθ . None of our conditions constrain the value of r, and we will choose r to be one. This leaves the phase e iθ . We are free to redefine the operators α (p ) and α+(p ) by α (p ) → e−iθα (p ) and α+(p ) → e +iθα+(p ) because this does not change the commutation relations for these operators. Thus we arrive at f (0, p ) = 1 and our free scalar field is

ϕ(x ) =

∫

d 3p [e−ip·xα(p ) + e ip·xα +(p )] . (2π )32ω p ⃗

In terms of our non-relativistically normalized operators, we have

ϕ(x ) =

∫

d 3p 3/2

(2π )

2ω p ⃗

⎡⎣e−ip·xa p ⃗ + e ip·xa +⎤⎦ . p⃗

4.9 The canonical commutation relations and causality There is another path to quantization of fields. In quantum systems with a finite numbers of degrees of freedom, the canonical commutation relations

[pj , qk ] = −iδjk are a crucial aspect of Schrödinger picture physics. Here pk = ∂L /∂qk̇ is the momentum variable conjugate to qk. In the Heisenberg picture, where time evolution is shifted from states to operators, the corresponding result is the equaltime commutation relation

⎡p (t ), q (t )⎤ = −iδ jk ⎣j ⎦ k

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together with quantum field the equal-time

⎡q (t ), q (t )⎤ = ⎡p (t ), p (t )⎤ = 0. These expressions generalize to ⎣ j ⎦ ⎣ j ⎦ k k theory. Defining a field π conjugate to ϕ by π (x ) = ∂L/∂(∂ 0ϕ(x )), commutation relations for a scalar field ϕ(x ) are

[π (t , x ⃗ ), ϕ(t , y ⃗ )] = − iδ 3(x ⃗ − y ⃗ ) [ϕ(t , x ⃗ ), ϕ(t , y ⃗ )] = 0 [π (t , x ⃗ ), ϕ(t , y ⃗ )] = 0 in an obvious generalization. These relations, defined at equal times, are often referred to as the canonical commutation relations. For a free scalar field, we have

L=

1 1 ∂μϕ∂ μϕ − m 2ϕ 2 2 2

so the canonically conjugate field is π (x ) = ∂ 0ϕ(x ). We can compute the canonical commutator for the scalar field we found in the previous section. We have [π (t , x ⃗ ), ϕ(t , y ⃗ )] = [∂ 0ϕ(t , x ⃗ ), ϕ(t , y ⃗ )] ⎡ d 3p ⎢ =⎢ −iω p ⃗ [e−ip·xα(p ) − e ip·xα+(p )], (2π )32ω p ⃗ ⎣

(

∫

=

∫

=

∫

=

∫

= −i

d 3p (2π )32ω p ⃗ d 3p (2π )32ω

∫ p⃗

d 3p (2π )32ω p ⃗

∫

∫

∫

)

d 3k (2π )32ω ⃗ k d 3k (2π )32ω ⃗ k d 3k (2π )32ω ⃗ k

∫

⎤ d 3k ⎡ −ik·y ik·yα+(k )⎤⎥ ( ) e α k + e ⎣ ⎦⎥ (2π )32ω ⃗ ⎦ k

(−iω p ⃗)⎡⎣e−ip·xα(p) − eip·xα+(p), e−ik·yα(k ) + eik·yα+(k )⎤⎦ (−iω p ⃗)(e−ip·xeik·y[α(p), α+(k )] − eip·xe−ik·y[α+(p), α(k )]) (−iω p ⃗)(e−ip·xeik·y + eip·xe−ik·y )((2π )32ω p1⃗ δ3(p ⃗ − k ⃗))

d 3p ip ⃗ ·(x ⃗−u ⃗ ) e (2π )3

= − iδ3(x ⃗ − y ⃗ )

where we have used the commutation relations, the equal-time condition x 0 = y 0 and rotational invariance to arrive at our final result. It is possible to derive the creation and annihilation operators and their commutation relations from the Klein–Gordon field, relativistic invariance and the canonical commutation relations. We can take either the Fock space construction or the canonical commutation relations as fundamental, and derive the other, so they are equivalent for free fields. Fundamentally, this is just an elaboration of the equivalence between [p, x ] = −i and [a, a + ] = 1 for a harmonic oscillator. For interacting theories, the canonical commutation relation remains the fundamental quantization condition for fields, just as [p, x ] = −i is in quantum mechanics. The equal-time commutation relations have an important generalization related to causality. Consider, for example the commutator [ϕ(t , x ⃗ ), ϕ(t , y ⃗ )] = 0. In special relativity, there is always an inertial frame where two spacelike-separated points x and y are contemporaneous, with x 0 = y 0 in that frame. This implies [ϕ(x ), ϕ(y )] = 0 if (x − y )2 = 0. In quantum physics, when two observables

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

commute, we say that they are simultaneously observable, in the sense that measuring one has no effect on the other. The behavior [ϕ(x ), ϕ(y )] = 0 when (x − y )2 = 0 is exactly what we would expect in a relativistic theory that is local and causal. The non-zero commutation relation [π (t , x ⃗ ), ϕ(t , y ⃗ )] = −iδ 3(x ⃗ − y ⃗ ) is not an exception, because it indicates the incompatibility of two different observables at the same point in space and time. We can explicitly check that [ϕ(x ), ϕ(y )] and [π (x ), π (y )] are both zero when (x − y )2 = 0, in the same way that we checked the [π (t , x ⃗ ), ϕ(t , y ⃗ )] commutator.

4.10 Equivalence to the functional integral formalism Using the functional integral formalism, we have found that a free scalar field theory is characterized completely by its two-point function. All other n point functions are given in terms of the two-point function using Wick’s theorem. For the Klein– Gordon theory, the two-point function is given by

G0(2)(x − y ) = 0 T [ϕ(x )ϕ(y )] 0 = i ΔF (x − y ) where

i ΔF (x − y ) =

∫

d 4p i e−ip·(x−y ). (2π )4 p 2 − m 2 + iϵ

This form is manifestly Lorentz invariant. This function can also be calculated in the operator formalism. For simplicity, we can take y = 0, and consider the two cases x 0 > 0 and x 0 < 0 separately. In either case, the only non-zero contribution comes when the earlier field creates a particle from the vacuum and the later field destroys it. The result is

i ΔF (x ) =

∫

d 3p 0 0 [θ (x 0)e i (p ⃗ ·x ⃗−ω p⃗x ) + θ ( −x 0)e i (p ⃗ ·x ⃗+ω p⃗x ) ]. 3 (2π ) 2ω p ⃗

We can obtain this result from the manifestly invariant form by integration over p0. Defining ω p ⃗ = + p ⃗ 2 + m2 , we write the denominator of the integrand as 2 2 (p0 ) − ω p ⃗ + iϵ = (p0 − ω p ⃗ + iϵ )(p0 + ω p ⃗ − iϵ )

so there are poles of p0 at p0 = ω p ⃗ − iϵ and p0 = −ω p ⃗ + iϵ . The iϵ prescription tells us to close the contour in the lower half-plane when x 0 > 0 a and in the upper halfplane when x 0 < 0. This is essentially the same integral we performed for the simple harmonic oscillator in chapter 1. From this, we see that the operator formalism and the functional formalism are completely equivalent for free scalar fields.

4.11 Continuous symmetries in quantum field theories In chapter 3, we discussed continuous symmetries in classical field theory. If a continued transformation on a set of fields ϕa → ϕa + δϕa can be shown to give rise to a change L → L + ∂μδF μ(x ) without using the equations of motion, then the current 4-18

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

J μ = π aμδϕ a − δF μ is conserved, ∂μJ μ = 0. The total charge

Q=

∫ d 3x J 0(t, x ⃗)

is independent of time. All of this structure persists in quantum field theories, but J μ(x ) is now a local operator, and Q is also an operator. Notice that if an infinitesimal unitary transformation of the form

U (ϵ ) = e iϵQ ≃ 1 + iϵQ leads to a simple transformation law for the fields:

U (ϵ )ϕ a (x )U +(ϵ ) ≃ (1 + iϵQ )ϕ a (x )(1 − iϵQ ) = ϕ a (x ) + iϵ[Q , ϕ a (x )]

∫ = ϕ a (x ) + iϵ ∫ = ϕ a (x ) + iϵ ∫ = ϕ a (x ) + iϵ ∫

= ϕ a (x ) + iϵ

d 3y[π bμ(x 0, y ⃗ )δϕ b(x 0, y ⃗ ) − δF μ(x 0, y ⃗ ) , ϕ a (x )] d 3y[π bμ(x 0, y ⃗ ) , ϕ b(x )]δϕ a (x 0, y ⃗ ) d 3y[π bμ(x 0, y ⃗ ) , ϕ a (x )]δϕ b(x 0, y ⃗ ) d 3y( −iδ 3(x ⃗ − y ⃗ )δ ab)δϕ b(x 0, y ⃗ )

= ϕ a (x ) + ϵδϕ a (x ) so the conserved charge generates the transformation law for the fields. Recall that spacetime translation gives rise to four conserved currents in the form of the stress-energy tensor T μν(x ), which for a single scalar field is given by

T μν = π μ∂ νϕ − g μνL = ∂ μϕ∂ νϕ − g μνL . The stress-energy tensor is symmetric, T μν(x ) = T νμ(x ), and conserved, ∂μT μν = 0. The four conserved charges are

H=

∫

d 3x ⃗ T 00 =

∫

⎡1 ⎤ 1 1 d 3x ⃗⎢ π 2 + (∇ϕ)2 + m 2ϕ 2⎥ ⎣2 ⎦ 2 2

Pj =

∫

d 3x ⃗ T 0j =

∫

d 3x ⃗[π 0 ]

The equal-time commutation relation hints at the mathematical complexity of quantum field theory. The Dirac delta function is not a normal function, but an object known as a distribution or generalized function. Similarly, quantum fields are sometimes referred to as operator-valued distributions. It is easy to show that the equal-time commutation relations imply that any conserved charge Q associated with a continuous symmetry acts as a generator of that symmetry

[Q , ϕ a (x )] = −iδϕ a (x ).

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The simplest case of an internal symmetry is the U(1) symmetry of a charged field. The Lagrangian of a free relativistic particle with a U(1) symmetry is

L = (∂μϕ+)(∂ μϕ) − m 2ϕ+ϕ − λ(ϕ+ϕ)2 where we now write ϕ+ as the Hermitian conjugate of the operator ϕ, whereas previously we used ϕ* for the complex conjugate of the classical field ϕ. Under an infinitesimal transformation ϕ → e iϵϕ, we have δϕ = iϵϕ and δϕ+=−iϵϕ+; L is invariant so δ L = 0. The conserved current is

J μ = π μδϕ + π +μδϕ+ = iϕ∂ μϕ+ − iϕ+∂ μϕ . When we quantize this model, we no longer have a Hermiticity condition on ϕ. This allows us to write

ϕ(x ) =

∫

ϕ+(x ) =

∫

d 3p 3/2

(2π )

2ω p ⃗

⎡b e−ip·x + c +e+ip·x⎤ ⎣ p⃗ ⎦ k⃗

and its adjoint is

d 3p 3/2

(2π )

2ω p ⃗

⎡c e−ip·x + b +e+ip·x⎤ . ⎣ p⃗ ⎦ k⃗

One can check that the commutation relations

⎡⎣b p ⃗ , b + ⎤⎦ = ⎡⎣c p ⃗ , c + ⎤⎦ = δ 3(p ⃗ − p ⃗ ′) , p⃗ ′ p⃗ ′ with all other commutators zero, imply the equal-time commutation relations, and vice versa. As an alternative, one can decompose the ϕ field in terms of two real fields as

ϕ(x ) =

1 ⎡ ⎣ϕ1(x ) + iϕ2(x )⎤⎦ 2

and find

1 a 1p ⃗ + ia p2⃗ ) ( 2 1 c p⃗ = (a 1p⃗ − ia p2⃗ ) 2

b p⃗ =

et cetera. The charge Q can be written as

Q=

∫ d 3x J 0(t, x ⃗) = ∫ d 3p (b p+⃗ b p⃗ − c p+⃗ c p⃗ )

which is just the total number of b-type particles minus c-type particles. We usually refer to the c-type particles as the antiparticles of the b-type particles. Note that the field ϕ destroys particles and creates antiparticles, while the ϕ+ operator creates particles and destroys antiparticles.

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Problems 1. Determine the Fourier transform of

1 2

k ⃗ + m2 in both three and one spatial dimensions. What does this tell us about the behavior of the Yukawa potential as the spacetime dimension d of the system is varied? 2. The canonical commutation relations express the quantum nature of quantum fields. Using the explicit representation of ϕ in terms of a and a +, (a) Show for a simple harmonic oscillator that [π (t ), ϕ(t )] = −i . It is sufficient to work at t = 0. (b) Show for a free scalar field, [π (t , x ⃗ ), ϕ(t , y ⃗ )] = −iδ 3(x ⃗ − y ⃗ ) . It is again sufficient to work at t = 0. 3. The Lagrangian for the Schrödinger equation is

L = iψ *∂tψ −

1 * (∇ψ )(∇ψ ). 2m

What is the so-called momentum operator conjugate to ψ, and what is the form of the canonical commutation relation? Suppose we define the operator a p ⃗ implicitly by

ψ (t , x ⃗ ) =

∫

d 3p ip ⃗ ·x ⃗−iEpt e a p⃗. (2π )3/2

Determine the form of Ep and use Fourier transforms to find the commutation relation for a p ⃗ and a p+⃗ .

Further reading [1] Itzykson C and Zuber J B 1980 Quantum Field Theory (International Series In Pure and Applied Physics) (New York: McGraw-Hill)

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 5 Interacting quantum fields

In quantum field theory, we make a distinction between interacting theories and noninteracting theories. An interacting theory has field equations which are non-linear, and therefore its Lagrangian contains terms cubic or higher in the fields. It is these higherorder terms which are responsible in perturbation theory for particle production when two bodies scatter. A non-interacting theory, in contrast, has terms in the Lagrangian which are linear and quadratic in the fields, but no higher. This class of models includes free-particle models, but it also includes models of particles moving in external classical fields, such as a Coulomb or other potential. Such models may be difficult to solve analytically, but the quantum fields can be expressed in terms of the wavefunction for single-particle states, even if those wavefunctions are not known exactly. In contrast, interacting field theories generally have scattering between two or more particles, and scattering can lead to particle production. There is also the possibility of bound states, where two or more particles are bound together into a stable composite particle. Typical examples include the pion, proton and neutron, where the strong force binds quarks and antiquarks into mesons like the pion and baryons like the proton and neutron. In some field theories, composite particles may not be stable, but eventually decay into other particles. For example, in quantum electrodynamics an electron and positron can form a hydrogen-like bound state, positronium, which has a lifetime measured in nanoseconds. Two fundamental experimental quantities in many areas of physics are scattering crosssections and decay rates. Perturbation theory gives a systematic method for calculating cross-section and decay rates using Feynman diagrams, but it also gives us a theoretical framework for understanding our results that is deeply appealing and intuitive. We will consider two models of interacting scalar field theories in this chapter. The first, the ϕ4 model, has a Lagrangian

L=

doi:10.1088/978-0-7503-3227-9ch5

1 μ 1 λ ∂ ϕ∂μϕ − m 2ϕ 2 − ϕ4 . 2 2 4!

5-1

ª IOP Publishing Ltd 2022

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

This model is the field theory generalization of the anharmonic oscillator. This model is ubiquitous throughout theoretical physics, and central to our understanding of many important concepts. In particle physics, a generalization of the ϕ4 model with an SU (2) × U (1) internal symmetry describes the self-interactions of the Higgs boson. The second model is a model with three fields: a complex field ψ, its Hermitian conjugate field ψ +, and a real scalar field ϕ. The Lagrangian of this model is

L = ∂ μψ +∂μψ − m 2ψ +ψ +

1 μ 1 ∂ ϕ∂μϕ − μ2 ϕ 2 − gψ +ψϕ . 2 2

This Lagrangian represents a particle and its antiparticle of mass m. associated with ψ and ψ +, interacting via the exchange of another particle of mass μ, associated with ϕ. This model can be interpreted as a model for the interactions of a nucleon and antinucleon interacting via meson exchange from which many lessons can be learned [1]. As we saw in chapter 4, such an exchange leads to a Yukawa potential of strength g and range μ−1.

5.1 Perturbation theory and Feynman diagrams In order to apply perturbation theory to a model with a single scalar field, we must separate the free and interacting parts of the Lagrangian. For example, in the ϕ4 model, we write

L = L0 + LI where the non-interacting part of the Lagrangian, L0 is given by

L0 =

1 μ 1 ∂ ϕ∂μϕ − m 2ϕ 2 2 2

and the interacting part LI is given by

LI = −

λ 4 ϕ. 4!

All the information about the model is encapsulated in the generating functional Z [J ], given by the functional integral

∫ =∫

Z [J ] =

[dϕ ]e i∫

d 4x(L+Jϕ )

[dϕ ]e i∫

d 4x(L 0+LI +Jϕ )

where J (x ) is a classical external source. Just as we saw in quantum mechanical models, we can expand the functional integral in a power series as

Z [J ] =

∫

[dϕ ]e i∫

d 4x(L 0+LI +Jϕ )

5-2

∞

1⎡ ⎢i n !⎣ n=0

∑

∫ d 4x L0⎤⎦⎥ . n

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

In the case of the ϕ4 model, this is

Z [J ] =

∫

d 4x(L 0+Jϕ )

[dϕ ]e i∫

∞

( −iλ)n n! n=0

=∑

∞

1⎡ ⎢ −iλ n! ⎣ n=0

∑

d 4x(L 0+Jϕ )

[dϕ ]e i∫

∫

∫

⎤n d 4x ϕ4(x )⎥⎦

∫

d 4x1 ϕ4(x1)…

∫

d 4xn ϕ4(xn).

Each term in this expansion can be evaluated in terms of the free field two-point functions i ΔF (x − y ) using either functional methods or Feynman diagram methods. As before, a simple formal expression for Z [J ] is given by noting that each instance of ϕ(xj ) in the interaction terms can be replaced by functional differentiation with respect to J (xj ) because

ϕ(xj )e i∫

d 4x(L 0+LI +Jϕ )

1 δ e i∫ i δJ (xj )

=

d 4x(L 0+LI +Jϕ )

and so

∫

d 4xj ϕ4(xj )e i∫

d 4x(L 0+LI +Jϕ )

∫

=

⎛ 1 δ ⎞4 ⎟ e i∫ d 4xj ⎜ ⎝ i δJ (xj ) ⎠

d 4x(L 0+LI +Jϕ )

.

Repeating this argument to all orders in perturbation theory we see that

Z [J ] =

−i λ e 4!

∫

⎛ ⎞4 d 4x⎜ 1 δ ⎟ ⎝ i δJ (xj ) ⎠

∫ [dϕ]ei∫ d x(L +Jϕ) 4

0

which generalizes easily to

Z [J ] = e

∫

i

⎛ ⎞ d 4x LI ⎜ 1 δ ⎟ ⎝ i δJ (xj ) ⎠

∫ [dϕ]ei∫ d x(L +Jϕ) 4

0

for any scalar Lagrangian of the form L = L0 + LI . For the case of the ϕ4 model, we know how to calculate Z [J ] in the noninteracting case where λ = 0. This is the lowest order of perturbation theory, and we have

⎡1 Z0[J ] = Z0[0] exp ⎢ ⎣2

⎤

∫ d 4xd 4y iJ (x) · iΔF (x − y ) · iJ (y )⎥⎦

so our master formula for Z [J ] is

Z [J ] = e

−i 4λ!

∫

(

d 4x

1 δ i δJ (x j )

4

) e ∫ d xd y iJ (x )·iΔ (x−y)·iJ (y) 1 2

4

4

F

or more generally

Z [J ] = e

i∫ d 4x LI

(

1 δ i δJ (x j )

)e ∫ d xd y iJ (x )·iΔ (x−y)·iJ (y). 1 2

5-3

4

4

F

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

If we calculate Z [J ] to order n in perturbation theory, then we can obtain all the npoint functions to that order using functional derivatives and the expansion ∞

Z [J ] = 1 +

1 n! n=1

∑

∫ d 4x1…d 4xnJ (x1)…J (xn)G (n)(x1, … , xn)

where G (n)(x1, … , xn ) = 〈0∣T [ϕ(x1)…ϕ(xn )]∣0〉 so each G (n) is a power series in λ. While it is possible to do all these calculations algebraically, it is usually much easier to use graphical methods, which we now develop.

5.2 Feynman diagrams in position space If we write out our formula for Z [J ] for the ϕ4 theory, we obtain

⎡ 1 −iλ Z [J ] = ∑ ⎢ ⎢ j = 0 j! ⎣ 4! ∞

∞

⎡

∫

∑ 1 ⎢⎣ 1 ∫

k=0

k! 2

⎛ 1 δ ⎞4 ⎤ ⎟⎥ d x⎜ ⎝ i δJ (xj ) ⎠ ⎥⎦

j

4

⎤k d 4xd 4y iJ (x ) · i ΔF (x − y ) · iJ (y )⎥ . ⎦

where all integrals over spacetime are understood to be independent integrals, even if an index appears to be repeated. In order to obtain a non-zero result at order λ j , we see that the number of functional integrals with respect to J must be less than or equal to the number of powers of J appearing in the sum over k, so we must have 4j ⩽ 2k . The first non-zero result is obtained for j = 1 and k = 2, which is obtained from

λ 4!

−i

∫

⎛ 1 δ ⎞4 1 ⎡ 1 d 4z⎜ ⎟ ⎢ ⎝ i δJ ( z ) ⎠ 2 ⎣ 2

∫

⎤2 d 4xd 4y iJ (x ) · i ΔF (x − y ) · iJ (y )⎥ . ⎦

The first functional differentiation gives us

−i

λ 4!

∫

⎛ 1 δ ⎞3 d 4z⎜ ⎟ ⎝ i δJ ( z ) ⎠

⎡1 iJ (y1)⎢ ⎣2

∫ d 4y1iΔF (z − y1) · ⎤2

∫ d 4x2d 4y2iJ (x2) · iΔF (x2 − y2) · iJ (y2)⎥⎦

canceling two factors of 1/2. The second functional differentiation gives us two terms

−i

λ 4!

∫

⎛ 1 δ ⎞2 d 4z⎜ ⎟ i ΔF (z − z ) ⎝ i δJ ( z ) ⎠

⎡1 ·⎢ ⎣2

and

−i

λ 4!

∫

⎛ 1 δ ⎞2 d 4z⎜ ⎟ ⎝ i δJ ( z ) ⎠

⎤

∫ d 4x2d 4y2iJ (x2) · iΔF (x2 − y2) · iJ (y2)⎥⎦

∫ d 4y1iΔF (z − y1) · iJ (y1) ∫ d 4y2iΔF (z − y2) · iJ (y2). 5-4

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After the third and fourth differentiations, we find that both terms give the same contribution

−i

λ 3 4!

∫ d 4ziΔF (z − z)iΔF (z − z)

This can be simplified to

λ −i VT [i ΔF (0)]2 8 where VT is the volume of spacetime. This is a contribution to the ground state energy of the interacting vacuum. This kind of contribution will occur infinitely many times in the expansion of Z [J ]. Let us now look at the case of similar contributions when there are more J’s than δ /δJ ’s, i.e., when k ⩾ 2j . These contributions come from the term in the expansion of Z [J ] of the form

1 ⎡⎢ λ −i j!⎢⎣ 4!

⎛ 1 δ ⎞4 ⎤ 1 ⎡ 1 d 4z⎜ ⎟⎥ ⎢ ⎝ i δJ (z ) ⎠ ⎥⎦ k! ⎣ 2 j

∫

⎤k

∫ d 4xd 4y iJ (x) · iΔF (x − y ) · iJ (y )⎥⎦ .

This term, when fully expanded, will give rise to many different terms. From one part of that expansion, 2j terms out of k can be chosen. The number of ways to do this is k! /(2j )!(k − 2j )!. However, those terms must be put into pairs to mimic the lowest-order calculation. The number of ways to do that pairing is (2j )! /j! but there are j! factorial ways to associate a given pair with a corresponding set of functional differentiations. The final result is

⎡1 ⎤j 1 1⎡ λ 2 [ (0)] − Δ i VT i F ⎢ ⎥⎦ ⎢⎣ (2k − j )! ⎣ 2 8 j!

∫

⎤ k − 2j d 4xd 4y iJ (x ) · i ΔF (x − y ) · iJ (y )⎥ ⎦

but this result can be summed over j and k: ∞

∞

∑∑ j = 0 k ⩾ 2j ∞ ∞

=∑∑ j = 0 k ⩾ 2j

⎡1 ⎤j 1⎡ λ 1 2 ⎢ ⎢⎣ − i VT [i ΔF (0)] ⎥⎦ j! 8 (2k − j )! ⎣ 2 ⎤j ∞ 1 ⎡1 1⎡ λ 2 ⎢⎣ − i VT [i ΔF (0)] ⎥⎦ ∑ ⎢⎣ j! 8 k! 2 k=0

⎡ λ 1 = exp ⎢ +− i VT [i ΔF (0)]2 + ⎣ 8 2

⎤ k − 2j

∫ d 4xd 4y iJ (x) · i ΔF (x − y ) · iJ (y )⎥⎦ ⎤k

∫ d 4xd 4y iJ (x) · i ΔF (x − y ) · iJ (y )⎥⎦ ⎤

∫ d 4xd 4y iJ (x) · i ΔF (x − y ) · iJ (y )⎥⎦

This is the simplest example of sums of disconnected diagrams exponentiating for this model. The two terms in the exponential represent the O(λ1J 0 ) and O(λ0J 2 ) contributions to iW [J ] = log Z [J ], respectively.

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If we complete the calculation of all the connected graphs contributing to iW [J ] up to O(λ ), we obtain λ 1 iW [J ] = − i VT [i ΔF (0)]2 + d 4xd 4y iJ (x ) · i ΔF (x − y ) · iJ (y ) 8 2 2 λ −i d 4x i ΔF (0) d 4x1i ΔF (x1 − x )iJ (x1) 4 4 λ −i dx dx1i ΔF (x1 − x )iJ (x1) + O(λ2 ) 4!

∫

(∫

∫

)

(∫

∫

)

where we again stress that repeated integration variables, such as x1 here, are independent, so the second line above involves three independent integrations over spacetime, and in the third line there are five independent integrations over spacetime. The two O(J 2 ) terms contribute to the connected n-point functions for n = 2 and n = 4. For n = 2, we have

Gc(2)(x1, x2 ) =

1 δ 1 δ iW [J ] i J (x1) i J (x2 ) J =0

which to O(λ ) is given by Gc(2)(x1, x2 ) = i ΔF (x1 − x2 ) − i

λ 2

∫ d 4x iΔF (0) ∫ d 4x1iΔF (x1 − x) ∫ d 4x2iΔF (x2 − x1).

Similarly, for n = 4, we obtain the lowest-order non-zero result

Gc(4)(x1, x2 , x3, x4) = − iλ ·

∫

∫

d 4x

∫

d 4x1i ΔF (x1 − x )

d 4x3i ΔF (x3 − x )

∫

∫

d 4x2i ΔF (x2 − x )

d 4x4i ΔF (x4 − x ).

Note that by functionally differentiating and setting each external line a fixed spacetime point xj, we remove any symmetry factors on a graph associated with permutations of the external lines. Thus the O(λ ) contribution to Gc(2)(x1, x2 ) has a symmetry factor of 2 associated with the internal loop. The O(λ ) contribution to Gc(4)(x1, x2, x3, x4 ) has no symmetry factor. We can also see that the n-point correlation functions are translation-invariant. For example, a shift in the integration variable x to x + x1 will show that the expression for Gc(2)(x1, x2 ) can be written Gc(2)(x1 − , x2 ).

5.3 Feynman diagrams in momentum space It is easier and more useful to calculate the connected n-point functions using Feynman diagrams in momentum space than in position space. Consider the expression Gc(2)(x1, x2 ) = i ΔF (x1 − x2 ) − i

λ 2

∫ d 4x iΔF (0) ∫ d 4x1iΔF (x1 − x) ∫ d 4x2iΔF (x2 − x1). 5-6

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We know that

i ΔF (x1 − x2 ) =

∫

d 4k i e−ik·(x1−x2) 4 2 (2π ) k − m 2 + iϵ

so to lowest order in λ we have (2) G˜c (p1 , p2 ) =

∫

d 4x1d 4x2e+ip1·x1+ip2·x2Gc(2)(x1, x2 )

=

∫

⎡ d 4x1d 4x2e+ip1·x1+ip2·x2⎢ ⎣

∫

⎤ d 4k i e−ik·(x1−x2)⎥ 4 2 2 (2π ) k − m + iϵ ⎦

d 4k i (2π )4δ 4(p1 − k )(2π )4δ 4(p2 + k ) 2 4 (2π ) k − m 2 + iϵ i . = (2π )4δ 4(p1 + p2 ) 2 p1 − m 2 + iϵ =

∫

From this expression, we see explicitly that the sum over the four-momenta entering the diagram is zero, p1 + p2 = 0, due to the delta function. This is general, and it is conventional to omit the overall four-momentum conserving delta function from the momentum space Feynman rules, but it will be reinstated when we work out formulae for cross-sections and decay rates. This factor is also conventionally omitted from expressions for n-point functions. For the lowest-order two-point functions, we simply write (2) G˜c (p ) =

i p 2 − m 2 + iϵ

to leading order in λ. (2) The O(λ ) contribution to G˜c (p ), obtained in the same way, is given by

λ =−i 2

∫

d 4k i 4 2 (2π ) k − m 2 + iϵ

⎛ ⎞2 i ⎜⎜ 2 ⎟⎟ 2 ⎝ p1 − m + iϵ ⎠

where we have again removed a factor of (2π )4δ 4(p1 + p2 ). Thus the two-point function at O(λ ) is given by (2) G˜c (p ) =

i λ −i 2 2 2 p − m + iϵ

∫

⎛ ⎞2 d 4k i i ⎜ ⎟ (2π )4 k 2 − m 2 + iϵ ⎝ p 2 − m 2 + iϵ ⎠

where the second term has the interpretation of the propagating particle interacting (4) with itself. The O(λ ) contribution to G˜c (p1 , p2 , p3 , p4 ) is extremely simple once the four-momentum conserving factor is removed: it is simply 4

(4) G˜c (p1 , p2 , p3 , p4 ) = −iλ ∏

p2 j=1 j

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The momentum space Feynman rules are easily stated: each propagator represented by a line, carries a four-momentum p, and contributes a factor of

i p − m 2 + iϵ 2

to the graph in which it appears. Each vertex carries a factor of −iλ , and the fourmomenta coming into a vertex is conserved. This comes from the integration over the spacetime location of each vertex, so in the case of the lowest-order graph for (4) G˜c (p1 , p2 , p3 , p4 ) we have p1 + p2 + p3 + p4 = 0, if all the momenta are taken as pointing inwards towards the vertex. All momenta associated with internal lines must be integrated over. For each graph, there is a symmetry factor associated with permutations of internal lines, just as in position space.

5.4 Scattering theory In scattering theory, we want to compute the amplitude for two particles, with momenta p1 and p2, to interact, which is to say collide, and produce some different state. The result of the collision could be another two-particle state, with momenta p1′ and p2′, or a state with three or more particles. We will begin by giving an intuitive description of how such amplitudes may be calculated using the example of ϕ4 theory at lowest order. Later we will develop the LSZ reduction formula, which is a rigorous, formal procedure for obtaining scattering amplitudes. We know that the path integral gives the amplitude for a state Ψ to evolve over a time T into a state Ψ′ as

Ψ′∣e−iTH ∣Ψ =

∫ [dϕ]eiS[ϕ]

where the boundary conditions of the field ϕ at t = ±T /2 represent the final and initial state Ψ′ and Ψ, respectively. Suppose we take the initial state ∣Ψ to be a twoparticle state ∣p1 , p2 and the final state ∣Ψ′ to be another two-particle state ∣p1′ , p2′ . If we had free fields, we could write these states as

∣Ψ = α +(p1 )α +(p2 )∣0 ∣Ψ′ = α +(p1′ )α +(p2′ )∣0 and of course Ψ′∣ = 0∣α (p2′ )α (p1′ ). We can obtain the creation and annihilation operators from a free field ϕ(x ) by Fourier transforms Explicitly

α( k ) = i

←→

∫ d 3x f * (x, k )(x) ∂0ϕ(x)

where

f (x , k ) = e−ik·x .

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From this, it seems like we can obtain our desired matrix element by using the Feynman rules supplemented by a rule for Wick contracting creation and annihilation operators with fields. These are

0∣ϕ(x )α +(k )∣0 =

∫

d 3p e−ip·x 0∣α(p )α +(k )∣0 (2π )32ω p ⃗

= e−ik·x and

0∣α(k )ϕ(x )∣0 = e ik·x . Our proposed formula for the lowest-order scattering amplitude for p1 + p2 → p1′ + p2′ can now be written as

i M(p1 + p2 → p1′ + p2′ ) = 0∣α(p2′ )α(p1′ ) ·

−iλ 4!

∫ d 4xϕ4(x)α+(p1)α+(p2 )∣0

where we have dropped time-ordering in this case because all four fields are at the same time. There has been a careless mixing of free field operators and interacting Heisenberg fields in this argument that makes it somewhat suspect. There is a formalism due to Dyson, called the interaction picture, in which these manipulations can be justified. The LSZ reduction formula, in contrast, avoids these problems by working with Heisenberg operators throughout. The scattering amplitude is often referred to as an S-matrix element. We need the concepts of in-states and out-states. An in-state is a Heisenberg picture state ∣α in , defined at t = 0 as Heisenberg states are, that ‘look like’ a Heisenberg state of free particles ∣α 0 if both are evolved backwards in time as t → −∞. Roughly speaking, we want

limt→−∞[e−itH ∣α

in

− e−itH0 ∣α 0 ] = 0.

Out-states are defined similarly, but the limit taken to define them is t → +∞. The Smatrix is defined as a unitary map from free-particle states to free-particle states given by 0

β ∣S ∣α

0

=

out

β ∣α

in .

The S-matrix element 0 β∣S ∣α 0 is the overlap between out β ∣α in between states that ‘look like’ incoming free-particle states and outgoing free-particle states. Using our rules for contracting fields with creation and annihilation operators, we obtain

i M(p1 + p2 → p1′ + p2′ ) = − iλ

∫

d 4x e ( 1

)

i p ′ +p2′ −p1 −p2 ·x

= − iλ(2π )4δ 4(p1′ + p2′ − p1 − p2 ) where the factor of 1/4! has been canceled by the 4! ways of performing the Wick contractions. It is customary to define the scattering matrix element iM (p1 + p2 → p1′ + p2′ ) without the factor of (2π )4δ 4(p1′ + p2′ − p1 − p2 ) which 5-9

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

conserves four-momentum, so i M(p1 + p2 → p1′ + p2′ ) = (2π )4δ (p1′ + p2′ − p1 − p2 ) iM (p1 + p2 → p1′ + p2′ ). This definition gives us the lowest-order result

iM (p1 + p2 → p1′ + p2′ ) = −iλ . There is a further step we can take: if we compare this result with our result 4

(4) G˜c (p1 , p2 , p3 , p4 ) = −iλ ∏

p2 j=1 j

i − m 2 + iϵ

we see that the scattering matrix element iM (p1 + p2 → p1′ + p2′ ) is just (4) G˜c (p1 , p2 , p3 , p4 ) with the external propagator factors removed:

⎡ 4 ⎛ p 2 − m 2 + iϵ ⎞ ⎤ (4) j ⎢ ⎥ ⎜ ⎟ ˜ ′ ′ ′ ′ iM (p1 + p2 → p1 + p2 ) = ∏ ⎜ ⎟Gc (p1 , p2 , −p1 , −p2 )⎥ ⎢ i ⎠ ⎣ j=1 ⎝ ⎦ p =−p ′ ,p =−p ′ 3

1

4

2

where p3 , p4 become −p1′ , −p2′ because p3 and p4 were oriented inward but p1′ and p2′ are oriented outward. The process of removing the external propagators is often referred to as amputation of the external legs. Implicit in this formula is a limit where all the external four-momenta are taken on-shell, after the external legs are removed, so p12 = p22 = p1′ 2 = p2′ 2 = m2 . This recipe is essentially the LSZ formula, to be discussed in chapter 6. That formula corrects this result with a factor associated with renormalization of the field, but this result is adequate for scattering calculated at low orders of perturbation theory. Note that our conventions are such that the matrix element iM is Lorentz-invariant. At this lowest order in perturbation theory, there is a single graph contributing to scattering, but a given S-matrix element is generally a sum over all the connected graphs which contribute to the scattering process.

5.5 A toy model of nucleons and pions We now turn to a scalar model which has some of the features of low-energy nuclear processes [1]. The Lagrangian

L = ∂ μψ +∂μψ − m 2ψ +ψ +

1 μ 1 ∂ ϕ∂μϕ − μ2 ϕ 2 − gψ +ψϕ . 2 2

represents a charged scalar particle and antiparticle of mass m. associated with ψ and ψ +, interacting via the exchange of another particle of mass μ, associated with ϕ. We can think of the charged particle as being a ‘scalar nucleon’ N with antiparticle N¯ , and the neutral particle associated with ϕ as a pion. Of course, this model is not realistic, because there are two nucleons, the proton and the neutron, both fermions, and there are three pions. Nevertheless, the model is instructive.

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The Feynman rules are a simple extension of the rules for ϕ4 . The momentum space propagator for ϕ is denoted by a dashed line, and given for a particle of fourmomentum k by

i . k 2 − μ2 + iϵ The ψ +ψϕ vertex carries with it a factor of −ig . One of the nice features of this model, which it shares with quantum electrodynamics, is that there are no symmetry factors to account for in individual Feynman diagrams, because at every vertex there is one and only one field of each of the three types. There is a new feature in the ψ propagator. There are several ways to see that the only non-zero propagator is

0∣T [ψ (x )ψ +(y )]∣0 =

∫

d 4p i e−ip·(x−y ). 4 2 (2π ) p − m 2 + iϵ

The other two possibilities, 0∣T [ψ (x )ψ (y )]∣0 and 0∣T [ψ +(x )ψ +(y )]∣0 , are identically zero due the conserved charge Q. Explicit computation with creation and annihilation operators also gives zero, or one can note that the real and imaginary components of the fields give canceling contributions. Physically, the propagator carries one unit of charge from y to x, so we write the propagator as a line with an arrow in the middle pointing from y to x. If x 0 > y 0 , we think of the propagator as representing a particle propagating from y to x. On the other hand, if y 0 > x 0, we think of the propagator as representing an antiparticle propagating from y to x. However, the Feynman propagator is non-zero for spacelike separation, and if x and y are spacelike-separated, we can change time-ordering by changing inertial frame. This leads to an interpretation in which an antiparticle moving forwards in time is equivalent to a particle moving backwards in time. This is known as the Feynman– Stueckelberg interpretation of antiparticles. It does not appear to have experimental consequences, but can be a useful part of the metaphorical language used with Feynman diagrams. ¯ ¯ scattering amplitude 5.5.1 The NN and NN

We want to compute the lowest-order scattering amplitude iM for a process where nucleons of four-momentum p1 and p2 scatter elastically into nucleons with fourmomentum p1′ and p2′ . At lowest order, the nucleons interact by exchanging a meson, so scattering requires two vertices and the two lowest-order graphs are thus of order g 2 , as shown in figure 5.1. In the first graph, a nucleon carries momentum p1 into the vertex, and a nucleon emerges with momentum p1′ . This fixes the momentum of the meson to be k = p1 − p1′ if we think of the momentum of the meson as coming out of the vertex. This implies that the momentum coming into the second vertex is p2 + k , which must be equal to p2′ . In this way, momentum conservation at the vertices ensures that

p1′ + p2′ = (p1 − k ) + (p2 + k ) = p1 + p2

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Figure 5.1. The O(g 2 ) graphs for N + N → N + N scattering. The first graph represents a t-channel process and the second a u-channel process.

which is conservation of total four-momentum. The Feynman amplitude for this graph, one of two at this order, is

iM1 = ( −ig )

i −ig 2 ( ) ig − = k 2 − μ2 + iϵ k 2 − μ2 + iϵ

but the kinematics allow us to write it as

iM1 =

−ig 2

(p1 − p1′ )2 − μ2 + iϵ

=

−ig 2 . (p2 − p2′ ) − μ2 + iϵ

However, there is a second graph which contributes at this order, the so-called crossed graph. Because the outgoing nucleons are indistinguishable, we must add the amplitude in which the nucleon with momentum p1 gives rise to a nucleon with momentum p2′. The amplitude for this graph is

iM2 =

−ig 2

(p1 − p2′ )2 − μ2 + iϵ

=

−ig 2 (p2 − p1′ ) − μ2 + iϵ

so the total amplitude at this order is the sum of the amplitudes

iM = iM1 + iM2 =

−ig 2

(p1 − p1′ )2 − μ2 + iϵ

+

−ig 2

(p1 − p2′ )2 − μ2 + iϵ

.

The amplitude for the scattering of antinucleons is exactly the same. The only difference between the two processes is that the arrows in the diagrams point in opposite directions. 5-12

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5.5.2 The NN¯ scattering amplitude In order to indicate which momentum is which, we can write this scattering process in a form like that of a chemical reaction: N (p1 ) + N¯ (p2 ) → N (p1′ ) + N¯ (p2′ ). Again there are two diagrams which contribute at O(g 2 ), as shown in figure 5.2. The first diagram is again meson exchange, and the matrix element is

iM1 =

−ig 2

(p1 − p1′ )2 − μ2 + iϵ

.

The second diagram cannot be the same as in NN scattering, because N and N¯ are distinct particles. The second graph is of a type called an annihilation diagram, where the NN¯ pair annihilate to form a virtual meson, which then gives rise to another NN¯ pair. The momentum of the meson is k = p1 + p2 = p1′ + p2′ and the second matrix element is

iM2 =

−ig 2 (p1 + p2 )2 − μ2 + iϵ

so the total matrix element at order g 2

iM = iM1 + iM2 =

−ig 2

(p1 − p1′ )

2

− μ2 + iϵ

+

−ig 2 (p1 + p2 )2 − μ2 + iϵ

5.5.3 Mandelstam variables and crossing symmetry The Mandelstam variables provide us with a Lorentz-invariant way to describe simple two-body scattering processes and their relationships. For a process where

Figure 5.2. The two O(g 2 ) graphs contributing to N + N¯ → N + N¯ scattering. The first graph represents a tchannel process and the second an s-channel process.

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particles initial momenta p1 and p2 give rise to particles with final momenta p1′ and p2′ , we define

s = (p1 + p2 )2 = (p1′ + p2′ )

2

t = (p1′ − p1 ) = (p2′ − p2 )

2

2

2

2

u = (p2′ − p1 ) = (p1′ − p2 )

which are manifestly Lorentz invariant. This convention is shown on the left-hand side of figure 5.3. An alternative convention is shown on the right-hand side of the figure. In this convention, the momenta are all oriented inwards, so that p1 + p2 + p3 + p4 = 0. If we work in the center of mass (CM) frame, we have by definition p1⃗ + p2⃗ = p1⃗′ + p2⃗′ so s = (E1 + E2 )2 = (E1′ + E2′)2 . By definition, E1 + E2 = E1′ + E2′ in the CM frame is the total energy ECM in the CM frame. This is fundamental in relativistic scattering, because s = ECM determines how much energy is available and thus what particles can be emitted. The other two Mandelstam particles also have simple physical interpretations. This is particularly easy when all four particles have the same mass, as in NN scattering. If we take the beam axis in the zˆ direction, we can write p1⃗ = pzˆ and p2⃗ = −pzˆ ; the energy of each of the two particles is E = p2 + m2 so ECM = 2E . The outgoing particles are restricted by kinematic to the same p and E, but need not have momenta along the beam axis. We define the scattering angle θ by p ⃗ ′ · p ⃗ = p2 cos θ . We now have

s = 4E 2 = 4(p 2 + m 2 ) 2

t = − (p1⃗ ′ − p1⃗ ) = −2p 2 (1 − cos θ ) 2

u = − − (p2⃗ ′ − p1⃗ ) = −2p 2 (1 + cos θ ). As we knew, large s means large ECM. We also see that small t processes are associated with small angle θ, while large t implies large-angle scattering. The three

Figure 5.3. Two-body scattering with two different conventions for the Mandelstam variables. In the first diagram, there are two incoming and two outgoing momenta. In the second, all momenta are oriented inward.

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variables in this case satisfy the linear constraint s + t + u = 4m2 , but it is easy to show that in the general case s + t+u is the sum of the squared masses of the incoming and outgoing particles. In the ultrarelativistic limit, we take all energies and momenta as large relative to any masses, so the Mandelstam variables become

s = 4p 2 t = − 2p 2 (1 − cos θ ) u = − 2p 2 (1 + cos θ ) with ECM = 2E = 2p. Using the Mandelstam variables, we can write our expression for NN scattering as

iMNN =

−ig 2 −ig 2 + t − μ2 + iϵ u − μ2 + iϵ

and we say that the first term represents a t-channel exchange process while the second represents a u-channel exchange process. In the case of NN¯ scattering, we have

iMNN¯ =

−ig 2 −ig 2 + t − μ2 + iϵ s − μ2 + iϵ

where the two terms come from t-channel and s-channel exchange, respectively. There is a relation between the NN and NN¯ scattering amplitudes which is called crossing symmetry. We orient all the momenta so that they are all going inward, as shown on the right-hand side of figure 5.3. The momenta must satisfy overall energymomentum conservation, so p1 + p2 + p3 + p4 = 0. Whether an external line represents an incoming (initial state) or outgoing (final state) particle is determined by the sign of the energy p j0 for each external line. If p j0 > 0, the particle j is incoming, and it is outgoing if p j0 < 0. For NN scattering, we take p10 , p20 > 0 and ¯ and define the fourp30 , p40 < 0. We can denote this process as 1 + 2 → 3¯ + 4, momenta of the outgoing particles to be p3¯ = −p3 and p 4¯ = −p4 so that p1 + p2 = p3¯ + p 4¯ . We identify p3¯ as p1′ and pr¯ as p2′ in our previous representation If we consider the NN scattering matrix element as a function iM (p1 , p2 , p3 , p4 ) of p1 ... p4 , then we have

iMNN = iM (p1 , p2 , −p3¯ , −p 4¯ ). On the other hand, the NN¯ scattering matrix element is obtained from exactly the same four-point function. This process has a particle and an antiparticle in both the ¯ and initial and final state, we can write this process as 1 + 3¯ → 2 + 4,

iMNN¯ = iM (p1 , −p 2¯ , p3 , −p 4¯ ). The function iM is exactly the same, but evaluated for a different range of fourmomenta. We can get a more succinct statement of crossing symmetry by

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considering iM as a function of the Mandelstam variables. For NN scattering, the Mandelstam variables are

sNN = (p1 + p2 )2 = (p3 + p4 )2 ≡ s tNN = (p1 + p3 )2 = (p2 + p4 )2 ≡ t uNN = (p1 + p4 )2 = (p2 + p3 )2 ≡ u and we use these variables to define what we mean when we write iM = iM (s, t , u ), so that iMNN = iM (s, t , u ) by definition. However, for NN¯ scattering, the Mandelstam variables are defined differently

sNN¯ = (p1 + p4 )2 = (p2 + p3 )2 = u tNN¯ = (p1 + p3 )2 = (p2 + p4 )2 = t uNN¯ = (p1 + p4 )2 = (p2 + p3 )2 = s so that we have iMNN¯ = iM (u, t , s ). Although this relation is easy to understand in terms of lowest-order scattering, it is an exact statement. We say that NN and NN¯ are related by a crossing symmetry where s ↔ u . The Mandelstam variables and crossing symmetry simplify calculations, but also promote better and faster communication: If someone asks if the s-channel makes an important contribution to a process, we know what is meant without elaboration. 5.5.4 Four more processes: Nϕ → Nϕ, N¯ ϕ → N¯ ϕ, NN¯ → ϕϕ and ϕϕ → NN¯ The exchanged particle need not be the meson, but can also be a nucleon. If we look at the Nϕ → Nϕ scattering process, there is an obvious s-channel process, for which the amplitude is

i (p1 + p2 ) − m 2 + iϵ i = ( − ig )2 s − m 2 + iϵ

iM1 = ( − ig )2

2

It is natural to describe this as a process in which the incoming nucleon and meson are annihilated at a vertex, giving rise to an off-shell, or virtual, nucleon, which in turn gives rise to an outgoing nucleon and meson. If we calculate the amplitude for this process in coordinate space, we can label the vertex where the incoming particles annihilate by x, and the vertex where the outgoing particles are created by y. The description given above seems to make sense when x 0 < y 0 The coordinate-space Feynman rules instruct us to integrate over all x and y, in particular over all x 0 and y 0 . This implies that there are contributions to this amplitude where y 0 < x 0 , where mere words apparently force us to say that the vertex at y spontaneously emits a nucleon, antinucleon and meson. The nucleon and meson are detected as outgoing particles, while the antiparticle races to its destiny at x, where it annihilates with incoming nucleon and meson. Of course, this artificial separation of one Feynman diagram into two separate kinds of processes is dependent on the inertial frame used. 5-16

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Figure 5.4. The O(g 2 ) graphs for N + ϕ → N + ϕ scattering. The first graph is an s-channel process, while the second is a u-channel process.

This non-covariant separation of processes occurs explicitly in older forms of perturbation theory based on the Hamiltonian formalism. We can draw the Feynman diagram for the second contribution to iM as a variation on the first, in which the incoming meson and outgoing nucleon are attached to a common vertex, as are the incoming nucleon and the outgoing meson. The four-momentum of the internal line connecting the two vertices is p2′ − p1 = p2 − p1′ , which tells us this is a u-channel process, with amplitude

iM2 = ( −ig )2

i u − m 2 + iϵ

This can be made much clearer by drawing the graph with x and y arranged horizontally, suggesting a spacelike separation of vertices, rather than vertically, suggesting a timelike separation, as shown in figure 5.4. The total amplitude at O(g 2 ) for Nϕ scattering is

⎡ ⎤ i i iM = iM1 + iM2 = ( −ig )2 ⎢ + ⎥. ⎣ s − m 2 + iϵ u − m 2 + iϵ ⎦ The amplitudes for N¯ ϕ → N¯ ϕ, NN¯ → ϕϕ and ϕϕ → NN¯ are similar, and there are crossing relations for the processes as there were for NN and NN¯ scattering.

5.6 The CPT theorem We know from the Standard Model that some of the fundamental symmetries which may seem obvious are in fact not symmetries of our Universe: neither parity P nor CP, the product of charge conjugation C with parity, are symmetries of the electroweak sector of the Standard Model. However, there is a symmetry which no relativistic quantum field theory can violate: this is CPT, where T is time-reversal 5-17

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symmetry. In our notation where all external momenta are ingoing, the amplitude the NN scattering process is iMNN = iM (p1 , p2 , −p3¯ , −p 4¯ ), while the amplitude for ¯ ¯ scattering, 3 + 4 → 1¯ + 2, ¯ is iMNN NN ¯ ¯ = iM ( − p1¯ , − p 2¯ , p3 , p4 ). Consider the scattering of charged scalars, which for concreteness we conceptualize as NN scattering. We use the convention where all momenta are oriented ¯ We do not assume any inward, so NN scattering is written as 1 + 2 → 3¯ + 4. particular model or any particular symmetries. Charge conjugation C takes particles into their antiparticles, and vice versa. In NN scattering, C takes 1 + 2 → 3¯ + 4¯ into 1¯ + 2¯ → 3 + 4. These amplitudes are not necessarily the same, because we have not assumed charge conjugation symmetry. Time reversal T reverses the direction of all velocities, and also incoming particles become outgoing, and outgoing particles become ingoing. On the other hand, the parity operator changes the direction of the velocities but does not exchange incoming and outgoing particles. Thus the product PT simply turns incoming particles into outgoing, and vice versa. Thus we have

C : 1 + 2 → 3¯ + 4¯ ⇒ 1¯ + 2¯ → 3 + 4 PT : 1¯ + 2¯ → 3 + 4 ⇒ 3 + 4 → 1¯ + 2¯ so CPT: 1 + 2 → 3¯ + 4¯ ⇒ 3 + 4 → 1¯ + 2¯ , which is to say it turns NN scattering ¯ ¯ scattering. In our notation where all external momenta are ingoing, the into NN amplitude of the NN scattering process is iMNN = iM (p1 , p2 , −p3¯ , −p 4¯ ), while the ¯ ¯ scattering, 3 + 4 → 1¯ + 2, ¯ is iMNN amplitude for NN ¯ ¯ = iM ( − p1¯ , − p 2¯ , p3 , p4 ). It is easy to check that the Mandelstam variables s, t, and u are the same for both processes, which implies the amplitudes must be identical. Thus we have the result that iM is invariant under CPT. The rigorous proof of this theorem is one of the triumphs of axiomatic field theory, which aims to deduce the properties of quantum field theories from a small set of axioms.

5.7 Cross-sections and decay rates Simple scattering processes are conceptualized as a collision between two objects. One of these objects is usually designated as the target. The frame in which the target is at rest is the LAB frame, and this is a convenient frame for beginning our discussion. The fundamental quantities measured in typical scattering experiments are cross-sections. The total cross-section is a measure of the size of the target as seen by the other object. Consider the scattering in classical mechanics of a point particle from a fixed sphere of radius R. The impact parameter b is the perpendicular distance between the incoming particle and a parallel line drawn through the center of the target, often called the beam axis. Any incoming particle with 0 ⩽ b ⩽ R will collide with the target and be scattered. The total cross-section σ for this scattering process is σ = πR2. On the other hand, suppose the incoming object is a sphere of radius R. We define the impact parameter using the motion of the object’s center. In this case, a collision will occur if 0 ⩽ b ⩽ 2R , and the total cross-section is now σ = 4πR2 . From this simple example, we see that the cross-section depends on the structure of both the target and the scatterer. In scattering off a microscopic particle, we generally send in an incoming beam of particles with number density ρ and velocity v. The flux 5-18

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ρv has dimensions of 1/([area][time]), and the number of collisions per unit time is given by σρv. Thus as we know the flux and measure the collision rate, we can obtain the total cross-section. Decay processes are similar, and the total decay rate plays a role similar to the total cross-section in two-body scattering. In quantum physics, the probability P (i → f ) of an initial state i producing a final state f is ∣i M∣2 . It is convenient to put our system in a spacetime box of volume VT, and to use single-particle states ∣p ⃗ rather than ∣p . Due to the spatial box, we now have discrete momentum states and our conventions are

⎡⎣a p ⃗ , a + ⎤⎦ = δ p ⃗ p ⃗ ′ p⃗ ′ and

p ⃗ ∣p ⃗ ′ = δ p ⃗ p ⃗ ′. A scalar field ϕ(x ) in a box can be written as

ϕ(x ) =

∑ k⃗

1 ⎡a e⃗ −ik·x + a +e+ik·x⎤ . ⎣ k ⎦ k⃗ 2E kV ⃗

As a check, note that this yields the correct canonical commutation relations. The Feynman rules get modified in two ways. First, there is a factor of

1 2E kV ⃗ when a field creates or destroys a state ∣k ⃗〉 . Second, the delta function that conserves overall four-momentum becomes 4 (2π )4δVT ( pin − pout ) ≡

∫VT d 4x ei(p

in −pout )·x

.

When this factor is squared in calculating probabilities, we obtain 2 4 4 ⎡⎣(2π )4δVT ( pin − pout )⎤⎦ = VT (2π )4δVT ( pin − pout )

For a decay process, we can take our initial state to be a single-particle state ∣k ⃗〉, but for two-particle scattering, we must take our initial state to be

∣i =

V ∣k1⃗ , k2⃗ .

This is the correct normalization for an incoming flux of particles normalized to one particle per unit volume. The state ∣k1⃗ , k2⃗ 〉 is normalized to one so each particle has a probability of 1 of being somewhere in the box, or a density of 1/V . For this state, there is zero probability of a collision as V → ∞. The factor of V ensures that there is a one target particle in the box, but a density of one particle per unit volume for the particles that collide with the target. Our formula for an S-matrix element using the Feynman rules is now

f ∣S − 1∣i =

4 4 iM VT fi (2π ) δVT (

⎡ 1 pin − pout )⎢ ⎢⎣ V

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∏ i

1 2Ei

⎤⎡ ⎥⎢∏ ⎥⎦⎢⎣ f

⎤ 1 ⎥ 2Ef V ⎥⎦

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

where the products are over particles in the initial and final states, respectively. We use S − 1 rather than S − 1 because we assume that some scattering process has taken place. Note that the Feynman rules assume the final state is the conventional one, ∣k1⃗ , k2⃗ 〉, so we have to correct for that in two-body scattering, but not for onebody decay. This gives rise to a factor of 1/ V . Each particle j in the final state f will end up in some set of single-particle states characterized by its momentum pj. The number of states in a small momentum-space volume d 3pj around pj is given by the standard density of states factor

V

d 3pj

.

(2π )3

However, we have seen that i M ∝ (2π )4δ 4(pin − pout ), where pin and pout are the total four-momentum of the initial and final states, respectively. The differential transition probability per unit time of obtaining a final state in a small region of the space of final states is given by

1 1 2 4 4 ∣M VT fi ∣ VT (2π ) δVT ( pin − pout ) T V

∏ i

1 2Ei

∏ f

d 3pf 1 V 2Ef V (2π )3

which simplifies to a form which is independent of the size of the box. After taking the limit V , T → ∞ , we finally obtain the differential transition probability per unit time

∣Mfi∣2 (2π )4δ 4( pin − pout ) ∏ i

1 2Ei

∏ f

3 1 d pf . 2Ef (2π )3

There are three distinct pieces to this expression, associated with the interaction, the initial state, and the final state. We can write our expression as

∣Mfi∣2

1 D Ei

∏2 i

where D is often called Lorentz-invariant phase space, and given by

D=

∏ (2 f

d 3pf

π )32Ef

(2π )4δ 4( pin − pout ).

2

Note that both ∣Mfi∣ and D are Lorentz invariant. 5.7.1 Decay rates For a decaying particle of energy Ei decaying into a region of phase space via the matrix element Mfi, the differential decay rate dΓ is

d Γ = ∣Mfi∣2

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1 D 2Ei

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and the total decay rate for this decay process is Γ=

∫ ∣Mfi∣2 21Ei D .

For most unstable particles, there is more than one decay process possible, and the total rate ΓT is just the sum of the rates for all possible decay processes. The lifetime τ of an unstable particle is 1/ΓT . Note that the formula is for the decay rate of a moving particle. If we write this as ΓE , where E is the energy of that particle, then the decay rate in the rest frame of the particle is Γμ, where μ is the mass of the particle. The energies in the two frames are related by E = γμ, so ΓE =

μ 1 Γμ = Γμ γ E

which is the expected relativistic result for the decay rate. As an example, consider the decay ϕ → NN¯ in our simple model, which is only possible when μ > 2m. The matrix element for the lowest-order decay process is iM = −ig . The final state factor is given by

∫

D=

∫

=

∫

d 3p1

d 3p2

(2π )32E1 d 3p

(2π )32E 2

(2π )34E 2

(

(2π ) 4δ 3(p1⃗ + p2⃗ )δ μ −

(

(2π )δ μ − 2 p ⃗ 2 + m 2

p1⃗ 2 + m 2 −

p2⃗ 2 + m 2

)

)

This reflects that in the rest frame of the meson, which is also the center of mass frame, the nucleons come out back to back, with momenta p ⃗ and −p ⃗ , each with energy E = p ⃗ 2 + m2 determined by 2E = μ. Solving for the magnitude p of p ⃗ , we find p=

μ2 − m2 . 4

Then ∫ D is given by

∫ D=∫

4πp 2 dp 2πδ μ − 8π 34E 2

(

We can use the formula δ ( f ( p )) =

1 δ (p − p0 ), ∣f ′(p0 )∣

where f ( p0 ) = 0, to find

∫

1 1 p2 4πE 2 ⎛ 2p ⎞ ⎜ ⎟ ⎝E⎠ p = 4πE 1 = μ 2 − 4m 2 8πμ

D=

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)

p2 + m2 .

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

Note that ∫ D approaches zero as μ approaches 2m from above. reflecting the decreasing amount of phase space as the mass μ decreases. Our expression for the decay rate Γ is

1 2 1 g μ2 − 4m 2 2μ 8πμ g2 = μ2 − 4m 2 16πμ

Γ=

5.7.2 Cross-sections The differential transition probability per unit time is just given in the LAB frame by the differential cross-section dσ times the number density ρ of incoming particles times the magnitude ∣v1⃗ ∣ of the incoming particle. We have already arranged for ρ to be one per unit volume when we considered initial states in a box. Thus in the LAB frame

dσ =

⎡ ⎤ 1 1 ⎢∏ ∣M ∣2 D⎥ ⎥⎦ ∣v1⃗ − v2⃗ ∣ ⎢⎣ i 2Ei

where in the LAB frame, the velocity of the target, v2⃗ , is zero and E2 = m2 . It turns out this formula is also correct for both particles moving along the beam axis, which we take to be the z-axis. In this case, we can write

p1z p − 2z E1 E2 = 4∣E2p1z − E1p2z ∣ = 4∣ϵμν12p1μ p2ν ∣

4E1E2∣v1⃗ − v2⃗ ∣ = 4E1E2

where we are using the totally antisymmetric four-tensor ϵμναβ . The quantity ϵμν12p1μ p2ν is the 12 component of an antisymmetric two tensor ϵμναβp1μ p2ν . While it is not invariant under all Lorentz transformations, it is invariant under boosts along the z-axis, which is our beam axis. This invariance makes sense because we think of dσ as a differential element of area transverse to the beam, which is therefore invariant under boosts along the beam axis. In the CM frame, we have p1⃗ = −p2⃗ , so that 4∣E2p1z − E1p2z ∣ = 4ET 4∣pi⃗ ∣ where ET is the total energy of the initial state and ∣pi⃗ ∣ is the magnitude of the momentum of either initial state particle. Therefore in the CM frame

dσ =

1 ∣M ∣2 D . 4ET ∣pi⃗ ∣

For two-body scattering, we can also put the density of final states D into a simple form in the CM frame. If we take the four-momenta of the final state

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particles to be k1 and k2, then in the CM frame we have k1⃗ = −k2⃗ as well as p1⃗ = −p2⃗ . We have

D=

∫

d 3k1 d 3k2 (2π )4δ 4(p1 + p2 − k1 − k2 ) (2π )32E1 (2π )32E2

where Ej is now the energy of a particle in the final state. The integration over k2⃗ is easy, reducing D to

D=

d 3k1 2πδ 4(E1 + E2 − ET ). (2π )34E1E2

∫

The integration over d 3k1 can be written as an integral over the magnitude k = ∣k1⃗ ∣ and the solid angle Ω:

d 3k1 = k 2dkd Ω where d Ω = sin θdθdϕ. Thus

D=

∫

k 2dkd Ω 2πδ(E1 + E2 − ET ). (2π )34E1E2

The energies E1 and E2 which appear in the delta function are functions of k, but we can write

k k ∂ (E1 + E2 ) = + E1 E2 ∂k k (E1 + E2 ) = E1E2 kET = E1E2 so that D becomes

1 k 2d Ω 2π 3 ⎛ (2π ) 4E1E2 kET ⎞ ⎜ ⎟ ⎝ E1E2 ⎠ 1 kd Ω = 16π 2 ET

D=

where it is understood that k is determined by E1 + E2 = ET . Putting all the factors together, we can obtain

dσ =

1 1 pf d Ω · ∣M ∣2 · 4ET pi 16π 2 ET

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where we have written pi for momentum of either initial particle and pf for the momentum of either final particle. Simplifying, we have

1 ⎛ pf ⎞ dσ = ⎜ ⎟ ∣M ∣2 . dΩ 64π 2ET2 ⎝ pi ⎠ This formula simplifies in the case of identical masses, where pf = pi

dσ 1 = · ∣M ∣2 . dΩ 64π 2ET2 This form is also valid in the ultrarelativistic limit, where all masses are effective zero. We can also write

1 dσ = · ∣M ∣2 . 64π 2s dΩ It is often convenient to calculate ultrarelativistic scattering behavior using the Mandelstam variables. The total cross-section is given by the integral over solid angle:

σ=

⎛

⎞

∫ d Ω⎜⎝ ddΩσ ⎟⎠

which in the CM frame is

σ=

1 64π 2s

∫ d Ω ∣M∣2 .

It is common to write differential cross-sections as derivatives with respect to variables other than θ and ϕ. For example, the Mandelstam variable t is often used instead of θ because it has a clear relation to θ in the CM frame, but it is Lorentz invariant.

Problems 1. In our simplified model of nucleons and mesons, draw the two O(g 2 ) Feynman diagrams for the scattering process N + N¯ → ϕ + ϕ, and then give the corresponding Feynman amplitude iM. Write iM in terms of the Mandelstam variables and give the channels in which scattering occurs. 2. Compute to lowest non-vanishing order in g the center of mass differential cross-section for NN¯ scattering in our simplified model of nucleons and mesons. 3. Compute the total cross-section for NN scattering in the center of mass to order g 4 . 4. The short-lived neutral kaon ( μ = 498 MeV) decays predominantly into two charged pions (m = 145 MeV) with Γ = 0.776 × 1010 s−1. The matrix element for this process is just a number, so we can use the Feynman rules of the 5-24

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scalar Yukawa theory to calculate it. Find the dimensionless ratio g/μ. Are we justified in thinking of this process as a weak interaction? Note: ℏ = 6.58 × 10−22 MeV s.

Reference [1] Coleman S 2018 Lectures of Sidney Coleman on Quantum Field Theory (Hackensack, NJ: World Scientific)

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IOP Publishing

A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 6 Renormalization

The previous chapter focused on scattering and particle decay at the lowest order of perturbation theory. As we move to higher-order in relativistic field theories, we encounter divergences associated with Feynman diagrams containing loops. The physical origin of this behavior is based on the change in a particle’s properties when it is in an interacting medium. For example, electrons inside a metal or semiconductor do not respond to an external force with the same acceleration they exhibit in vacuum. Because of their interaction with the ion lattice and with other electrons, the mass m of an electron in free space must be replaced by an effective mass meff which can differ from m by an order of magnitude. Similarly, the speed of light inside a material is not c, but c/n, where n if the refractive index of the medium. These are quantum effects, and can be calculated using field theoretic methods. The vacuum of relativistic quantum field theories is in principle no different than the ground state of an interacting many-body system in condensed matter physics. In metaphorical language, they are rich in interactions of virtual particles, manifesting in perturbation theory as vacuum graphs. When those virtual particles interact with physical particles, properties of the physical particles such as the mass are changed. There are two differences, however, between condensed matter physics and particle physics. In condensed matter physics, we can take particles like electrons outside of material, effectively turning off the interaction and allowing us to compare the mass outside the material with the effective mass within. In particle physics, we cannot turn off the electric charge of an electron, and measure its mass without electric charge. On the other hand, there are also excitations in condensed matter physics, such as holes and phonons, which do not exist outside the material. In general, we must think carefully about what it means to measure something like a mass.

doi:10.1088/978-0-7503-3227-9ch6

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ª IOP Publishing Ltd 2022

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

6.1 Mass renormalization The ϕ4 field theory is a traditional model for studying renormalization. We have calculated the O(λ ) correction to the connected two-point function in four dimensions to be Gc(2)(p) =

⎡ iλ i i + ⎢− p 2 − m 2 + iϵ p 2 − m 2 + iϵ ⎣ 2

∫

⎤ d 4k i i . ⎥ 2 4 2 2 (2π ) k − m + iϵ ⎦ p − m2 + iϵ

The integral over k, associated with a loop in the corresponding Feynman diagram, is ultraviolet divergent, meaning it is divergent for large loop momentum. The numerator contains four powers of k, while the denominator has two powers, so this divergence is called a quadratic divergence. This divergence occurs in the same way in the two-point function at every order in perturbation theory, and can be summed up:

Gc(2)(p ) =

∞ ⎡⎛ i iλ ∑ ⎢⎜− p 2 − m 2 + iϵ n = 0 ⎣⎝ 2

∫

⎤n ⎞ d 4k i i ⎥ ⎟ (2π )4 k 2 − m 2 + iϵ ⎠ p 2 − m 2 + iϵ ⎦ ⎤ ⎥ 1 ⎥ ⎥ ⎞ d 4k i i ⎥ ⎟ (2π )4 k 2 − m 2 + iϵ ⎠ p 2 − m 2 + iϵ ⎦⎥

⎡ ⎢ i ⎢ = 2 ⎛ p − m 2 + iϵ ⎢ ⎢ 1 − ⎜ − iλ ⎢⎣ ⎝ 2 i . = 4 iλ d k 1 i + ϵ p2 − m2 − 2 (2π )4 k 2 − m 2 + iϵ

∫

∫

The effect of the sum of these graphs is a shift in the mass of the particle, as determined by the pole in the propagator, from its non-interacting value of m2 to a new value:

m2 → m2 −

iλ 2

∫

d 4k 1 . 4 2 (2π ) k − m 2 + iϵ

The analogous effect in quantum mechanics is a shift in the difference between the energies of the ground state and the first excited state due a perturbation. This kind of shift is familiar, and occurs in most quantum systems. The novelty of relativistic field theory is that the shift is formally infinite, which we address below. This change of the mass is the lowest-order contribution to the particle’s self-energy, so called because it represents the modification of the propagator by interactions. Let us define a one-particle-irreducible (1PI) graph as a connected graph which does not become disconnected if any one internal line is cut. The subgraph we have summed to obtain the lowest-order self-energy,

−

iλ 2

∫

d 4k i 4 2 (2π ) k − m 2 + iϵ

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is one-particle irreducible because the graph does not fall into two pieces if the single propagator line is cut. On the other hand, the graph

⎡ iλ i ⎢− p 2 − m 2 + iϵ ⎣ 2

∫

⎤ d 4k i i ⎥ (2π )4 k 2 − m 2 + iϵ ⎦ p 2 − m 2 + iϵ

is not 1PI because cutting either of the two external lines divides the graph into two disconnected pieces. Examples of 1PI and non-1PI graphs are shown in figure 6.1. The external legs are conventionally drawn in 1PI graphs, but are not part of the 1PI graph. We define the sum of all 1PI corrections to the propagator to be the self-energy −i Σ(p2 ). It generally inherits the spacetime symmetry of the model, so in a relativistic model, it is a function of p2 . In non-relativistic field theories, it is a function of the energy ω and the three-momentum p ⃗ . In the ϕ4 model, we have to lowest order in perturbation theory

− i Σ( p 2 ) = −

iλ 2

∫

d 4k i + O(λ2 ). (2π )4 k 2 − m 2 + iϵ

Figure 6.1. Examples of 1PI and non-1PI graphs. The first two graphs are 1PI, because cutting any internal propagator leaves the graph connected. The third graph, formed by combining the first two, is not 1PI, because cutting the single line connecting the two subgraphs does result in a disconnected graph.

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We can sum up the self-energy contributions to the interacting propagator in a geometric series just as we did its lowest-order contribution:

Gc(2)(p ) =

∞ ⎡ ⎤n i i 2 i p − Σ ⎢ ⎥ ( ) ∑ p 2 − m 2 + iϵ ⎦ p 2 − m 2 + iϵ n = 0 ⎣

⎡ ⎤ ⎢ ⎥ i 1 = 2 ⎢ ⎥ 2 2 p − m + iϵ ⎢ 1 − Σ(p ) 2 2 p − m + iϵ ⎥⎦ ⎣ i . = 2 2 p − m − Σ(p 2 ) + iϵ This summation is known as Dyson’s series. It can be written in a very compact form as

Gc(2)(p ) = G0(2)(p ) + G0(2)(p )( −i Σ(p 2 ))Gc(2)(p ) as shown in figure 6.2. This form of Dyson’s series can also be written in coordinate space using convolutions. If we knew the self-energy of a particle exactly, we would also know the exact interacting propagator for that particle. Typically, we know the self-energy to some low-order of perturbation theory. In most non-relativistic manybody theories, the energy and momentum integrations do not diverge, and the particle’s effective mass meff in the interacting medium is modified from its value in free space, as in the case of an electron in a metal. In this case, we can determine the effective mass by locating the pole in the propagator. This is given by solving 2 2 m eff − m 2 − Σ(m eff ) = 0.

This procedure does not give us anything sensible in relativistic field theories, for a good reason: we cannot turn off the interaction or take the particle out of the interacting medium. The mass we find experimentally is the pole mass mP or physical mass; we have no direct access to any mass appearing in the Lagrangian. Pole masses are physical, and appear in many places in field theory. For example, we have seen that the pole mass of an exchanged particle determines the range of the associated potential. The process of

Figure 6.2. Graphical representation of Dysonʼs formula. The heavier line indicates the full propagator Gc(2)(p ) while the lighter line is the free propagator G 0(2)(p ).

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1

renormalization, a hallmark of relativistic quantum field theories, is the process of rewriting perturbation theory so that only physically measurable quantities, like the mass of an electron or its charge, appear in the perturbative expansion. Let us see how this can be done for the lowest-order self-energy of the ϕ4 theory. We must calculate the lowest-order self-energy

−i Σ (1)(p 2 ) = −

iλ 2

d 4k i 4 2 (2π ) k − m 2 + iϵ

∫

in such a way that the divergence is made explicit. As a first step, we perform a Wick rotation on the integral over k. A Wick rotation changes a Minkowski space calculation into an equivalent one in four-dimensional Euclidean space. We once again note that the denominator k 2 − m2 + iϵ has poles at k 0 = +ω k ⃗ − iϵ and k 0 = −ω k ⃗ + iϵ . We are free to deform the contour of integration for k 0 as long as the contour does not cross the poles, so we rotate the contour counter-clockwise, replacing k 0 with ik 4 , so dk 0 becomes idk 4 , and k 4 runs from −∞ to +∞. The self-energy is

−i Σ (1)(p 2 ) = −

iλ 2

id 4kE i 2 4 (2π ) −k E − m 2 + iϵ

∫

2

where k 2 has been replaced by −k E2, given by k E2 = (k 4 )2 + k ⃗ . Now

−i Σ (1)(p 2 ) = −

iλ 2

∫

d 4kE 1 . 2 4 (2π ) k E + m 2

The iϵ in the propagator is no longer needed and has been dropped. Our next step is to regularize this integral, which is to make it finite so it can be manipulated, and to make clear the dependence on m and any other parameters. There are several different useful regularization schemes; for the moment, we will use a simple and intuitive regularization, a four-dimensional Euclidean cutoff. This means that we will integrate over Euclidean momenta satisfying k E2 < Λ2, where Λ has dimensions of mass. We assume Λ ≫ m. Because the integrand depends only on the magnitude of kE , all we need to perform the integral is the decomposition of d 4kE into a radial integration over k = kE times the hyperspherical area of a unit sphere, Ω 4 : d 4kE → Ω 4k 3dk . When we study dimensional regularization, we will derive the general form for Ωd in any dimension, but for now we only need that Ω 4 = 2π 2 . We now can write

− i Σ (1)(p 2 ) = −

iλ 2

∫0

Λ

2π 2k 3dk 1 4 2 (2π ) k + m 2 Λ2

k2 k + m2 ⎛ Λ2 + m 2 ⎞⎤ iλ ⎡ 2 2 ⎢ m =− Λ − log ⎜ ⎟⎥ ⎝ m 2 ⎠⎦ 32π 2 ⎣ ⎛ Λ2 ⎞⎤ iλ ⎡ 2 2 ⎢ m =− Λ − log ⎜ 2 ⎟⎥ ⎝ m ⎠⎦ 32π 2 ⎣ =−

iλ 32π 2

∫0

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where in the last step we have used Λ ≫ m. Our result is quadratically divergent and also has a subleading logarithmic divergence, both of which are independent of p2 . The propagator denominator is now

p2 − m2 −

⎛ Λ2 ⎞⎤ λ ⎡ 2 ⎢Λ − m 2 log ⎜ 2 ⎟⎥ . 2 ⎝ m ⎠⎦ 32π ⎣

Because we cannot set λ to zero, and we can only measure the physical mass mP determined by the pole, we have

m P2 = m 2 +

⎛ Λ2 ⎞⎤ λ ⎡ 2 2 ⎢ Λ − m log ⎜ 2 ⎟⎥ . ⎝ m ⎠⎦ 32π 2 ⎣

We say that m P2 is the sum of the square of the bare mass m and a perturbative correction. The bare mass is conceived of as the mass the particle would have if λ = 0, or the mass parameter which appears in the original Lagrangian. If m P2 is to be finite as the cutoff Λ → ∞, then the bare mass must diverge in that same limit.

6.2 Coupling constant renormalization We also see divergences in scattering. We continue with the ϕ4 theory in four dimensions, and consider two-body scattering, p1 + p2 → p1′ + p2′. We know that to lowest order in perturbation theory we have iM = −iλ . There are three Feynman graphs which contribute at order λ2 , as shown in figure 6.3. In each graph, there are two vertices connecting two external lines and two internal lines. The three different graphs are differentiated by whether p1 shares a vertex with p2, p1′ or p2′. As we will see shortly, those three possibilities correspond to exchange of two virtual particles in the s, t or u channel. Let us consider the t-channel process, where p1 comes into one vertex and p1′ comes out. We perform this calculation in detail, and begin with the contribution to the scattering amplitude iM, which includes a factor of (2π )4δ 4(p1 + p2 − p1′ − p2′ ). We write the amplitude as i A(p1 , p2 , p3 , p4 ) = i A(p1 , p2 , −p1′ , −p2′ ), adhering to our previous notation where p1 + p2 + p3 + p4 = 0. We have

i A(p1 , p2 , −p1′ , −p2′ ) =

( −iλ)2 2

∫

d 4k1 d 4k2 (2π )4δ 4(p1 + k1 − p1′ − k2 )(2π )4δ 4 (2π )4 (2π )4

(p2 + k2 − p2′ − k1) ·

(2π )4δ 4(p2 + k2 − p2′ − k1)

where the symmetry factor of 2 comes from the permutation symmetry of the two internal lines. Integrating over k2, we see that the first delta function sets so that the second delta function becomes k2 = p1 − p1′ + k1 4 δ (p1 + k2 − p1′ − p2′ ). Relabeling k1 as k, we can now write

i A(p1 , p2 , −p1′ , −p2′ ) = iA(p1 , p2 , −p1′ , −p2′ )(2π )δ 4(p1 + k2 − p1′ − p2′ )

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Figure 6.3. Scattering in a ϕ4 model to O(λ2 ). In addition to the tree-level vertex, there are three one-loop diagrams, which are t-channel, u-channel and s-channel processes, respectively. Because these graphs are divergents, a counterterm, shown as a small cross, is also required. See section 6.4 below for an explanation of counterterms.

where

iA(p1 , p2 , −p1′ , −p2′ ) =

( −iλ)2 2

∫

d 4k i i . (2π )4 (k + p − p ′ )2 − m 2 + iϵ k 2 − m 2 + iϵ 1 1

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The amplitude iA(p1 , p2 , −p1′ , −p2′ ) contributes to the scattering amplitude iM. We say that this is a one-loop contribution to the scattering amplitude, and the associated diagram is a one-loop diagram, because there is one independent internal momentum loop which must be integrated over. Looking at the large-k behavior of the amplitude, we see that it behaves as ∫ d 4k /k 4 , indicating a likely logarithmic ultraviolet divergence. Because we are working within a Lorentz-invariant framework, iA can 2 2 only depend on the external momenta as a function of (p1 − p1′ ) = (p2 − p2′ ) = t so we can write it simply as iA(t ). The contribution of the other two O(λ2 ) graphs are readily seen to be iA(s ) and iA(u ), so the scattering amplitude at one loop is given by

iM = −iλ + iA(s ) + iA(t ) + iA(u ) + O(λ3). Of course, we would like to perform the integral over k and show, for example, that iA(t ) is indeed a function of t and logarithmically divergent. We will accomplish this in three steps, which are typical for these kinds of calculations of Feynman amplitudes containing loops. First, we must choose a regularization method for the momentum integral. This modification of the integral makes it temporarily nondivergent, and allows us to perform operations which would be mathematically dubious for a divergent integral. We could use a four-dimensional cutoff, where we restrict the integration range to k < Λ . This would work in this case, but it does have the unpleasant feature of not respecting Lorentz invariance. An alternative which was used extensively for QED is Pauli–Villars regularization, which modifies the propagators for internal lines in a Lorentz-invariant way which softens the largemomentum behavior. A typical example is the modification

i i i → 2 − 2 . k 2 − m 2 + iϵ k − m 2 + iϵ k − Λ2 + iϵ For any finite value of Λ, the propagator behaves as 1/k 4 for large momentum, which softens any divergences in a given integral. In the limit Λ → ∞, we formally regain the usual propagator. This modification of the propagator has the obvious interpretation of a propagator for a particle of mass m minus a propagator for a particle of mass Λ. The minus sign, which is necessary to achieve the better ultraviolet behavior, also leads to an interpretational problem: the ‘Λ particle’ has a negative Hilbert space norm. No known regularization scheme is without difficulties, or is satisfactory for every field theory. The most widely used regularization scheme available today is dimensional regularization, developed by ’t Hooft and Veltman [1] to handle the renormalization of non-Abelian gauge theories. For four-dimensional field theories, it consists of changing the dimension of the loop integrations from d = 4 dimensions to d = 4 − ϵ dimensions, where ϵ is a small number. Conceptually, we view the limit ϵ → 0 in dimensional regularization as equivalent to the limit Λ → ∞ for a four-dimensional cutoff or for Pauli–Villars regularization. The key step is the replacement

∫

d 4k → μϵ (2π )4

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∫

d 4−ϵk (2π )4−ϵ

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

where the mass scale μ keeps the mass dimension of the momentum integration fixed at 4, independent of ϵ. The amplitude we wish to calculate is now

iA(p1 , p2 , −p1′ , −p2′ ) =

( −iλ)2 ϵ μ 2

∫

d 4−ϵk i 4 −ϵ (2π ) (k + p1 − p1′ )2 − m2 + iϵ

i k − m 2 + iϵ 2

where the ϵ = 4 − d is completely distinct from the −iϵ attached to m2 . Fortunately for our sanity, the latter will remain firmly attached to m2 , and play no part in the analytic continuation to non-integral dimensions. This continuation to d = 4 − ϵ introduces a new parameter μ; we will discuss its interpretation after we have completed our calculations. Our next step is to combine the denominators of the propagators. There is a standard technique for this due to Feynman:

1 = ab

1

∫0 dx [ax + b(11 − x)]2 .

This result is easily proved by simply integrating over x, and generalized to a 2 product of three or more factors. Here we take a = (k + p1 − p1′ ) − m2 + iϵ and b = k 2 − m2 + iϵ , giving a denominator term

(

2

ax + b(1 − x ) = x k 2 + 2k · (p1 − p1′ ) + (p1 − p1′ ) − m 2 + iϵ 2

)

2

+ (1 − x )(k − m + iϵ ) 2

= k 2 + 2xk · (p1 − p1′ ) + x(p1 − p1′ ) − m 2 + iϵ . If we now make the change of variable k → k − x(p1 − p1′ ), then the denominator term becomes 2

k 2 + (x − x 2 )(p1 − p1′ ) − m 2 + iϵ so

iA(p1 , p2 , −p1′ , −p2′ ) =

λ2 ϵ μ 2

∫

1

∫0

dx

d 4−ϵk (2π )4−ϵ

1 . 2 ⎡k 2 (x x 2 ) p ⎤2 2 ′ p m i + − − − + ϵ ( ) 1 1 ⎣ ⎦ 2

It is now clear that the integral is a function of t = (p1 − p1′ ) , so we will now write it as iA(t ). We will now derive a procedure for evaluating integrals of the form

I0(α ) ≡ μϵ

∫

d 4−ϵk 1 (2π )4−ϵ (k 2 − m 2 + iϵ )α

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where α is a positive integer. Under a Wick rotation, I0(1) becomes

I0(1) = −iμϵ

∫

d 4−ϵkE 1 . 2 4 −ϵ (2π ) k E + m 2 − iϵ

We introduce a representation of the denominator which turns the integral over kE in a Gaussian integral

I0(1) = −iμϵ

∫

d 4−ϵkE (2π )4−ϵ

∞

∫0

2

2

dt e−t(kE +m −iϵ)

which can be carried out to give ∞ iμϵ π d /2 −t(m 2−iϵ) dt e t d /2 (2π )d 0 ∞ iμϵ 2 =− dt t −d /2e−t(m −iϵ) d /2 0 (4π ) ϵ ⎛ d iμ d⎞ 2 ⎜ ⎟(m − iϵ ) 2 −1. =− Γ − 1 (4π )d /2 ⎝ 2⎠

I0(1) = −

∫

∫

For our purposes, we want I0(2) which can be obtained from

∂ I0(1) ∂m 2 ⎛ ⎞ d iμϵ d ⎞⎛ d =− Γ⎜1 − ⎟⎜ − 1⎟(m 2 − iϵ ) 2 −2 d /2 ⎝ ⎠ (4π ) 2 ⎠⎝ 2 ϵ ⎛ d iμ d⎞ = Γ⎜2 − ⎟(m 2 − iϵ ) 2 −2 d /2 ⎝ ⎠ (4π ) 2

I0(2) =

after using z Γ(z ) = Γ(z + 1) with z = d /2 − 1. We now have a regularized expression for iA(t ):

iA(t ) =

⎛ϵ⎞ iλ2μϵ (4π )−2+ϵ/2 Γ⎜ ⎟ ⎝2⎠ 2

1

∫0

dx (m 2 − (x − x 2 )t − iϵ )−ϵ/2

We want the behavior of this integral as ϵ → 0, specifically the parts which diverge or are finite and non-zero as ϵ → 0. The key result needed is the behavior of the gamma function near zero:

⎛ϵ⎞ 2 Γ⎜ ⎟ = − γ + O ( ϵ ) ⎝2⎠ ϵ where γ ≈ 0.577 2 is the Euler–Mascheroni constant. For the other factors, we write

μϵ = e ϵ log μ = 1 + ϵ log μ + O(ϵ 2 ) ϵ (4π )ϵ/2 = 1 + log (4π ) + O(ϵ 2 ) 2 ϵ −ϵ /2 2 2 = 1 − log (m 2 − (x − x 2 )t − iϵ ) + O(ϵ 2 ). (m − (x − x )t − iϵ ) 2 6-10

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Putting all this together, we have

iA(t ) =

iλ2 ⎡ 2 ⎢ − γ + log (4π ) − 2(4π )2 ⎣ ϵ

1

∫0

⎛ m 2 − (x − x 2 )t − iϵ ⎞⎤ dx log ⎜ ⎟⎥ μ2 ⎝ ⎠⎦

where we have dropped all the terms that vanish as ϵ → 0. To simplify our expressions, we define 1

f (t , μ 2 ) =

∫0

⎛ m 2 − (x − x 2 )t − iϵ ⎞ dx log ⎜ ⎟ μ2 ⎝ ⎠

so

iA(t ) =

⎤ iλ2 ⎡ 2 − γ + log (4π ) − f (t , μ2 )⎥ 2⎢ ⎣ ⎦ 2(4π ) ϵ

The complete scattering amplitude to order λ2 is given by

iM (s , t , u ) = −iλ + iA(s ) + iA(t ) + iA(u ) which can be written as

iM (s , t , u ) = −iλ + −

⎤ 3iλ2 ⎡ 2 − γ + log (4π )⎥ 2⎢ ⎦ 32π ⎣ ϵ

iλ2 {f (s , μ2 ) + f (t , μ2 ) + f (u , μ2 )} . 32π 2

The second term isolates the contribution which diverges as ϵ → 0, while the third term is finite and contains the dependence of iM on s, t and u. The physical scattering amplitude iM is divergent as ϵ → 0 if λ is kept finite in the limit. As was the case with the mass, renormalization is necessary to obtain a finite answer. The question is, how do define the renormalized, physical coupling, which is to say, how do we measure it? In the case of the electric charge, a familiar coupling constant, the definition was assumed from the earliest experiments. Suppose we scatter two non-relativistic electrons so that s ≃ 4m e2 and t and u are very small on the scale of the electron mass me. By fitting the results of scattering data to the Rutherford formula for the cross-section, we can determine e 2 . We can do the same here: we declare that our definition for the physical value λP is that

−iλP = iM (4m 2 , 0, 0) . This implies that − iλP = − iλ +

⎤ 3iλ2 ⎡ 2 iλ 2 − γ + log (4 π ) − f 4m 2 , μ 2 ) + f (0, μ 2 ) + f (0, μ 2 )} + O(λ3) ⎢ ⎥ ⎦ 32π 2 { ( 32π 2 ⎣ ϵ

which is the beginning of a formal power series expansion of λP in powers of λ. We can also write λ as a formal expansion in powers of λP , which is −iλ = −iλP −

⎤ 3iλ P2 ⎡ 2 iλ P2 − γ + log (4 π ) + f 4m 2 , μ 2 ) + f (0, μ 2 ) + f (0, μ 2 )} + O(λ P3 ). ⎢ ⎥ ⎦ 32π 2 { ( 32π 2 ⎣ ϵ

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Now we rewrite iM (s, t , u ) in terms of λP : iM (s, t, u ) = −iλP −

iλ P2 3 2 2 2 2 2 2 {f (s, μ ) − f (4m , μ ) + f (t, μ ) + f (u, μ ) − 2f (0, μ )} + O(λ P ). 32π 2

By this rewriting we have accomplished several things. One is obvious: the divergence associated with ϵ → 0 is removed. The second is more subtle: the dependence on μ, introduced by dimensional regularization is also removed, because, for example, f (s, μ2 ) − f (4m2 , μ2 ) is independent of μ. Other definitions of the physical coupling are also possible. For example, we could define a new coupling

iM (0, 0, 0) = −iλP ′. This point, with s = t = u = 0, is not directly accessible in two-body scattering, but can be inferred from physical scattering data by analytic continuation. Of course, there is only one coupling constant, however, it is defined, so our new coupling constant can be written in terms of the old one:

−iλP ′ = −iλP −

iλ P2 f (0, μ2 ) − f (4m 2 , μ2 )} + O(λ P3) 2{ 32π

and vice versa. One convenient regularization scheme is to choose the definition so that only the term

⎤ 3iλ2 ⎡ 2 − γ + log (4π )⎥ 2⎢ ⎦ 32π ⎣ ϵ is removed. This is known as the MS-bar (MS ) renormalization prescription. The original scheme of this type was known as minimal subtraction, or MS, in which only the term

3iλ2 2 32π 2 ϵ was removed. It was later found more natural to also remove the −γ + log 4π as well. In MS-bar,

−iλ MS = −iλ +

⎤ 3iλ2 ⎡ 2 − γ + log (4π )⎥ 2⎢ ⎦ 32π ⎣ ϵ

so that

iM (s , t , u ) = −−iλ MS −

2 iλ MS 3 f (s , μ2 ) + f (t , μ2 ) + f (u , μ2 )} + O(λ MS ). 2{ 32π

Of course, these different definitions of the coupling constant represent the same physics. By comparing the different expressions for iM (s, t , u ), we see that all three are equivalent, and we can move between them if needed. However, the MS regularization scheme is usually the simplest and easiest in most cases. In the case of non-Abelian gauge theories, the MS scheme is standard.

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6.3 Field renormalization In addition to mass and coupling constant renormalization, there is another necessary renormalization, renormalization of the field. We have seen that Dyson’s series gives us an expression for the full propagator of the interacting theory as (2) G˜c (p ) =

i . p 2 − m 2 − Σ(p 2 ) + iϵ

Suppose we know that the pole is at p2 = m P2 . We can expand the propagator denominator around p2 = m P2 as

p 2 − m 2 − Σ(p 2 ) + iϵ = p 2 − m 2 − Σ(m P2 ) − (p 2 − m P2 )Σ′(m P2 ) + Σ(p 2 ) where the prime indicates differentiation with respect to p2 . The remainder term Σ(p2 ) represents the higher-order terms in the expansion of Σ(p2 ) and satisfies Σ(m P2 ) = 0 and Σ′(m P2 ) = 0. We identify m2 + Σ(m P2 ) = m P2 and write (2) G˜c (p ) =

i i = 2 2 p 2 − m 2 − Σ(p 2 ) + iϵ − Σ′ m p − m P2 ) − Σ(p 2 ) + iϵ 1 ( ( P ))(

so the residue near the pole is

Zϕ =

i 1 − Σ′(m P2 )

which we define to be the field renormalization constant. Near the pole, we can write (2) G˜c (p ) ≈

iZϕ p 2 − m p2 + iϵ

+ O(p 2 − m P2 ) .

The change in the residue from i to iZϕ can be eliminated by defining a renormalized field ϕR by ϕR = Zϕ1/2ϕ. Then the renormalized propagator, defined using renormalized fields, behaves as (2) G˜ cR (p ) ≈

i + O(p 2 − m P2 ) . 2 p − m p + iϵ 2

As we have seen, in the ϕ4 model, the first-order contribution to Σ(p2 ) is independent of p2 , so Zϕ = 1 to order λ. However, at order λ2 , there is a divergent contribution to Σ′(m P2 ), which must be removed. Even in theories without divergences in Σ′(m P2 ), there is generally a non-zero finite part which must be taken into account. As with the renormalization of masses and coupling constants, there are various way to define the renormalized field. Using dimensional regularization, it is natural to renormalize the field using minimal subtraction, which does produce a propagator residue of i at the pole. There is a finite multiplicative factor between propagators renormalized using two different renormalization schemes, but the physics is ultimately the same.

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6.4 Renormalization: a systematic process Our current procedure for renormalization is cumbersome. For example, the physical mass is defined implicitly in terms of the self-energy, and expanding in the bare coupling means a tedious process of re-expansion in the renormalized coupling. Here we develop a systematic process for renormalization in which we start from a finite mass and coupling. At this point, we switch notation and write the bare mass as m0; the notation mB is also used. The physical mass mP will be written simply as m, but is sometimes written as mR, for renormalized mass. The bare coupling will be λ 0 and the renormalized coupling simply λ. Similarly, the renormalized field will be ϕ, sometimes written as ϕR , and the bare field will be ϕ0, also written as ϕB . With this convention, our original Lagrangian is

L=

1 1 λ (∂ϕ0)2 − 2 m02ϕ02 − 40! ϕ04 . 2

The process of renormalization is the writing of the perturbative expansion of a field theory so that no divergences appear, and the perturbative expansion is written solely in terms of physically defined parameters. We begin by writing the Lagrangian as

L=

1 1 1 1 λ λ (∂ϕ)2 − m 2ϕ 2 − ϕ4 + δϕ(∂ϕ)2 − δmm 2ϕ 2 − δλϕ4 . 2 4! 2 4! 2 2

The extra terms in L involving δϕ , δm , and δλ are referred to as counterterms Their role is to remove all divergences encountered in a perturbative expansion. Just like the λϕ4 term in the Lagrangian, counterterm are treated as perturbations. The counterterm parameters δϕ , δm , and δλ are determined order by order in perturbation theory by the renormalization conditions, and each one is a power series in λ. We can write the complete Lagrangian as the sum of a free Lagrangian, an interaction and counterterms

L = L0 + LI + L .CT with LI + L.CT treated as a perturbation of the free Lagrangian. We now have two expressions for the same Lagrangian. Comparing the two, we see that we must have

1 1 (∂ϕ0)2 = 2 (1 + δϕ)(∂ϕ)2 2 so that Zϕ = 1 + δϕ . Defining Zm = 1 + δm and Zλ = 1 + δλ , we see that

L=

1 1 λ Zϕ(∂ϕ)2 − Zmm 2ϕ 2 − Zλ ϕ4 2 4! 2

and also

L=

1 1 λ (∂ϕ0)2 − 2 ZmZϕ−1m2ϕ02 − ZλZϕ−2 4! ϕ04 2

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so that m 02 = ZmZϕ−1m2 and λ 0 = ZλZϕ−2λ . The Feynman rules for the counterterms are simple. Just as the Feynman rule for the ϕ4 vertex is −iλ, the rule for the coupling counterterm is −iδλλ . We can combine the field and mass counterterms in a single rule: i (δϕp2 − δmm2 ). The procedure for renormalization is this: we calculate the lowest-order corrections to the self-energy and the scattering amplitude to lowest order in perturbation theory. Counterterms are added so that the propagator and scattering amplitude satisfy the renormalization conditions. At this order, the propagator is

i p − m − Σ(p ) + δ ϕ(1)p 2 − δm(1)m 2 2

2

2

where δ ϕ(1) and δm(1) are the lowest-order contributions to δϕ and δm . They are chosen so that the propagator is free of divergences. If our choice is to have m2 be the physical mass, and the propagator to have residue i at the pole, then

δm(1)m 2 = −Σ(m 2 ) and

δ ϕ(1) =

∂Σ(p 2 ) ∂p 2

. p 2 =m 2

As we have seen, in the case of ϕ4 , δ ϕ(1) is zero, but δm(1) is divergent. A graphical representation of the self-energy to one loop is shown in figure 6.4. Similarly, the lowest-order expression for the scattering amplitude is now

iM (s , t , u ) = −iλ + iA(s ) + iA(t ) + iA(u ) − iδλλ where λ is now the renormalized coupling, and previously calculated function A now depends on the renormalized coupling as well. Because A is divergent and of order λ2 , we must choose δ λ(1) so that the sum of the five terms is finite and satisfies the renormalization condition we have chosen. When we calculate the self-energy or scattering amplitude at the next order, we must include the contributions of δ ϕ(1) and δm(1). Their role is to cancel the divergences in subgraphs, so that δ ϕ(2) and δm(2) are chosen to handle the new divergences which appear at this next order in perturbations theory. The same procedure is followed to all orders in perturbation theory.

Figure 6.4. The two graphs contributing to the one-loop self-energy in a ϕ4 model. The first graph is divergent; this divergence is removed by the second graphs, which is a counterterm.

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6.5 Renormalizability In order to better understand how renormalization handles ultraviolet divergences, we define a quantity known as the superficial degree of divergence D for any connected graph. Consider a graph in ϕ4 theory in d = 4 dimensions with E external legs, I internal lines, and V vertices. The number of internal momenta integrations after all delta functions have been satisfied is defined to be the number of loops L of the graph. The graphs we have examined so far are all one-loop graphs. The number of loops is related to I and V: each internal line momentum carries kj with it an integration over d 4kj , but each vertex carries with it a four-dimensional delta function that reduces the number of actual integrations. However, one delta function will always remain to conserve overall momentum. Thus we have

L = I − (V − 1) = I − V + 1. The number of vertices V is related to E and I, because each external line connects once to a vertex, and each internal line connects twice. For ϕ4 , there are four lines connecting to each vertex, so E + 2I = 4V . The superficial degree of divergence D is obtained by noting that each independent loop integration carries with it a factor of d 4k , and each scalar propagator a factor behaving as 1/k 2 in the ultraviolet. Thus

D = 4L − 2I = 4(I − V + 1) − 2I = 2I − 4V + 4 = 4 − 4E . We see that D decreases with E, and D < 0 indicates a superficially convergent graph. The superficial degree of divergence is the relevant estimator of the divergence of a graph because a complicated graph may contain divergent subgraphs, whose divergences are handled by the renormalization program. For example, we can construct a graph with E = 6 containing a subgraph with a logarithmic divergence related to coupling constant renormalization. This divergence is removed by renormalization at lower order than that of the complete graph. For ϕ4 , we need only consider the case of E even, because the symmetry ϕ → −ϕ only allows graphs with E even. The interesting cases are: E = 0 with D = 4; E = 2 with D = 2; and E = 4 with D = 0. Graphs with E = 0 are vacuum loops, and we have seen that the O(λ ) graph has two closed loops, each with a single propagator, so D = 4 is correct. For two-body scattering, we have E = 4, and we have seen that the one-loop divergences are logarithmic, with D = 0. Graphs with E = 2 contain two divergences, one associated with mass renormalization and the other associated with field renormalization. For E > 4, our formula indicates that D is negative. We often say that such graphs are convergent, but this is misleading. Graphs with E > 4 can have subgraphs with E ⩽ 4, and those subgraphs will have divergences, but the counterterms from lower orders in perturbation theory will remove those subgraph divergences.

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It is useful to generalize this calculation in two ways. First, we take our interaction to involve n fields rather than 4, so that we have a ϕ n interaction. Second, we change the dimension of our theory from four dimensions to an arbitrary dimensionality d. Our equation for the number of loops is unchanged

L = I − (V − 1) = I − V + 1 but the relation between E, I, and V becomes

E + 2I = nV . The superficial degree of divergence is now D = dL − 2I because propagators have the same ultraviolet behavior and dimension, but the dimensionality of the loop momental integrals changes. We now have

D = dL − 2I = d (I − V + 1) − 2I = (d − 2)I − dV + d ⎛d − 2⎞ ⎟(nV − E ) − dV + d =⎜ ⎝ 2 ⎠ ⎡ ⎛d − 2⎞ ⎤ ⎛d − 2⎞ ⎟ − d ⎥V − ⎜ ⎟E . = d + ⎢n⎜ ⎝ 2 ⎠ ⎣ ⎝ 2 ⎠ ⎦ To understand the implications of this formula, let us consider a ϕ6 interaction in d = 4. In that case, our formula becomes

D = 4 + 2V − E . For larger values of E, D decreases. However, for fixed E, D is always increasing with V. For E = 8 and V = 2, we find D = 0, indicating a logarithmic divergence. The divergent one-loop graph is essentially the same as the logarithmically divergent one we evaluated for ϕ4 , but now there are 2 internal legs and 4 external legs at each vertex. This divergence with E = 8 means that we must have a ϕ8 counterterm to make the eight-point function G˜ (8) finite. That means we must have a renormalization condition for a ϕ8 interaction defining a ϕ8 coupling constant. Our attempt to renormalize a ϕ6 model in four dimensions has made necessary a ϕ8 coupling as well. It is easy to see that the same graph with one ϕ8 vertex and one ϕ6 vertex will induce a divergence in the ten-point function, and a graph with two ϕ8 vertices will give rise to a graph with 12 external legs and a log divergence. A ϕ6 interaction in d = 4 induces divergences requiring the addition to the Lagrangian of all ϕ2n terms and the corresponding definition of infinitely many coupling constants. Field theories which require the specification of an infinite number of couplings are called non-renormalizable. Theories of this type are not meaningless and do not lack predictive power, but they do have a different character from renormalizable theories which require only a finite number of coupling constants as input. QED is a renormalizable model, requiring as input the electron mass and the fine structure constant. The Standard Model is also a renormalizable field theory: although it has a

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large number of parameters, that number is finite. On the other hand, the Lagrangian describing low-energy nuclear physics is non-renormalizable: in principle it requires the input of an infinite number of parameters. In practice, the number of parameters required at low orders in perturbation theory is finite, and gives useful information. In theory, the parameters of low-energy nuclear physics are determined by QCD, quantum chromodynamics. This is the sector of the Standard Model describing the strong interactions of quarks and gluons. QCD is renormalizable and has only a few parameters: the gauge coupling gS, analogous to e and the quark masses. The determination of the parameters of nuclear physics from QCD is a matter of ongoing research. It is natural to ask if fundamental field theories must be renormalizable. While opinions vary, the answer is that we do not know. What we can do is determine what field theories are renormalizable. The key condition is that the superficial degree of divergence D not increase as the number of vertices V increase. Thus renormalizability requires

⎛d − 2⎞ ⎟−d⩽0 n⎜ ⎝ 2 ⎠ or

n⩽

2d . d−2

In four dimensions, we must have n ⩽ 4, as we have already seen. In d = 3, we have the restriction n ⩽ 6, so interaction up to ϕ6 are allowed. In two dimensions, any finite values of n are allowed. Some two-dimensional models where the interaction can be defined by an infinite series in ϕ are renormalizable. For example, the sineGordon model, defined by

L=

1 α (∂ϕ)2 − 2 (1 − cos βϕ) 2 β

is renormalizable provided β 2 < 8π . If we consider the ϕ3 interaction, it is renormalizable provided d ⩽ 6. Although the case d = 2 turns out to be rather rich, the allowed interactions in scalar field theories for d > 2 is actually rather small. There is a simple connection between renormalizability and the dimensions of the coupling constant, which is often used as a shortcut for determining renormalizability. Consider a scalar field in d dimensions with action

S=

⎡

⎤

∫ d dx⎢⎣ 12 (∂ϕ)2 − 12 m2ϕ2 − λnn! ϕn⎥⎦.

In natural units, the action must be dimensionless because it appears in the exponential of the path integral. Looking at either the kinetic term or the mass term, we see that the units of ϕ are given by

[ϕ ] = M

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It follows that the dimensions of λ n are

[λn ] = M

( ).

d −n d −2 2

Comparing this with our formula for D, we see that if the mass dimension of λ n is positive or zero, the interaction is renormalizable, but if the mass dimension is negative, it is non-renormalizable. This reflects that fact that a coupling constant with negative mass dimension will have ever-increasing powers of the momenta in the numerator as V increases. As an easy application of this idea, consider the gravitational energy of two masses at a distance r from one another: E = Gm1m2 /r tells us that [G ] = M−2 suggesting that gravity constructed as a perturbation theory in G is non-renormalizable. As we have seen, in our usual system of units, the action has the same units as ℏ, and appears in the path integral as exp (iS /ℏ). It is often useful to consider a dimensionless version of ℏ as a tool for understanding various expansions in field theory. In perturbation theory, a vertex will always carry a factor of ℏ. The propagator is obtained as the functional inverse of a part of the Lagrangian quadratic in the fields, so every propagator carries with it a factor of 1/ℏ. Recalling that the number of loops L is given by L = I − V + 1, we see that the power P of ℏ associated with a graph is given by

P = L − 1. Graphs with no loops, L = 0, have P = −1. These are often referred to as tree graphs. One-loop graphs have P = 0, two-loop graphs have P = 1 and so on. The one-loop and higher corrections can thus be viewed as quantum corrections to treelevel processes,

6.6 Matrix elements and the LSZ reduction formula We can now correct our formula from chapter 5 for the scattering process p1 + p2 → p1′ + p2′, which was

⎡ 4 ⎛ p 2 − m 2 + iϵ ⎞ ⎤ j ⎟G˜c(4)(p , p , −p ′ , −p ′ )⎥ . iM (p1 + p2 → p1′ + p2′ ) = ⎢∏ ⎜⎜ 1 2 1 2 ⎥ ⎟ ⎢ i ⎝ ⎠ ⎣ j=1 ⎦ p =−p ′ ,p =−p ′ 3

1

4

2

This formula assumes that the propagators are normalized like free fields, with residue i at the pole mass. This is necessary because subsequent calculations with, e.g., densities of states assume that normalization. The need for such an adjustment appears in several different ways. Suppose we are using a renormalized four-point function where the renormalization conditions on the propagators are not a specification of the pole mass and the requirement that the residue at the pole mass be i. Then we must replace m2 by the pole mass m p2 in the formula above, and perform a finite re-scaling of the fields, ϕ → Zϕ1/2ϕ, to bring the two-point 6-19

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function to the necessary normalization. Thus the correct form for the scattering amplitude is ⎡ 4 ⎛ ⎤ p j2 − m p2 + iϵ ⎞ (4) ⎟G˜ (p , p , − p ′ , − p ′ )⎥ iM (p1 + p2 → p1′ + p2′ ) = ⎢∏ ⎜Zϕ1/2 1 2 ⎥ ⎢ ⎜ ⎟ c 1 2 i ⎠ ⎣ j=1 ⎝ ⎦p

. ′ ,p =−p ′ 4 2

3 =−p1

(4)

Now suppose we are working with the bare four-point function G˜0c . In this case, the field strength renormalization factor Zϕ1/2 is divergent, but the formula is exactly the same. A third case arises when composite operators create and destroy bound states. A well-known example involves the pion. There are three pion states, which we write as π a(k ) where a = 1, 2, 3. They form an isotriplet under isospin, which is an internal SU (2) symmetry. From the a = 1, 2 states we form the physical π ± particles, and the a = 3 state corresponds to the π 0. There are three axial currents A μa (x ) which create and destroy pions; these currents are fundamental in the theory of weak interactions. Lorentz invariance restricts the form of the relevant matrix element to have the form

0 A μa (x ) π b(p ) = ip μfπ e−ip·z where fπ is known as the pion decay constant; it is ubiquitous in pion physics. The pion decay constant can be determined experimentally and also obtained from lattice gauge theory simulations of quarks. For our purposes here, we observe that

0 ∂ μA μa (x ) π b(p ) = m π2fπ e−ip·x . This tells us that the operator

1 ∂ · Aa (x ) fπ m π2 can be used as a pseudoscalar field which creates canonically normalized pions in the LSZ formula, with 1/fπ mπ2 playing the role of Zϕ1/2 .

Problems 1. Prove the following generalization of Feynman’s trick:

1 = (n − 1)! a1…an

1

∫0

dx1…dxn

(

n

)

δ ∑ j = 1xj − 1

[x1a1 + ⋯+xnan ]n

.

Hint: use the Schwinger representation

1 = a

∞

∫0

dt e−ta .

and introduce a new variable r = t1 + ⋯ + tn . Then make the change of variable t j = rxj . 6-20

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2. Consider the Euclidean integral

I=

∫

d 4pE 1 . 4 2 (2π ) (p + m 2 )2 E

Calculate I using a 4d cutoff for regularization ( pE < Λ ). You will need the hyperspherical volume element d 4pE = p3 dpd Ω 4 → 2π 2p3 dp = π 2p2 d (p2 ). How does the divergence structure and finite part compare to what you obtain from dimensional regularization? 3. Calculate the O(λ ) to the self-energy

−i Σ (1)(p 2 ) = −

iλ 2

∫

d 4k i 4 2 (2π ) k − m 2 + iϵ

using dimensional regularization. Compare the divergent and finite parts of your result to the result using a four-dimensional UV cutoff.

Bibliography [1] ’t Hooft G and Veltman M J G 1972 Regularization and renormalization of gauge fields Nucl. Phys. B 44 189–213

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A Multidisciplinary Approach to Quantum Field Theory, Volume 1 An introduction Michael Ogilvie

Chapter 7 Symmetries and symmetry breaking

Symmetries give physics some of the most powerful tools physicists have available. Our understanding of the dynamics of complicated systems is often marred by ignorance, but symmetries offer a path to understanding that does not depend on the detailed dynamics, but on the symmetries obeyed by the dynamics. An understanding of the consequences of symmetries is crucial to understanding modern quantum field theory. Symmetries in quantum field theory may be classified according to whether they are discrete or continuous, and whether they are spacetime symmetries or internal symmetries. In this chapter, we explore both discrete and continuous internal symmetries and their spontaneous breaking. We explain why spontaneous symmetry breaking cannot occur in a system with a finite number of degrees of freedom, but can occur in systems with an infinite number of degrees of freedom. Goldstone’s theorem is introduced, and the interplay between renormalization and symmetry breaking is discussed.

7.1 Internal symmetries We have already seen a well-known example of a discrete internal symmetry. The ϕ4 Lagrangian

L=

1 1 λ (∂ϕ)2 − m 2ϕ 2 − ϕ4 2 4! 2

is invariant under the discrete transformation ϕ(x ) → −ϕ(x ). This is an internal symmetry: no spacetime coordinates are changed. A moment’s thought shows that the symmetry implies invariance under a p ⃗ → −a p ⃗ and similarly for its Hermitian conjugate. If this symmetry holds in the quantum field theory, then n-point functions transform as G (n)(x1 ... xn ) → ( −1) nG (n)(x1 ... xn ) so n-point function with n odd must be zero. Similarly, S-matrix elements must transform as p1′ ... pn′ S p1 ... pm → doi:10.1088/978-0-7503-3227-9ch7

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ª IOP Publishing Ltd 2022

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

( −1) n+m p1′ ... pn′ S p1 ... pm so that the number of particles can change in a collision only by a factor of 2. Put a little differently, the number of particles is conserved modulo 2. This symmetry is known as a Z(2) symmetry because the group Z(2) consists of +1 and −1. This discrete symmetry group is part of the class of Z (N ) groups. The elements of the group Z (N ) consist of the N complex roots of unity, Z (N ) = {z ∈  ∣z N = 1} = {e 2πin /N ∣N = 0 ... N − 1}. We can easily make a similar model with a continuous symmetry. Taking an N-component scalar field where it is understood that the vector notation refers to an internal symmetry, we write the fields as

⎛ ϕ1 ⎞ ⎜ ⎟ ⃗ ϕ=⎜⋮⎟ ⎜ ⎟ ⎝ ϕN ⎠ and the Lagrangian as

L=

1 1 2 2 λ ⃗2 2 2 ϕ . (∂ϕ ⃗ ) − m ϕ ⃗ − 2 2 4

( )

This model has an invariance under N-dimensional rotations: L is unchanged under the transformation ϕ ⃗ → Rϕ ⃗ , where R is an N × N real matrix obeying RT R = I . The set of such matrices form the group O(N ), the group of N × N orthogonal matrices. The use of λ/4 for the parameter controlling the strength of the coupling is more useful for N > 1 than the choice λ /4! we made for N = 1. This model is often referred to simply as the O(N ) model. The O(2) model, as we have seen, can be put into a complex form by defining a complex field Φ as

Φ=

1 (ϕ1 + iϕ2 ). 2

The Lagrangian can then be written as

L = ∂Φ*∂Φ − m 2Φ*Φ − λ(Φ*Φ)2 which has a U (1) symmetry, Φ → e iθ Φ corresponding to SO(2), the group of 2 × 2 rotation matrices with determinant 1. The groups SO(N ) of special orthogonal matrices differ from the groups O(N ), which includes improper rotations. Improper rotations obey RT R = I , but have det R = −1. For N odd, improper rotations are obtained by a reflection ϕ ⃗ → −ϕ ⃗ combined with a rotation. For N = 2, we can use the transformation (ϕ1, ϕ2 ) → (ϕ1, −ϕ2 ) to obtain all improper rotations. This corresponds to the symmetry Φ → Φ+ for the complex form of the O(2) model, and is the symmetry associated with charge conjugation. Scalar models with O(N ) internal symmetry have many uses in different areas of physics. Because the O(2) model has a complex form with particles and antiparticles, it provides a starting point for scalar electrodynamics, where the complex scalar field couples to the photon field. Scalar field theories are used in condensed matter physics to describe the behavior of ferromagnets, with the symmetry of the fields determined by the symmetry of the interactions of the magnetic domains. An interaction which

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favors the domains to be aligned along one particular axis have a Z(2) symmetry. Interactions between domains which are rotationally symmetric lead to field theory with an O(3) symmetry. Because both the symmetry group and the dimensionality of magnetic systems can vary, the number of interesting applications of field theory in such systems is very large. There are also systems which have approximate symmetries. Typically, this occurs when some small interaction term does not respect the symmetry. For example, we can add a small linear term to the one-component ϕ4 model:

L=

1 1 λ (∂ϕ)2 − m 2ϕ 2 − ϕ4 + Jϕ 2 4! 2

where J is a constant. This additional term breaks the symmetry under ϕ → −ϕ. There is no longer the symmetry which prevents an even/odd number of particles from turning into an odd/even number in a collision. However, if J is sufficiently small, the rate of this violation of the Z(2) symmetry is small. This is an important effect in particle physics, where there is a hierarchy of interaction strengths. The lightest mesons, the pions, provide us with an example of an approximate symmetry. There are three real pion fields, π 1, π 2 and π 3, which transform under an internal symmetry group O(3). This symmetry is a part of the isospin symmetry found in the low-energy sector of QCD. This symmetry is not exact because the u and d quarks, which combine with their antiquarks to make the pions as bound states, have different masses and different charges. These effects come from the electroweak sector of the Standard Model and are small compared to the strong interactions of QCD which give rise to pions as bound states. These symmetry breaking effects give rise to mass difference between the charged pions

π ±=

1 1 (π ± π 2 ) 2

and the neutral pion

π 0 = π 3. The masses of the pions are slightly different. The current values are

m π ± = 139.570 61(24) MeVc−2 m π 0 = 134.977 0(5) MeVc−2 indicating that isospin symmetry is a good approximate symmetry here. 7.1.1 Introduction to spontaneous symmetry breaking In addition to exact symmetries and approximate symmetries, there are also spontaneously broken symmetries. In spontaneous symmetry breaking, the symmetry group of the Lagrangian is not the symmetry group of the ground state. It is hard to exaggerate the importance of this concept in modern physics. We can approach spontaneous symmetry breaking from the intersection of statistical physics with everyday life. Spontaneous symmetry breaking is closely related to phase transitions, 7-3

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and the most common phase transitions in everyday life are the transitions from liquid water to steam and from water to ice. Molecules of water are, on average, distributed uniformly in the liquid phase and gaseous phase, but in the solid phase they are not. The ice in our refrigerator has a microcrystalline structure, and it is possible to make large crystals of water. The crystalline phase is invariant under a set of discrete translational symmetries of the crystalline lattice, but it does not have the full translational invariance that the liquid and gas phases have. This is an example of spontaneous symmetry breaking. The Hamiltonian for ice is the same as the Hamiltonian for water and steam, but the equilibrium state of ice does not have the same symmetries of the Hamiltonian. Another example of spontaneous symmetry breaking in our kitchens is the refrigerator magnet. These objects have permanent magnetic dipole moments, pointing in a fixed direction, yet the underlying microscopic physics of most magnets at the atomic level has a symmetry prohibiting a permanent dipole moment. One way to describe this in the language of quantum field theory is to say that the generator of some symmetry Q commutes with H, [Q, H ] = 0, but Q is not a symmetry of the vacuum: Q 0 ≠ 0. This leads to multiple possible ground states. We are faced with a situation where a symmetry of the Lagrangian (and Hamiltonian) is not manifested in nature. Let us consider a classical particle in a one-dimensional potential of the form

V (x ) =

λ 2 (x − v 2 )2 4!

plotted in figure 7.1. This potential is known as the double-well potential. It has two global minima at x± = ±v with V (x± = 0) and a local maximum at x = 0 with V (0) = λv 4 /4!. It has a Z(2) symmetry under x → −x because V ( −x ) = V (x ). If we expand the potential, we have

V (x ) =

λv 4 λv 2 2 λ − x + x4 4! 12 4!

Figure 7.1. The double-well potential V (x ) =

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λ (x 2 4!

− v 2 )2 .

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

so this is a form of the potential for the ϕ4 model where we have taken m2 to be negative, m2 = −λv2 /6 and added a constant λv 4 /4!. The point x = 0 is a point of unstable equilibrium because V ′′(0) < 0, while x± are stable points of equilibrium with V ′′(x± ) > 0. If we consider the same potential using quantum mechanics, we can imagine expanding around either minimum to obtain a local quadratic approximation to the potential

V (x ) ≃

λ (x ± v )2 6

from which we can obtain two Gaussian wave functions ψ±(x ) centered at x± = ±v, respectively. These two wave functions are related of course: we can write ψ±(x ) = ψ0(x − x± ) = ψ0(x ∓ v ) where ψ0 is a Gaussian centered at x = 0. These two wave functions, which are both positive, are mapped into one another by the symmetry x → −x , and the expectation value of the Hamiltonian is the same for both states. This looks like spontaneous symmetry breaking, because

ψ+ x ψ+ = +1 and

ψ+ x ψ+ = −1. However, this is in conflict with a standard theorem of quantum mechanics which states that the ground state is unique and positive. Perturbation theory misses the important nonperturbative phenomenon of tunneling between the two vacua. The possibility of tunneling is indicated by the non-zero overlap t between the two wave functions. If both states are correctly normalized to 1, then the overlap between the states will be some number between 0 and 1

t=

∫ dx ψ+*(x)ψ−(x) = ∫ dx ψ0(x − a)ψ0(x + a).

A better approximation to the true ground state is given by

ψe(x ) =

1 ⎡ ⎣ψ+(x ) + ψ−(x )⎤⎦ 2

which is invariant under parity and is positive everywhere. There is another state

ψo(x ) =

1 ⎡ ⎣ψ+(x ) − ψ−(x )⎤⎦ 2

which is odd under parity. In lowest-order perturbation theory, these two states have the same energy, but tunneling between the two sides of the potential lifts the degeneracy. The even wave function ψe(x ) is a reasonable approximation to the true ground state, and ψo(x ) yields a slightly higher expectation value for the energy. Parity symmetry is unbroken: the expectation value of x in the ground state is zero.

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This mechanism, where tunneling between would-be symmetry breaking ground states restores the symmetry which seemed broken, is seen in all quantum mechanical systems with a finite number of degrees of freedom. This mechanism can fail, however, when the number of degrees of freedom is infinite. Now let’s consider a model defined on a three-dimensional cubic spatial lattice of lattice spacing a using the same double-well potential. At every lattice site x ⃗ there is a field ϕ(x ⃗ ) instead. We assume nearest-neighbor fields ϕ(x ⃗ ) and ϕ(y ⃗ ) are coupled by a quadratic term which is a lattice approximation to (∇ϕ )2 /2

1 ⎛ ϕ(x ⃗ ) − ϕ(y ⃗ ) ⎞2 ⎟ ⎠ a

∑ 2 ⎜⎝ nn

where the sum is over all nearest-neighbor pairs. The Hamiltonian is

H=

1 ⎛ ϕ(x ⃗ ) − ϕ(y ⃗ ) ⎞2 ⎟ + ⎠ a

∑a3 2 ⎜⎝ nn

∑a3V (ϕ(x ⃗)) . x⃗

As before, minimizing the potential energy V at each site leads to an expansion around ϕ(x ) = ±v. The quadratic term linking neighboring sites will give a large contribution to the ground state energy unless we choose the same wave function at each site. Thus we can construct two approximate ground states, which are product states formed from the same wave functions as before

Ψ± =

∏ ψ±(ϕ(x ⃗)) . x⃗

These states are degenerate in energy per site. If we again appeal to tunneling to prevent spontaneous symmetry breaking, we must evaluate the overlap of the two wave functions. For a lattice of finite extent with N sites, the overlap integral is

Ψ+∣Ψ− = t N . The tunneling amplitude rapidly goes to zero as N → ∞. This is the crucial difference between systems with a finite number of degrees of freedom: tunneling is suppressed as the number of degrees of freedom goes to infinity. This is the reason spontaneous symmetry breaking can occur in quantum field theories but not in quantum mechanics. In order to obtain symmetry breaking in a finite system, the symmetry must be broken explicitly. In the case of our ϕ4 lattice model, we can add a small symmetrybreaking field term to the potential as

V (ϕ ) =

λ 2 2 (ϕ − v 2 ) − hϕ 4!

where h is a constant. If h > 0, this will cause the global minimum of V (ϕ ) to be positive. As h → 0+, this minimum value of the potential approaches the value +v. Similarly, as h → 0−, the minimum approaches −v. However, for any finite value of N, the vacuum expected value 0 ϕ(x ⃗ ) 0) will be zero due to tunneling. We have

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two different limits to consider, h → 0 and N → ∞, and they do not commute. Spontaneous symmetry breaking for ϕ4 means

lim lim 0 ϕ(x ⃗ ) 0) ≠ 0 h→0N→∞ while

lim lim 0 ϕ(x ⃗ ) 0) = 0. N → ∞h → 0 This non-commutativity of limits is a hallmark of spontaneous symmetry breaking.

7.2 Spontaneous symmetry breaking and perturbation theory Let us consider perturbation theory for the single-component ϕ4 model, which has the discrete symmetry ϕ → ϕ. Our starting point for perturbation theory is to choose a minimum of the potential and expand V around that value. If we take the positive minimum, we write ϕ(x ) = v + δϕ(x ) and expand the potential around v we obtain

1 1 1 (2) V (v)(δϕ)2 + V (3)(v)(δϕ)3 + V (4)(v)(δϕ)2 4! 2 2 1 1 1 2 = λv (δϕ)2 + λv(δϕ)3 + λ(δϕ)4 . 24 6 6

V (v + δϕ) = V (v) +

Inserting this expansion into the Lagrangian, we have

L=

1 1 1 1 (∂δϕ)2 − λv 2(δϕ)2 − λv(δϕ)3 − λ(δϕ)4 . 24 6 6 2

We regard v as the lowest-order approximation to 0 ϕ(x ⃗ ) 0), and take δϕ as representing quantum fluctuations around this classical minimum. The Lagrangian tells us that we have a particle of mass-squared λv2 /3 and cubic and quartic interaction terms However, there is something a bit strange about perturbation theory using λ as an expansion parameter, because the mass-squared and the two couplings are all proportional to λ. It is conceptually easier to organize perturbation theory as a loop expansion, where the expansion parameter for any n-point function is the number of loops L. As discussed in chapter 6, we can count the number of loops by changing iS → iS /ℏ in the functional integral, with the understanding that ℏ = 1 in our natural units. Then the power of ℏ associated with any graph is L − 1. Thus graphs with no loops, which are called tree graphs, have a factor of ℏ−1, graphs with one loop are of order ℏ0 , two loops graphs are of order ℏ, and so on. The value of the loop expansion applied to spontaneous symmetry breaking is that the loop expansion is not affected by shifts in the field. We are free to shift ϕ(x ) within the functional integral. Explicitly, we have

∫ [dϕ]eiS[ϕ+v]ℏ = ∫ [dϕ]eiS[ϕ]ℏ for any v, so the loop expansion can be used where an expansion in λ seems problematic.

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It is useful to look at a different, yet familiar representation of the ϕ4 potential, where we write

V (ϕ ) =

1 2 2 1 m ϕ + λϕ4 . 2 4!

The extrema of V are given by solving

V ′(ϕ) = m 2ϕ +

λ 3 ϕ =0 6

and their stability by checking the sign of

V ′′(ϕ) = m 2 +

λ 2 ϕ . 2

For m2 > 0, the stable solution ϕ¯ is given by ϕ¯ = 0, but for m2 < 0, the stable solutions are 2

6m ϕ¯ = ± − . λ At first, it seems funny to consider m2 < 0, because we may have learned to associate the coefficient of the quadratic term in the Lagrangian with the particle mass. However, that association assumes that ϕ = 0 is the minimum of the potential. In fact, m2 is just some real coefficient; writing it as m2 is just a vestigial habit we have picked up. In the case m2 < 0, we see that we can identify v as

v=

−

6m 2 λ

and V ′′(ϕ ), which is the tree-level mass when m2 < 0, is given by

V ′′(ϕ ) = −2m 2 . The behavior of the vacuum expected value of ϕ and the particle mass are very interesting as m2 crosses zero. At tree level, we have

⎧0 m2 > 0 ⎪ ϕ¯ = ⎨ 6m 2 2 m 0 ⎩−2m m < 0

so that the physical particle mass is zero when m2 = 0. In the language of critical phenomena, this field theory has a second-order phase transition at m2 = 0, which marks the boundary between the unbroken phase where ϕ¯ = 0 and the broken phase

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where ϕ¯ ≠ 0. This point in parameter space, here given by m2 = 0, is called the critical point. The expectation value of ϕ, given here at tree level by ϕ¯ , is known as the order parameter, because it indicates whether the symmetry is broken or unbroken. The typical indicators of a second-order phase transition are that the mass of at least one particle goes to zero at the critical point, and the order parameter indicates a change from unbroken to broken symmetry at the critical point.

7.3 Broken continuous symmetries and Goldstone bosons Spontaneously broken continuous symmetries have many of the features of broken discrete symmetries, but also give rise to something new: Goldstone bosons in the broken-symmetry phase. As an example, let us consider the O(2) scalar field theory whose Lagrangian can be written as

1 1 λ ⃗2 2 2 2 L = (∂ϕ ⃗ ) − m 2ϕ ⃗ − ϕ 2 2 4 1 1 1 λ 2 2 2 = (∂ϕ1) + (∂ϕ2 ) − m 2(ϕ12 + ϕ22 ) − (ϕ12 + ϕ22 ) . 2 2 2 4

( )

Minimizing the potential leads to the equation

m 2ϕj + λϕj (ϕ12 + ϕ22 ) = 0 for j = 1, 2 so either ϕj = 0 or ϕ12 + ϕ22 = −m2 /λ . As before, the critical point occurs at m2 = 0. However, the continuous symmetry manifests itself in the infinite number of solutions of the equation ϕ12 + ϕ22 = −m2 /λ . Just as we could have chosen either of two solutions in the discrete case, here we can choose from an infinite number of solutions. We choose

−m2 ϕ¯1 = + ≡v λ and thus take ϕ¯2 = 0. If we make the shift ϕ1 → v + ϕ1 in L, we find to quadratic order that

L=

1 1 (∂ϕ )2 + 2 (∂ϕ2 )2 + m2ϕ12 + 0 · ϕ22 . 2 1

The particle associated with ϕ1 has a mass −2m2 , and the particle associated with ϕ2 has a mass of zero. We say that in the broken phase, this model has a Goldstone boson: a scalar particle which has zero mass as a consequence of the spontaneous breaking of a continuous symmetry. Sometimes people say that ϕ2 ‘is’ the Goldstone boson, but this is dependent on the choice of ϕ¯1 = v. In fact, the presence of a Goldstone boson does not depend on the representation of the fields. In this case, we do not need to use a Cartesian representation of the fields. We can instead use a polar representation of the fields where

(ϕ1(x), ϕ2(x)) = (ρ(x) cos θ(x), ρ(x) sin θ(x)) 7-9

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or equivalently

ϕ1 + iϕ2 = ρ(x )e iθ(x ). Written in terms of ρ and θ, the Lagrangian becomes

1 1 1 λ L = (∂ρ)2 + ρ 2 (∂θ )2 − m 2ρ 2 − ρ 4 . 2 4 2 2 Note that θ does not appear in the potential energy. The expected value of ρ is determined from

m 2ρ + λρ3 so

m2 > 0

0 ρ¯ =

. −m2 2 m 0, corresponding to particles and E < 0, corresponding to antiparticles. For E > 0, we can write the equation of motion for uL(0) as

⎛ σ⃗ · p⃗ ⎞ ⎜ ⎟uL(0) = −uL(0) ⎝ E ⎠ or equivalently

(σ ⃗ · pˆ )uL = −uL(0) in any reference frame. Because the intrinsic spin of a spin-1/2 particle is given by S ⃗ = σ ⃗ /2, we see that massless particles of type L are left-handed in the sense that their spin along the direction of motion is always opposite to the direction. Similarly, massless particles of type R are right-handed, because their spins are always along the direction of motion. It is easy to check that in both cases, the antiparticle solutions with E < 0 have the opposite behavior: antiparticles of type L are right-handed, and

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antiparticles of type R are left-handed. The helicity h of a particle is defined to be its spin along its direction of motion: h ≡ pˆ · S ⃗ . The helicity is Lorentz-invariant for massless particles, but frame-dependent for massive particles. In the case of massive particles, it is easy to find a boost which changes the sign of h; in the rest frame of a massive particle, h is not well-defined. Nevertheless, at ultrarelativistic energies where the mass of particles can be neglected, helicity is approximately conserved. The neutrinos of the Standard Model are all left-handed Weyl fermions. This fact is at the heart of parity violation in the weak interactions.

8.3 The Dirac equation We have seen that we can only make massless field equations using either the (1/2, 0) or (0, 1/2) spinor representations of the Lorentz group, which are both two-dimensional. This leaves open the possibility of combining the two representations to obtain a mass term for the Lagrangian of a massive spin-1/2 particle. Because the mass m is a scalar, it must multiply some term quadratic in both fields. Although u L+uL and u R+uR are both the timelike components of a four-vector, it is easy to check that both u L+uR and u R+uL are scalars under boosts and rotations. Thus we can write a free Dirac Lagrangian as

LD = LL + LR + Lm = u L+i (∂ 0 − σ ⃗ · ∇)uL + u R+i (∂ 0 + σ ⃗ · ∇)uR − m(u R+uL + u L+uR ) . This can be written in matrix form as

⎛ i (∂ 0 + σ ⃗ · ∇)⎞ uL −m LD = ( u R+ u L+ )⎜ ⎟ u . −m ⎝ i ( ∂ 0 − σ ⃗ · ∇) ⎠ R

( )

(8.1)

Again using our trick of varying u L+, R independently of uL,R , we obtain the equations of motion

i (∂ 0 + σ ⃗ · ∇)uR − muL = 0 i (∂ 0 − σ ⃗ · ∇)uL − muR = 0. From these two equations, we obtain

∂ 2uL = (∂ 0 + σ ⃗ · ∇)(∂ 0 − σ ⃗ · ∇)uL = (∂ 0 + σ ⃗ · ∇)(imuR ) = im(imuL ) = − m 2uL and similarly for uR. This is what we were looking for: a four-component equation linear in spacetime derivatives. It can be written in block form as

⎛ −m i (∂ 0 + σ ⃗ · ∇)⎞ uL ⎜ ⎟ u = 0. −m ⎝ i ( ∂ 0 − σ ⃗ · ∇) ⎠ R

( )

In order to arrive at a more compact notation, we first define σ μ = (1, σ ⃗ ) and σ¯ μ = (1, −σ ⃗ ) and then define the gamma matrices 8-10

A Multidisciplinary Approach to Quantum Field Theory, Volume 1

γμ =

⎛ 0 σ μ⎞ ⎜ ⎟. ⎝ σ¯ μ 0 ⎠

Exercise: show that the gamma matrices defined above satisfy the anticommutation relations {γ μ, γ ν} = 2g μν . The equation of motion can now be written as

(iγ μ∂μ − m)ψ = 0 where

u ψ = uL . R

( )

Examining our expression for LD , equation (8.1), we see that it is written using (u R+ u L+ ), not ψ +=(u L+ u R+ ). We can remedy this by noting that in block form

⎛ ⎞ γ 0 = ⎜0 1 ⎟ ⎝1 0 ⎠ so

LD = ψ +γ 0(iγ μ∂μ − m)ψ . It is conventional to define the Dirac adjoint ψ¯ of ψ to be ψ¯ = ψ +γ 0 so

LD = ψ¯ (iγ μ∂μ − m)ψ . As with ψ +, we can vary ψ¯ independently of ψ for the purpose of obtaining field equations. Although we were led to a particular form for the gamma matrices, other useful equivalent forms exist. It is the anticommutation relation {γ μ, γ ν} = g μν which is important in creating a covariant relativistic equation which implies the Klein– Gordon equation. We have been using the Weyl representation of the gamma matrices where

⎛0 1 ⎞ ⎟ γ0 = ⎜ ⎝1 0 ⎠ ⎛ ⎞ γ ⃗ = ⎜ 0 σ⃗⎟ ⎝−σ ⃗ 0 ⎠ ⎛ ⎞ γ5 = ⎜−1 0 ⎟ . ⎝ 0 1⎠ As we have seen, the Weyl representation is natural from the point of view of representation theory. It is particularly useful in cases where the mass is negligible,

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either m = 0 or at energies E ≫ m, because the two components uL and uR decouple. Another commonly used representation is the Dirac representation, where

⎛1 0 ⎞ ⎟. γ0 = ⎜ ⎝ 0 −1⎠

(8.3.2)

⎛ 0 σ⃗⎞ ⎜ ⎟ ⎝− σ ⃗ 0 ⎠

(8.3.3)

γ⃗ =

⎛ ⎞ γ5 = ⎜ 0 1 ⎟ . ⎝1 0 ⎠

(8.3.4)

The Dirac representation, as we will see, is very useful in understanding the spin and particle content of the Dirac equation and natural in the non-relativistic limit.

8.4 Solutions of the Dirac equation In the case of the Klein–Gordon equation, the free field solutions were simple, and could be written as e−ip·x . The free field solutions of the Dirac equation are fourcomponent spinors, and thus more complicated. We will begin by looking for solutions of the form

ψ (x ) = e−ip·xu(p ) where u(p ) is a four-component spinor depending on p. Plugging this ansatz into the Dirac equation (iγ · ∂ − m )ψ = 0, we find

(γ · p − m)u(p ) = 0. Assuming that m ≠ 0, we can go to the rest frame where p = (E , 0) and the equation becomes

(γ 0E − m)u = 0. It is at this point that it becomes convenient for interpretational purposes to use the Dirac representation of the gamma matrices. Using equation (8.3.2), we find two solutions with E = m, ⎛0⎞ ⎛1 ⎞ ⎜ ⎟ ⎜0⎟ u = ⎜ ⎟ u = ⎜1 ⎟ ⎜0⎟ ⎜0⎟ ⎝0⎠ ⎝0⎠ and two solutions with E = −m,

⎛0⎞ ⎜ ⎟ u = ⎜0⎟ ⎜1 ⎟ ⎝0⎠

⎛0⎞ ⎜ ⎟ u = ⎜0⎟ ⎜0⎟ ⎝1 ⎠

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The spin associated with these solutions can be obtained using the spin matrix operator, given by

1 ⎛σ ⎞ Σ⃗ = ⎜ ⃗ 0 ⎟ 2 ⎝0 σ⃗⎠ which is obtained from the rotation generators σ jk via σ jk = ϵjkl 2σ l . We see that we have two positive-energy solutions with spin in the z direction of ±1/2, and two similar negative-energy solutions. We could find solutions with p ⃗ ≠ 0 by boosting the p ⃗ = 0 solutions. Alternately, we can solve the Hamiltonian equation

Hψ = Eψ which in momentum space is

(α⃗ · p ⃗ + βm)u(p ) = Eu(p )

Exercise: derive this form for H starting from LD . In the Dirac representation, this equation takes the form

⎛ m σ⃗ · p⃗⎞ ⎜ ⎟u(p ) = Eu(p ). ⎝σ⃗ · p⃗ − m ⎠ If we write u in block form as

⎛ uA(p ) ⎞ u (p ) = ⎜ ⎟ ⎝ uB(p )⎠ then we obtain two coupled equations:

(σ ⃗ · p ⃗ )uB(p ) = (E − m)uA(p ) (σ ⃗ · p ⃗ )uA(p ) = (E + m)uA(p ) which allow us to solve for one of the two-spinor fields in terms of the other. For E > 0, we define u As(p ) to be χ 1(p ) or χ 2 (p ), where

⎛ ⎞ χ 1 = ⎜1 ⎟ ⎝0⎠

⎛ ⎞ χ 2 = ⎜0⎟. ⎝1 ⎠

Then the full solution is

⎛ χs ⎞ ⎟ ⎜ u s (p ) = N ⎜ σ ⃗ · p ⃗ s ⎟ χ ⎝E + m ⎠

(8.4.1)

where we leave the normalization constant N undetermined for the moment. For E < 0, we can find similar solutions, with u Bs = χ s and 8-13

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u As =

σ⃗ · p⃗ s σ⃗ · p⃗ χ = χs. E−m − E −m

We denote these two solutions with E < 0 as u 3(p ) and u 4(p ) respectively. These solutions are all eigenstates of H, and orthogonal: u r+(p )u s(p ) = 0 if r ≠ s . Exercise: check the orthogonality relations. The solutions u1,2(p )e−ip·x are E > 0 solutions. In the quantum field theory of the free Dirac field, these appear in the Fourier decomposition of the operator ψ multiplying a fermion destruction operator, just as e−ip·x multiplies a boson destruction operator for a free bosonic field. As we saw in the case of a scalar field, there must also be a term in the Fourier decomposition proportional to e +ip·x which creates particles. However, our solutions so far all proportional to e−ip·x . We define the desired solutions with E > 0 by

v 2,1(E , p ⃗ )e+ip·x ≡ u 3,4(E , −p ⃗ )e−i [− E t−(−p ⃗ )·x ⃗]. We know that the u(p ) solutions satisfy (γ · p − m )u(p ) = 0. If we define p0 = E > 0, then u 3.4( −p0 , −p ⃗ ) satisfies

( −γ · p − m)u 3.4( −p0 , −p ⃗ ) = 0 which is equivalent to

(γ · p + m)u 2.1(p0 , p ⃗ ) = 0. We know that u r+u r and vr+vr are the timelike components of four-vectors. The standard normalization is

u r+u s = 2Eδ rs v r+v s = 2Eδ rs . Using the solutions equation (8.4.1) we find that

u r+u r = N

2

= N

2

= N

2

⎡ (σ ⃗ · p ⃗ )2 ⎤ ⎢1 + ⎥ (E + m )2 ⎦ ⎣ ⎡ (E + m )2 + E 2 − m 2 ⎤ ⎢ ⎥ (E + m )2 ⎣ ⎦ ⎡ 2E ⎤ ⎢ ⎥ ⎣ (E + m ) ⎦

so that N can be taken to be E + m . Exercise: show that N = E + m is also the correct normalization for v. These relations can be used in turn to prove that

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u¯ ru s = + 2mδ rs v¯ r v s = − 2mδ rs u¯ rv s = 0. This leads to the key identities

∑u s( p)u¯ s( p) = γ · p + m s

∑u s( p)u¯ s( p) = γ · p − m.

(8.4.2)

s

One way to prove these identities is to divide both sides of the equations by 2m and show that both sides of each equation are projection operators onto the same spinor subspace.

8.5 The free Dirac field In section 4.8, we wrote the free scalar field in terms of creation and annihilation operators and single-particle wave functions as

ϕ(x ) =

∫

ϕ(x ) =

∫

d 3p [e−ip·xα(p ) + e ip·xα +(p )] . (2π )32ω p ⃗

and

d 3p (2π )3/2 2ω p ⃗

⎡⎣e−ip·xa p ⃗ + e ip·xa +⎤⎦ . p⃗

The commutation relations are

⎡⎣a p ⃗ , a + ⎤⎦ = δ 3(p ⃗ − p ⃗ ′) p⃗ ′ [a p ⃗ , a p ⃗ ′] = 0 ⎡⎣a +, a + ⎤⎦ = 0 p⃗

p⃗ ′

We have

0 ϕ(x ) k ⃗ = 0

∫

d 3p (2π )3/2 2ω p ⃗ d 3p

=

∫

=

e−ik·x . (2π )3/2 2ω p ⃗

(2π )3/2 2ω p ⃗

⎡⎣e−ip·xa p ⃗ + e ip·xa +⎤⎦ k ⃗ p⃗

e−ip·xδ 3(p ⃗ − k ⃗ )

The free scalar field consists of a wave function e−ip·x multiplying an annihilation operator a p ⃗ plus the Hermitian conjugate of the product, integrated over all momenta in such a way that the canonical commutation relations hold.

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We would like to repeat this construction for Dirac fields. Because Dirac fields are complex, we will have two sets of creation and annihilation operators, one of particles and one for antiparticles, as is the case for a complex scalar field. Spin adds an additional degree of complexity: a particle can be in one of two spin states, represented by the single-particle spin solutions u r(p ). So our single-particle states are given by

u s(p ⃗ )e−ip·x where we have written u s(p ⃗ ) to remind us that the energy is fixed by p ⃗ . This motivates us to propose

ψ (x ) =

∑∫ s

d 3p 3/2

(2π )

b (p ⃗ )u 1/2 [ s

(2E p ⃗)

s

(p ⃗ )e−ip·x + d s+(p ⃗ )v s (p ⃗ )e+ip·x ]

and of course ψ +(x ) is the Hermitian conjugate

ψ +(x ) =

d 3p

∑∫

b +(p ⃗ )u s+(p ⃗ )e+ip·x 1/2 [ s

(2π )3/2 (2E p ⃗ )

s

+ ds(p ⃗ )v s+(p ⃗ )e−ip·x ] .

Suppose we assume that the creations and annihilation operators obey the same canonical commutation relations as the corresponding scalar operators, generalized to include the extra index for spin: + [br (p ⃗ ) , bs (p ⃗ ′)] = δrsδ 3(p ⃗ − p ⃗ ′) + [d r(p ⃗ ) , d s (p ⃗ ′)] = δrsδ 3(p ⃗ − p ⃗ ′)

where all other commutators are zero. Let us calculate the Hamiltonian. The momentum variable conjugate to ψ is

π (x ) =

∂L = iψ¯ (x )γ 0 = iψ +(x ). ∂(∂ 0ψ (x ))

The Hamiltonian density H is given by

H = π (x )i ∂ 0ψ (x ) − L = ψ (x )+i ∂ 0ψ (x ) − ψ¯ (x )(iγ · ∂ − m)ψ (x ). However, the free field ψ satisfies the Dirac equation (iγ · ∂ − m )ψ (x ) = 0 so the calculation of H reduces to

H=

∫ d 3x ψ +(x)i ∂0ψ (x).

Using the orthogonality relations u r+u s = +2Eδ rs , vr+v s = −2Eδ rs and u r+v s = 0, we find

H=

∑∫ s

d 3p E p ⃗[bs+(p ⃗ )bs (p ⃗ ) − ds(p ⃗ )d s+(p ⃗ )] . (2π )3

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We can write ds(p ⃗ )d s+(p ⃗ ) as d s+(p ⃗ )ds(p ⃗ ) plus an infinite constant, but it is the minus sign that is problematic, because the Hamiltonian is unbounded from below. If, however, we change the commutation relations to anticommutations relations

{ br(p ⃗ ) , bs+(p ⃗ ′)} = δrsδ 3(p ⃗ { dr(p ⃗ ) , d s+(p ⃗ ′)} = δrsδ 3(p ⃗

− p ⃗ ′) − p ⃗ ′)

this problem disappears. Discarding the infinite constant, we have

H=

∑∫ s

d 3p E [b +(p ⃗ )bs (p ⃗ ) + ds(p ⃗ )d s+(p ⃗ )] . 3 p⃗ s (2π )

The anticommutation relations imply Fermi statistics and the Pauli exclusion principle. For example, a two-particle state p1⃗ , r1, p2⃗ r2 with two distinct momenta and/or spin is antisymmetric because

p1⃗ , r1, p2⃗ r2 ≡ br+1 (p1⃗ )br+2 (p2⃗ ) 0 = −br+1 (p1⃗ )br+2 (p2⃗ ) 0 = − p2⃗ r2 , p1⃗ , r1 . This connection between spin and statistics is more general. The spin-statistics theorem states that particles with integer spin are bosons, with symmetric wave functions under two-particle exchange, while particles with half-integer spin are fermions, with antisymmetric wave functions under two-particle exchange. This result was first stated for free fields by Fierz in 1939 and treated systematically in 1940 by Pauli. Later work extended the theorem to interacting systems, culminating in a general proof using the methods of axiomatic field theory. With x 0 = y 0, we have

{ ψa(x), ψ¯b(y )} = ∑ ∫ s

d 3p

1/2 ∑ ∫ (2π )3/2 (2E p ⃗ ) r

d 3q 1/2

(2π )3/2 (2E p ⃗ )

× {bs (p ⃗ )u s(p ⃗ )e−ip·x + d s+(p ⃗ )v s (p ⃗ )e+ip·x , bs+(q ⃗ )u¯ r(q ⃗ ) e+iq·y + ds(q ⃗ )v¯ r (p ⃗ )e−iq·y} d 3p

=∑

∫

r, s

1/2

(2π )3/2 (2E p ⃗ )

∫

d 3q 1/2

(2π )3/2 (2E p ⃗ )

× [u as(p ⃗ )u¯ br(q ⃗ )e−ip·xe+iq·y + vas (p ⃗ )e+ip·xv¯ br (p ⃗ )e+iq·y ]δrsδ 3(p ⃗ − q ⃗ ) d 3p [u ar(p ⃗ )u¯ br(p ⃗ )e−ip·(x−y ) + vas (p ⃗ )v¯ br (p ⃗ )e+ip·(x−y )] =∑ 3 (2 ) 2 E π ( ) ⃗ p r

∫

=

∫

d 3p [(γ · p + m)abe−ip·(x−y ) + (γ · p − m)abe+ip·(x−y )] (2π )3(2E p ⃗ )

=

∫

d 3p 2γab0 E pe⃗ ip ⃗ ·(x ⃗−y ⃗ ) (2π )3(2E p ⃗ )

= γab0 δ 3(x ⃗ − y ⃗ )

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where we have canceled the terms proportional to m and used the vanishing of the terms proportional to p ⃗ upon integration. Multiplication by another γ 0 shows that

{ψa(t, x ⃗), ψb+(t, y ⃗)} = δabδ 3(x ⃗ − y ⃗)

(8.5.1)

the canonical anticommutation relations in position space. Recalling that π (x ) = iψ +(x ), we can also write

{ψa(t , x ⃗ ), π b(t , y ⃗ )} = iδabδ 3(x ⃗ − y ⃗ ) . The form of the canonical anticommutation relations given in equation (8.5.1) can be applied to interacting systems as well as free systems. For non-relativistic systems, the implementation is particularly simple, and similar to the treatment of nonrealistic scalar.

8.6 Dirac bilinears Because we know LD is a Lorentz scalar, we see immediately that ψψ ¯ is also a scalar, and ψγ ¯ μψ is a four-vector under proper Lorentz transformations. However, parity P:(t , x ⃗ ) → (t , −x ⃗ ) is an improper Lorentz transformation and must be treated separately. Parity transforms vectors such as a velocity v⃗ as P: v⃗ → −v⃗ , but axial vectors like angular momentum L⃗ do not change: P: L⃗ → L⃗. Thus parity transforms the operators J ⃗ and K⃗ as P: J ⃗ → J ⃗ and P: K⃗ → −K⃗ , respectively. Thus parity ± + transforms J ⃗ =J ⃗ ± iK⃗ into one another: P: J ⃗ ↔J −. This in turns implies that parity changes a representation characterized by (j+ , j− ) into quantum numbers (j− , j+ ). In the particular case of (1/2, 0) and (0, 1/2), K⃗ = ±iσ ⃗ /2, so parity acts to turn a lefthanded field ϕL into a right-handed one ϕR . For our Dirac field ψ, parity acts as

P: ψ (t , x ⃗ ) → γ 0ψ (t , −x ⃗ ) because γ 0 interchanges uL and uR. Exercise: Verify that ψψ is invariant under parity. Prove the identity ¯ 0 0 0 0 and use it to show γ (γ , γ ⃗ )γ = (γ , − γ ⃗ )

P: (ψγ ¯ 0ψ , ψγψ ¯ ⃗ ) → (ψγ ¯ 0ψ , −ψγψ ¯ ⃗ ) as expected for a four-vector.The composite fields ψψ ¯ μψ are examples of ¯ and ψγ Dirac bilinears. The set of all 4 × 4 matrices form a 16-dimensional inner product space using Tr[A+B ] as an inner product. From a single pair of Dirac fields, we expect to extract 16 independent objects which can be decomposed into subsets transforming among themselves just as we did in section 8.2.1 for the tensor xj pk . We have already identified ψψ ¯ μψ as a four-vector, so that ¯ as a Lorentz scalar and ψγ leaves 11 to resolve. There is a fifth gamma matrix which plays a crucial role in further decomposition. It is defined by

γ5 ≡ γ 5 ≡ iγ 0γ1γ 2γ 3

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It is easy to prove from the anticommutation relations that: γ5+ = γ5 so γ5 is Hermitian; γ52 = 1; {γ μ, γ 5} = 0. Exercise: prove all these properties. It is the last property, {γ μ, γ 5} = 0, which warrants referring to this matrix as a fifth gamma matrix. In fact, if for some reason we wanted a Dirac equation in five dimensions, we could use γ5 to construct it, with ψ remaining a four-component spinor. In the Weyl representation of the gamma matrices we have been using, γ5 is both block-diagonal and diagonal

⎛ ⎞ γ5 = ⎜− 1 0 ⎟ ⎝ 0 1⎠ but in the Dirac representation it is given by

⎛ ⎞ γ5 = ⎜ 0 1 ⎟ . ⎝1 0 ⎠ It is straightforward to show that ψγ ¯ 5ψ is a scalar under Lorentz transformations, while ψγ ¯ 5γ μψ transforms as a Lorentz four-vector. However, under parity, we find

P: ψγ ¯ 5ψ → ψγ ¯ 0γ5γ 0ψ = −ψγ ¯ 5ψ and

P: ψγ ¯ 5γ μψ → (ψγ ¯ 0γ5γ 0γ 0ψ , ψγ ¯ 0γ5γγ⃗ 0ψ ) = ( −ψγ ¯ 5γ 0ψ , +ψγ ¯ 5γψ ⃗ ). We refer to ψγ ¯ 5ψ as a pseudoscalar and ψγ ¯ 5γ μψ as an axial vector. This accounts for five more of the sixteen Dirac bilinears. The remaining six are formed using the set of matrices

σ μν ≡

i μ ν [γ , γ ] 2

which satisfy σ μν = −σ νμ. Just as ψγ ¯ μψ transforms as a vector, ψσ ¯ μνψ transforms as an antisymmetric two-tensor, with six independent components. These are the operators used in constructing Lagrangians containing Dirac fields. With additional fermion fields, more such bilinears can be constructed. For example, with two fermions, we can form composite fields like ψ¯1γ μψ2 but all such operators have one of the five different Lorentz transformation behaviors. Our results are summarized as

Type Scalar Pseudoscalar Vector Axial vector Tensor

Operator ψψ ¯ ψγ ¯ 5ψ ψγ ¯ μψ ψγ ¯ 5γ μψ ψσ ¯ μνψ

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In addition to their role in constructing Lagrangians, these composite fields create and destroy states consisting of one fermion and one antifermion. For example, the π + meson is a pseudoscalar meson composed of an up quark and a down anti-quark. A local operator which creates such a particle is u¯γ5d .

8.7 Chiral symmetry and helicity We have seen that for massless particles, we can define the helicity h, which is the projection of the spin onto the direction of motion. For a left-handed field uL, h = −1/2 and for a right-handed field uR, h = +1/2. This behavior coincides with the action of γ5 /2 in the Weyl representation where γ5 is given by

⎛ ⎞ γ5 = ⎜− 1 0 ⎟ ⎝ 0 1⎠ acting on

u ψ = uL . R

( )

We can use this to define states which are similar to left-handed and right-handed states even when the particle mass is non-zero. We define projection operators

1 PL = (1 − γ5) 2 1 PR = (1 + γ5) 2 which in the Weyl representation are given by

⎛ PL = ⎜1 ⎝0 ⎛ PR = ⎜ 0 ⎝0

0 ⎞⎟ 0⎠ 0 ⎞⎟ . 1⎠

Like all projection operators, they satisfy PL2 = PL and PR2 = PR . Furthermore, PL + PR = I , the identity operator. Noether’s theorem applies to fermion fields as well as bosonic fields, and the Dirac field is complex. The phase transformation ψ → e iθψ induces a transformation ψ¯ → ψ¯ e−iθ which together give a global symmetry of the Dirac Lagrangian LD . This in turn gives us a current j μ = ψγ ¯ μψ which is conserved, satisfying ∂μj μ = 0. Parallel to the case of a complex scalar field, upon quantization the conserved charge

Q=

∫ d 3x j 0 (t, x ⃗)

gives the number of fermions minus antifermions. If the mass m of a Dirac fermion is zero, there is an additional symmetry. A continuous chiral transformation

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ψ → e iθγ5ψ induces a change

ψ¯ → ψ +e−iθγ5γ 0 = ψ +γ 0e+iθγ5 = ψ¯ e+iθγ5. A mass term in the Lagrangian of the form −mψψ ¯ would not be invariant under this transformation.

8.8 Charge conjugation and coupling to the electromagnetic field The Dirac Lagrangian is invariant under a global phase transformation ψ → e iθψ , so there is a conserved current j μ = ψγ ¯ μψ which is associated with conservation of fermion number. For charged fermions, the corresponding electric current is ej μ = eψγ ¯ μψ . Classically, such a current couples to the electromagnetic field via a term LI = −eψγ ¯ μψAμ. This changes the Dirac equation to

L = ψ¯ (iγ · ∂ − m − eγ · A)ψ . This is often referred to as minimal coupling; in chapter 1 of volume 2, we will derive this interaction from local gauge invariance. Charge conjugation C is a linear operator that turns particles into antiparticles and vice versa. For particles with electric charge, it turns particles with charge e into antiparticles with charge −e . As we have seen, for a complex scalar field, C turns ϕ into ϕ+; for free fields, C interchanges the creation and annihilation operators of particles with those of antiparticles. If we take the complex conjugate of the Dirac equation interacting with an electromagnetic field

(iγ · ∂ − m − eγ · A)ψ = 0 and take its complex conjugate, we obtain

( −iγ *·∂ − m − eγ *·A)ψ *=0. If we can find a unitary matrix U acting on four-spinors such that U +γ μU = γ u*, then ψC ≡ Uψ will satisfy

(iγ · ∂ − m + eγ · A)ψc = 0 which is the equation ψC must satisfy: the net effect of charge conjugation is to flip the sign of e. The key gamma matrix identity in finding U is

γ u*=γ 0γ μγ 0 = γ 0(γ 0, γ ⃗ )γ 0 = (γ 0, −γ⃗ ) . We define U to be some 4 × 4 matrix C times γ 0 , U = Cγ 0, and require −1 (Cγ 0) γ μ(Cγ 0) = −γ μ*.

The explicit form of the matrix C in the Dirac and Weyl representations looks a bit odd: it is C = iγ 2γ 0 , suggesting that somehow the y axis has been singled out from x and y. It occurs here because in both the Dirac and Weyl representation, only γ 2 has imaginary non-zero entries.

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Exercise: check that C = iγ 2γ 0 satisfies (Cγ 0 )−1γ μ(Cγ 0 ) = −γ μ*. With this set of definitions, we have

ψC = Cγ 0ψ *=Cψ¯ t ψ¯C = ψ tC so that the final form of the transformation has a simple appearance. We have seen that a free complex scalar field ϕ, which annihilates particles and creates antiparticles, can be decomposed into two real fields as ϕ1 = (ϕ + ϕ+ )/ 2 and ϕ2 = −i(ϕ − ϕ+ )/ 2 . The fields ϕ1 and ϕ2 each create and annihilate particles of type 1 and 2, respectively. It is natural to ask if something similar can occur with Dirac fermions. Ettore Majorana hypothesized the existence of such particles in 1937. In his honor, real fermion fields and particles are called Majorana fermions. Naively, it seems like we should impose a condition on spinor fields like ψ = ψ * analogous to the condition ϕ = ϕ* for scalars, and this can be done for nonrelativistic fermions. However, this condition will be frame-dependent for relativistic fermions unless it is implemented using a gamma matrix representation which has * ; otherwise Lorentz transformations will not preserve the real σμν matrices, σμν = σμν condition ψ = ψ *. A frame-independent condition is to require that the fermion field be invariant under charge conjugation: ψC = ψ where ψC = iγ2γ 0ψ *.

8.9 Functional integration for fermions Conceptually, we would like to model functional integration for fermions on the case of free complex scalars. In that case, we have an integral of the form

Z=

∫ [dϕ][dϕ*] exp{i ∫ d 4x [ϕ*(x)D(x)ϕ(x )]}.

In this formula, D(x ) is some Hermitian operator like −∂ 2x + m2 and the notation [dϕ ][dϕ*] indicates integration over the real and complex parts of the scalar field ϕ = (ϕ1 + iϕ 2) / 2 . Functional integration is then implemented by finding the orthonormal eigenvectors fn (x ) and real eigenvectors λ n of D

Dfn = λnfn and writing an arbitrary field configuration as

ϕ(x ) =

∑cnϕn(x) n

where the cn’s are complex. Then Z may be written as

⎡

Z=

∫ ∏ dcndcn*exp ⎢⎢⎣i∑λn n

n

⎤ cn 2 ⎥ . ⎥⎦

Convergence as usual requires the addition of a small imaginary part to the eigenvalues. Up to an irrelevant overall factor, our final result is 8-22

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Z=

∏ n

1 = det[D]−1. λn

The canonical anticommutation relations make clear that we cannot implement a path integral for fermions in exactly the same way that we did for scalar bosons. We implement fermions in the path integral formalism using Grassmann variables. A Grassmann algebra A is a set of elements such that, if a, b ∈ A, then

{a , b} = 0. The case b = a implies a = 0. We want integration over such variables to be a linear operator such that for any function f 2

∫ da f (a) = ∫ da f (a + r) where r is any real number. Because a 2 = 0, any function f (a ) can be expanded as

f (a ) = f0 + f1 a . If we choose to define the normalization of integration over a to be 1,

∫ da a = 1 then invariance under shifts a → a + b implies

∫ da 1 = 0 For two Grassmann variable a and a, ¯ we have the following integration table

∫ dada¯ 1 = 0 ∫ dada¯ a = 0 ∫ dada¯ a¯ = 0 ¯ = 1. ∫ dada¯ aa Note in the last and only nontrivial result, the order of the integrand matters:

¯ ) = −1. ∫ dada¯ aa¯ = ∫ dada¯ (−aa We can use these formulae to calculate the integral of functions of a and a¯ rather easily. For example

¯ ) = λ. ∫ dada¯ e λaa¯ = ∫ dada¯ (1 + λaa Suppose we have a quadratic fermion action of the form

S=

∫ d 4x ψ¯ (x)Dxψ (x) 8-23

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where Dx is now the Dirac operator iγ · ∂x − m or some similar operator on spinors. We will decompose our classical fields as

ψ (x ) =

∑ψn(x)an n

and

ψ¯ (x ) =

∑ψ¯n(x)a¯ n n

where ψn and ψ¯n are classical spinor fields satisfying

Dψn = λnψn and

∫ d 4x ψ¯n(x)ψm(x) = δnm. As in the case of scalars, we generally give the eigenvalues to have a small positive imaginary part iϵ . Even though it is not needed for convergence of the functional integral, it is still required to get the correct interpretation of the Feynman propagator. We can now calculate the functional determinant

Z=

∫ [dψ ][dψ¯ ] exp i ∫ d 4x [ψ¯ (x)D(x)ψ (y )].

Substituting the expansion of ψ and ψ¯ and using the orthonormality properties of the eigenfunctions ψn and ψ¯n , we find

Z=

∫

⎡ ⎤ ∏n danda¯ n exp ⎢i∑λna¯ nan⎥ ⎢⎣ n ⎥⎦

=

∫

∏n danda¯ n(1 + iλna¯ nan)

= ∏n iλn = det (Dx) where we have again discarded the irrelevant factors of i. This is, of course, the inverse of the result for complex scalars, and is another manifestation of antisymmetry and Fermi statistics. We will discuss the interpretation of this result in detail shortly, but note that the determinant represents one-loop vacuum energy diagrams, as it does in the case of scalars. In order to construct perturbation theory for fermions, we must be able to calculate n-point functions for free fermions. For Dirac fermions, such n-point functions must consist of an equal number of ψ and ψ¯ fields, so n-must be even. The fermion two-point function is given by

1 Z

∫ [dψ ][dψ¯ ]ψ (x)ψ¯ (y ) exp{i ∫ d 4x [ψ¯ (x)Dψ (x )]} 8-24

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which we take as the definition of the vacuum-expected value of the time-ordered product 1 0 T [ψ (x )ψ¯ (y )] 0 = [dψ ][dψ¯ ]ψ (x )ψ¯ (y ) exp i d 4x [ψ¯ (x )Dψ (x )] . Z The same arguments we used for time-ordering of scalar fields for bosons applies for fermions as well. However, because the variables inside the functional integral are anticommuting c-numbers, ψ (x )ψ¯ (y ) = −ψ¯ (y )ψ (x ) and we see that the time-ordered product is

{∫

∫

}

⎧= 0 ψ (x )ψ¯ (y ) 0 x0 > y0 0 T [ψ (x )ψ¯ (y )] 0 = ⎨ . ⎩= − 0 ψ (y )ψ¯ (x ) 0 y 0 > x 0 In the specific case where D is the Dirac operator iγ · ∂ − m, we define the Feynman propagator for fermions as

iSF (x − y ) =

1 Z

∫ [dψ ][dψ¯ ]ψ (x)ψ¯ (y ) exp{i ∫ d 4x [ψ¯ (x)(iγ · ∂ − m)ψ (x )]}.

This is actually a 16-component object, because each of the spinors has four components. We will generally avoid indicating the components as much as possible. As with bosons, it is convenient to obtain an explicit form for the fermion propagator by introducing four-component Grassmann spinors sources η¯(x ) and η(x ) coupled to ψ and ψ¯ , respectively. The addition of these sources modifies the fermion action to

∫ d 4x [ψ¯ (x)Dxψ (x) + η¯(x)ψ (x) + ψ¯ (x)η(x)]. Note that the operators η¯(x )ψ (x ) and ψ¯ (x )η(x ) commute within the functional integral. We wish to use the same differentiation of sources technique with fermions that was so useful with scalars. However, differentiation with respect to Grassmann variables is anticommutative, because if

δ a=1 δa then we must have

δ δ δ δ =− δa δb δa δb because then

δ δ δ δ (ba ) = ( −ba ) δa δb δa δb δ δ (ba ) =− δa δb δ δ (ab). = δb δa

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In fact, the rules for differentiating Grassmann variables are exactly the same as the rules for integration:

∫

da a = 1 δ a = 1. δa

We can eliminate the linear source couplings using

DxD −1(x , y ) = δ 4(x − y ). to shift the fermion fields

∫ ψ¯ (x ) → ψ¯ (x ) − ∫ ψ (x ) → ψ (x ) −

d 4y D −1(x , y )η(y ) d 4y η¯(y )D −1(y , x )

in the functional integral to obtain

∫ d 4x [ψ¯ (x)Dxψ (x)] + ∫ d 4xd 4yη¯(x)D −1(x, y )η(x). This allows us to write

Z [η , η¯ ] =

∫

{∫

[dψ ][dψ¯ ] exp i

d 4x[ψ¯ (x )Dxψ (x )] − i

∫

⎡ = Z [0, 0] exp ⎣⎢ −i

∫

⎤ d 4xd 4yη¯(x )D −1(x , y )η(x )⎦⎥

⎡ = det [D] exp ⎢⎣ −i

∫

⎤ d 4xd 4yη¯(x )D −1(x , y )η(x )⎥⎦ .

d 4xd 4yη¯(x )D −1(x , y )η(x )

We can now calculate the two-point function

0 T [ψ (x )ψ¯ (y )] 0 =

⎤ 1 ⎡1 δ 1 δ Z [η , η¯ ]⎥ ⎢ ⎦η,η¯ = 0 Z [0, 0] ⎣ i δη¯(x ) i δη(y )

= i D −1(x , y ). In the specific case of Dirac fermions where D = iγ · ∂ − m + iϵ , we have

(iγ · ∂ − m)SF (x − y ) = δ 4(x − y ) so

(γ · p − m)S˜F (p ) = 1 or

S˜F (p ) =

1 . γ·p−m

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The denominator of the propagator is in fact a matrix. We can write this as

S˜F (p ) =

(γ · p + m ) (γ · p − m)(γ · p + m) (γ · p + m ) p 2 − m 2 + iϵ

where we have inserted +iϵ where it is needed to define how positive and negative energy poles are treated. This can be treated as an instance of the rule that an m carries around a small −iϵ term so that S˜F (p ) can be written as

S˜F (p ) =

1 . γ · p − m + iϵ

iS˜F (p ) =

i γ · p − m + iϵ

The Feynman propagator is

in momentum space and

0 T [ψ (x )ψ¯ (y )] 0 = iSF (x − y ) d 4p i e−ip·x = (2π )4 γ · p − m

∫

in coordinate space. Wick’s theorem can be used for the calculation of n-point functions of free fermion fields, but the anticommuting properties add a sign factor which must be included. For relativistic two-point functions, we have by definition

0 T [ψ (x1)ψ¯ (x2 )] 0 = iSF (x1 − x2 ) but

0 T [ψ¯ (x2 )ψ (x1)] 0 = −iSF (x1 − 2). For four-point functions, we have

0 T [ψ (x1)ψ¯ (x2 )ψ (x3)ψ¯ (x4)] 0 = iSF (x1 − x2 ) · iSF (x3 − x4) − iSF (x1 − x4) · iSF (x3 − x2 ) but, for example,

0 T [ψ (x1)ψ¯ (x2 )ψ¯ (x4)ψ (x3)] 0 = − iSF (x1 − x2 ) · iSF (x3 − x4) + iSF (x1 − x4) · iSF (x3 − x2 ). One way to proceed is to first move any product of fermions into a canonical form of alternating ψ’s and ψ¯ ’s, ψψψψ ¯ ¯ ..., with any necessary change of overall sign relative to the original expression, and then consider all even and odd permutations and pairings and their relative sign changes.

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8.10 Feynman rules and scattering for a Yukawa field theory We can now consider a Yukawa field theory in which fermions and antifermions exchange a real scalar particle. The Lagrangian is given by

1 1 (∂ϕ)2 − μ2 ϕ 2 + ψ¯ (iγ · ∂ − m)ψ − gϕψψ ¯ . 2 2

L=

This model differs from our previous scalar field model in that charged scalar fields ψ and ψ * are now replaced by Dirac fermion fields ψ and ψ¯ , and the field ϕ interacts with the scalar fermion bilinear ψψ ¯ . In field theories with fermions, not only do the propagators change, but the possible forms of Lorentz-invariant interactions becomes larger, and the Feynman rules must be augmented to properly implement Fermi statistics. We can develop perturbation theory for this model by separating the Lagrangian as

L0 =

1 1 (∂ϕ)2 − μ2 ϕ 2 + ψ¯ (iγ · ∂ − m)ψ 2 2

and

LI = −gϕψψ ¯ . We then can write the generating function as

⎤ d 4x(L + Jϕ + ηψ ¯ + ψη ¯ )⎥⎦ ⎡ ⎛1 δ 1 δ ⎞⎤ 1 δ [dψ ][dψ¯ ][dϕ ] , , d 4x L0⎜ = exp ⎢i ⎟⎥ ⎝ i δJ (x ) i δη(x ) i δη¯(x ) ⎠⎦ ⎣ ⎡ ⎤ [dψ ][dψ¯ ][dϕ ] exp ⎣⎢i d 4x(L0 + Jϕ + ηψ × ¯ + ψη ¯ )⎦⎥

Z [J , η , η¯ ] =

∫

⎡ [dψ ][dψ¯ ][dϕ ] exp ⎢⎣i

∫

∫

∫

∫

⎡ = exp ⎢i ⎣

∫

∫

⎛1 δ 1 δ ⎞⎤ 1 δ , , d 4x L0⎜ ⎟⎥Z0[J , η , η¯ ]. ⎝ i δJ (x ) i δη(x ) i δη¯(x ) ⎠⎦

The free generating functional Z0 is

⎡ Z [J , η , η¯ ] = exp ⎢ ⎣

⎞⎤

⎛

∫ d 4xd 4y⎜⎝i¯η(x)iSF (x − y )iη(x) + 12 iJ (x)iΔF (x − y )iJ (y )⎟⎠⎥⎦

which is normalized so that Z0[0, 0, 0] = 1. The ϕ propagator remains

˜ F (k ) = iΔ

i k − μ2 + iϵ 2

in momentum space, and the charged particle propagator is

iS˜F (p ) =

i . γ · p − m + iϵ

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The trilinear vertex remains

−ig which is a consequence of the scalar nature of the interaction. However, there is an implicit 4 × 4 identity matrix δab present implying that the spinor indices a of ψ¯ and b of ψ are set equal and summed over. Suppose instead the interaction were 5 5 . −gϕψγ ψb . Then the Feynman rule for the interaction would be −igγab ¯ 5ψ = −gϕψ¯aγab The factors associated with external legs follow the same logic as for scalar fields. For a covariantly normalized state k , we found in chapter 5 that

0 ϕ(x ) k = e−ik·x k ϕ(x ) 0 = e−ik·x . The generalization for Dirac fields is

0 ψ (x ) k , s = e−ik·xu s(p ⃗ ) k , s ψ (x ) 0 = e+ik·xv s (k ⃗ ) 0 ψ¯ (x ) k , s = e+ik·xv¯ s (p ⃗ ) k , s ψ¯ (x ) 0 = e−ik·xu¯ s(k ⃗ ) which are associated, respectively, with the destruction of a fermion, the creation of an antifermion, the destruction of an antifermion, and the creation of a fermion. The coordinate-space Feynman rules associate a spacetime point with every interaction, a propagator with every propagator connecting two spacetime points, and a coordinate-space wave function with every external line. The transition from fermion coordinate-space Feynman rules to momentum space rules proceeds in the same manner as for scalars. The momentum-space Feynman rules associate a momentum with every propagator, a four-momentum conserving delta function with every interaction, and a momentum-space wave function with every external line. 8.10.1 Nucleon–nucleon scattering at order g 2 There are two diagrams for the process N (p ) + N (k ) → N (p′) + N (k′) at order g 2, corresponding to t and u channel scattering. We need to calculate the momentum space matrix element associated with

1 ( −ig )2 2

∫ d 4x1 ∫ d 4x2 p′, k′

T [ψ¯ (x2 )ψ (x2 )ϕ(x2 )ψ¯ (x2 )ψ (x2 )ϕ(x2 )] p , k

For notational simplicity we assume that each particle has a spin index s = 1, 2 in addition to its momentum, but do not explicitly indicate it. The momenta are constrained to conserve overall four-momentum: p + k = p′ + k′ The matrix element for the t-channel process is

iM1 = ( −ig )2 u¯(p′)u(p )

i u¯(k′)u(k ) (p′ − p ) − μ2 + iϵ 2

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where we have chosen the ordering of the particles in the initial and final states so that the overall sign is positive. Exercise: convince yourself that a ψγ ¯ 5ψϕ interaction would lead to the replacement u¯(p′)u(p ) → u¯(p′)γ5u(p ) and so on in matrix elements. In order to obtain the corresponding u-channel process, we have to interchange the order of ψ¯ (x1) and ψ¯ (x2 ), which leads to a relative minus sign. Thus the u-channel contribution is

iM2 = −( −ig )2 u¯(k′)u(p )

i u¯(p′)u(k ). (k′ − p ) − μ2 + iϵ 2

The two contributions to NN scattering at O(g 2 ) are shown in figure 8.1. At this order, we have

iM = iM1 + iM2 and there will be interference terms when we calculate M 2 . In order to obtain a simpler first calculation, we can imagine that we have two different kinds of nucleons, which are distinguishable but have the same mass. We can call them ‘proton’ and ‘neutron’ and write the scattering process as n(p ) + p(k ) → n(p′) + p(k′). Our new interaction is

(

)

LI = −g ψ¯nψn + ψ¯pψp ϕ . This is not supposed to be a realistic model of nuclear interactions, just a model to illustrate a calculational technique. This coupling of fermions and scalar no longer allows the u-channel process at order g 2 , so iM1 is the only contribution to the

Figure 8.1. The two graphs contributing to NN scattering at order g 2 .

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scattering process at this order. In order to calculate M1 2 = M1*M1, we have to take the complex conjugate of u¯(p′)u(p ) and u¯(k′)u(k ). This kind of calculation is simplified by noting that u¯(p′)u(p ) can be regarded as a 1 × 1 matrix, so

[u¯(p′)u(p )]* = [u¯(p′)u(p )]+ = [u +(p′)γ 0u(p )]+ = u +(p )γ 0u(p′) = u¯(p )u(p′). This simple result holds for some but not all fermion bilinears so it is best to work out the correct result each time unless you are sure. Of course, a similar result holds for the complex conjugate of u¯(k′)u(k ). At this point a significant simplification occurs if we do not know the spin polarization of the incoming or outgoing particles. In that case, we must average over all incoming spins and sum over all outgoing spins. This procedure is usually referred to as spin averaging, even though it is only the initial spins which are averaged over. In M1 2 , we must average over an index r for p, and sum over an index s for p′: 2

4.

1 ∑ ∑ u¯ as(p′)u ar(p)u¯ br(p)u bs(p′) 2 r, s = 1a, b = 1 where we have been explicit about the spin labels r and s as well as the spinor indices a and b. This can be written as 2

4.

1 ∑ ∑ u bs(p′)u¯ as(p′)u ar(p)u¯ br(p) 2 r, s = 1a, b = 1 This is the trace of the product of two 4 × 4 matrices, with each matrix constructed as an outer product: ⎡ 2 ⎞⎤ ⎞⎛ 2 1 ⎢⎛⎜ tr ⎜∑u s(p′)u¯ s(p′)⎟⎟⎜⎜∑u r(p )u¯ r(p )⎟⎟⎥ 2 ⎢⎣⎝ s = 1 ⎠⎥⎦ ⎠⎝ r = 1 and we can use the identity equation (8.4.2) to write this as 1 tr[(γ · p′ + m)(γ · p + m)]. 2 This can be further simplified by the trace identities

tr[1] = 4 tr[γ μ ] = 0 ⎡1 ⎤ tr[γ μγ ν ] = tr⎢ {γ μγ ν}⎥ = 4g μν ⎣2 ⎦ so that

1 tr[(γ · p′ + m)(γ · p + m)] = 2(p · p′ + m 2 ) . 2

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With this result we can complete the calculation of the spin-averaged matrix element ¯ 2 , finding that M 1

M¯1

2

= 4g 2(p · p′ + m 2 )(k · k′ + m 2 )

1 2 (t − μ 2 )

¯ Exercise: Replace p · p′ and k · k′ using s, t and u so that M 1 completely in terms of Mandelstam variables.

2

is written

There are many more useful gamma matrix identities, all independent of the gamma matrix representation. The basic tools used to derive them are the anticommutation relation and the cyclicity of the trace. Noting that the set of matrices −γ μ also satisfy the anticommutation relation {γ μ, γ ν} = 2g μν , we see that the trace of odd number of gamma matrices must vanish. Other helpful identities include

γ μγμ = 4 γ μγ νγμ = − 2γ ν γ μγ νγ ργμ = 4g νρ γ μγ νγ ργ σγμ = − 2γ σγ ργ ν Exercise: Prove all these identities. For the trace of the product of four gamma matrices, we have

tr[γ μγ νγ ργ σ ] = tr[(2g μν − γ νγ μ)γ ργ σ ] = 8g μνg ρσ − tr[γ νγ μγ ργ σ ] = 8g μνg ρσ − tr[γ ν(2g μρ − γ ργ μ)γ σ ] = 8g μνg ρσ − 8g μρg νσ + tr[γ νγ ργ μγ σ ] = 8g μνg ρσ − 8g μρg νσ + tr[γ νγ ρ(2g μσ − γ σγ μ)] = 8g μνg ρσ − 8g μρg νσ + 8g μσg νρ − tr[γ νγ ργ σγ μ ] = 8g μνg ρσ − 8g μρg νσ + 8g μσg νρ − tr[γ μγ νγ ργ σ ] where we have applied cyclicity of the trace to put the last term in the final line into the form of our original expression. Thus we find

tr[γ μγ νγ ργ σ ] = 4g μνg ρσ − 4gg νσ + 4g μσg νρ. This procedure can be extended to the trace of any even number of gamma matrices. There are additional identities involving the γ5 matrix. For example, we have

tr[γ5] = 0 tr[γ5γ μγ ν ] = 0 tr[γ5γ μγ νγ ργ σ ] = 4iϵ μνρσ .

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Exercise: Prove these identities. As in our scalar Yukawa model, there will be two diagrams at order g 2 for both NN¯ scattering and Nϕ scattering. Note that as a consequence of the fermion field anticommutation relations, the two diagrams have a relative minus sign for NN¯ scattering, as is the case for NN scattering. However, there is no such sign for Nϕ scattering, because the bosonic operators associated with ϕ commute with the fermion operators. 8.10.2 Loop diagrams with fermions We have seen the effect of Fermi statistics on tree-level scattering amplitudes. A more subtle change in the Feynman rules occurs with fermion loops. Consider the one-loop fermion contribution to the self-energy of the bosonic field ϕ. In coordinate space, this involves a term of the form

( −ig )2 2

∫ d 4x1d 4x2 ϕ(x1)ψ¯ (x1)ψ (x1)ϕ(x2)ψ¯ (x2)ψ (x2)

before Wick contractions are performed. To make a loop with the fermion fields, we must move ψ¯ (x1) past the other three fields and then contract it with ψ (x2 ). This gives rise to a minus sign, and the fermion fields reduce under Wick contraction to

−ψa(x1)ψ¯b(x2 )ψb(x2 )ψ¯a(x1) → −tr[iSF (x1 − x2 )iSF (x2 − x1)] so we reduce our expression to

−

( −ig )2 2

∫ d 4x1d 4x2 ϕ(x1)tr[iSF (x1 − x2)iSF (x2 − x1)]ϕ(x2).

The minus sign occurs for any fermionic loop. For our Yukawa-interaction model, we reduce

( −ig )n n!

n

∏∫

d 4xj ϕ(xj )ψ¯ (xj )ψ (xj )

j=1

by first noting that there are (n − 1)! ways of ordering the spacetime points, because the choice of a first point doesn’t matter in a circle. Thus there is a residual combinatorial factor of 1/n, which is the symmetry factor associated with discrete rotations of the points on the loop. We have to move ψ¯ (x1) past the 2n − 1 fields ψ (x1)⋯ψ (xn ) which gives us a factor of −1. The final form is

−

( −ig )n n

∫ d 4x1⋯d 4xn ϕ(x1) ... ϕ(xn)tr[iSF (x1 − x2) ... iSF (xn − x1)].

In the next section, we will learn a different way to compute these effects. From the n = 4 loop diagram, we learn another facet of this simple model: with the change from charged scalars to charged fermions, the model is no longer renormalizable as it stands. The four-point function for the scalar field ϕ picks us a

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divergent contribution from the n = 4 fermion loop diagram. Its momentum space behavior in the UV is given by

⎛

⎞4

∫ d 4p⎜⎝ γ 1· p ⎟⎠

which is logarithmically divergent. Therefore, a ϕ4 counterterm is required, and also a corresponding renormalization condition to define the coupling. Thus the complete renormalizable Lagrangian is 1 1 λ L = (∂ϕ)2 − μ2 ϕ 2 + ψ¯ (iγ · ∂ − m)ψ − gϕψψ ¯ − ϕ4 . 2 4! 2

8.11 Interpreting the boson and fermion functional determinants We saw in chapter 4 that the generating function Z0[J ] for a real free scalar field could be written as a functional determinant for J = 0

Z [J = 0] = det [ −∂ 2 − m 2 − iϵ ]−1/2 ⎡ 1 ⎤ d 4k = exp ⎢ − VT log (k 2 − m 2 + iϵ )⎥ 4 (2π ) ⎣ 2 ⎦

∫

which could in turn be interpreted as a vacuum energy term ⎡ d 3k 1 ⎤ ωk ⎥ e−iE0T = exp⎢ −iTV ⎣ (2π )3 2 ⎦

∫

after integration over k 0 . Here we extend this idea to interacting theories using our interacting model of charged scalar fields exchanging a neutral scalar

LS = ∂ μψ +∂μψ − m 2ψ +ψ +

1 μ 1 ∂ ϕ∂μϕ − μ2 ϕ 2 − gψ +ψϕ 2 2

and the corresponding model of Dirac fermions exchanging a neutral scalar

LD = ψ¯ (iγ · ∂ − m)ψ +

1 μ 1 ∂ ϕ∂μϕ − μ2 ϕ 2 − gψψϕ ¯ . 2 2

Omitting sources for simplicity, the generating function for the scalar model is given by

∫ =∫

ZS[0] =

[dϕ ][dψ ][dψ + ]e i∫

d 4x LS

[dϕ ] det [ −∂ 2 − m 2 − iϵ − gϕ ]−1e i∫

d 4x Lϕ0

after integration over ψ and ψ +. We have defined the Lagrangian Lϕ0 to be

Lϕ 0 =

1 μ 1 ∂ ϕ∂μϕ − μ2 ϕ 2 . 2 2

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Similarly we have for Dirac fermions

∫ =∫

ZD[0] =

[dϕ ][dψ ][dψ + ]e i∫

d 4x Lϕ0

[dϕ ] det [iγ · ∂ − m + iϵ − gϕ ]e i∫

d 4x Lϕ0

after integration over ψ and ψ¯ . This result differs from the complex scalar result in the appearance of the Dirac operator rather than the Klein–Gordon operator, and the determinant replaces the inverse determinant. In both cases, all the interactions associated with vacuum graphs are contained in the functional determinants. Our problem now is the interpretation of those determinants. For any particular field configuration ϕ(x ), the calculation of the functional determinant requires the calculation of the infinite product of all eigenvalues of a differential operator which depends on ϕ(x ). Except for some special cases, this is typically impractical, and there is the further hurdle of the integration over all field configurations of ϕ. We are therefore forced to fall ty -20back on perturbation theory, in these cases a power series in the coupling constant g. Consider a matrix M diagonalizable by a similarity transformation S so that M = SDS −1, where D = djδjk is diagonal. Then

det [M ] = det D = ∏j dj ⎤ ⎡ ⎢ = exp ∑ log dj ⎥ ⎥⎦ ⎢⎣ j = exp [Tr log M ] where we use the diagonal matrix log D to define log M = S (log D )S −1. Ruthlessly extending this to the infinite-dimensional case, we have

det[ −∂ 2 − m 2 − iϵ − gϕ ]−1 = exp[ −Tr log( −∂ 2 − m 2 − iϵ − gϕ)] and

det [iγ · ∂ − m + iϵ − gϕ ] = exp[ +Tr log (iγ · ∂ − m + iϵ − gϕ)]. The trace in the infinite-dimensional case must be an interpreted as an the sum over a complete set of states in function space, and for the Dirac operator there is a further trace over spinor indices, which we will continue to indicate as a lower-case trace. In order to interpret this expression, we will factor the functional determinant as

⎡ ⎤ 1 det [iγ · ∂ − m + iϵ − gϕ ] = det [iγ · ∂ − m + iϵ ] det ⎢1 + ( − gϕ ) ⎥ . ⎣ ⎦ iγ · ∂ − m + iϵ We recognize the first factor as the functional determinant of a free fermion, and the second term can be written as

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⎡ ⎡ ⎤ ⎛ ⎞⎤ 1 1 det ⎢1 + ( − gϕ )⎥ = exp ⎢ + Tr log ⎜1 + ( − gϕ ) ⎟ ⎥ ⎣ ⎦ ⎝ ⎠⎦ iγ · ∂ − m + iϵ iγ · ∂ − m + iϵ ⎣ ⎡ = exp ⎢ + Tr ⎢⎣ ⎡ = exp ⎢ − Tr ⎢⎣ ⎡ exp ⎢ − Tr ⎢⎣

⎞n ⎤ ( − 1)n+1 ⎛ 1 ( − gϕ ) ⎟ ⎥ ⎜ ⎝ iγ · ∂ − m + iϵ ⎠ ⎥⎦ n n=1 ∞

∑

⎞n ⎤ 1⎛ i ( − igϕ )⎟ ⎥ ⎜ ⎠ ⎥⎦ n ⎝ iγ · ∂ − m + iϵ n=1 ∞

∑

⎤ 1 (iSF ( − igϕ ))n⎥ . ⎥⎦ n n=1 ∞

∑

To understand the meaning of these formal manipulations, we write out the n = 1, n = 2 and n = 3 terms in coordinate space:

−Tr (iSF ( − igϕ)) = −

∫ d 4x1 tr[iSF (x1, x1)(−igϕ(x1))],

1 Tr (iSF ( − igϕ)iSF ( − igϕ)) 2 1 =− d 4x1d 4x2 tr[iSF (x1, x2 )( −igϕ(x2 ))iSF (x2 , x1)( −igϕ(x1))] 2 −

∫

1 Tr (iSF ( − igϕ )iSF ( − igϕ ))3 3 1 =− d 4x1d 4x2d 4x3 tr[iSF (x1, x2 )( − igϕ(x2 ))iSF (x2 , x3)( − igϕ(x3))iSF (x3, x1)( − igϕ(x1))] . 3 −

∫

It should be apparent that these expressions represent fermion loops, with fields ϕ(xj ) coupling at the vertices. We can define an effective action S[ϕ ] which is a functional of the field ϕ alone

Seff [ϕ ] =

∫ d 4x, Lϕ0 − i logdet[iγ · ∂ − m + iϵ − gϕ]

where the −i in front of the determinant compensates for the overall i multiplying Seff :

∫ =∫

Z=

[dϕ ]e iSeff [ϕ] [dϕ ]e i∫

d 4x, Lϕ0+Tr log [iγ·∂−m +iϵ−gϕ]

where we have used logdet[iγ · ∂ − m + iϵ − gϕ ] = Tr log [iγ · ∂ − m + iϵ − gϕ ]. The effects of the fermions are incorporated in Seff by non-local interactions between the bosonic fields. Effective actions of this form have many uses. For example, if we are interested in the low-energy physics of light mesons, the effects of heavy nucleons may be included by approximating the effects of the nucleon determinant.

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8.12 The linear sigma model of mesons and nucleons The linear sigma model of mesons and nucleons incorporates many features of both nuclear physics and QCD. It is formed from four bosonic fields, the pseudoscalar mesons π j and the scalar meson σ, and two Dirac fermion fields p and n, associated with the proton and the neutron, respectively. The proton and the neutron behave as a doublet

Ψ=

( np)

under the internal symmetry group of isospin, which is an SU (2) symmetry group. Under an isospin transformation, ψ transforms as

Ψ → exp( −iθnˆ · τ ⃗ /2)Ψ where the τ ⃗ ’s generate the isospin group; we take them to be a copy of the Pauli matrices, acting on isospin indices. By analogy with angular momentum, we say that ψ transforms according to the I = 1/2 representation of isospin. Note that the Dirac ¯ τ ⃗Ψ ¯ Ψ transforms as an isoscalar with I = 0 while the triplet of bilinears Ψ bilinear Ψ transforms as an isotriplet with I = 1. Isospin is an approximate symmetry of the strong nuclear force which has its origin in the very small masses of the u and d quark on the scale of the strong interactions, so that both may be taken to be approximately zero. The pion fields represent another I = 1 isotriplet, conveniently written as π⃗ and the σ is an I = 0 isoscalar. In constructing interactions between the nucleons and mesons, we can have Yukawa interactions which respect isospin symmetry: an interaction between isoscalars, σψψ ¯ ⃗ · π ⃗ . However, there ¯ and another interaction between isotriplets, ψτψ are more symmetries to be taken into account. Recall from section 8.7 that massless Dirac fermions enjoy a chiral symmetry under γ5 in which the phases of the lefthanded and right-handed components ψL and ψR transform oppositely. If there is more than one massless Dirac field present, this symmetry extends to transformations which combine a chiral rotation with an internal symmetry transformation, which acts on Ψ as

Ψ → exp( −iθγ5nˆ · τ ⃗ /2)Ψ . This transformation is known as a chiral isospin transformation. When combined with non-chiral isospin transformations, we can transform the left- and right-handed components of Ψ independently. These transformation from the product group SU (2)L × SU (2)R . This is a symmetry of QCD when the u and d quarks are taken to be massless, and we want our model of strong nuclear interactions to inherit this property of SU (2)L × SU (2)R . The key is a corollary to our analysis of the Lorentz group: SU (2)L × SU (2)R is isomorphic to the O(4) group. If we posit that σ + iτ ⃗ · πγ⃗ 5 transforms as

σ + iτ ⃗ · πγ⃗ 5 → exp( −iθnˆ · τ ⃗ /2)(σ + iτ ⃗ · πγ⃗ 5) exp( +iθnˆ · τ ⃗ /2) under normal isospin transformations (which leave σ invariant) and

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σ + iτ ⃗ · πγ⃗ 5 → exp( +iθγ5nˆ · τ ⃗ /2)(σ + iτ ⃗ · πγ⃗ 5) exp( +iθγ5nˆ · τ ⃗ /2) under chiral isospin transformations, then we can construct SU (2)L × SU (2)R -invariant interaction of the form

an

¯ (σ + iτ ⃗ · πγ⃗ 5)Ψ . Ψ Moreover, the combination σ 2 + π ⃗ 2 is invariant under O(4) symmetry. Remembering our previous treatment of purely bosonic linear sigma model, we can write our Lagrangian as

¯ iγ · ∂Ψ − gπNN Ψ ¯ (σ + iτ ⃗ · πγ⃗ 5)Ψ + 1 (∂σ )2 + 1 (∂π ⃗ )2 − λ (σ 2 + π ⃗ 2 − v 2 )2 . L=Ψ 2 4 2 The bosonic potential favors spontaneous symmetry breaking. Taking σ = v at tree level, we see that the nucleons acquire a common mass, given by gv. This symmetry breaking breaks SU (2)L × SU (2)R down to SU (2), the normal isospin transformations which do not mix σ and the π⃗ fields. The pions are the Goldstone bosons associated with the symmetry breaking.

Problems i

1. Find the explicit block form of the matrices σ μν ≡ 2 [γ μ, γ ν ] in the Weyl representation and show that they are the generators of Lorentz transformations. 2. Check in detail that the object ϕL+(1, nˆ · σ ⃗ )ϕR transforms as a four-vector under a boost in any direction nˆ . You will need to connect the parameter η⃗ parametrizing the boost with the usual β and γ. 3. Determine the transformation law under parity of the components of the antisymmetric two-tensor ψσ ¯ μνψ , organizing them as ψσ ¯ 00ψ , ψσ ¯ 0jψ and jk . Explain why this is the correct transformation law for this object. ψσ ¯ ψ 4. Derive the completeness relations

∑u (s )(p)u¯ (s )(p) = γ · p + m s

∑v(s )(p)v¯ (s )(p) = γ · p − m s

for spinors. One way to do this is to prove that both sides both sides of the completeness relations are ±2m Λ ±, where the Λ ± are projection operators. To do this, you need to show that both sides satisfy the relations Λ ±2 = Λ ± and Λ + + Λ− = 1. An even easier way to proceed is to show that the two sides of the equations have the same action on basis states u(p ) and v(p ). Take your pick of methods. 5. Show that the conserved charge associated with invariance under ψ → e iθψ is given by

Q≡

¯ 0ψ = ∫ d 3p ⃗ ∑⎡⎣a ps+⃗ a ps ⃗ − b ps+⃗ b ps ⃗⎤⎦ ∫ d 3x ⃗ ψγ s

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6. Show that the Lagrangian

L = ψ¯ (iγ μ∂μ)ψ −

1 2 g [(ψψ ¯ )2 − (ψγ ¯ 5ψ )2 ] 2

is invariant under the continuous transformation ψ → exp (iθγ5)ψ but the Lagrangian

L = ψ¯ (iγ μ∂μ)ψ −

1 2 g (ψψ ¯ )2 2

is only invariant under the discreet transformation ψ → γ5ψ . It is sufficient to prove invariance for θ infinitesimal, but more illuminating for θ arbitrary. If you need help, consult the famous paper by Gross and Neveu 1974 Phys. Rev. D 10 3235. 7. Consider the scattering process N (p ) + ϕ(k ) → N (p′) + ϕ(k′) in our Yukawa model with a −gϕψψ ¯ coupling. (a) Draw the two Feynman diagrams contributing at O(g 2 ). (b) Give the momentum of the exchanged particle in each case, and indicate for each diagram whether it is an s, t, or u channel process. (c) Write down the invariant matrix element for each process. 8. In the modern version of Fermi’s theory of the weak interactions, muon decay μ−(p ) → e−(p′) + ν¯e(k′) + νμ(k ) is described the Feynman amplitude

M=

G [u¯(k )γ μ(1 − γ5)u(p )][u¯(p′)γ μ(1 − γ5)v(k′)] . 2

This is an interaction of the form (V μ − Aμ )(Vμ − Aμ). The neutrinos are massless, and the electron mass can be neglected because mμ > 200me . Show that the spin-averaged amplitude squared is given by

¯ M

2

=

1 ∑ M 2 spins

2

= 64G 2(k · p′)(k′ · p ).

9. The charged pion, π −, decays weakly into either an electron or a muon plus an anti-neutrinos, via either π −(q ) → μ− +ν¯μ(k ) or π −(q ) → e−(p ) + ν¯e(k ). Naively, one expects decay to an electron to dominate, because the electron is much lighter and phase space is therefore larger. Nevertheless, the π − decays to a muon 99.9% of the time, which subsequently decays to an electron plus neutrinos. In the context of the V−A theory of the weak interactions, the pseudoscalar pion can only couple to the axial vector, taking the form

0 Aμ (x ) q = fπ qμe−iq−x where fπ is a constant we can take to be defined by this matrix element. The overall matrix element is

M=

G fπ qμ[u¯(p )γ μ(1 − γ5)v(k )] . 2

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(a) Show using (γ · k )v(k ) = 0 and the corresponding result for u¯(p ) that the matrix element can be reduced to

M=

G fπ mμu¯(p )(1 − γ5)v(k ) 2

showing that the V−A dynamics enhances the matrix element by a factor of the lepton mass. (b) Calculate M* and show that

¯ M

2

= 4G 2f π2 (p · k ).

(c) The final result for the decay rate is

G 2f π2 mπ ml2 ⎛ m2⎞ ⎜1 − l2 ⎟ . 8π mπ ⎠ ⎝ 2

Γ(π −→l −+ν¯l ) = Calculate the ratio

Γ(π −→e−+ν¯e ) Γ(π −→μ− +ν¯μ) to show that muon decay does dominate. 10. Consider the linear sigma model for pions and nucleons ¯ iγ · ∂Ψ − gπNN Ψ ¯ (σ + iτ ⃗ · πγ⃗ 5)Ψ + 1 (∂σ )2 + 1 (∂π ⃗ )2 − λ (σ 2 + π ⃗ 2 − v 2 )2 . L=Ψ 2 4 2

(a) Taking spontaneous symmetry breaking to occur by having σ0 = 0 σ 0 ≠ 0, check explicitly that minimization of the potential term

1 λ(π ⃗ 2 + σ 2 − v 2 ) 2 4 leads to three Goldstone bosons. What is the mass of the σ? (b) What is the mass of the nucleon?

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