Relativistic Quantum Field Theory, Volume 2: Path Integral Formalism [Concise ed.] 1643277057, 9781643277059

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Relativistic Quantum Field Theory, Volume 2 Path integral formalism

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland Kent State University, Kent, Ohio, USA

Morgan & Claypool Publishers

Copyright ª 2019 Morgan & Claypool Publishers 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. Rights & Permissions To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact [email protected]. ISBN ISBN ISBN

978-1-64327-708-0 (ebook) 978-1-64327-705-9 (print) 978-1-64327-706-6 (mobi)

DOI 10.1088/2053-2571/ab3108 Version: 20191101 IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print) A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA, 94901, USA IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

This book is dedicated to my wife, Dr Veronica Antocheviz Dexheimer Strickland, and our amazing daughter Emily.

Contents Preface

xi

Acknowledgements

xiv

Author biography

xv

Units and conventions

xvi

1

Path integral formulation of quantum mechanics

1-1

1.1 1.2 1.3

The transition probability amplitude Derivation of the quantum mechanical path integral Path integral in terms of the Lagrangian 1.3.1 Summary 1.3.2 Caveats and clarifications Computing simple path integrals 1.4.1 Free particle in the path integral formalism 1.4.2 Quantum harmonic oscillator in the path integral formalism 1.4.3 Path integral for deviations from the classical solution 1.4.4 Connection to our usual understanding of the quantum harmonic oscillator 1.4.5 Generalization to an arbitrary potential and the WKB approximation Calculating time-ordered expectation values Adding sources Asymptotic states and vacuum–vacuum transitions Generating functional and Green’s function for quadratic theories 1.8.1 Quantum harmonic oscillator Euclidean path integral and the statistical mechanics partition function 1.9.1 Connection to statistical mechanics References

1.4

1.5 1.6 1.7 1.8 1.9

1-1 1-2 1-6 1-7 1-7 1-8 1-9 1-10 1-12 1-15 1-16 1-17 1-19 1-20 1-22 1-23 1-25 1-25 1-26

2

Path integrals for scalar fields

2-1

2.1 2.2

Generating functional for a free real scalar field Interacting real scalar field theory 2.2.1 Perturbative expansion of λϕ4 2.2.2 The two-point function 2.2.3 The four-point function

2-1 2-4 2-4 2-6 2-7

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Relativistic Quantum Field Theory, Volume 2

2.3 2.4 2.5 2.6 2.7

Generating functional for connected diagrams The self-energy The effective action and vertex functions Generating function for one-particle irreducible graphs 2.6.1 Scalar Schwinger–Dyson equation Interacting complex scalar fields References

2-8 2-11 2-13 2-17 2-18 2-19 2-20

3

Path integrals for fermionic fields

3-1

3.1

3.2 3.3 3.4

Finite-dimensional Grassmann algebra 3.1.1 Derivatives of Grassmann variables 3.1.2 Integrals over Grassmann variables 3.1.3 Gaussian integrals of Grassmann variables 3.1.4 Infinite-dimensional Grassmann algebra Path integral for a free Dirac field Path integral for an interacting Dirac field Fermion loops

3-1 3-2 3-3 3-3 3-5 3-5 3-7 3-7

4

Path integrals for abelian gauge fields

4-1

4.1 4.2 4.3

Free abelian gauge theory The photon propagator Generating functional for abelian gauge fields in general Lorenz gauge 4.3.1 Faddeev–Popov gauge fixing preview Generating functional for QED in general Lorenz gauge General Lorenz-gauge QED generating functional to O(e2) QED effective action and vertex functions Ward-Takahashi identities 4.7.1 Relation between the electron-photon vertex and inverse electron propagator 4.7.2 Electron self-energy and the sub-leading vertex 4.7.3 Extension to higher orders 4.7.4 Implications for renormalization

4-1 4-2 4-4

4.4 4.5 4.6 4.7

4-4 4-4 4-5 4-9 4-10 4-11 4-13 4-14 4-15

5

Groups and Lie groups

5-1

5.1

Group theory basics 5.1.1 Subgroups 5.1.2 The group center

5-1 5-1 5-2 viii

Relativistic Quantum Field Theory, Volume 2

5.2

5.3

5.4 5.5 5.6

5.7 5.8

Examples 5.2.1 Finite groups 5.2.2 Infinite discrete groups 5.2.3 Continuous compact groups Representations of groups 5.3.1 Equivalent representations 5.3.2 Reducible and irreducible representations 5.3.3 Product representations The group U(1) The group SU(2) The group SU(3) 5.6.1 Quadratic Casimir invariants 5.6.2 Cartan sub-algebra and ladder operators 5.6.3 Irreducible representations 5.6.4 The singlet –D00 5.6.5 The triplet and anti-triplet –D10 and D01 5.6.6 Higher dimensional representations 5.6.7 Product representations The group SU(N) The Haar measure 5.8.1 General method for constructing invariant Haar measures 5.8.2 Haar measure example - SU(2) References

6

Path integral formulation of quantum chromodynamics

6.1

The Fadeev–Popov method 6.1.1 General Lorenz gauge 6.1.2 Summary QCD Feynman rules 6.2.1 Bare quark propagator 6.2.2 Bare gluon propagator 6.2.3 Bare ghost propagator 6.2.4 Quark-gluon vertex 6.2.5 Gluon self-interactions 6.2.6 Ghost-gluon vertex 6.2.7 Closed ghost and quark loops Simple example application of the QCD Feynman rules

6.2

6.3

ix

5-2 5-2 5-2 5-3 5-3 5-5 5-5 5-6 5-6 5-7 5-9 5-11 5-12 5-12 5-13 5-13 5-14 5-14 5-16 5-16 5-17 5-18 5-19 6-1 6-5 6-9 6-11 6-12 6-12 6-13 6-13 6-13 6-14 6-14 6-15 6-15

Relativistic Quantum Field Theory, Volume 2

6.4 6.5

Becchi, Rouet, Stora, and Tyutin symmetry Slavnov–Taylor identities References

7

Renormalization of QCD

7.1 7.2 7.3 7.4

Divergences in scalar field theories Divergences in Yang–Mills theory Dimensional regularization refresher One-loop renormalization of QCD 7.4.1 Quark wave function and mass renormalization 7.4.2 Gluon wave function renormalization 7.4.3 Vertex renormalization 7.4.4 Collection of results 7.4.5 The renormalized QCD Lagrangian The one-loop QCD running coupling References

7.5

8

6-16 6-18 6-22 7-1

Topological objects in field theory

8.1 8.2 8.3

The kinky sine-Gordon model Two-dimensional vortex lines Topological solutions in Yang–Mills 8.3.1 Static solutions 8.4 The instanton 8.5 The Potryagin index 8.6 Explicit solution for a q = 1 instanton 8.7 Quantum tunneling, θ-vacua, and symmetry breaking 8.8 Quantum anomalies 8.8.1 The chiral anomaly in the Schwinger model 8.8.2 Understanding the anomaly 8.8.3 The chiral anomaly in 3+1d 8.9 An effective Lagrangian for the anomaly 8.10 Instantons and the chiral anomaly 8.11 Perturbation theory for the chiral anomaly References

x

7-1 7-4 7-5 7-6 7-6 7-7 7-9 7-11 7-11 7-12 7-15 8-1 8-1 8-5 8-7 8-7 8-9 8-11 8-12 8-13 8-15 8-16 8-18 8-19 8-21 8-23 8-23 8-27

Preface In introductory quantum mechanics one learns how to quantize a system given a fixed number of non-relativistic bosonic or fermionic particles. One uses a formalism in which the Hamiltonian is promoted from being a number to an operator resulting in the Schrödinger equation. The Hamiltonian operator itself is decomposed into kinetic and potential energy contributions and the potential energy form is typically taken as an external input. A question then arises: how does one arrive at the potential in the first place based on first principles? For example, how do we know (beyond experiment) that the Coulomb potential is the appropriate potential for charged particles? Are there quantum corrections to this potential? In addition, since potentials are not well-defined relativistically (instantaneous interactions), how can we generalize quantum mechanics to the relativistic case and satisfy causality? In classical physics, fields are introduced in order to construct physical laws that are causal and local. Classical forces described by, for example Coulomb’s law or Newton’s universal gravitation, require action at a distance. As a result, the force felt by a body changes instantaneously if any other body’s position (or charge, etc) changes, no matter how far away the other object is. In the classical field theories of Maxwell (electromagnetic field) and Einstein (gravitational field), the interactions between objects are mediated by a field that acts locally and causality is restored. So it seems that, since they are causal and local, we should figure out how to quantize classical field theories. Another inconsistency in quantum mechanics was our somewhat haphazard treatment of wave-particle duality. For example, both electrons and photons act simultaneously like waves and particles and physically they share many common features. They both undergo wave-like diffraction from obstacles, but can also act like discrete particles (photoelectric effect, Compton scattering, etc). Despite this, in classical theory, electrons are simply postulated to exist as matter, while photons are interpreted as ripples in the electromagnetic field. Is it possible that electrons are themselves ripples in an ‘electron field’? As we will learn over the course of these three volumes, the answer is a definitive yes. In general, the field is the fundamental object and particles are derived concepts that appear only after quantization of the field (e.g. the Higgs field gives birth to the Higgs boson). In quantum mechanics, we take classical number-valued quantities and promote them to operators acting in a Hilbert space. As we will see, at least in ‘canonical quantization’, the rules for quantizing a field are only slightly different. The basic degrees of freedom in quantum field theory (QFT) are operator-valued functions of space and time and, since space and time are continuous, we are dealing with an infinite number of degrees of freedom, so we will need to (re-)learn how to deal with systems with a large number of degrees of freedom (many-body theory). Once we are done, we will be able to properly define QFTs that can be used in a variety of different contexts, for example high energy theory, condensed matter, cosmology, quantum gravity, etc.

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Relativistic Quantum Field Theory, Volume 2

Beyond this, Dirac taught us that a consistent theory of relativistic electrons requires the existence of anti-electrons (aka positrons). As a consequence, it is possible to create particle–antiparticle pairs once the energy available exceeds twice the electron rest mass E > 2mc 2 in a process called pair production. And, of course, the reverse can happen, which is called pair annihilation. And one can easily see both of these types of events using modern particle detectors. The conclusion we must draw from this is that particles are not indestructible objects; they can be created and destroyed and may only live for a short amount of time. They are merely excitations surfing on a quantum field. But the story is more fantastical then this. If pair production has an energy threshold, then one could argue that as long as the energies available do not exceed this threshold (E > 2mc 2 ), then non-relativistic theories would be self-consistent; however, at this point the Heisenberg uncertainty principle comes into play. Let us say that we wanted to measure the position of a particle with a given spatial resolution L . The Heisenberg uncertainty principle tells us that the uncertainty in the momentum is Δp ≳ 1/L . In a relativistic setting, energy and momentum are connected, therefore, we also have an uncertainty in the energy ΔE ≳ 1/L . However, when the energy plus its uncertainty exceeds E + ΔE > 2mc 2 , then it is possible to create purely quantum-mechanical particle–antiparticle pairs. Equating the two, we obtain a threshold distance L 0 = 1/(2m ) = λ Compton /(4π ) with λ Compton = 2π /m. From this exercise we learn that the spontaneous production of particle–antiparticle pairs is important when a particle of mass m is localized in space to a distance, which is on order of less than its Compton wavelength. A similar argument holds if one considers localizing particles in time. The energy uncertainty increases as the time interval is made shorter and one eventually turns on the possibility of pair production. The consequence of this is that as one considers shorter and shorter time (or space) intervals, one sees more and more particles! Why is this relevant? Since microscopic particles interacting with one another can resolve the other particles’ microscropic dynamics (they behave like observers since they exchange quanta in order to interact), they ‘see’ their partners as being surrounded by an ensemble of particles and antiparticles that flit in and out of existence. In QFT, these ephemeral particles are called virtual particles. The ‘cloud’ of virtual particles that surround particles can modify its observable properties and must be taken into account to consistently understand them. In figure 0.1, I show a typical Feynman graph for the splitting of an electron (solid line with the arrows) into virtual photons (sinusoidal lines) and virtual electron– positron pairs (closed loops). In this figure, time progresses from the left to the right. Starting from the incoming electron line we see the radiation of a virtual photon, which then splits into a virtual electron–positron pair, and so forth. In QFT, during this time, all possible configurations of the various intermediate particles are sampled, but some configurations are more probable than others. We will learn how to quantify this over the course of these three volumes. A very similar phenomenon to pair production exists in condensed matter systems. When one considers the conduction of electrons in metals, for example, one finds that there is a valence band of electrons that are locally bound to atoms xii

Relativistic Quantum Field Theory, Volume 2

Figure 0.1. A typical Feynman graph for the splitting of a electron (solid line with the arrows) into virtual photons (sinusoidal lines) and virtual electron–positron pairs (closed loops).

and a conduction band in which electrons are able to move around. Because of the spin-statistics theorem, fermions obey the Pauli exclusion principle and ‘stack up’ to an energy called the Fermi energy. It is possible for a photon (or an energetic phonon) to excite one of the electrons that have an energy below the Fermi energy (in the Fermi sea) to above the Fermi sea. In metals, this process requires very little energy, however, in semiconductors the material is gapped. This introduces a lower limit on the amount of energy required to excite an electron into the conduction band which is called the gap. The processes of electron–hole pair production and annihilation can be described using Feynman graphs similar to the relativistic case and the mathematical machinery used to describe many body states in materials is very similar to what is encountered in relativistic QFT. This three-volume series began as lecture notes for a two-semester introductory course in QFT and quantum chromodynamics (QCD) and, as such, is written rather informally. The text is intended to be used in an introductory graduate-level course in QFT or an advanced undergraduate course. Volume 1 introduces classical fields and the method of canonical quantization in order to have a bridge to the language and formalisms used by students focusing on condensed matter physics. Volume 2 builds upon what was learned in volume 1, but starts anew using the modern path integral formalism and focuses on applications to quantum electrodynamics and chromodynamics. Volume 3 continues with discussions of applications to particle physics phenomenology, the weak interaction, the Higgs mechanism, and finite temperature field theory. Michael Strickland 30 June 2019

xiii

Acknowledgements I thank the students who provided feedback on the lectures notes that formed the basis for this book.

xiv

Author biography Michael Strickland Dr Michael Strickland is a professor of physics at Kent State University. His primary interest is the physics of the quark-gluon plasma (QGP) and high-temperature QFT. The QGP is predicted by QCD to have existed until approximately 10−5 seconds after the Big Bang. The QGP is currently being studied terrestrially by experimentalists at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and the Large Hadron Collider (LHC) at CERN. Dr Michael Strickland has published research papers on various topics related to the QGP, QFT, relativistic hydrodynamics, and many other topics. In addition, he has co-written a classic text on the physics of neural networks.

xv

Units and conventions • I will use natural units in which ℏ = c = 1. In these units, masses, energies, momenta, and temperatures have the same ‘energy-like’ units. I will typically use eV, MeV, or GeV as the standard unit for these energy-like quantities. For example, the mass of an electron is me = 0.511 MeV , where this is understood implicitly to denote me = 0.511 MeV/c 2 but, since c = 1, we typically suppress the division by c 2 when writing the mass. • Another consequence of using natural units is that the vacuum electric permeability and magnetic permeabilities are both equal to one, ε0 = μ0 = 1, which greatly simplifies the presentation of the formulas of electromagnetism. A similar simplification happens in thermodynamic formulas since kB = 1. • In natural units, spatial and temporal separations have units of inverse energy units (e.g. GeV−1). In some places in the notes we will use a bracket notation to indicate the dimension of a quantity, for example [m] = 1 tells us that masses have units of energy to the power one and lengths [L] = −1 indicating that length scales scale like energy to the power −1. • If the argument of a function, for example exp, should be dimensionless, then the argument should be equivalent to an energy times a length scale (E × L), since lengths have units of inverse energies in natural units. • In some cases you might need to switch from energy scales to explicit distance scales or vice versa. For this purpose, the following combination is very useful

ℏc = 0.197 326 938 GeV fm, where fm stands for femtometers (1 fm = 10−15 m). For purposes of estimation, one can use the approximate value ℏc ≈ 0.2 GeV fm . For example, if I were to tell you that a length (or time) scale is L = 0.2 fm , you can divide by ℏc to find the equivalent in inverse GeV, for example L = (0.2 fm)/(ℏc) ≃ 1 GeV−1. Similarly, we can work in reverse to convert energy (mass) scales to inverse distance scales. For example, from the electron mass me = 0.511 MeV we can determine the Compton wavelength λ Compton = 2π /m and then convert from GeV−1 to meters, that is λ Compton,e ≃ 12 296 GeV−1 × ℏc ≃ 2.426 × 10−12 m . • Finally, in the three volumes in this series the Minkowski space metric is assumed to be mostly negative which means that (g μν )Minkowski ≡ η μν = diag(1, −1, −1, −1).

xvi

IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 1 Path integral formulation of quantum mechanics

In the standard path of ‘second quantization’, one derives classical equations of motion, formulates the appropriate Poisson brackets, and then simply makes the somewhat ad hoc assumption that one can replace Poisson brackets by commutators or anti-commutators for bosons and fermions, respectively. In the process, we transmute classical c -valued fields into operator-valued functions of space and time. One would like any alternative path to field quantization in which one does not have to invoke arbitrary mappings. Such a path is given to us by path integral quantization. It will provide us with a rigorous and systematic way to approach quantum field theory, which will reproduce known results coming from the path of canonical second-order quantization but it will also allow us to formulate things such as vertex functions and so on in a more intuitive manner. For more materials on this subject, I point the reader to refs. [1, 2].

1.1 The transition probability amplitude In order to proceed, we first remind ourselves of the path integral formulation of non-relativistic quantum mechanics. We will do this for a one-dimensional system, but the derivation is straightforwardly extended to an arbitrary number of spatial dimensions. We start with a quantum wavefunction in the Schrödinger picture represented in the spatial basis

ψ (x , t ) = 〈x∣ψ (t )〉S .

(1.1)

Next, we remind ourselves of a familiar relationship from introductory quantum mechanics which tells us how to evolve the Schrödinger-picture wave function given the wavefunction specified at a time ti

ψ (t )

S

= e−iH (t −ti ) ψ (t ) S ,

(1.2)

where, as usual, I have taken ℏ = 1 and H is understood to be an operator. Note that ∣ψ (ti )〉S = ∣ψ (ti )〉H where the H indicates the Heisenberg picture. doi:10.1088/2053-2571/ab3108ch1

1-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

Defining

x(t ) ≡ e iH (t−ti ) x ,

(1.3)

〈x(t f )∣ψ (ti )〉S = 〈x∣e−iH (t f −ti )∣ψ (ti )〉 = 〈x∣ψ (t f )〉S = ψ (xf , t f ) .

(1.4)

we see that

Using this and inserting a complete set of states

ψ (xf , t f ) = 〈x(t f )∣ψ (ti )〉S =

〈x(t f )∣x(ti )〉  〈x( ti ) ∣ψ ( ti  )〉S ∫ dxi     ≡ K (xf , t f ; xi, ti )

(1.5)

ψ (xi, ti )

∫ dxi K (xf , tf ; xi , ti )ψ (xi , ti ) .

=

(1.6)

The function K introduced above is sometimes called the propagator and tells us how to compute the wavefunction as a function of the final time (and position) given the wavefunction at the initial time. The wavefunction ψ (xf , t f ) gives the probability amplitude to find the particle at the point xf at time t f and K (xf , t f ; xi , ti ) is the transition probability amplitude. The probability for the particle to transition from (xi , ti ) to (xf , t f ) is P (xf , t f ; xi , ti ) = ∣K (xf , t f ; xi , ti )∣2 . Of course, we need not stop at inserting one complete set of states. Let us consider what happens when we insert another complete set of states

ψ (xf , t f ) = =

∫ dxi 〈x(tf )∣x(ti )〉 ψ (xi , ti ) 〈x(t f )∣x(t1)〉  〈x(t )∣x(t )〉 ψ (xi , ti ) ∫ dxi ∫ dx1   1i K (xf , t f ; x1, t1)

=

(1.7)

(1.8)

K (x1, t1; xi, ti )

∫ dxi ∫ dx1 K (xf , tf ; x1, t1)K (x1, t1; xi , ti )ψ (xi , ti ) .

(1.9)

From this we learn that

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

∫ dx1 K (xf , tf ; x1, t1)K (x1, t1; xi , ti ) .

(1.10)

1.2 Derivation of the quantum mechanical path integral Equation (1.10) can be applied iteratively by inserting n intermediate points to obtain

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

∫ dx1 ∫ dx2⋯∫ dxn Kf ,n K n,n−1 K n−1,n−2⋯K1,i , 1-2

(1.11)

Relativistic Quantum Field Theory, Volume 2

where I have introduced the shorthand notation Kk,l ≡ K (xk , tk ; xl , tl ). To further n compress this formula, we write ∫ dx1 ∫ dx2⋯∫ dxn = ∏ j =1 ∫ dxj to obtain n

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

(

∏ ∫ j=1

)

dxj K f ,n K n,n−1 K n−1,n−2⋯K1,i .

(1.12)

In this formula, we imagine that a particle propagates from the initial position to the final position and we compute the total propagator by integrating over all possible intermediate positions at a finite number of intermediate times. This is illustrated for three different possible paths in figure 1.1. In the next step, we are going to take n → ∞. With this in mind, let us focus on a single element of the path

Kj ,j −1 = xj ∣e−iHt j eiHt j−1∣xj −1 = xj ∣e−iH Δt ∣xj −1 .

(1.13)

In the limit n → ∞, holding the initial and final times fixed, we have Δt → 0, so we can Taylor expand the exponential to obtain

Kj ,j −1 ≃ xj 1 − iH Δt xj −1 ≃ δ(xj − xj −1) − i Δt xj H xj −1 ,

(1.14)

where we have discarded terms of O((Δt )2) and higher. Note that, above, we have assumed that the Hamiltonian is time-independent, which is true for any closed energy-conserving system. To proceed, let us consider a Hamiltonian of the standard form

Figure 1.1. Three possible paths that could be followed by a particle that starts at spacetime point (xi , ti ) and ends and (xf , t f ).

1-3

Relativistic Quantum Field Theory, Volume 2

H=T+V=

p2 + V (x ) , 2m

(1.15)

and evaluate the expectation value appearing on the second line of equation (1.14). First, let us look at the contribution from the kinetic energy

〈xj ∣T ∣xj −1〉 =

∫

dp′ 2π

∫

dp 〈xj ∣p′〉 2π  ip ′ x j

e

p2 p 〈p∣xj −1〉 p′    2m    e−ipxj−1 2πδ(p − p ′)

=

∫

p2 2m

(1.16)

dp ip(xj −xj−1) p 2 e , 2π 2m

where, in the first equality, I have inserted two complete sets of momentum states in order to make the evaluation of the kinetic energy operator trivial. Turning to the potential energy contribution, we obtain

⎛ xj + xj −1 ⎞ ⎟ ∣x 〈xj ∣V ∣xj −1〉 ≃ 〈xj ∣ V ⎜ 〉 ⎝ ⎠ j −1 2 ≡ V (x )

(1.17)

≃ V (xj −1)〈xj ∣xj −1〉 ≃ V (xj −1)δ(xj − xj −1), where we have introduced the average position xj −1 ≡ (xj + xj −1)/2 in order to account for the fact that we could act either to the left or the right with the position operator. Strictly speaking, this is not necessary, since we will soon take the continuum limit, but this does remove some ambiguity in the evaluation for the time being. Next, we express the delta function appearing on the last line of equation (1.17) as

δ(xj − xj −1) =

dp ip(xj −xj−1) e , 2π

(1.18)

dp ip(xj −xj−1) e V (x ) . 2π

(1.19)

∫

to obtain

〈xj ∣V ∣xj −1〉 =

∫

Using equation (1.18) for the delta function appearing in the second line of equation (1.14) and collecting the kinetic and potential energy terms, we can express equation (1.14) as

Kj ,j −1 ≃

∫

⎤ ⎛ p2 ⎞ dp ip(xj −xj−1)⎡ ⎢1 − i Δt⎜ e + V (xj )⎟ + O((Δt )2 )⎥ . ⎝ 2m ⎠ 2π ⎣ ⎦

(1.20)

The term in the square brackets is simply the first-order expansion of the exponential of the scalar Hamiltonian. If one expands the propagator to higher orders in Δt , one

1-4

Relativistic Quantum Field Theory, Volume 2

finds that, in the square brackets above, one generates an exponential, allowing us to write

Kj ,j −1 ≃

∫

dpj −1 2π

eipj−1 (xj −xj−1)−i ΔtH (pj−1 , xj−1) ,

(1.21)

where I have relabeled the momentum integration variable to pj −1. Now that we know the expression for an infinitesimal element, we can multiply all of the propagators appearing in equation (1.12) to obtain n

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

∫ ∏ dxj Kf ,n K n,n−1⋯K1,i j=1

⎛ n = ⎜⎜∏ ⎝ j=1

∫

⎞⎛ dxj ⎟⎟⎜ ⎠⎝

∫

dpn 2π

∫

dpn−1 ⋯ 2π

∫

dp0 ⎞ ip (xn+1−xn)−i ΔtH (p , xn) n ⎟e n 2π ⎠ (1.22)

× eipn−1 (xn−xn−1)−i ΔtH (pn−1 , xn−1)⋯eip0 (x1−x0)−i ΔtH (p0 , x0) n ⎞⎛ n ⎛ n ⎞ i∑ [pℓ (x ℓ+1−x ℓ )−ΔtH (pℓ , x ℓ )] ⎟ ⎜ ⎜ ⎟ dxj ⎟⎜∏ dpk ⎟e ℓ =0 = ⎜∏ ⎠ ⎠⎝ k = 0 ⎝ j=1

∫

∫

where it is understood that x0 = xi and xn+1 = xf . Note that there is one more term in the product and sum over momentum since, for n intermediate points in x , we require n + 1 propagators to connect xi to xf . Focusing on the argument of the exponential above, we can write n

⎡ ⎛ xℓ + 1 − xℓ ⎞ ⎤ ⎟ − H (p , x ) = ℓ ⎥ ℓ ⎠ ⎦ Δt

∑Δt⎢⎣pℓ ⎝⎜ Δt → 0 lim

ℓ=0

∫t

tf

dt[px ̇ − H (p(t ), x(t ))].

(1.23)

i

Introducing the notations n

Dx ≡ lim

n →∞

∏∫

dxj ,

(1.24)

dpj ,

(1.25)

j=1

and n

Dp ≡ lim

n →∞

∏∫ j=0

we can express the continuum propagator compactly as

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

⎛

∫ Dx Dp exp ⎜⎝i ∫t

i

tf

⎞ dt[px ̇ − H ]⎟ . ⎠

(1.26)

Exercise 1.1 Show that if equation (1.20) is expanded to second order in Δt that one generates an expression, which is consistent with the Taylor series of the exponentiated Hamiltonian as stated below equation (1.20).

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Relativistic Quantum Field Theory, Volume 2

1.3 Path integral in terms of the Lagrangian In the case that the Hamiltonian is of the form H = (p2 /2m ) + V (x ) it is possible to perform the pj integrations and reduce the path integral to integrations only over the coordinate x . To see this, we back up one step and look at the terms involving the momentum in one of the terms contributing to the sum in equation (1.23) ∞

Ij ≡

∫−∞

⎛⎡ p j2 ⎤⎞ ⎥⎟ . exp ⎜i ⎢pj (xj +1 − xj ) − Δt ⎜⎢ ⎥⎦⎟ 2π 2 m ⎝⎣ ⎠

dpj

(1.27)

This is a Gaussian integral that can be evaluated using

⎛ ⎞1/2

∫ dx e−ax +bx = eb /4a⎝ πa ⎠ 2

2

⎜

⎟

.

(1.28)

Identifying a = i Δt /2m and b = i (xj +1 − xj ) gives

⎛ m ⎡ xj +1 − xj ⎤2 ⎞⎛ m ⎞1/2 ⎟ . I j = exp ⎜i Δt ⎢ ⎥ ⎟⎜ 2 ⎣ Δt ⎦ ⎠⎝ 2πi Δt ⎠ ⎝

(1.29)

p j2 m ⎡ xj +1 − xj ⎤2 , ⎢ ⎥ = 2 ⎣ Δt ⎦ 2m

(1.30)

Recognizing that

we obtain

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

⎛ n ⎛ m ⎞(n+1)/2 ⎜ ⎜ ⎟ ⎜∏ ⎝ 2πi Δt ⎠ ⎝ j=1

∫

⎛ ⎞ ⎜ ⎟ ⎞ ⎤⎟ n ⎡ p2 ⎜ j dxj ⎟⎟ exp ⎜i Δt ∑ ⎢ − V (xj )⎥ ⎟ , (1.31) ⎢ ⎥⎦ ⎟ 2 m ⎜ ⎠ j=0 ⎣  ⎟ ⎜ Tj − Vj = Lj ⎝ ⎠

where Lj is the Lagrangian evaluated at point j . Defining

⎛ m ⎞(n+1)/2 ⎜ ⎟ , n →∞ ⎝ 2πi Δt ⎠

N ≡ lim

(1.32)

and taking the continuum limit as before, we obtain

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

⎛

∫ Dx exp ⎜⎝i ∫t

tf

i

⎞ dt L(x , x )̇ ⎟ . ⎠

(1.33)

Finally, we note that the time-integral of the Lagrangian defines the action

S [x , x ]̇ =

∫t

tf

dt L(x , x )̇ ,

i

1-6

(1.34)

Relativistic Quantum Field Theory, Volume 2

where S [x , x ]̇ indicates that the action is a functional of x and x .̇ This allows us to write the propagator in the rather compact form

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

∫ Dx

eiS[x, x]̇ .

(1.35)

This formula tells us that the propagator is computed from a sum over all paths weighted by the complex exponential of the action S . The action S is minimized on the classical trajectory for the particle and hence nearby paths will have nearly the same phase factor, leading to constructive interference between the terms contributing to the path integral. If, however, the path is ‘far’ from the classical path, then exp(iS ) will rapidly oscillate, leading to a very small probably amplitude for that configuration. In natural units, ‘far’ is measured by when the action differs from the classical one by order one. In standard SI units, the argument of the exponential above would instead be iS /ℏ, allowing us to see that, in these units, paths whose action differs from the classical one by more than order ℏ will suffer these oscillations. What this means in practice is that there is a quantum ‘blurriness’ of particle propagation which extends to paths whose action is within a window of a few multiples of ℏ from the classical path. Paths that are completely different from the classical one are allowed, however, they are unlikely. 1.3.1 Summary Let us summarize what we have learned thus far. We have found that, in the general case, the quantum mechanical propagator can be expressed in terms of an infinite dimensional path integral

K (qf , t f ; qi , ti ) =

⎛

∫ DqDp exp ⎜⎝i ∫t

tf

i

⎞ dt[pq ̇ − H ]⎟ , ⎠

(1.36)

and that, in the case that H = p2 /2m + V (q ), one has1

K (qf , t f ; qi , ti ) = N

⎛

∫ Dq exp ⎜⎝i ∫t

tf

i

⎞ dt L(q , q )̇ ⎟ = N ⎠

∫ Dq eiS[q, q]̇ ,

(1.37)

where we have switched notation to generalized coordinates and it is understood that, in both cases, the path integral is constrained at the end points to enforce the boundary condition q(ti ) = qi and q(t f ) = qf . 1.3.2 Caveats and clarifications • We glossed over something in the definition of the normalization factor in equation (1.32). By inspection, one can see that, in the limit n → ∞ (Δt → 0), this quantity diverges. The fact that it is infinite does not matter physically since this infinity can be absorbed by normalization of the quantum wavefunction. In practice this is done by normalizing to a known path integral 1

See the caveats and clarifications below for a qualifying remark.

1-7

Relativistic Quantum Field Theory, Volume 2

such as the path integral for a freely propagating particle. In fact, as we will see in the context of quantum field theory, this normalization constant always cancels in physical quantities, for example expectation values, etc. Formally speaking, one should have in mind the discrete form and only take the continuum limit at the end of the calculation, however, in practice it suffices to simply denote this normalization abstractly as one and proceed ignoring this subtlety. • As noted during the derivation, obtaining equation (1.37) relied on the Hamiltonian being over the form H = p2 /2m + V (x ). This is not necessarily always the case, for example, if there are internal frictional forces proportional to the velocity. However, as it turns out, equation (1.37) is more general. It holds for a classical Lagrangian of the form

L(q , q )̇ =

1 ∑q ̇ Aij qj̇ + 2 i, j i

∑Bi(q)qi̇ − V (q) ,

(1.38)

i

where A is a symmetric, non-singular matrix which does not depend on q . In this case, one can show that

∫ Dp

⎛ exp ⎜i ⎝

∫t

tf

i

⎞ ⎛ dt[pq ̇ − H ]⎟ ∝ exp ⎜i ⎠ ⎝

∫t

i

tf

⎞ dt L(q , q )̇ ⎟ ∏ (det A)−1/2 , (1.39) ⎠ t

where ∏t implies a product over all discrete time points. If A is independent of q , then the product at the end is a constant, which can be absorbed into the overall normalization constant. If, however, A does depend on q , then one can use

∏ t

⎛ 1 ⎞ (det A)−1/2 = exp ⎜⎜ − ∑ log(det A)⎟⎟ , ⎝ 2 t ⎠

(1.40)

to obtain

K (qf , t f ; qi , ti ) = N

∫ Dq

⎛ exp ⎜i ⎝

∫t

i

tf

⎡ ⎤⎞ i dt⎢L(q , q )̇ + δ(0)log(det A(q ))⎥⎟ . (1.41) ⎣ ⎦⎠ 2

1.4 Computing simple path integrals So far we found that, in the general case, the quantum mechanical propagator can be expressed in terms of an infinite dimensional path integral

K (qf , t f ; qi , ti ) =

⎛

∫ Dq Dp exp ⎜⎝i ∫t

i

tf

⎞ dt[pq ̇ − H ]⎟ , ⎠

(1.42)

and that, in the case that the classical Lagrangian can be expressed in the form

1-8

Relativistic Quantum Field Theory, Volume 2

L(q , q )̇ =

1 ∑q ̇ Aij qj̇ + 2 i, j i

∑Bi(q)qi̇ − V (q) ,

(1.43)

i

one has

K (qf , t f ; qi , ti ) = N

∫ Dq

⎛ exp ⎜i ⎝

∫t

tf

i

⎞ dt L(q , q )̇ ⎟ = N ⎠

∫ Dq eiS[q, q]̇ ,

(1.44)

where we have switched notation to a generalized coordinate q and it is understood that, in both cases, the path integral is constrained at the end points to enforce the boundary condition q(ti ) = qi and q(t f ) = qf . In the next two sections, I would like to cover three additional topics in this context before proceeding to quantum field theory: (1) a practical example of the computation of the path integral for a free particle, a harmonic oscillator, and a general discussion of the quadratic approximation, (2) how to calculate time-ordered expectation values in the path integral formalism, and (3) how to describe asymptotic states. Before embarking on item (1), I will note that typically one does not have to actually evaluate a path integral explicitly to make use of the formalism. As you will see, the actual analytic evaluation can be a bit tricky, even for simple cases. 1.4.1 Free particle in the path integral formalism The Hamiltonian for a free particle is H = p2 /2m and the corresponding Lagrangian is L = mq 2̇ /2. In the discrete formulation, the propagator is given by

⎛ n 1⎜ K (qf , t f ; qi , ti ) = lim ⎜∏ Δt → 0 A ⎝ j=1

∫

⎛ n q − qj ⎤2 ⎞ dqj ⎞ ⎟, ⎟ exp ⎜i Δt ∑ m ⎡⎢ j +1 ⎜ ⎣ Δt ⎥⎦ ⎟ 2 A ⎟⎠ ⎠ ⎝ j=0

(1.45)

where A = 2πi Δt /m , Δt = (t f − ti )/(n + 1), and it is understood that q0 = qi and qn+1 = qf . To evaluate this integral we can use the the ‘Feynman slice’ method, which makes use of the fact that an integral over a Gaussian gives another Gaussian and then we can proceed iteratively. To start, let us consider the integral over q1 since it only appears in the first and second time slices ∞

I1 =

⎜

∞

= =

⎞

⎛

∫−∞ dq1 exp ⎝ 2imΔt ⎡⎣(q1 − q0)2 + (q2 − q1)2⎤⎦⎠ ⎛

⎟

⎡ ⎛

⎞2

∫−∞ dq1 exp ⎜⎜⎝ 2imΔt ⎢⎣2⎜⎝q1 − 12 (q0 + q2)⎟⎠ ⎛ im 1 ⎞ A exp ⎜ (q2 − q0)2 ⎟ . ⎝ ⎠ 2Δt 2 2

For the n = 2 term, there are two terms entering

1-9

+

⎤⎞ 1 (q2 − q0)2 ⎥⎟⎟ 2 ⎦⎠

(1.46)

Relativistic Quantum Field Theory, Volume 2

I2 = =

∞

⎛

⎡

⎤⎞

∞

⎛

⎡ ⎛

∫−∞ dq2 exp ⎜⎝ 2imΔt ⎢⎣ 12 (q2 − q0)2 + (q3 − q2)2⎥⎦⎟⎠ ⎞2

∫−∞ dq2 exp ⎜⎜⎝ 2imΔt ⎢⎣ 32 ⎜⎝q2 − 13 q0 − 23 q3⎟⎠

=A

+

⎤⎞ 1 (q3 − q0)2 ⎥⎟⎟ 3 ⎦⎠

(1.47)

⎛ im 1 ⎞ 2 exp ⎜ (q3 − q0)2 ⎟ . ⎝ 2Δt 3 ⎠ 3

From here on, the pattern should be clear, with the nth integral given by

In = =

∞

⎛

⎡

∞

⎛

⎡

⎤⎞

∫−∞ dqn exp ⎜⎝ 2imΔt ⎢⎣ 1n (qn − q0)2 + (qn+1 − qn)2⎥⎦⎟⎠ ⎛

∫−∞ dqn exp ⎜⎜⎝ 2imΔt ⎢⎣ n +n 1 ⎜⎝qn −

⎞2 1 n qn+1⎟ q0 − ⎠ n+1 n+1

⎤⎞ 1 (qn+1 − q0)2 ⎥⎟ + ⎦⎠ n+1 ⎛ im 1 ⎞ n exp ⎜ (qn+1 − q0)2 ⎟ . =A ⎝ 2Δt n + 1 ⎠ n+1

(1.48)

Putting the pieces together, we obtain

⎛ im 1 ⎞ 1 123 n exp ⎜ (qn+1 − q0)2 ⎟ ⋯ ⎝ 2Δt n + 1 ⎠ Δt → 0 A 2 3 4 n+1 ⎛ im 1 ⎞ 1 exp ⎜ (qf − qi )2 ⎟ = lim ⎝ 2Δt n + 1 ⎠ Δt → 0 A n + 1 2⎞ ⎛ m im (qf − qi ) ⎟⎟ , exp ⎜⎜ = 2πiT T ⎝ 2 ⎠

K (qf , t f ; qi , ti ) = lim

(1.49)

where, in the last line, we used the fact that (n + 1)Δt = t f − ti ≡ T . 1.4.2 Quantum harmonic oscillator in the path integral formalism Next, we compute the path integral for a harmonic oscillator with a Hamiltonian of the form

H=

p2 mω 2 2 + q , 2m 2

(1.50)

m 2 mω 2 2 q̇ − q . 2 2

(1.51)

and an associated Lagrangian

L=

1-10

Relativistic Quantum Field Theory, Volume 2

Since the path integral is dominated by paths close to the classical solution, we will change variables to

q(t ) = q (t ) + ε(t ) ,

(1.52)

where q (t ) is a solution to the classical equation of motion

d 2q + ω 2q = 0. dt 2

(1.53)

The full solution is subject to the boundary conditions q(ti ) = qi and q(t f ) = qf and hence it follows that

ε(ti ) = ε(t f ) = 0.

(1.54)

To proceed, we expand the action in terms of this new set of variables to find S [q, q ]̇ = S [q , q ̇] + S [ε, ε]̇ +

∫t

tf

i

dt

d ⎛ dq ⎞ ⎜mε ⎟ − dt ⎝ dt ⎠

∫t

i

tf

⎛ d 2q ⎞ dt mε⎜ 2 + ω 2q ⎟ . (1.55) ⎝ dt ⎠

The third term above vanishes because of the boundary condition (1.54) and the fourth term vanishes because q is a solution to the classical equation of motion (1.53). The fact that these two terms vanish is generic and independent of our assumption that the system is a harmonic oscillator. The fact that the first two terms are both simple actions for a harmonic oscillator is specific to this case and does not happen in general. As a result of this factorization, we can rewrite the path integral (1.44) in this case as

K (qf , t f ; qi , ti ) = NeiS[q , q ]̇

∫ Dε eiS[ε, ε]̇

= NeiS[q , q ]̇

⎛ Dε exp ⎜i ⎝

∫

∫t

i

tf

⎡m mω 2 2⎤⎞ dt⎢ ε 2̇ − ε ⎥⎟ , ⎦⎠ ⎣2 2

(1.56)

where we have used the fact that the classical path is fixed by the equations of motion and therefore can be treated as constant with respect to the integration over paths. To begin, let us evaluate the classical action

m tf dt(q 2̇ − ω 2q 2 ) 2 ti tf ⎫ m⎧ = ⎨[qq ̇]ttif − dt(qq ̈ + ω 2q 2 )⎬ ⎭ ti 2⎩ m = [qq ̇]ttif , 2

S [q , q ̇] =

∫

∫

(1.57)

where, in going from the first to second lines, we integrated the kinetic energy term by parts and, in going from the second to third lines, we used the fact that, for the harmonic oscillator, q ̈ = −ω 2q . The last line above can be evaluated by inserting a

1-11

Relativistic Quantum Field Theory, Volume 2

general solution of the form q (t ) = A sin(ωt ) + B cos(ωt ) subject to the boundary conditions q(ti ) = qi and q(t f ) = qf . The final result is

S (qf , t f , qi , ti ) =

mω ⎡ 2 ⎤ q + q f2 cos(ωT ) − 2qi qf ⎦ 2 sin(ωT ) ⎣ i

(

)

(1.58)

where T ≡ t f − ti . Exercise 1.2 Show that equation (1.55) is correct. Exercise 1.3 Find the transition probability amplitude, K for a particle moving in one dimension and subject to a constant external force, which has a Lagrangian 1 L = mq 2̇ + Fq . 2 Exercise 1.4 Show that equation (1.58) is correct.

1.4.3 Path integral for deviations from the classical solution What remains is to evaluate the path integral over the deviations from the classical solution in equation (1.56). We will do so using the ‘determinant method’. Our goal is compute

J (T ) ≡

⎛

∫ Dε exp ⎜⎝i ∫t

tf

dt

i

⎞ m 2 [ε ̇ − ω 2ε 2 ]⎟ , ⎠ 2

(1.59)

subject to the boundary condition εi = εf = 0. Notationally, we have indicated that the quantity J only depends on T since the path integral that remains has trivial boundary conditions and, by time-translation invariance, can only depend on the difference of the initial and final times. We begin integrating the first term in the argument of the integral by parts to obtain

i

∫t

i

tf

dt

m 2 [ε ̇ − ω 2ε 2 ] = −i 2

∫t

i

tf

dt

⎤ m ⎡ d2 i ε⎢ 2 + ω 2⎥ε = ⎦ 2 ⎣ dt 2

∫t

tf

dt εOˆ ε ,

(1.60)

i

where

⎤ ⎡ d2 Oˆ ≡ −m⎢ 2 + ω 2⎥ . ⎦ ⎣ dt

(1.61)

On a formal level, we now simply have to evaluate the Gaussian integral with the result being proportional to (det Oˆ )−1/2 , but since this is our first time around, let us see how this works in detail and also determine the proportionality constant. Since the fluctuation vanishes at the endpoints, we can write the function as a linear combination of the form

1-12

Relativistic Quantum Field Theory, Volume 2

ε( t ) =

∑cnφn(t ) ,

(1.62)

n

where φn(t ) are the orthonormal eigenfunctions of Oˆ

⎛ nπ (t − ti ) ⎞ 2 ⎟ n ∈ , sin ⎜ ⎝ ⎠ T T

(1.63)

Oˆ φn(t ) = λnφn(t ) ,

(1.64)

⎡⎛ nπ ⎞2 ⎤ λn = m⎢⎜ ⎟ − ω 2⎥ , ⎣⎝ T ⎠ ⎦

(1.65)

φn(t ) = which satisfy

with

and are, as advertised above, orthonormal

∫t

tf

dt φn(t )φm(t ) = δnm .

(1.66)

i

The functions φn(t ) form a complete basis, therefore, we can use the coefficients cn as a discrete set of integration variables. Transforming to these variables, we have

∫ Dε = J ∏ ∫ n

dcn , 2πi

(1.67)

where J is related to the Jacobian of the transformation and we have introduced some explicit factors of 2πi by hand for future convenience. Note that the Jacobian is independent of cn and, hence, can be treated as a constant. Using the basis states introduced above, we have

∫t

tf

dt εOˆ ε =

i

∑ ∫t m, n

tf

dt cmcn(φmOˆ φn) =

i

∑λncn2 .

(1.68)

n

As a result, we have

J (T ) =

∫ Dε

⎛ = J ⎜⎜∏ ⎝ n

⎛i exp ⎜ ⎝2 ∞

∫−∞

⎞ dt εOˆ ε⎟ ⎠ i ⎛i ⎞ dcn ⎞ ⎟⎟ exp ⎜⎜ ∑λncn2⎟⎟ . 2πi ⎠ ⎝2 n ⎠

∫t

tf

(1.69)

Each of the integrals is now independent and given by a standard Gaussian integral, with the result being

⎛ ⎞−1/2 J (T ) = J ⎜⎜∏ λn⎟⎟ = J(det Oˆ )−1/2 . ⎝ n ⎠

1-13

(1.70)

Relativistic Quantum Field Theory, Volume 2

The above expression gives us a concrete understanding of what the determinate of the operator Oˆ implies; it is given by the product of the eigenvalues of the operator. At this point we would like to take the continuum limit, but this proves to be difficult and potentially ill-defined since both the Jacobian and the product of eigenvalues are potentially divergent. We can simplify our lives by normalizing to the free-particle case since the result above should reduce to it in the limit that ω → 0, which will allow us to efficiently determine the normalization. Before proceeding we note that, although the operator Oˆ depends on ω, both the eigenfunctions and the Jacobian do not depend on ω. As a result, we have

Jω(T ) = J(det Oˆω)−1/2

(1.71)

J0(T ) = J(det Oˆ 0)−1/2 .

(1.72)

and

Next, we consider the following ratio 1/2 1/2 ⎛ Jω(T ) ⎛ det Oˆ 0 ⎞ λn(ω = 0) ⎞ ⎜ ⎟ ⎟ = ⎜∏ =⎜ . J0(T ) ⎝ det Oˆω ⎠ λn(ω) ⎟⎠ ⎝ n

(1.73)

Computing the eigenvalue product, one finds

∏ n

λn(ω = 0) = λ n (ω )

∏ n

⎛ nπ ⎞2 ⎜ ⎟ ⎝T ⎠ = ⎛ nπ ⎞2 2 ⎜ ⎟ − ω ⎝T ⎠

∏ n

n →∞ ωT 1 , ⎯⎯⎯⎯→ ⎛ ωT ⎞2 sin(ωT ) ⎟ 1−⎜ ⎝ nπ ⎠

(1.74)

where, in the last step, we took the continuum limit (n → ∞). Using equations (1.49) and (1.73) we then obtain

Jω(T ) = J0(T )

ωT = sin(ωT )

mω , 2πi sin(ωT )

(1.75)

which gives us our final result for the propagator of a quantum harmonic oscillator

K (qf , t f ; qi , ti ) =

mω ⎡(q 2+q 2 )cos(ωT )−2q q ⎤ mω i i f⎦ e 2 sin(ωT ) ⎣ i f . 2πi sin(ωT )

(1.76)

Note also, that by matching to the free particle case, we have fixed N = 1. One can verify explicitly that in the limit ω → 0 the result above reduces to equation (1.49). Exercise 1.5 An alternative method to obtain equation (1.75) is to use the fact that J (T ) defined in equation (1.59) is a transition amplitude from point qi = 0 to point qf = 0 due to the trivial boundary conditions on the fluctuation integral, that is J (T ) = K (0, t f ; 0, ti ). Based on this insight, use equations (1.10) and (1.58) to

1-14

Relativistic Quantum Field Theory, Volume 2

obtain equation (1.75) up to an overall constant. Then fix the remaining constant by requiring that the result reduces to that for a free particle in the limit ω → 0. 1.4.4 Connection to our usual understanding of the quantum harmonic oscillator Since the result does not seem immediately useful, I will note that the propagator obtained above gives the full spacetime evolution of the wavefunction for an arbitrary initial condition via equation (1.6). As it turns out, we can also extract the energy eigenvalues directly from equation (1.76). To obtain this result, we note that for a system with a constant energy, it is possible to also obtain the propagator in terms of the eigenfunction basis, that is

K (qf , t f ; qi , ti ) = 〈qf , t f ∣qi , ti〉 = 〈qf ∣e−iH (t f −ti )∣qi 〉 = ∑ 〈qf ∣ψn〉〈ψn∣e−iH (t f −ti )∣ψm〉〈ψm∣qi 〉

(1.77)

m, n

= ∑ψn*(qf )ψn(qi )e−iEn(t f −ti ) . n

Defining the partition function

⎛

∞

Z (T ) ≡

⎞

∞

∫−∞ dq K (q, T ; q, 0) = ∑⎝∫−∞ dq ∣ψn(q)∣2 ⎠e−iE T = ∑e−iE T . ⎜

⎟

n

n

n

(1.78)

n

This result is general for any system with a time-independent energy. Computing the partition function from equation (1.76) we obtain

mω 2πi sin(ωT )

Z (T ) =

∞

⎞1/2 1 ⎛ 1 ⎜ ⎟ 2i ⎝ cos(ωT ) − 1 ⎠

=

mω

∫−∞ dq ei sin(ωT ) [cos(ωT )−1]q

2

(1.79)

1 1 e−iωT /2 = = 2i sin(ωT /2) 1 − e−iωT = ∑e−i (2n+1)ωT /2 . n

Comparing equations (1.78) and (1.79) we find

⎛ 1⎞ En = ω⎜n + ⎟ , ⎝ 2⎠

(1.80)

which is the correct result. Note that one can also extract the probability amplitudes for all states by Taylor expanding K (q, −iT ; q, 0) around T = ∞.

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Relativistic Quantum Field Theory, Volume 2

Exercise 1.6 Above I demonstrated that the transition amplitude K (qf , t f ; qi , ti ) for the harmonic oscillator contains the full information about the energy spectrum. It also contains all of the wavefunctions! To see this, start from equation (1.76) and take T = t f − ti = −iτ = i log(ε )/ω, qi = qf = q , and then Taylor-expand around ε = 0. Following this procedure, show that the coefficient of the O( ε ) term gives the ground state probability distribution. Then extract the n = 3 and n = 7 states. I recommend using Mathematica for this exercise, otherwise, you will spend a great deal of time doing tedious and mind-numbing work. Finally, can you prove why this works? 1.4.5 Generalization to an arbitrary potential and the WKB approximation The preceding discussion relied heavily on the fact that the potential was quadratic and hence the action split into two separate contributions, the classical action and the fluctuation action. In the case of a general potential, V (q ), equation (1.55) becomes instead

∫t

S [q , q ]̇ = S [q , q ̇] + Seff [ε , ε ,̇ q ] +

tf

dt

i

−

∫t

i

tf

⎛ 2 d q dV dtε⎜⎜m 2 + dq ⎝ dt

d ⎛ dq ⎞ ⎜mε ⎟ dt ⎝ dt ⎠

⎞ ⎟, ⎟ q=q ⎠

(1.81)

where Seff is the effective action determined by

Leff =

1 2 1 d 2V mε ̇ − 2 2 dq 2

ε 2 + O(ε 3 ) .

(1.82)

q=q

As before, the third and fourth terms in equation (1.81) vanish due to boundary conditions and the classical equations of motion. One can now ask when is it appropriate to throw out terms of O(ε3) and higher in equation (1.82). Restoring ℏ by going back to SI units, we first note that in the limit ℏ → 0 one should obtain the classical case. In the path integral, the action enters as S /ℏ. The fluctuation ε contributes at leading order quadratically to the path integral and, as a result, the integral is dominated by contributions ε ∼ h . As a result, it makes sense to rescale the fluctuation field

ε→

(1.83)

ℏ ε˜ ,

such that the action divided by ℏ has the form

S [q , q ]̇ S (0)[q , q ̇] + S (2)[ε˜ , ε˜ ,̇ q ] + = ℏ ℏ

1-16

∞

∑ ℏn/2S (n)[ε˜ , ε˜ ,̇ q ] , n=3

(1.84)

Relativistic Quantum Field Theory, Volume 2

where the superscript ‘(n)’ indicates truncation at nth-order in the fluctuation. Typically (but not always), the odd terms in the sum over n above vanish by symmetry, leaving a fluctuation contribution of the form

K (qf , t f ; qi , ti ) = NeiS

(0)[q ,

̇ ℏ q ]/

⎛

∫ Dε˜ exp ⎜⎝i ∫t

i

tf

⎞ (2) dt L eff (ε˜ , ε˜ ,̇ q )⎟ ⎠

(1.85)

× (1 + O(ℏ)). If we stop at this order and ignore the terms of O(ℏ) and higher, this corresponds to the semiclassical WKB approximation2. Note that in the small ℏ limit the classical contribution exp(iS (0)[q , q ̇]/ℏ) oscillates wildly when the action changes from its minimum, while the remaining multiplicative factors depend smoothly on ℏ.

1.5 Calculating time-ordered expectation values Given a Schrödinger picture operator OS, the time-dependent Heisenberg-picture operator is

OˆH = eiHt OˆS e−iHt .

(1.86)

For example, the generalized position operator in the Heisenberg picture is

qˆ(t ) = eiHt qˆ e−iHt ,

(1.87)

where I have dropped the subscripts H and S in favor of simply labeling the operators as functions of t or not. The time-dependent position basis vectors in the Heisenberg picture are obtained from the time-independent basis vectors in the Schrödinger picture via

q(t ) = e iHt q ,

(1.88)

qˆ(t ) q(t ) = q(t ) q(t ) .

(1.89)

and are eigenstates of qˆ(t )

Next, consider a function F (q ) which defines an operator on a Hilbert space

Fˆ (qˆ ) ≡

∫ dq q

q F (q ).

(1.90)

2 For the quantum harmonic oscillator this is exact since all higher order terms vanish identically and the semiclassical approximation is exact.

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Relativistic Quantum Field Theory, Volume 2

Sandwiching Fˆ between our intial and final states gives

〈qf , t f ∣Fˆ (qˆ(t ))∣qi , ti〉 = 〈qf ∣e−iHt f Fˆ (qˆ(t ))eiHti∣qi 〉 = 〈qf ∣e−iH (t f −t )Fˆ (qˆ )e−iH (t −ti )∣qi 〉

∫ dq 〈qf ∣e−iH (t −t )Fˆ (qˆ)∣q〉〈q∣e−iH (t−t )∣qi〉 = ∫ dq F (q(t )) 〈qf ∣e−iH (t −t )∣q〉〈q∣e−iH (t −t )∣qi 〉 = ∫ dq F (q(t )) K (qf , t f ; q , t ) K (q , t ; qi , ti )

=

f

i

f

= N2 ⎛ ×⎜ ⎝

⎛

∫ dq F (q(t ))⎜⎝∫q

i

qa,f =qf

a,i=q

∫q

qb,f =q

b,i=qi

(1.91)

⎞ Dqa eiS[qa , qȧ ]⎟ ⎠

⎞ Dqb eiS[qb, qḃ ]⎟ , ⎠

where, as usual ∣q, t 〉 stands for ∣q(t )〉. We first recognize that we can absorb q into the path integration, that is

N2

⎛

∫ dq ∫ Dqa ∫ Dqb = A1 ⎜⎝∫ =

1 A

n dq ⎞⎛ ⎟⎜⎜∏ A ⎠⎝ i = 1

2n + 1

∏ ∫ k=1

∫

⎛ n dqi ⎞⎜ ⎟⎟ ∏ A ⎠⎜⎝ j = 1

dqk =N A

∫

dqj ⎞ ⎟ A ⎟⎠

(1.92)

∫ Dqc

where A = 2πi Δt /m with Δt = (t f − ti )/(2n + 2) and the final path integration over qc now covers from qi → qf . Secondly, we see that the arguments of the exponentials of the action appearing in the last line of equation (1.91) simply add and result in a single integral over the Lagrangian that spans ti → t f which defines S [qc , qċ ] = S [qa , qȧ ] + S [qb, qḃ ]. As a result, we have

〈qf , t f ∣Fˆ (qˆ(t ))∣qi , ti〉 = N

∫ Dq F (q(t )) eiS[q, q]̇ ,

(1.93)

where we have relabeled qc → q in the end. Now let us consider two functions F (q(t1)) and G (q(t2 )) and work in reverse, this time glossing over the details concerning the normalization factors until the end. We start with

〈FG〉 ≡ N

∫ Dq F (q(t1)) G(q(t2)) eiS[q, q]̇ .

(1.94)

If t1 < t2 , we can use the composition property demonstrated above to obtain

〈FG〉 ∝

∫ Dqa DqbDqc dq1 dq2 F (q(t1)) G(q(t2)) ei(S[q , q ̇ ]+S[q , q ̇ ]+S[q , q ̇ ]) , a

1-18

a

b

b

c

c

(1.95)

Relativistic Quantum Field Theory, Volume 2

where we have broken the integration in to three segments ti < t1 < t2 < t f with segments a , b, and c covering ti → t1, t1 → t2 , and t2 → t f , respectively. With this time ordering we can write

〈FG〉 ∝

∫ dq1 dq2 〈qf , tf ∣q2, t2〉 G(q2(t2)) 〈q2, t2∣q1, t1〉

F (q1(t1)) 〈q1, t1∣qi , ti〉 = 〈qf ∣e−iH (t f −t2)Gˆ (qˆ2 )e−iH (t2−t1)Fˆ (qˆ1)e−iH (t1−ti )∣qi 〉

(1.96)

= 〈qf , t f ∣Gˆ (qˆ2(t2 ))Fˆ (qˆ1(t1))∣qi , ti〉. If, instead, we have t1 > t2 then a similar exercise gives

〈FG〉 ∝ 〈qf , t f ∣Fˆ (qˆ1(t1))Gˆ (qˆ2(t2 ))∣qi , ti〉 .

(1.97)

Combining the two results, we obtain the following nice result

〈qf , t f ∣ T Fˆ (t1)Gˆ (t2 ) ∣qi , ti〉 = N

∫ Dq F (q(t1)) G(q(t2)) eiS[q, q]̇ ,

(1.98)

where T is the familiar time-ordering operator

T Oˆ 1(t1)Oˆ 2(t2 ) = θ (t2 − t1)Oˆ (t2 )Oˆ (t1) + θ (t1 − t2 )Oˆ (t1)Oˆ (t2 ) ,

(1.99)

and we have restored the proper normalization factor. Note that we have suppressed the explicit appearance of q1 and q2 on the left-hand side of equation (1.98) in order to make the formula more compact. As we can see, this proves that the path integral of the scalar version of our operators automatically gives the time-ordered expectation value! This is quite handy. As one might guess, this pattern continues as one adds more operators, with the general result being

〈qf , t f ∣ T Oˆ 1(t1)Oˆ 2(t2 )⋯Oˆn(tn) ∣qi , ti〉 =N

∫ Dq O1(q(t1)) O2(q(t2))⋯On(q(tn)) eiS[q, q]̇ .

(1.100)

This is a quite nice result since we see that the path integral automatically takes care of the time ordering for us. Exercise 1.7 Show that equation (1.97) is correct.

1.6 Adding sources Next I would like to introduce the method of adding sources. In physics, we frequently consider a system that is subject to a source, which is an external perturbation, and study the response of the system. We can do this in the path

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Relativistic Quantum Field Theory, Volume 2

integral formalism by making the system interact with an arbitrary external forcing source

L(q , q )̇ → L(q , q )̇ + J (t )q(t ) ,

(1.101)

and define a generalized propagator that is a functional of the source

K (qf , t f ; qi , ti∣J ) = N

⎛

∫ Dq exp ⎜⎝i ∫t

tf

i

⎞ dt[L(q , q )̇ + J (t )q(t )]⎟ . ⎠

(1.102)

The functional K contains all information about the system, which is evidenced by the fact that functional derivatives of K generate all time-ordered n-point functions

δ nK 1 i n δJ (t1)δJ (t2 )⋯δJ (tn)

=N J =0

∫ Dq q(t1) q(t2)⋯q(tn)

tf ⎛ ⎞ exp ⎜i dt L(q , q )̇ ⎟ ⎝ ti ⎠ ˆ ˆ ˆ = 〈qf , t f ∣ T q(t1)q(t2 )⋯q(tn) ∣qi , ti〉.

∫

(1.103)

For more complicated operators, one can build them up as power series, for example ∞

〈qf , t f ∣ exp(qˆ(t1)) ∣qi , ti〉 =

1 δ nK i nn! δJ n(t1) n=0

∑

= N

∫

J =0

⎛ Dq exp(q(t )) exp ⎜i ⎝

∫t

i

tf

⎞ dt L(q , q )̇ ⎟ . ⎠

(1.104)

1.7 Asymptotic states and vacuum–vacuum transitions Looking forward to the next chapter where we will apply this methodology to quantum field theory, we now want to consider a slightly different setup, namely that we have states with no excitations (the system is in its ground state) at ti = −∞ and t f = ∞ and that we turn on a source J for a finite time in the interval t and t′. Before taking the limit we introduce finite (but large) times T and T ′ which satisfy −T < t < t′ < T ′ and then, in the end, we will take T → ∞. We are interested in evaluating

〈Q′,T ′∣Q,T 〉J = N

∫ Dq

⎛ exp ⎜i ⎝

∫T

T′

⎞ dt[L(q , q )̇ + J (t )q(t )]⎟ , ⎠

(1.105)

where the superscript J indicates that we are calculating this in the presence of a source. Inserting two intermediate spacetime points

〈Q′,T ′∣Q,T 〉J =

∫ dq ∫ dq′ 〈Q′, T ′∣q′, t′〉〈q′,t′∣q,t〉J 〈q, t∣Q, T 〉 ,

1-20

(1.106)

Relativistic Quantum Field Theory, Volume 2

where we have made it explicit that the source only acts in the interval t → t′. Looking at the first expectation value in the integral we have ′

′

〈Q′ , T ′∣q′ , t′〉 = 〈Q′∣e−iHT eiHt ∣q′〉 ′

′

= ∑〈Q′∣e−iHT ∣ψn〉〈ψn∣eiHt ∣q′〉 n iE n(t ′ −T ′)

= ∑ψn*(q′)ψn(Q′)e

(1.107)

,

n

where ψn are a complete set of energy eigenstates. Using the same method, we find

〈q , t∣Q , T 〉 =

∑ψn*(Q )ψn(q)e−iE (t−T ) . n

(1.108)

n

Motivated by the discussion above, we would like to somehow single out only the ground state from each of these sums when we take T → −∞ and T ′ → ∞. To accomplish this, we make a rotation of the time integration T ′ → T ′e−iδ and T → T e−iδ where δ ⩽ π /2 is an arbitrary angle (which will we take to 0+ in the limiting sense if we require physical, Minkowski space results)3. With this rotation, we have, for example ′

′

′

′ cos δ+iT ′ sin δ )

∑ψn*(q′)ψn(Q′)eiE (t −T ) T ′→∞

lim 〈Q′ , T ′∣q′ , t′〉 = lim T ′→∞

n

n

∑ψn*(q′)ψn(Q′)eiE (t −T T ′→∞

= lim

n

(1.109)

n

′

′

= ψ0*(q′)ψ0(Q′)eiE0(t −T ) , and, likewise, limT →−∞〈q, t∣Q, T 〉 = ψ0*(Q )ψ0(q )exp( −iE0(t − T )). Combining these results, the limiting form of equation (1.106) becomes

lim

T →−∞e−iδ T ′→∞e−iδ

〈Q′,T ′∣Q,T 〉J

′ = ψ0*(Q )ψ0(Q′)e−iE0(T −T )

(1.110)

∫ dq ∫

dq′ ψ0*(q′)〈q′,t′∣q,t〉J ψ0(q ).

The double integral on the right-hand side takes an initial ground state at time t , propagates it to t′, and then projects it back onto the ground state, hence it is the expectation value of the vacuum to vacuum transition amplitude. Solving for the double integral, we obtain

lim ∫ dq ∫ dq′ ψ0*(q′)〈q′,t′∣q,t〉J ψn(q) = T →−∞ e

−iδ

T ′→∞e−iδ

3

〈Q′ , T ′∣Q , T 〉J . ′ ψ0*(Q )ψ0(Q′)e−iE0(T −T )

This maps to a counterclockwise rotation of the end points of the time-integration contour by δ .

1-21

(1.111)

Relativistic Quantum Field Theory, Volume 2

Note that the left-hand side does not depend on Q or Q′ and hence neither can the right-hand side. As a result, we can choose any Q and Q′ we like. Let us choose Q = Q′ = 0. We will return to this point soon, but for now we will just take this as a prescription. This gives

lim ∫ dq ∫ dq′ ψ0*(q′)〈q′,t′∣q,t〉J ψ0(q) = T →−∞ e

−iδ

T ′→∞e−iδ

〈0, T ′∣0, T 〉J . ′ ψ0*(0)ψ0(0)e−iE0(T −T )

(1.112)

The quantity in the denominator of the right-hand side is just a numerical factor. Based on this, we define the vacuum to vacuum transition amplitude Z [J ]

Z [J ] ∝ 〈0, ∞∣0, −∞〉J ∝

lim

T →−∞e−iδ T ′→∞e−iδ

〈0, T ′∣0,T 〉J .

(1.113)

In terms of the path integral, we have

Z [J ] = N

∫ Dq ei∫

∞

dt [L(q , q )̇ +J (t )q(t )+(1/2)iεq 2]

−∞

∝ 〈0, ∞∣0, −∞〉J .

(1.114)

To normalize Z [J ] we require that, in the absence of sources, the ground state to ground state transition amplitude returns identity as it should, that is Z[0] = 〈0∣0〉 = 1. This gives

∫ Dq ei∫ dt [L(q, q)̇ +J (t )q(t )] Z [J ] = . i∫ dt L(q , q )̇ D q e ∫

(1.115)

Above, we have suppressed the explicit time limits and additional imaginary convergence factor. These should be implicit in what follows. Using the results obtained in the previous section, it should be clear at this point that functional derivatives of Z [J ] generate time-ordered vacuum expectation values. In general, one has

G (t1, t2, ⋯ , tn) = 〈0∣T qˆ(t1)qˆ(t2 )⋯qˆ(tn)∣0〉 =

1 δ nZ i n δJ (t1)δJ (t2 )⋯δJ (tn)

. (1.116) J =0

As a result of this identity, Z [J ] is referred to as the generating functional for n-point functions (correlations functions).

1.8 Generating functional and Green’s function for quadratic theories One of the nice things about the generating functional is that, since the boundaries in time are at ±∞, the integrals involved become easier to deal with. Before dealing with the actual path integral, let us first consider a d -dimensional quadratic integral of the form

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Relativistic Quantum Field Theory, Volume 2

I≡

⎛

⎞−1/2

∫ d dx ei(1/2A x x +j x ) = ⎜⎝det 2Aπi ⎟⎠ ij i j

i i

e−iGijji jj /2 ,

(1.117)

where G = A−1 is, as we shall see, the analogue of the Green’s function. For this integral, the analogue of the generating funcitonal is

∫ d dx ei(1/2A x x +J x ) −iG J J /2 Z0(J ) ≡ =e , ∫ d dx ei /2A x x ij i j

i i

ij i j

(1.118)

ij i j

where the subscript 0 above is to emphasize the assumption of a quadratic form for the integral. From this, it follows that the ‘2-point function’

∫ d dx x1x2 e(i /2)A x x 〈x1x2〉 = , ∫ d dx e(i /2)A x x ij i j

(1.119)

ij i j

is given by

〈xkxℓ〉 ≡

1 ∂ 2Z0(J ) i 2 ∂Jk ∂Jℓ

.

(1.120)

J =0

Using equation (1.117) we see that

〈xkxℓ〉 = iG kℓ .

(1.121)

Therefore, the ‘2-point function’ is the ‘Green’s function’ of the integral I . Higher n-point functions (aka moments or correlation functions) can be calculated similarly. For this theory, the odd n-point functions are zero by symmetry. For the four-point function, for example, one finds

〈xkxℓxmxn〉 = 〈xkxℓ〉〈xmxn〉 + 〈xkxm〉〈xℓxn〉 + 〈xkxn〉〈xmxℓ〉 .

(1.122)

For a general 2n -point function, one obtains a sum over all possible (2n − 1) !! pairings, that is

〈x1⋯x2n〉 =

∑

〈xi1xi2〉⋯〈xi2n−1xi2n〉

(1.123)

P(x1, ⋯ , x2n)

where P (x1, ⋯ , x2n ) represents all possible pairings. This is related to Wick’s theorem, which we learned about in volume 1 in the context of canonical quantization. 1.8.1 Quantum harmonic oscillator Switching now to the generating functional for the quantum harmonic oscillator, using equation (1.115) one finds in this case

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Relativistic Quantum Field Theory, Volume 2

⎤⎞ q 2 + Jq⎥⎟ ⎦⎠ Z [J ] = ⎛ ⎡m mω 2 2⎤⎞ Dq exp ⎜i dt ⎢ q 2̇ − q ⎥⎟ ⎦⎠ ⎣2 2 ⎝ ⎛ i ⎞ = exp⎜ − dt dt′ J (t )G (t , t′)J (t′)⎟ , ⎝ m ⎠ ⎛ exp ⎜i ⎝

∫ Dq

⎡

∫ dt ⎢⎣ m2 q2̇ − m2ω

∫

2

∫

(1.124)

∫ ∫

where Z [J ] is now a functional of the time-dependent source, the integration limits are understood to be from ±∞, and now G is really a proper Green’s function since it is the functional inverse of the operator −∂ t2 − ω 2 , that is

⎛ d2 ⎞ ⎜ 2 + ω 2⎟G (t , t′) = −δ(t − t′) . ⎝ dt ⎠

(1.125)

Above, we have integrated by parts to change ∞

⎡⎛

⎞2

⎟ ∫−∞ dt m2 ⎢⎣⎜⎝ dq dt ⎠

⎤ − ω 2q 2 ⎥ = − ⎦

⎡

∞

2

⎤

∫−∞ dt m2 q⎢⎣ dtd 2 + ω2⎥⎦q,

(1.126)

where we have used the fact that qi = qf = 0 at infinity. Recall that with the infinite time interval we have to introduce the iε prescription implicitly. For the harmonic oscillator with a time-independent ω, one has

G (t , t′) =

1 −iω∣t−t′∣ e . 2iω

(1.127)

Having computed the generating functional (1.124), we can use it to calculate the n-point functions. For example, the two-point function

〈0∣Tq(t )q(t′)∣0〉 =

1 δ 2Z [J ] i 2 δJ (t )δJ (t′)

= J =0

i G (t , t′) , m

(1.128)

from which we see that the time-ordered vacuum expectation value 〈0∣Tq(t )q(t′)∣0〉 is a Green’s function. Similarly, one can show that for quadratic theories the higher n-point functions can be expressed in terms of products of two-point functions. Exercise 1.8 Show that equation (1.117) is correct. Exercise 1.9 Show that equation (1.122) is correct. Exercise 1.10 Show that equation (1.124) is correct.

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Relativistic Quantum Field Theory, Volume 2

1.9 Euclidean path integral and the statistical mechanics partition function Instead of computing the transition amplitude 〈qf , t f ∣qi , ti 〉 = 〈qf ∣exp( −iH (t f − ti )∣qi 〉 as before, we could instead compute the imaginary time evolution

KE ≡ 〈qf ∣e−βH ∣qi 〉 β ∈ + ,

(1.129)

where E denotes ‘Euclidean’, which I will explain further below. This corresponds to an imaginary time interval t f − ti = −iβ . One can show that using the same time slicing procedure used previously, that the imaginary-time amplitude can be written as a path integral of the form

KE (qf , qi ; β ) ≡ N

∫ DqE e−S ,

(1.130)

E

where

SE [q , q ]̇ =

∫0

β

⎡1 ⎤ dτ ⎢ mq 2̇ + V (q )⎥ , ⎣2 ⎦

(1.131)

with q ̇ = dq /dτ . The imaginary-time amplitude is an analytic continuation of the normal (Minkowski space) transition amplitude to imaginary time

KE (qf , qi ; β ) = K (qf , qi ; −iβ ) .

(1.132)

Equation (1.129) can be obtained directly from equation (1.37) by a change of variables t = −iτ , dt = −idτ , and dx /dt = idx /dτ . With this change of variables, the Minkowski spacetime interval ds 2 = dt 2 − d x2 → −dsE2 with dsE2 = dτ 2 + dx 2 . This explains why we call KE the Euclidean path integral. Exercise 1.11 Using discretized Euclidean spacetime, show that equation (1.130) is correct.

1.9.1 Connection to statistical mechanics Using the Euclidean form and inserting a complete set of orthonormal energy eigenstates as we did in equation (1.77), we obtain

KE (qf , qi ; β ) =

∑ψn*(qf )ψm(qi )e−βE

.

(1.133)

∫ dq KE (q, q; β ) = ∑e−βE .

(1.134)

n

n, m

As before we can define the partition function

Z (β ) ≡

n

n

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Relativistic Quantum Field Theory, Volume 2

The right-hand side above is the statistical mechanical partition function for a system of bosons with temperature T = 1/β (kB = 1 in natural units). Hence, we observe the equivalence

Z (β ) =

∫ dq KE (q, q; β ) = N ∫q(0)=q(β ) DqE e−S . E

(1.135)

As a consquence, the thermal statistical partition function for bosons is equal to a functional integral over a compact Euclidean time: τ ∈ [0, β ] with periodic boundary conditions. We will return to this connection later in volume 3, when we discuss finite temperature field theory.

References [1] Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw-Hill) [2] Shankar R 2011 Principles of Quantum Mechanics 2nd edn (New York: Plenum)

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 2 Path integrals for scalar fields

In this chapter, we begin our discussion of path integral quantization of quantum field theory by starting with a simple one-component scalar field. Before setting off, let us remind ourselves what we just learned conceptually: it is possible to formulate quantization without the use of the Schrödinger equation, operators, or commutators using the idea of the path integral. As we saw previously, in the context of quantum mechanics, the propagator K contains all information necessary: energy eigenvalues, probability distribution functions, space–time evolution, etc. Based on this success, it is natural to consider applying the path integral method to fields. For further materials related to this topic, I refer to the reader to refs. [1–4].

2.1 Generating functional for a free real scalar field By analogy with what we saw previously, it is natural to generalize the q(t ) → ϕ(x μ ) and hence Dq(t ) → Dϕ(x μ ) and define a vacuum-to-vacuum transition amplitude (generating functional) for real scalar fields via

∫ Dϕ ei∫ [L(ϕ, ∂ ϕ)+Jϕ+iεϕ ] Z [J ] ≡ , ∫ Dϕei∫ [L(ϕ, ∂ ϕ)+iεϕ ] 2

μ

x

x

μ

2

(2.1)

where L is the Lagrangian density and now the path integral maps to a integral over all possible values of the field ϕ at all points in space and time and ∫ = ∫ d 4x . x Instead of imagining that we make slices in time, we instead imagine that we break up spacetime time into infinitesimal hypercubes and integrate over all possible field values in each tiny hypercube. As previously, this can also be written compactly in terms of the field action S = ∫ L x

doi:10.1088/2053-2571/ab3108ch2

2-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

∫ Dϕei(S[ϕ, ∂ ϕ]+∫ [Jϕ+iεϕ ]) Z [J ] ≡ . ∫ Dϕei(S[ϕ, ∂ ϕ]+∫ iεϕ ) 2

μ

x

μ

(2.2)

2

x

For a free scalar field, the Lagrangian density is the Klein–Gordon Lagrangian density

L=

1 1 ∂μϕ ∂ μϕ − m 2ϕ 2 . 2 2

(2.3)

A simple integration by parts allows us to express the corresponding action as

S [ϕ , ∂μϕ ] = −

1 ϕ(□ + m 2 )ϕ . 2

∫x

(2.4)

As a result, equation (2.2) becomes

⎛

Z0[J ] =

⎡1 ⎤⎞ 2 ⎢⎣ ϕ(□ + m − iε )ϕ − Jϕ⎥⎦⎟ ⎠ 2 , ⎛ ⎞ 1 2 ⎜ ⎟ ϕ(□ + m − iε )ϕ Dϕ exp −i ⎝ ⎠ x 2

∫ Dϕ exp⎜⎝−i ∫x ∫

(2.5)

∫

where the subscript ‘0’ now indicates that this is the generating functional for our free scalar field theory. Next we can use the fact that ⎛ ⎡

⎤⎞

⎡

⎤

∫ Dx exp⎜⎝−⎢⎣ 12 (x, Ax) + (B, x) + C ⎥⎦⎟⎠ = (det A)−1/2 exp ⎢⎣ 12 (B, A−1B ) − C ⎥⎦,

(2.6)

where A can be an operator and ( f , g ) ≡ ∫ fg . As a result, the numerator of the x generating functional (2.5) becomes ⎡1

⎤

∫ Dϕ e−i∫ ⎣⎢ 2 ϕ(□+m −iε)ϕ−Jϕ⎦⎥ = [det i(□ + m2 − iε)]− 2 e−(1/2)∫ ∫ J (x)D (x−y)J (y), 2

1

x

x

y

F

(2.7)

where DF = i (□ + m2 − iε )−1 is a Green’s function for the Klein–Gordon equation defined by

(□ + m 2 − iε )DF (x − y ) = −iδ 4(x − y ).

(2.8)

I remind you that its integral definition is (see volume 1) −ikμx μ

4

∫ (2dπk)4 k 2 −e m2 + iε .

DF (x ) = i

(2.9)

Likewise, the denominator of equation (2.5) becomes

∫ Dϕe−i∫ d ×( 2 )ϕ(□+m −iε)ϕ = [det i(□ + m2 − iε)]−1/2 . 4

1

2

2-2

(2.10)

Relativistic Quantum Field Theory, Volume 2

Putting the pieces together, we obtain

⎛ 1 Z0[J ] = exp ⎜ − ⎝ 2

⎞

∫x ∫y J (x)DF (x − y )J (y )⎟⎠.

(2.11)

As discussed in the previous chapter, this is the generating functional for n -point functions for our theory which is, in this case, a free scalar field. All free n -point functions can be obtained from functional derivatives of Z0

G 0(x1, ⋯ , xn) = 〈0∣T ϕ(x1)⋯ϕ(xn)∣0〉 =

1 δ nZ0[J ] i n δJ (x1)⋯δJ (xn)

.

(2.12)

J =0

For this theory, the one-point function vanishes (as do all odd n-point functions by symmetry)

G 0(x1) = 〈0∣T ϕ(x1)∣0〉 = 〈0∣ϕ(x1)∣0〉 =

1 δZ0[J ] i δJ (x1)

= 0.

(2.13)

J =0

The two-point function returns i times the Feynman propagator

G 0(x1, x2 ) = 〈0∣T ϕ(x1)ϕ(x2 )∣0〉 = −

δ 2Z0[J ] δJ (x1)δJ (x2 )

= DF (x1 − x2 ) .

(2.14)

J =0

As mentioned previously, for this theory, the three-point function vanishes. After some trivial algebra, the four-point function can be shown to be

G 0(x1, x2 , x3, x4) = 〈0∣T ϕ(x1)ϕ(x2 )ϕ(x3)ϕ(x4)∣0〉 =

δ nZ0[J ] δJ (x1)δJ (x2 )δJ (x3)δJ (x4)

= DF (x1 − x2 )DF (x3 − x4) + DF (x1 − x3)DF (x2 − x4) + DF (x1 − x4)DF (x2 − x3).

J =0

(2.15)

This is the same result we obtained using Wick’s theorem in volume 1. We could continue to higher n-point functions and, by doing so, we could generate expressions for all of them without appeal to Wick’s theorem. Exercise 2.1 Show that equation (2.6) is correct. Exercise 2.2 Show that equation (2.15) is correct.

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Relativistic Quantum Field Theory, Volume 2

2.2 Interacting real scalar field theory While formally nice, free field theories are quite boring, so let us look again at an interacting scalar field theory in which the interaction only depends on the local value of the field. In this case, the Lagrangian density can be split into free and interacting parts

L(ϕ , ∂μϕ) = L0(ϕ , ∂μϕ) + Lint (ϕ) .

(2.16)

With this modification, we can apply equation (2.2) without any further changes. As mentioned in section 1.6, we can use the Taylor series to expand the action in a power series and then replace the field by a functional derivative with respect to the sources using the map

ϕ(x ) →

1 δ . i δJ ( x )

(2.17)

For our general interacting scalar field theory, this translates into the following expression for the generating functional

⎡ Z [J ] = N exp ⎢i ⎣

⎞⎤

⎛

∫x Lint⎜⎝ 1i δJδ(x) ⎟⎠⎥⎦Z0[J ],

(2.18)

where the leading exponential is now a functional operator and N is a normalization that guarantees that Z[0] = 1. Making everything explicit, we have ⎡ ⎛ 1 δ ⎞⎤ ⎛ 1 ⎞ exp ⎢i ∫ L int ⎜ ⎟⎥exp⎜ − ∫ ∫ J (x )DF (x − y )J (y )⎟ ⎠ ⎝ i δJ (z ) ⎠⎦ ⎝ 2 x y ⎣ z . Z [J ] = ⎧ ⎡ ⎛ 1 δ ⎞⎤ ⎛ 1 ⎞⎫ ⎨exp ⎢i ∫ L int ⎜ ⎟⎥exp⎜ − ∫ ∫ J (x )DF (x − y )J (y )⎟⎬ ⎠⎭ ⎝ i δJ (z ) ⎠⎦ ⎝ 2 x y ⎣ z ⎩ J =0 ⎪

⎪

⎪

⎪

(2.19)

2.2.1 Perturbative expansion of λϕ4 Next we specialize to λϕ4 theory, which has a Lagrangian density

L=

1 1 λ ∂μϕ∂ μϕ − m 2ϕ 2 − ϕ4 , 2 2 4!

(2.20)

such that

Lint = −

λ 4 ϕ . 4!

(2.21)

If we are interested in the O(λ ) correction to the generating functional, we can make a Taylor expansion of the exponential functional operator and truncate at O(λ )

⎡ iλ exp ⎢ − ⎣ 4!

⎛

⎞4 ⎤

⎛

⎞4

∫z ⎜⎝ 1i δδJ (z)⎟⎠ ⎥⎦ ≃ 1 − 4iλ! ∫z ⎜⎝ 1i δδJ (z)⎟⎠ 2-4

+ O(λ2 ).

(2.22)

Relativistic Quantum Field Theory, Volume 2

The first term, which is O(λ0 ), will give the generating functional for a free scalar theory. To compute the O(λ ), we must evaluate the fourth functional derivative of Z0[J ]. Skipping the intermediate steps, one finds (homework)

⎧ ⎛ 1 δ ⎞4 ⎡ ⎜ ⎟ Z0[J ] = ⎨3DF (0)2 − 6DF (0)⎢⎣ ⎝ i δJ ( z ) ⎠ ⎩

⎤2

∫x DF (z − x)J (x)⎥⎦

⎡ +⎢ ⎣

⎤4 ⎫ DF (z − x )J (x )⎥ ⎬Z0[J ]. ⎦⎭ x

(2.23)

∫

We can map this to a set of diagrams by replacing DF(x − y ) by a straight line and DF(0) by a closed loop with the result being

In these diagrams, external lines (if they exist) are attached to sources as can be seen by examining the corresponding integral expressions in equation (2.23). The first diagram, the double-bubble diagram, is our first example of a vacuum diagram because it contains no external lines. The factor of three in the first diagram is related to the number of ways of taking four lines and joining two sets in pairs. The second tadpole diagram and the overall factor associated with it can be generated by taking a bare four-vertex and combining two legs in all possible ways. These are the familiar symmetry factors we encountered in volume 1. Using functional methods, these numbers are automatically generated for you. Finally, the last diagram is simply the bare four-field interaction with no contractions1. Next, we obtain the denominator of equation (2.19) by taking the limit J → 0 in the above expressions. In terms of diagrams, this translates into the elimination of all diagrams with external legs. The result is

To obtain the final result, we must compute the ratio

where, in going from the first to second lines, we have Taylor-expanded the denominator and then discarded terms of O(λ2 ) and higher. Note that, in the end, the double-bubble vacuum diagram is canceled between the numerator and the 1 Keep in mind that these are position-space diagrams. The diagrams here are a slight departure from our momentum-space conventions from volume 1.

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Relativistic Quantum Field Theory, Volume 2

denominator. A similar cancellation of vacuum diagrams happens at all orders of perturbation theory. Exercise 2.3 Show that equation (2.23) is correct. 2.2.2 The two-point function Next, we compute the two-point function to O(λ ) using

G (x1, x2 ) = 〈0∣T ϕ(x1)ϕ(x2 )∣0〉 = −

δ 2Z [J ] δJ (x1)δJ (x2 )

.

(2.27)

J =0

The first term in (2.26) simply returns DF (x1 − x2 ), which is the free propagator. The term in equation (2.26) involving the bare four-vertex will go to zero if we take two functional derivatives and then set J = 0. So the only term we need to be concerned with at this order is the tadpole diagram. Evaluating the functional derivatives necessary gives

⎛ ⎞ δ2 DF (z − x )DF (z − y )J (x )J (y )⎟ ⎜Z0[J ]DF (0) ⎠ δJ (x1)δJ (x2 ) ⎝ x, y , z δ ⎛ δZ 0 DF (0) DF (z − x )DF (z − y )J (x )J (y ) = ⎜ δJ (x2 ) ⎝ δJ (x1) x, y , z ⎞ DF (z − x1)DF (z − y )J (y )⎟ + 2Z0[J ]DF (0) ⎠ y, z

∫

∫

∫

=

δ 2Z0 DF (0) DF (z − x )DF (z − y )J (x )J (y ) δJ (x1)J (x2 ) x, y , z δZ 0 DF (0) DF (z − x2 )DF (z − y )J (y ) +2 δJ (x1) y, z δZ 0 DF (0) DF (z − x1)DF (z − y )J (y ) +2 δJ (x2 ) y, z

∫

(2.28)

∫

∫

+ 2Z0[J ]DF (0)

∫z DF (z − x1)DF (z − x2).

Setting J = 0, we see that only the last term is non-zero. Multiplying it by an overall factor of −iλ /4 and adding this to the free result, we obtain

where we have used the fact that DF (x1 − x2 ) = DF (x2 − x1) and DF (0) = DF (z − z ) to rewrite the second term in a form similar to what we obtained in volume 1.

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Relativistic Quantum Field Theory, Volume 2

Figure 2.1. Space–time Feynman diagrams for the time-ordered product of four scalar fields. The open circles labelled 1, 2, 3, and 4 correspond to the space–time points x1, x2, x3, and x4, respectively. Solid lines indicate a scalar Feynman propagator DF (xi − xj ).

2.2.3 The four-point function Now let us consider the four-point function obtained using

G (x1, x2 , x3, x4) = 〈0∣T ϕ(x1)ϕ(x2 )ϕ(x3)ϕ(x4)∣0〉 =−

δ 4Z [J ] δJ (x1)δJ (x2 )δJ (x3)δJ (x4)

.

(2.30)

J =0

At O(λ0 ), we obtain equation (2.15) from equation (2.26). This maps to the diagrams shown in figure 2.1.These diagrams can be summarized by the following representative diagram

where the subscript P indicates that one should take all distinct combinations of the four external indices to generate the three diagrams in figure 2.1 and divide by the number of permutations. Since these diagrams represent the case of no scattering, they will not contribute to the S -matrix. Next we consider the contributions of O(λ ). For this, we first need to take two more functional derivatives of equation (2.28). As before, there will be many terms generated that will go to zero when we set J = 0 in the end. One class of nonvanishing terms result from two functional derivatives acting on the sources inside the integral, for example acting on the first line of the second equality in equation (2.28). This results in a contribution of the form

2DF (x1 − x2 )DF (0)

∫z

DF (z − x3)DF (z − x4) ,

(2.32)

which maps to a disconnected diagram in which one particle propagates from 1 to 2 using the free propagator and the other particle propagates from 3 to 4 with a propagator with one tadpole insertion. Collecting all such terms and multiplying the overall factor of −iλ /4 one obtains

2-7

Relativistic Quantum Field Theory, Volume 2

−

iλ DF (0) [DF (x1 − z )DF (z − x2 )DF (x3 − x4) 2 z + DF (x1 − z )DF (z − x3)DF (x2 − x4) + DF (x1 − z )DF (z − x4)DF (x2 − x3) + DF (x2 − z )DF (z − x3)DF (x1 − x4) + DF (x2 − z )DF (z − x4)DF (x1 − x3) + DF (x3 − z )DF (z − x4)DF (x1 − x2 )].

∫

(2.33)

These six contributions can be summarized by

These once again represent disconnected contributions, which will not contribute to scattering, but instead simply encode the leading-order correction to the propagators entering into the four-point function. Looking again at equation (2.26), we see that there will be another contribution coming from four functional derivatives acting on the four sources attached to the bare vertex. This results in a contribution of the form

This diagram is connected and contributes to scattering, however, at this order in the perturbative expansion we obtain this trivial bare vertex. In order to obtain true vertex corrections, we would have to go to O(λ2 ) and higher. Note also that in this case there is only one permutation since all possible permutations can be deformed into a diagram that looks like the bare vertex, so the P above is a bit of overkill. Collecting all of the contributions, we finally obtain

We can rewrite this slightly as

In this form, we see that the coefficients 72 and 24 are simply the number of ways to connect four external lines to a bare vertex in such a way as to create the corresponding diagram’s topology. With this, we come full circle to the Feynman rules for λϕ4 presented in volume 1, namely that we should insert a DF for every line, a factor of −iλ for every vertex, and then multiply by the corresponding symmetry factor of the diagram in question, Nsym = n! (4! ) n /(NextNint ).

2.3 Generating functional for connected diagrams As we saw in the previous discussion, functional derivatives of Z [J ] generate the full time-ordered n -point functions of the theory, however, these contain disconnected

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Relativistic Quantum Field Theory, Volume 2

contributions that do not play any role in, for example, physical scattering. Using some insight from statistics we can introduce connected correlation functions (cumulants) which automatically subtract off the ‘trivial’ disconnected correlations. For this purpose, we introduce a new functional W [J ], which is related to Z [J ] by

eiW [J ] = Z [J ]

(2.38)

W [J ] = −i log(Z [J ]) .

(2.39)

or

From W [J ] we can define the irreducible or connectedn-point functions, Gc , via

Gc(x1, ⋯ , xn) = 〈0∣T ϕ(x1)⋯ϕ(xn)∣0〉c =

δ nW [J ] i n−1 δJ (x1)⋯δJ (xn) 1

,

(2.40)

J =0

where subscript ‘c’ above indicates ‘connected’ Green’s functions. These functional derivatives of W [J ] generate only connected Feynman diagrams. For our scalar field theory, this first becomes relevant at the level of the four-point function, but let us begin by considering the two-point function to see how this works in a simple case. Using equation (2.39) one finds

Gc(x1, x2 ) =

δ 2W δJ (x1)δJ (x2 )

J =0

⎡ 1 δZ ⎤ 1 δZ δ 2Z =⎢ 2 − ⎥ . (2.41) Z δJ (x1)δJ (x2 ) ⎦J =0 ⎣ Z δJ (x1) δJ (x2 )

When J = 0, the first term vanishes due to the fact that δZ /δJ ∣J =0 = 0. Note, however, if we were working in a theory that had a non-vanishing vacuum expectation value for the field (non-vanishing one-point function), then the first term would serve to subtract off the product of the average field value squared, which is how one defines the cumulant at this order. Since δZ /δJ ∣J =0 vanishes in this case, one has

Gc(x1, x2 ) = −

δ 2Z = G (x1, x2 ) δJ (x1)δJ (x2 )

and, hence, Gc is equivalent to the propagator to all orders in λ . For the connected four-point function we need

2-9

(2.42)

Relativistic Quantum Field Theory, Volume 2

⎡ 1 δ 4W δ 2Z δ 2Z ∣J =0 = i ⎢ 2 ⎣ Z δJ (x1)δJ (x2 ) δJ (x3)δJ (x 4 ) δJ (x1)δJ (x2 )δJ (x3)δJ (x 4 ) +

1 δ 2Z δ 2Z Z 2 δJ (x1)δJ (x3) δJ (x2 )δJ (x 4 )

+

1 δ 2Z δ 2Z 2 Z δJ (x1)δJ (x 4 ) δJ (x2 )δJ (x3)

−

⎤ 1 δ 4Z ]J =0⎥ ⎦ Z δJ (x1)δJ (x2 )δJ (x3)δJ (x 4 )

(2.43)

= i [G (x1, x2 )G (x3, x 4 ) + G (x1, x2 )G (x3, x 4 ) + G (x1, x2 )G (x3, x 4 ) − G (x1, x2, x3, x 4 )].

Looking at the first term in the second equality above and using the leading order correction to G , we can represent it diagrammatically as

Continuing in this manner one finds that the first three terms in equation (2.43) generate all of the disconnected four-point diagrams. Combining all terms one obtains

We can express this in a kind of summary diagram for the dressed (or complete) fourpoint function

where a shaded circle on a line indicates the full propagator, vertex, etc. The dressed quantity represents the all orders in perturbation theory expression for the quantity. Expanding the dressed propagators to O(λ ), we would obtain equation (2.36). This pattern continues to higher n-point functions. For example, the six-point function has an expansion of the form

2-10

Relativistic Quantum Field Theory, Volume 2

In general, all contributions to G (n) can be expressed as a sum of Gc(n) and products of Gc(m) with m < n , where here m and n indicate the order of the Green’s function. As a result, the connected (irreducible) Green’s functions form the building blocks of the perturbative expansion.

2.4 The self-energy Equation (2.19) can be used to determine all n-point functions of our scalar field theory (connected + disconnected) to an arbitrary order in λ . Since the connected n-point functions are the true building blocks, it makes sense to focus on them. For example, through O(λ3), the dressed connected two-point function is

where we have not listed the numerical prefactors and factors of i . Note that, to truly be called the dressed propagator we should not truncate at a finite order in the coupling. It is just impossible to draw all of the diagrams here. As we shall see, the effect of the terms of O(g ) and higher is to change the physical mass away from the bare mass m that appears in the Lagrangian and are hence called self-energy corrections. We notice that some of the diagrams are just ‘multiples’ of previous diagrams and can be cut into two separate pieces that have appeared at lower orders by cutting a single line. These types of diagrams are called one-particle reducible. For example, the diagram can be cut into and by cutting the central line. However, some diagrams, such as cannot be turned into two copies of previous diagrams by cutting one line and, as a result, they are called one-particle irreducible diagrams. As we can see in equation (2.48) many of the diagrams are, in fact, one-particle reducible and since they are just multiples of previous orders, there is a way to iteratively generate them. In fact, it is possible to sum all diagrams by introducing a special quantity called the proper self energy, which is related to a sum of all one-particle irreducible diagrams above. To define it precisely we notice that in all diagrams above there is one incoming and one outgoing line. We refer to these as the legs of the diagram. We 2-11

Relativistic Quantum Field Theory, Volume 2

can ‘chop off the legs’ of each the diagrams by multiplying left and right by an inverse free propagator (G0 )−1. After removing the legs, we can visually indicate some placeholders for them using dotted lines, for example

The momentum-space proper self energy Σ(p ) is defined as the sum of all one-particle irreducible diagrams with the legs cut off, which is in momentum-space

The momentum-space dressed propagator (two-point function) can be written in terms of the bare propagator G0(p ) = i /(p2 − m2 ) and the proper self energy

Σ( p ) Σ( p ) Σ( p ) G 0(p ) + ⋯ G 0(p ) G 0(p ) + G 0(p ) i i i ⎛ ⎞ Σ Σ Σ = G 0⎜1 + G 0 + G 0 G 0 + ⋯⎟ ⎝ ⎠ i i i −1 ⎛ Σ ⎞ = G 0⎜1 − G 0⎟ ⎝ i ⎠

Gc(2)(p ) = G 0(p ) + G 0(p )

(2.51)

−1

= i ( iG0−1 − Σ) . One can verify through explicit calculation that the iterations on the first line generate all terms in the connected two-point function. We will construct a more rigourous proof and functional definition of Σ in the next section. Plugging in the explicit expression for the bare propagator, we obtain

Gc(2)(p ) =

i . p − m 2 − Σ( p ) 2

(2.52)

As we see the proper self-energy Σ represents the change in the mass from its bare 2 by the position of value to its physical value. Defining the physical pole mass m pole the pole in the dressed propagator

Gc(2)(p ) =

i , 2 p − m pole 2

(2.53)

2 with m pole ≡ m2 + Σ(p ). This seems simple enough, but as we learned in volume 1, the self-energy is divergent and one must come up with a way to regulate and renormalize this divergence in practice. We will return to this later. Finally, we remind the reader that Gc = G so the subscript above is superfluous, however, we will apply a similar logic to higher n-point functions and in that case the distinction will be important.

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Relativistic Quantum Field Theory, Volume 2

Based on the discussion above, we have learned that

i (Gc(2))−1 = iG0−1 − Σ .

(2.54)

We therefore see that the connected Green’s function can be written entirely in terms of the bare propagator and the proper self-energy, which is a sum of one-particle irreducible diagrams. The quantity on the right is our first example of a vertex function. Defining Γ (2)(p ) = iG0−1 − Σ = p2 − m2 − Σ(p ) we see that

Gc(2)(p )Γ (2)(p ) = i .

(2.55)

We will now discuss how to generalize this to higher n-point functions by constructing a new generating functional Γ which is called the effective action. This will allow us to more formally define the one-particle irreducible building blocks that one can use to construct all n-point functions.

2.5 The effective action and vertex functions In this section, for generality, we will consider a general real scalar theory with a potential U (ϕ ). The effective action Γ[ϕ ] is a functional of the field and is defined through a Legendre transformation of the generating function for connected Green’s functions W [J ]. Its introduction is similar in spirit to the introduction of the internal energy and its relation to the free energy in statistical mechanics, with W [J ] playing the role of the free energy. The definition of Γ[ϕ ] is

W [J ] = Γ[Φ] +

∫x

J (x )Φ(x ),

(2.56)

which is similar again, in the statistical mechanics analogy, to switching thermodynamic variables using F (T , V ) = U (S , V ) − TS . Using equation (2.56) one has

δW [J ] = Φ(x ), δJ ( x )

(2.57)

δ Γ[Φ] = − J (x ) . δ Φ( x )

(2.58)

and

From the first relation we see that since

Φ( x ) =

δW [J ] = Gc(1)(x ; J ) = 〈ϕ(x )〉J , δJ ( x )

(2.59)

Φ can be identified with the local vacuum-expectation value of the field in the presence of a source J . We have indicated that J ≠ 0 in the third equality above by adding an auxiliary argument to Gc . Additionally, we define Φ0 ≡ limJ →0Φ. For our scalar λϕ4 theory with m2>0 one has Φ0 = 0, however, there are cases where it might not be zero, so we will keep it general for now. 2-13

Relativistic Quantum Field Theory, Volume 2

We begin by defining

Γ (n)(x1, … , xn) ≡

δ nΓ[Φ] , δ Φ(x1)⋯δ Φ(xn)

(2.60)

which are the one-particle-irreducible vertex functions. To proceed, we notice that

iGc(2)(x1, x2 ; J ) =

δ 2W [J ] δ Φ(x1) . = δJ (x1)δJ (x2 ) δJ (x2 )

(2.61)

Similarly, from equation (2.60), we have

Γ (2)(x1, x2 ) =

δ 2 Γ[Φ] δJ (x1) . =− δ Φ(x1)δ Φ(x2 ) δ Φ(x2 )

(2.62)

As a result, we see that

Gc(2)(x1, x2 ; J ) = i [Γ (2)(x1, x2 )]−1 .

(2.63)

Note that to obtain the physical propagator, we need to take J = 0 and therefore Φ = Φ0, that is

Gc(2)(x1, x2 ) = lim i [Γ (2)(x1, x2 )]−1. Φ→Φ 0

(2.64)

From here on, we will suppress the auxiliary J argument in Gc(2)(x1, x2; J ), with the understanding it will eventually be taken zero. Prior to that, it will serve as a way to derive a functional differential equation that can be used to bootstrap from the twopoint function to any n-point function. Equation (2.63) says that Gc(2) and Γ (2) quantities are, up to a factor of i , functional inverses of one another. To see this, explicitly consider

∫z

Gc(2)(x1, z )Γ (2)(z , x2 ) = i

∫z

δ Φ(x1) δJ (z ) δ Φ(x1) = iδ 4(x1 − x2 ) . (2.65) =i δJ (z ) δ Φ(x2 ) δ Φ(x2 )

If we Fourier transform this relation on the left and right, we obtain equation (2.55). Next, we take a functional derivative of equation (2.65) with respect to J (y ) and use the fact that the right-hand side does not depend on J to obtain

∫z

⎡ δ ⎤ Gc(2)(x1, z )⎥Γ (2)(z , x2 ) = − ⎢ ⎣ δJ ( y ) ⎦

⎡ δ ⎤ Γ (2)(z , x2 )⎥ Gc(2)(x1, z )⎢ ⎣ δJ ( y ) ⎦ ⎡ ⎤ δ Φ(y′) δ = − Γ (2)(z , x2 )⎥ (2.66) Gc(2)(x1, z )⎢ ⎣ δJ (y ) δ Φ(y′) ⎦ y ′,z

∫z ∫

= −i

∫y ′,z Gc(2)(x1, z )Gc(2)(y′, y )Γ (3)(z, x2 , y′).

Using the fact that

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Relativistic Quantum Field Theory, Volume 2

δ δ 3W [J ] Gc(2)(x1, z ) = −i = iGc(3)(x1, z , y ) , δJ ( y ) δJ (x1)δJ (z )δJ (y )

(2.67)

we obtain

∫z

Gc(3)(x1, z , y )Γ (2)(z , x2 ) = −

∫y′,z Gc(2)(x1, z)Gc(2)(y′, y )Γ(3)(z, x2, y′) .

(2.68)

Multiplying left and right by Gc(2)(x2, x3) on the left and right and integrating over x2 , we obtain

∫z

Gc(3)(x1, z , y )

∫x

Γ (2)(z , x2 )Gc(2)(x2 , x3)

2

=−

∫x , y′,z Gc(2)(x1, z)Gc(2)(y′, y )Gc(2)(x2, x3)Γ(3)(z, x2, y′) .

(2.69)

2

Applying equation (2.65), this becomes

Gc(3)(x1, x3, y ) = i

∫x , y′,z Gc(2)(x1, z)Gc(2)(y′, y )Gc(2)(x2, x3)Γ(3)(z, x2, y′) .

(2.70)

2

Finally, relabeling x2 → x2′, x3 → x2 , y → x3, y′ → x′3, z → x′1, and using the symmetry Gc(x , y ) = Gc(y, x ), we finally obtain

Gc(3)(x1, x2 , x3) = i

∫x′ , x′ , x′ 1

2

Gc(2)(x3,

Gc(2)(x1, x′1)Gc(2)(x2 , x′2 )

3

(2.71)

(3)

x′3)Γ (x′1, x′2 , x′3).

This formula tells us that the connected three-point function can be obtained from the three-point vertex function by attaching dressed propagators to it. We can write this formula symbolically as

Gc(3) = iG 3Γ (3) ,

(2.72)

where G here simply indicates the two-point function. This relation can be shown diagrammatically as

where an open circle will be used to indicate a connected n-point function, that is

and a diagonal hatched circle will be used to indicate an n-point vertex, that is

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Relativistic Quantum Field Theory, Volume 2

To go to higher-order n-point functions we note that from equations (2.67) and (2.72) we have (symbolically)

δ (2) Gc = Gc(3) = iG 3Γ (3) . iδJ

(2.76)

This can be portrayed graphically as

This type of construction can be extended to higher n-point functions/vertices. Focusing on the connected n-point function first, we note that, in general, one has

δ (n−1) = Gc(n), Gc iδJ

(2.78)

which translates into a graphical rule of the form

Next, we consider functional derivatives of an arbitrary vertex function with respect to J . Symbolically, one has

1 δΦ δ (n−1) Γ = Γ (n) = Γ (n)G , iδJ i δJ

(2.80)

which has a graphical representation

With these rules, we can take an arbitrary number of functional derivatives of equation (2.72) or, equivalently, equation (2.73). Let us consider what happens graphically at the level of the four-point function

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Relativistic Quantum Field Theory, Volume 2

where we have applied the functional derivative to each of the dressed propagators on the legs and finally the vertex function itself. As we can see, the vertices and the dressed propagator can be used as the fundamental building blocks to construct higher n-point functions. Note that, if the theory does not contain a three-vertex at the level of the Lagrangian, then all diagrams in parentheses above are zero and the last diagram in equation (2.82) defines the dressed four-point function entirely in terms of the four-vertex with dressed propagators on the legs. As one more example, let us see what happens at the level of the five-point function. Applying δ /(iδJ ) to the left- and right-hand sides of equation (2.82) we obtain

Continuing in this manner, we can generate all connected n -point functions and write them in terms of the dressed propagator and vertex functions. These types of diagrams are called tree diagrams.

2.6 Generating function for one-particle irreducible graphs We would now like to demonstrate that the tree decomposition reduces all diagrams to their one-particle irreducible contents. To start, we separate the effective action into free and interacting parts

Γ[Φ] ≡ Γ0[Φ] + Γint[Φ] =

1 m2 2 ∂μΦ∂ μΦ − Φ + Γint[Φ] 2 2

i = ΦG0−1Φ + Γint[Φ]. 2

(2.84)

We then perform the same decomposition for the generating functional for connected diagrams W [J ]

W [J ] ≡ W0[J ] + Wint[J ] =

2-17

i JG 0J + Wint[J ]. 2

(2.85)

Relativistic Quantum Field Theory, Volume 2

In both relations the integrations necessary are implicit. Using equation (2.56) we have

Γint[Φ] = Wint[J ] +

i i JG 0J − ΦG0−1Φ − J Φ . 2 2

(2.86)

Using equation (2.58) we have −1 J = −Γ (1) = −Γ (1) int − iG0 Φ ,

(2.87)

which, upon insertion in equation (2.86), gives

Γint[Φ] = Wint[J ] +

i Γint + i ΦG0−1 G 0 Γint + iG0−1Φ 2

(

) (

i ΦG0−1Φ + Φ Γint + iG0−1Φ 2 i (1) = Wint[J ] + Γ (1) intG 0 Γ int , 2 −

(

)

) (2.88)

or, equivalently,

Wint[J ] = Γint[Φ] −

i (1) Γ intG 0Γ (1) int , 2

(2.89)

which shows that the sum of all connected diagrams in Wint[J ] is composed of all oneparticle irreducible connected diagrams in Γint[Φ] plus all one-particle reducible diagrams coming from the second term (since we can cut at G0 , the second contribution is obviously reducible). 2.6.1 Scalar Schwinger–Dyson equation Finally, I would like to derive equation (2.51), which says that the dressed propagator can be expressed as a geometric series in the proper self-energy using the effective action formalism. Once again using the decomposition Γ[Φ] = Γ0[Φ] + i Γint[Φ] = ΦG0−1Φ + Γint[Φ] and taking two functional derivatives of this with 2 respect to Φ, we obtain

Γ (2) = iG0−1 + Γ (2) int .

(2.90)

Next, we notice that equation (2.63) can be written as

(

G = i [Γ (2)]−1 = G 0 1 − iG 0Γ (2) int

−1

)

.

(2.91)

Expanding and collecting terms, this can also be written as an integral equation for G

G = G 0 + iG 0Γ (2) intG.

2-18

(2.92)

Relativistic Quantum Field Theory, Volume 2

From equation (2.84) and our previous definition Γ (2) = iG0−1 − Σ we see that

Σ ≡ −Γ (2) int .

(2.93)

G = G 0 − iG 0ΣG,

(2.94)

This gives

which upon iteration reproduces the geometric series we handwavingly introduced earlier in equation (2.51). Thus, up to a sign, the self-energy is given by the interacting part of the second derivative of the effective action and iterating it generates the full dressed propagator. Exercise 2.4 Show that, keeping track of the spacetime labels for the incoming and outgoing lines, the diagram topologies shown in equation (2.82) are generated.

2.7 Interacting complex scalar fields The extension of the results presented thus far to complex scalar fields is relatively straightforward. As we saw in volume 1, in this case the Lagrangian can be written as

L = ∂μψ †∂ μψ − m 2ψ †ψ − U (∣ψ ∣2 ) ,

(2.95)

where I have indicated the complex conjugate by a dagger since, for a scalar, the Hermitian conjugator and complex conjugate are the same. Since the complex scalar field has two degrees of freedom, one can use the real and imaginary parts as the degrees of freedom; however, as we discussed in the first half of the course, it is more convenient to treat ψ and ψ † as the independent variables. Associated with these two, we introduce a complex source function J and the interacting generating functional becomes

∫ Dψ †Dψ ei(S[ψ , ∂μψ , ψ , ∂μψ ]+∫x [J ψ +Jψ ]) Z [J , J ] ≡ , † † ∫ Dψ †Dψ eiS[ψ , ∂μψ , ψ , ∂μψ ] †

†

†

†

†

where an

∫x

(2.96)

iε∣ψ ∣2 is implicit in the exponentials in both the numerator and

denominator. If U = 0, then we have a free complex scalar field and, using the same techniques we used for a real scalar field, we obtain

Z0[J , J † ] = exp( −

∫x ∫y

J †(x )DF (x − y )J (y )),

and

2-19

(2.97)

Relativistic Quantum Field Theory, Volume 2

⎡ ⎛1 δ 1 δ ⎞⎤ , exp⎢i ∫ Lint ⎜ ⎟⎥Z0[J , J † ] z i δJ (z ) i δJ †(z ) ⎠⎦ ⎝ ⎣ . Z [J , J † ] = ⎧ ⎫ ⎡ ⎛1 δ ⎪ ⎪ 1 δ ⎞⎤ ⎨ , exp⎢i ∫ Lint ⎜ ∣J =0 ⎟⎥Z0[J , J † ]⎬ ⎪ ⎪ ⎝ i δJ (z ) i δJ †(z ) ⎠⎦ ⎣ z ⎩ ⎭

(2.98)

The time-ordered Green’s functions can be obtained from this via

G (x1, … , xn; y1, … , ym ) = 〈0∣Tψ (x1)⋯ψ (xn)ψ †(y1)⋯ψ †(ym )∣0〉 =

1 i

n +m

δ nZ0[J ] δJ †(x1)⋯δJ †(xn)J (y1)⋯δJ (xm )

.

(2.99)

J =0

Exercise 2.5 Starting from equation (2.95) and assuming an interaction potential of the form U = g(ψ †ψ )2 , write down the leading order perturbative correction to the non-interacting generating functional.

References [1] [2] [3] [4]

Kleinert H 2016 Particles and Quantum Fields (Singapore: World Scientific) Srednicki M 2007 Quantum Field Theory (Cambridge: Cambridge University Press) Ryder L H 2013 Quantum Field Theory (Cambridge: Cambridge University Press) Rivers R J 1988 Path Integral Methods in Quantum Field TheoryCambridge Monographs on Mathematical Physics (Cambridge: Cambridge University Press)

2-20

IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 3 Path integrals for fermionic fields

If we want to apply the path integral formalism to fermionic fields, we are immediately faced with a conceptual problem because thus far the path integral itself has been formulated in terms of classical numbers (c -numbers) and, as we know, fermionic fields are anti-commuting objects. In order to formulate the path integral for fermionic fields, we are going to have to use an algebra that corresponds to anti-commuting c -numbers. In mathematics, such objects form a so-called Grassmann algebra G. Before embarking on formalities, I mention that, geometrically, the algebra of Grassmann variables is that of the cross-product (exterior product), that is a × b = −b × a and a × a = 0, so, although it seems strange at first, it is really nothing dramatically new. A Grassmann algebra can be viewed as an algebra where the product is the exterior product.

3.1 Finite-dimensional Grassmann algebra If {θ1, ⋯ , θn} are generators of an n-dimensional Grassmann algebra G , then their defining relation is

{θi , θj} = θiθj + θjθi = 0.

(3.1)

An immediate consequence of this is that

θi2 = 0.

(3.2)

Due to this, if we Taylor-expand a function of Grassmann variables f (θi ), then it only contains a finite number of terms n

f (θ1, … , θn) = c0 +

n

∑ ∑

ci1⋯ikθ1⋯θk ,

(3.3)

k = 1i1⋯ ik = 1

with the rule θi2 = 0 enforced. For example, if G is one-, two-, or three-dimensional, we have doi:10.1088/2053-2571/ab3108ch3

3-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

f (θ ) = c0 + c1θ , f (θ1, θ2 ) = c0 + c1θ1 + c2θ2 + c12θ1θ2, f (θ1, θ2, θ3) = c0 + c1θ1 + c2θ2 + c3θ2 + c12θ1θ2 + c13θ1θ3 + c23θ2θ3,

(3.4)

respectively. 3.1.1 Derivatives of Grassmann variables Differentiation with respect to Grassmann variables is a linear operation defined by

∂ θj = δij . ∂θi

(3.5)

⎧ ∂ ⎫ ⎨ , θj ⎬ = δij , ⎩ ∂θi ⎭

(3.6)

This implies that

which, for a one-dimensional algebra, reduces to

{ } ∂ ,θ ∂θ

= 1,

(3.7)

which can be checked by applying the left-hand side to the one-dimensional function f (θ ) introduced in equation (3.4). The anti-commutation relations (3.1) also imply that

⎧ ∂ ∂ ⎫ ⎨ , ⎬ = 0. ⎩ ∂θi ∂θj ⎭

(3.8)

As a consequence, the operator ∂ θi is nilpotent, that is ∂ 2θi = 0, which reflects the fact that the function f (θ1, …, θn ) is, at most, order one in each variable. Due to the anti-commuting nature of Grassmann variables, when taking the derivative of a product of two Grassmann variables, one can introduce two different types of derivatives, left derivatives and right derivatives, where here ‘left’ implies that it always acts on the first element and ‘right’ implies that it always acts on the second element. The left derivative is defined by

∂ ∂ ∂L (θiθj ) = (θiθj ) − (θjθi ) = δikθj − δjkθi , ∂θk ∂θk ∂θk

(3.9)

and the right derivative is defined by

∂R (θiθj ) = δjkθi − δikθj . ∂θk

(3.10)

These represent the Grassmann version of the product rule. Finally, I note that the chain rule exists for Grassmann variables

3-2

Relativistic Quantum Field Theory, Volume 2

∂ L,Rg ∂f ∂ L,R . f (g ) = ∂θk ∂g ∂θk

(3.11)

3.1.2 Integrals over Grassmann variables For Grassmann variables, integration and differentiation are identical

∫ dθif (θ1, …, θn) ≡ ∂∂θi f (θ1, …, θn).

(3.12)

For a one-dimensional Grassmann algebra, this implies that

∫ dθi(c0 + c1θ ) = ∂∂θ (c0 + c1θ ) = c1

(3.13)

which gives us two rules

∫ dθ = 0, ∫

(3.14)

dθ θ = 1,

which straightforwardly generalizes to the n-dimensional case

∫ dθi = 0, ∫ dθi θi = 1.

(3.15)

3.1.3 Gaussian integrals of Grassmann variables As a result of the above discussion, if ω and ω are independent Grassmann quantities, then

∫ dω = ∫ dω = 0, ∫ dω ω = ∫ dω ω = 1.

(3.16)

If we construct a Gaussian function from these two independent Grassmann quantities, one has

e−ωω = 1 − ωω,

(3.17)

owing to the fact that ω 2 = ω 2 = 0. As a result,

∫ dω dωe−ωω = ∫ dω dω1 − ∫ dω dωωω = −1. 3-3

(3.18)

Relativistic Quantum Field Theory, Volume 2

Next, let us generalize this to two independent (commuting) two-dimensional Grassmann variables ω ω = ω1 , (3.19) 2

( )

and

ω=

⎛ ω1 ⎞ ⎜ ⎟. ⎝ ω2 ⎠

(3.20)

Their scalar product is

ω T ω = ω1ω1 + ω2ω2

(3.21)

and the scalar product squared is

(ω Tω)2 = (ω1ω1 + ω2ω2 )(ω1ω1 + ω2ω2 ) = ω1ω1ω2ω2 + ω2ω2ω1ω1 = 2ω1ω1ω2ω2 .

(3.22)

Higher powers of the scalar product are zero. As a result, one has

e−ω

Tω

= 1 − ω1ω1 − ω2ω2 + ω1ω1ω2ω2 .

(3.23)

Integrating over ω and ω , we obtain

∫ dω dωe−ω ω = ∫ dω1dω1dω2dω2(1 − ω1ω1 − ω2ω2 + ω1ω1ω2ω2) = 1 T

(3.24)

where we have used the fact that the first three terms integrate to zero and the last term integrates to one. This is similar to the result we obtained for a one-dimensional Gaussian Grassmann integral. This continues to be true as we consider higher dimensional vectors of Grassmann variables: only the final term involving the product of all components (barred and unbarred) survives the integration over Grassmann variables. Next, let us consider making a change of variables to ξ and ξ defined by

ω = Mξ, ω = Nξ ,

(3.25)

where M and N are 2 × 2 matrices of c -numbers. The product of ω1ω2 , for example, becomes

ω1ω2 = (M11ξ1 + M12ξ2 )(M21ξ1 + M22ξ2 ) = (M11M22 − M12M21)ξ1ξ2 = (det M )ξ1ξ2.

(3.26)

This tells us how to compute the Jacobian for the transformation, since we require that integral of a non-vanishing product is the same before and after the integration, that is

∫ dω1dω2ω1ω2 = ∫ dξ1dξ2ξ1ξ2, 3-4

(3.27)

Relativistic Quantum Field Theory, Volume 2

which, according to equation (3.26), requires

dω1dω2 = (det M )−1 dξ1dξ2.

(3.28)

Note that this is the inverse of the normal rule for a c -number change of variables. Considering, once more, the Gaussian integral (3.24) we have T

∫ dω dωe−ω ω = (det MN )−1 ∫ dξ dξe−ξ N Mξ = 1. T

T

(3.29)

Using the det MN = det NM = det N T M and defining A = N TM , we find T

∫ dξ dξe−ξ Aξ = det A.

(3.30)

Once again, we note, that this formula holds also for all dimensionalities of the Grassmann space. 3.1.4 Infinite-dimensional Grassmann algebra To define the fermionic path integral, we take the limit n → ∞ in order to construct an infinite dimensional Grassmann algebra that satisfies

{θ (x ), θ (y )} = 0, ∂ L,R θ (x ) = δ(x − y ), ∂θ (y )

∫ dC (x) = 0,

(3.31)

∫ dC (x)C (x) = 1. Additionally, we allow the Grassmann variables to be complex, that is θ (x ) = R[θ (x )] + i ℑ[θ (x )] Exercise 3.1 Show that equation (3.6) is correct.

3.2 Path integral for a free Dirac field Next we consider the Lagrangian density for a free Dirac field, which has the form

L = iψ γ μ∂μψ − mψ ψ ,

(3.32)

where ψ and ψ are Grassmann fields. Following the discussion of a complex scalar field, we treat ψ and ψ as independent and introduce independent complex Grassmann-variable sources η and η for ψ and ψ , respectively. With this, we can define the generating functional for free Dirac fields

3-5

Relativistic Quantum Field Theory, Volume 2

Z0[η , η ] ⎛

⎞

∫ Dψ Dψ exp ⎝i∫x [ψ (x)(i ∂ − m)ψ (x) + η (x)ψ (x) + ψ (x)η(x)]⎠ ⎜

=

⎟

⎛

⎞

.

(3.33)

∫ Dψ Dψ exp ⎝i ∫x ψ (x)(iγ μ∂μ − m)ψ (x)⎠ ⎜

⎟

Introducing the inverse free Dirac propagator SF−1 = i ∂ − m, we can write this more compactly as

Z0[η , η ] =

∫ Dψ Dψ exp (i ∫x [ψ SF−1ψ + ηψ + ψ η ]) ∫ Dψ Dψ exp (i ∫x ψ SF−1ψ )

(3.34)

.

We recall from volume 1 that the Dirac propagator SF (x ) is defined via

iSF (x ) = (i ∂ − m)DF (x ),

(3.35)

SF−1(x )SF (x ) = −i (i ∂ − m)(i ∂ + m)DF (x ) = i (□ 2 + m 2 )DF (x ) = δ 4(x ).

(3.36)

and satifies

In momentum space, it is

iSF (p ) =

i . p − m + iε

(3.37)

To evaluate the free generating functional Z0[η , η ], we begin by evaluating the numerator of equation (3.34) since the denominator is the numerator with the sources set to zero. This can be done as follows ⎛

⎞

∫ Dψ Dψ exp ⎝i∫x ⎡⎣ ψ SF−1ψ + ηψ + ψη⎤⎦⎠ ⎜

=

⎟

⎞

⎛

∫ Dψ Dψ exp ⎜⎝i ∫x,y ⎡⎣ (ψ + η SF )SF−1(SF η + ψ ) − η SF η⎤⎦⎟⎠

⎛ = exp ⎜ − i ⎝

⎞ ⎛ η (x )SF (x − y )η(y )⎟ Dψ ′Dψ ′ exp⎜ − ⎝ ⎠ x,y ⎛ ⎞ = det − iSF−1 exp ⎜ − i η (x )SF (x − y )η(y )⎟ , ⎝ ⎠ x,y

∫

(

∫

)

⎞ ψ ′ − iSF−1 ψ ′⎟ ⎠ x

∫ (

(3.38)

)

∫

where, in going from the first to the second line, we have made a change of variables to ψ ′ = ψ + η SF and ψ ′ = SF η + ψ and, in going from the second to the third line, we have used the infinite dimensional version of equation (3.30).1,2 From this, we see 1

Compared to equation (3.30) we do not have to indicate the transpose in the Gaussian form since this is implicit in the definition of ψ = ψ †γ 0 . 2 For lack of space, I have been a bit sloppy/abstract with the argument of the exponential on the second line of equation (3.38) in that there should be some delta functions and correct arguments of x and y placed here and there; however, the third line is indicated correctly.

3-6

Relativistic Quantum Field Theory, Volume 2

that the denominator of equation (3.34) is simply det ( −iSF−1). Putting the pieces together, we obtain

⎛ Z0[η , η ] = exp ⎜ −i ⎝

⎞

∫x,y η (x)SF (x − y )η(y )⎟⎠.

(3.39)

Note that since η and η are Grassmann variables one has

∫x,y η (x)SF (x − y )η(y ).

Z0[η , η ] = 1 − i

(3.40)

Based on this, we can define the two-point function for the Dirac field (time-ordered vacuum expectation value)

G (x1, x2 ) ≡ 〈0∣Tψ (x1)ψ (x2 )∣0〉 = − =−

δ 2Z0[η , η ] δη (x1)δη(x2 )

η=η = 0

δ δ ⎡ ⎢1 − i δη (x1) δη(x2 ) ⎣

⎡ δ = i⎢ ⎣ δη (x1)

⎤

∫x,y η (x)SF (x − y )η(y )⎥⎦

η=η = 0

(3.41)

⎤

∫x η (x)SF (x − x2)⎥⎦

η=η = 0

= iSF (x1 − x2 ).

3.3 Path integral for an interacting Dirac field In analogy to the real and complex scalar field examples covered previously, we can define the generating functional for interacting Dirac fields via

⎡ 1 δ 1 δ ⎤ exp ⎢i ∫ Lint i δη , i δη ⎥ Z0[η , η ] x ⎦ ⎣ Z [η , η ] = ⎫ ⎧ ⎤ ⎡ ⎨exp ⎢i ∫ Lint 1 δ , 1 δ ⎥ Z0[η , η ]⎬ i δη i δη ⎦ ⎣ x ⎭ ⎩

( )( ) ( )( )

.

(3.42)

η=η = 0

The rest of the machinery we developed for scalar fields, for example connected Green’s functions, effective action, vertex functions, etc can be extended straightforwardly to the case of fermions. We will return to this in our discussion of the path integral formulation of QED itself.

3.4 Fermion loops When we covered the Feynman rules for QED in volume 1, we introduced a special rule that every closed fermion loop in a diagram, for example

, results in a

multiplicative factor of −1 in the transition matrix. This is related to Fermi statistics, but I would like to show where this comes from using the path integral formulation. 3-7

Relativistic Quantum Field Theory, Volume 2

As we will see the factor of −1 comes from the relative sign between the two terms in the product rule for functional derivative with respect to Grassmann variables. For example, the left functional derivative gives

δL [η(x2 )η(x3)] = δ 4(x1 − x2 )η(x3) − δ 4(x1 − x3)η(x2 ). δη(x1)

(3.43)

If we expand equation (3.42) to O(e2), there is a term of the form

−

1 2

∫x,y,x′,y′ η (x)SF (x − y )η(y )η (x′)SF (x′ − y′)η(y′),

(3.44)

which maps to a fermionic loop and will give a non-vanishing contribution to

δ2 δ2 Z [η , η ], δηi (z )δηj (z ) δηk (z′)δηℓ (z′)

(3.45)

where i , j , k , and ℓ are spinor indices. Applying this to equation (3.44) using the left functional derivative, we obtain

−

1 δ2 δ2 2 δηi (z )δηj (z ) δηk (z′)δηℓ (z′) =− − =−

∫x,y,x′,y′ η (x)SF (x − y )η(y )η (x′)SF (x′ − y′)η(y′)

1 δ2 δ ⎡ ⎢ 2 δηi (z )δηj (z ) δηk (z′) ⎣

∫x,x′,y′ η (x)SF·ℓ(x − z′)η (x′)SF (x′ − y′)η(y′) ⎤

∫x,y,x′ η (x)SF (x − y )η(y )η (x′)SF·ℓ(x′ − z′)⎥⎦ ⎡ 1 δ2 ⎢ 2 δηi (z )δηj (z ) ⎣

∫x′,y′ SFkℓ(z′ − z′)η (x′)SF (x′ − y′)η(y′)

−

∫x,y′ η (x)SF·ℓ(x − z′)SFk·(z′ − y′)η(y′)

−

∫y,x′ SFk·(z′ − y )η(y )η (x′)SF·ℓ(x′ − z′)

+

∫x,y η (x)SF (x − y )η(y )SFkℓ(z′ − z′)⎥⎦

(3.46)

⎤

1 = − ⎡⎣ SFkℓ(z′ − z′)SFij (z − z ) − SFiℓ(z − z′)SFkj (z′ − z ) 2 − SFkj (z′ − z )SFiℓ(z − z′) + SFij (z − z )SFkℓ(z′ − z′)⎤⎦ = SFiℓ(z − z′)SFkj (z′ − z ) where, in going from the next-to-last line to the last line, we have used the fact that there can be no fermionic tadpoles since SF (0) = SF (x − x ) = −SF (x − x ) = 0. The last line maps to a position-space diagram with topology

3-8

. If we had performed

Relativistic Quantum Field Theory, Volume 2

the same evaluation with bosons, we would have (1) obtained an additional contribution with topology and (2) the overall sign on the last line above would have been − instead of +. The relative sign difference is the origin of the ‘−1 for every fermionic loop’ rule for Feynman diagrams and can be seen to come from the anti-commuting nature of the fermionic field which results from the ‘unusual’ product rule for derivatives with respect to Grassmann variables. Exercise 3.2 Show that one obtains the same result discussed above if one uses the right functional derivative instead of the left one.

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 4 Path integrals for abelian gauge fields

Having covered fermions, the next logical step is to discuss abelian gauge fields so that we can write down the generating functional for quantum electrodynamics. I will begin by reviewing what we know about abelian gauge fields from volume 1. I will postpone the formal definition of abelian until a future lecture, where we will see that it is related to a statement about commutation relations for the generators of the gauge symmetry of the model. For now, it is enough to know that the electromagnetic field is abelian.

4.1 Free abelian gauge theory The Lagrangian density for the electromagnetic field can be expressed in terms of the field strength tensor F μν = ∂ μAν − ∂ νAμ where Aμ = (ϕ, A), with the result being

1 L = − FμνF μν. 4

(4.1)

This Lagrangian density (4.1) is invariant under a local gauge transformation

Aμ (x ) → Aμ (x ) + ∂μλ(x ),

(4.2)

where λ(x ) is an arbitrary local scalar function of spacetime that dies off sufficiently rapidly as ∣x∣ → ∞. As we discussed in volume 1, there are four degrees of freedom in the vector potential Aμ, however, in the vacuum, only the two of them that are physical can be mapped to the two polarization states of the photon. The extra degrees of freedom are fixed by (1) enforcing Gauss’ law, which allows one to write A0 in terms of the other three components and by (2) enforcing a gauge-fixing condition such as the Lorenz-gauge condition ∂μAμ = 0. I also remind you that even after enforcing the Lorenz-gauge condition, there remains a residual gauge freedom associated with three-dimensional gauge transformations. We will return to this issue when we discuss the Faddeev–Popov procedure. doi:10.1088/2053-2571/ab3108ch4

4-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

Applying Euler’s equations to the Lagrangian density (4.1) results in the equations of motion for the vector potential

∂μF μν = 0,

(4.3)

which can be expressed as (ημν □ − ∂μ∂ν)Aν = 0. Correspondingly, as we demonstrated in volume 1, after an integration by parts, equation (4.1) can be written as

L=

1 μ A ημν□ − ∂μ∂ν − iε Aν , 2

(

)

(4.4)

where we have inserted the iε for convergence. 1

Exercise 4.1 Show that adding a mass term for the photon of the form 2 m2AμAμ to equation (4.1) violates gauge invariance.

4.2 The photon propagator Based on our prior examples of scalar and spinor theories, we would naturally be led to construct a photon propagator D Fμλ which is the inverse of the quadratic operator, that is

(ημλ□ − ∂μ∂λ − iε )D Fλν(x ) = δμνδ 4(x ).

(4.5)

However, if we act on this equation with ∂ μ we find

(∂λ□ − □∂λ)D Fλν(x ) = 0 × ∂λD Fλν = ∂ νδ 4(x ).

(4.6)

This tells that D Fμν has no inverse since attempting to isolate DF results in an infinity. This problem is related to gauge fixing and we can construct an invertible photon propagator by adding a term to the Lagrangian that enforces a physical gauge constraint

1 L → − FμνF μν + Lgf , 4

(4.7)

where Lgf enforces a particular gauge condition. In what follows, we will use the manifestly Lorentz-invariant general Lorenz gauge, which has a gauge fixing term of the form

Lgf = −

1 (∂μAμ )2 , 2ξ

(4.8)

resulting in

1 1 (∂μAμ )2 . L = − FμνF μν − 4 2ξ

4-2

(4.9)

Relativistic Quantum Field Theory, Volume 2

The equation of motion for the Lagrange multiplier λ = 1/ξ gives the Lorenz gauge condition. In constructing the Lagrangian above, we have generalized the covariant Lorenz gauge. The end result is that different values of ξ will map to specific gauge choices in the family of Lorenz (covariant) gauges. Taking ξ = 1 defines the standard Lorenz gauge and is referred to as Feynman gauge. Taking ξ = 0 gives the Landau gauge and taking ξ = 3 gives the Yennie gauge. Integrating the first term by parts, we can write this as

1 1 μ L = Aμ ημν□ − ∂μ∂ν − iε Aν + A (∂μ∂ν)Aν 2 2ξ ⎤ ⎛ 1⎞ 1 ⎡ = Aμ ⎢ημν□ − ∂μ∂ν⎜1 − ⎟ − iε⎥Aν . ⎝ 2 ⎣ ξ⎠ ⎦

(

)

(4.10)

We now require the inverse operator that solves

⎡ ⎤ ⎛ 1⎞ ⎢ημλ□ − ∂μ∂λ⎜1 − ⎟ − iε⎥D Fλν(x ) = δμνδ 4(x ). ⎝ ξ⎠ ⎦ ⎣

(4.11)

Transforming to momentum space we obtain

⎡ ⎤ ⎛ 1⎞ −⎢ημλ p 2 − pμ pλ ⎜1 − ⎟ + iε⎥D Fλν(p ) = δμν. ⎝ ξ⎠ ⎦ ⎣

(4.12)

In the vacuum, D Fμν( p ) can only be a linear combination of a term proportional to the metric tensor and a quadratic form constructed from the four-momentum, that is

D Fμν( p ) = Aη μν + Bp μ p ν .

(4.13)

Plugging this into equation (4.12), one obtains A = −1/p2 and B = (1 − ξ )/p4 , which gives the following general form for the photon propagator

D Fμν(p ) =

1 ⎡ μν p μ pν ⎤ − η + − ξ (1 ) ⎢ ⎥. p 2 + iε ⎣ p 2 + iε ⎦

(4.14)

Note that if we attempt to remove the gauge fixing term from equation (4.9) by taking ξ → ±∞, then we obtain B → ∓∞ and the propagator is ill-defined, as we explained before. Also note that in Feynman gauge, ξ = 1, the second term in square brackets above is zero, leading to a particularly simple result.

D Fμν(p ) = −

η μν p 2 + iε

(Feynman gauge).

Exercise 4.2 Show that equation (4.10) is correct.

4-3

(4.15)

Relativistic Quantum Field Theory, Volume 2

4.3 Generating functional for abelian gauge fields in general Lorenz gauge Based on the preceding discussion, we can make an educated guess for the generating functional for quantum electromagnetism

⎤⎞ 1 (∂μAμ )2 + JμAμ + iεAμ Aμ ⎥⎟ ⎦⎠ 2ξ ⎝ . (4.16) ⎛ ⎤⎞ ⎡ 1 1 ∫ DAμ exp ⎜i ∫ d 4x⎢− FμνF μν − (∂μAμ )2 + iεAμ Aμ ⎥⎟ ⎦⎠ ⎣ 4 2ξ ⎝ ⎛

⎡ 1 ⎣ 4

∫ DAμ exp ⎜i ∫ d 4x⎢− FμνF μν − Z0[J μ ] =

Performing the quadratic path integral over Aμ, we obtain

⎛i Z0[J μ ] = exp ⎜ ⎝2

⎞

∫x ∫y Jμ(x)D Fμν(x − y )Jν(y )⎟⎠.

(4.17)

In both expressions above, I have put a subscript ‘0’ to emphasize that, so far, our photons are non-interacting. 4.3.1 Faddeev–Popov gauge fixing preview In writing the above expressions, we have ignored something very important, namely that when we perform the path integral over Aμ, we have to avoid overcounting gauge-equivalent copies of the fields. Naively, the path integral over gauge fields contains another hidden infinity related to the integration over infinitely-many gauge-transformed versions of the same gauge field. As it turns out, equation (4.17) is correct in the general Lorenz gauge but not in a general gauge. In a general gauge, for QED it becomes necessary to introduce Grassmann-valued auxiliary (ghost) fields. And, as we will see below, for a non-abelian gauge field this procedure is necessary already in general Lorenz gauge. For now, let us postpone this discussion a bit and focus on our lucky case a bit more, namely abelian gauge fields in a general Lorenz gauge and return to generalities later.

4.4 Generating functional for QED in general Lorenz gauge We are now in a position to write down the generating functional for QED. As we learned in volume 1, we can write the QED Lagrangian by taking the source for the gauge field to be charged leptons (electrons, anti-electrons, muons, etc). In the end, we were able to write the QED Lagrangian density as

LQED = ψ (i D − m)ψ −

1 FμνF μν, 4

(4.18)

where D is the gauge covariant derivative

Dμ ≡ ∂μ − ieAμ .

4-4

(4.19)

Relativistic Quantum Field Theory, Volume 2

The QED Lagrangian density is invariant under local gauge transformations in which the vector potential and Dirac field transform as

Aμ → Aμ +

1 ∂μλ(x ), e

(4.20)

ψ → eiλ(x )ψ .

(4.21)

The generating functional can be written compactly in terms of two Grassmannvalued sources η and η and and a vector source J μ

Z [η , η , J μ ] ⎤⎞ 1 (∂μAμ )2 + ηψ + ψ η + JμAμ ⎥⎟ ⎦⎠ 2ξ ⎝ ⎣ . ⎛ ⎤⎞ ⎡ 1 ∫ D[ψ , ψ , Aμ ]exp ⎜i ⎢LQED − (∂μAμ )2 ⎥⎟ ⎦⎠ 2ξ ⎝ x ⎣ ⎛

⎡

∫ D[ψ , ψ , Aμ ]exp ⎜i ∫x ⎢LQED − =

(4.22)

∫

Expanding out LQED

LQED = ψ (i ∂ − m)ψ −

1 FμνF μν +  eψ A ψ, 4 Lint

(4.23)

we see that the first two terms look like non-interacting spinors and abelian gauge fields, respectively, and we have identified the interaction term. As a result, we can write

⎡ Z [η , η , J μ ] = N exp ⎢i ⎢⎣

⎞⎤

⎛

∫x Lint⎜⎝ 1i δηδ , 1i δηδ , 1i δδJμ ⎟⎠⎥⎥⎦Z0[η, η , J μ]

⎛ δ3 ⎞ exp⎜ −eγ μ ∫ ⎟Z0[η , η , J μ ] x δηδηδJ μ ⎠ ⎝ = , ⎧ ⎫ ⎛ δ3 ⎞ μ μ ⎨exp⎜ −eγ ∫ ⎟Z0[η , η , J ]⎬ x δηδηδJ μ ⎠ ⎝ ⎩ ⎭η=η =J μ=0

(4.24)

where Z0 is

⎛i Z0[η , η , J μ ] = exp ⎜ ⎝2

⎞

∫x,y Jμ(x)DFμν(x − y )Jν(y ) − i∫x,y η (x)SF(x − y )η(y )⎟⎠. (4.25)

4.5 General Lorenz-gauge QED generating functional to O(e 2 ) Following the examples from the scalar and spinor cases, we can now write down the bare n-point functions for QED. To proceed, we expand

4-5

Relativistic Quantum Field Theory, Volume 2

⎛ exp ⎜ −eγ μ ⎝

3

⎞

3

2

δ e δ + ⎟ = 1 − eγ μ∫ ∫x δηδηδ μ μ 2 J ⎠ x δηδηδJ

2 ⎛ δ3 ⎞ × + O(e 3). ⎜γ μ ⎟ μ x,y ⎝ δηδηδJ ⎠

(4.26)

∫

The one at leading order generates the free n-point functions. To obtain the O(e ) corrections to the free n-point function we need

The last term vanishes due to the absence of fermionic tadpoles and the first term can be interpreted as a diagram of the form

where above it is understood that there are appropriate sources attached to each of the lines. Continuing to work through O(e ), we find that the bare electron propagator and photon propagators are

The bare three-point function comes from equation (4.27)

4-6

Relativistic Quantum Field Theory, Volume 2

where in the last line we have introduced the bare QED vertex function Γ 0ν = γ ν . In diagrams, we insert a factor of ieγ ν for every bare QED vertex. Next, let us consider the terms generated at O(e 2 ). At this order we need

δ3 δ3 1 Z0 Z0 δη(z2 )δJ ν(z2 )δη (z2 ) δη(z1)δJ μ(z1)δη (z1) = − iSF (z2 − z1)D Fμν(z2 − z1)SF (z1 − z2 ) ⎡ ⎤⎡ ⎤ − SF (z2 − z1)D Fμν(z2 − z1)⎢ SF (z1 − y1)η(y1)⎥⎢ η (y2 )SF (y2 − z2 )⎥ ⎣ y1 ⎦⎣ y2 ⎦ ⎡ ⎤⎡ ⎤ − ⎢ η (y1)SF (y1 − z1)⎥⎢ SF (z2 − y2 )η(y2 )⎥D Fμν(z2 − z1)SF (z1 − z2 ) ⎣ y1 ⎦⎣ y2 ⎦

∫

∫

∫

∫

⎡ + SF (z2 − z1)⎢ ⎣

⎤2 γμD Fμν(z1 − y1)Jν(y1)⎥ SF (z1 − z2 ) ⎦ y1

∫

⎡ ⎤2 ⎡ ⎤ SF (z1 − y′1)η(y′1)⎥ − iSF (z2 − z1)⎢ γμD Fμν(z1 − y1)Jν(y1)⎥ ⎢ ⎣ y1 ⎦ ⎣ y ′1 ⎦ ⎡ ⎤ η (y′2 )SF (y′2 − z2 )⎥ ×⎢ ⎣ y ′2 ⎦ ⎤⎡ ⎤ ⎡ SF (z1 − y′2 )η(y′2 )⎥ η (y′1)SF (y′1 − z1)⎥⎢ − i⎢ ⎣ y ′1 ⎦⎣ y ′2 ⎦

∫

∫

∫

∫

∫

⎤2

⎡ ×⎢ ⎣

1

⎡ + i⎢ ⎣

⎤2 ⎡ η (y1)SF (y1 − z1)⎥ D Fμν(z2 − z1)⎢ ⎦ ⎣ y1

∫y γμD Fμν(z1 − y1)Jν(y1)⎥⎦ SF (z1 − z2) ∫

⎡ −⎢ ⎣

⎤2 SF (z1 − y′1)η(y′1)⎥ ⎦ y ′1

∫

⎤2

⎤2 ⎡

∫y η (y1)SF (y1 − z1)⎥⎦ ⎢⎣∫y′ γμD Fμν(z1 − y′1)Jν(y′1)⎥⎦

⎡ ×⎢ ⎣

1

1

⎤2

∫y″ SF (z1 − y″1)η(y″1)⎥⎦ , 1

4-7

(4.32)

Relativistic Quantum Field Theory, Volume 2

where, for compactness, a squared quantity implies a duplication while changing the labels appropriately, for example ⎡ ⎤2 ⎡ ⎤⎡ ⎤ S ( z − y ′ ) η ( y ′ ) SF (z1 − y′1)η(y′1)⎥⎢ SF (z2 − y′2 )η(y′2 )⎥ . (4.33) ⎢ F 1 1 1 ⎥ ≡ ⎢ ⎣ y ′1 ⎦ ⎣ y ′1 ⎦⎣ y ′2 ⎦

∫

∫

∫

The first line on the right-hand side of equation (4.32) maps to the following diagram

The second and third lines map to diagrams of the form

The fourth line maps to

The fifth and sixth lines map to diagrams of the form

The seventh line maps to a diagram of the form

Finally, the eighth line maps to two disconnected vertices

Putting all the pieces together, we obtain 4-8

Relativistic Quantum Field Theory, Volume 2

where the integrals are over the positions of all vertices and, as in the case of a scalar field theory, we see that vacuum diagrams such as equation (4.34) cancel in the ratio. The first term on the right-hand side above is just the free generating functional and the second term generates the bare QED vertex. The first two graphs in the second term contribute to the electron and photon two-point functions. The third and fourth terms contribute to electron-(anti)-electron and (anti)-electron-photon scattering (e±e± → e±e±, e±e∓ → e±e∓, and e±γ → e±γ ). The final term gives a disconnected contribution to the electron-photon scattering, which would contribute to a six-point function. Exercise 4.3 Referring to the discussion surrounding equation (4.27), prove why there can be no fermionic tadpoles.

4.6 QED effective action and vertex functions The arguments presented in the context of scalar fields in chapter 2 carry over directly to QED. We can introduce a generating functional for connected diagrams as before

W [η , η , J μ ] = −i log Z [η , η , J μ ],

(4.41)

and the effective action through the Legendre transformation

W [η , η , J μ ] = Γ⎡⎣ Ψ , Ψ , Aμ⎤⎦ +

∫x ⎡⎣ Ψ(x)η(x) + η (x)Ψ(x) + A μ(x)J μ(x)⎤⎦,

(4.42)

with

δW = Ψ(x ) = 〈ψ (x )〉η,η ,J δη(x ) δW = Ψ(x ) = 〈ψ (x )〉η,η ,J δη (x ) δW = Aμ(x ) = 〈Aμ (x )〉η,η ,J δJμ(x ) Finally, note that relations such as

δΓ = −η(x ), δ Ψ( x ) δΓ = −η (x ), δ Ψ( x ) δΓ = −J μ(x ). δ A μ(x ) =i

straightforward modifications, for example 4-9

(4.43)

also extend to QED with some

Relativistic Quantum Field Theory, Volume 2

∫x′ ,x′ ,x′ Gc(ψψ )(x′1, x1)Gc(ψψ )(x2, x′2 )

A) Gc(,ψψ (x1, x2 , x3) = i μ

1

×

2

3

) Gc(,AA μν (x3,

x′3)Γ

(ψψA)ν

(4.44)

(x′1, x′2 , x′3),

where the QED vertex functions are defined via

Γ (μm1,,n⋯,ℓ,) μℓ ≡

δm δn δℓ δ Ψ(x1)⋯δ Ψ(xm ) δ Ψ(x′1)⋯δ Ψ(x′n ) δ Aμ1(x″1)⋯δ Aμℓ(x″ℓ ) × Γ⎡⎣ Ψ , Ψ , Aμ⎤⎦ .

(4.45)

4.7 Ward-Takahashi identities Although the QED Lagrangian density is invariant under local gauge transformations, the gauge fixing term and source term appearing in the definition of the generating functional Z in equation (1.22) are not. This leads to an apparent dependence of physical observables obtained from Z on the choice of gauge. Of course, physical results cannot depend on the gauge choice (choice of ξ in the case of general Lorenz gauge), so there must be some non-trivial relations among the n-point functions that ensure this. To derive the necessary relationships, we consider an infinitesimal local gauge transformation. Subject to an infinitesimal gauge transformation of the form (4.21) with Λ(x ) ≡ λ(x )/e

Aμ → Aμ + ∂μΛ(x ), ψ → eie Λ(x )ψ ≃ ψ + ie Λ(x )ψ , ψ → e−ie Λ(x )ψ ≃ ψ − ie Λ(x )ψ ,

(4.46)

one finds that the action appearing in the argument of the exponential in the generating functional

S≡

⎤

⎡

∫x ⎢⎣LQED − 21ξ (∂μAμ)2 + ηψ + ψ η + JμAμ⎥⎦,

(4.47)

has a variation

δΛS =

⎤

⎡

∫x ⎢⎣− 1ξ (∂μAμ)□Λ + ieΛ(ηψ − ψ η) + J μ∂μΛ⎥⎦,

(4.48)

where we have discarded the term that is second order in □Λ since Λ is infinitesimal. This variation leads to the argument of Z picking up a factor exp(iδΛS ). Since Λ is infinitesimal, we can expand the exponential to obtain

⎤

⎡

∫x ⎢⎣− 1ξ □(∂μAμ) + ie(ηψ − ψ η) − ∂μJ μ⎥⎦Λ + O(Λ2),

1+i

(4.49)

where we have integrated by parts twice in the first term and once in the last term. 4-10

Relativistic Quantum Field Theory, Volume 2

If we require that Z is the same before and after the gauge transformation, this implies that

⎡i ⎤ ⎛ δ δ ⎞ δ ⎢ □∂μ + e⎜η − η ⎟ − ∂μJ μ⎥Z = 0, δη ⎠ ⎝ δη δJμ ⎣ξ ⎦

(4.50)

where we have rewritten, for example, Aμ = −iδ /δJμ, to turn this into a functional differential equation. Rewriting this in terms of the generating functional for connected graphs, W = −i log Z , we obtain

⎛ δW δW ⎞ 1 δW − □∂μ + ie⎜η −η ⎟ − ∂μJ μ = 0. δη ⎠ ⎝ δη ξ δJμ

(4.51)

Finally, we can express this in terms of the effective action Γ = W − ∫ ⎡⎣Ψη + η Ψ + A μJ μ⎤⎦ x to obtain

⎛ δΓ ⎞ 1 δΓ δΓ + ∂μ⎜ − ie Ψ − □∂μAμ + ie Ψ ⎟ = 0. ξ δΨ δΨ ⎝ δAμ ⎠

(4.52)

This identity can be used to generate an entire hierarchy of identities by taking various functional derivatives of it and then setting all sources to zero (expectation values go to their source-free values, which is zero for vacuum QED). Exercise 4.4 Show that equations (4.51) and (4.52) are correct.

4.7.1 Relation between the electron-photon vertex and inverse electron propagator As an example, let us consider what happens if we take functional derivatives of equation (4.52) with respect to Ψ(y2 ) and Ψ(y1) and then set Ψ = Ψ = Aμ = 0, which in Lorenz gauge gives

⎞ ∂ ⎛ δ 3Γ[0] ⎜ ⎟ ∂xμ ⎝ δ Ψ(y2 )δ Ψ(y1)δ A μ(x ) ⎠ ⎡ δ Γ[0] ⎤ δ Γ[0] = ie⎢ δ 4(x − y2 )⎥ , δ 4(x − y1) − δ Ψ(x )Ψ(y1) ⎣ δ Ψ(x )Ψ(y2 ) ⎦

(4.53)

where the argument ‘0’ of Γ above serves to remind us that we are in the limit of zero external sources. We introduce the proper vertex function, Γμ(p2 , q, p1 ), as the Fourier transform of the spatial three-vertex

(2π )4δ 4(p2 − p1 − q )ie Γμ(p2 , q , p1 ) 3

≡

∫x,y ,y ei(p ·y −p ·y −q·x) δ Ψ(y )δδΨΓ(y[0])δ A μ(x) . 2

1

2

1

1

2

2

4-11

1

(4.54)

Relativistic Quantum Field Theory, Volume 2

Note that the ie in the definition above is convention and the (2π )4δ 4(p2 − p1 − q ) simply expresses energy-momentum conservation at the vertex. Likewise, using the fact that the second functional derivative of the effective action generates an inverse −1 propagator, we introduce S˜F as the inverse of the exact electron propagator −1

(2π )4δ 4(p1 − p2 )iS˜F (p1 ) ≡

∫y ,y ei(p ·y −p ·y ) δ Ψ(δyΓ)[0]Ψ(y ) . 2

1

2

1

1

2

2

(4.55)

1

Next, we multiply both sides of equation (4.53) by exp(i [p2 · y2 − p1 · y1 − q · x ]) and integrating over

and relabeling the final arguments, we obtain the

∫x,y ,y , 1 2

Fourier-transformed equation −1 −1 q μΓμ(p , q , p + q ) = S˜F (p + q ) − S˜F (p ).

(4.56)

This is our first Ward-Takahashi identity and it relates contractions of the photon momentum with the proper photon-electron vertex (three-point function) with a difference of the inverse electron propagators (two-point functions). This relation must be satisfied at all orders of perturbation theory. It has a graphical representation

−1

It is straightforward to verify that the bare electron propagator S˜F → SF−1 = p − m and vertex Γ ν → Γ 0ν = γ ν satisfy this relation

q μΓ ν0 = p + q − m − ( p − m) .  q q

⏟

(4.58)

In the limit of vanishing, photon momentum equation (4.56) becomes the Ward identity −1

∂S˜F = Γμ(p , 0, p ). ∂p μ

(4.59)

Once again, it is straightforward to verify that the bare quantities satisfy the Ward identity, that is

∂SF−1 = γμ = Γ0,μ(p , 0, p ). ∂p μ

Exercise 4.5 Show that equation (4.56) follows from equation (4.53).

4-12

(4.60)

Relativistic Quantum Field Theory, Volume 2

4.7.2 Electron self-energy and the sub-leading vertex Following our discussion in the context of scalar fields, we can introduce the proper self-energy −1 S˜F (p ) = SF−1(p ) − Σ(p ).

(4.61)

The left-hand side of the Ward identity can be written in terms of the bare vertex and Σ −1 ∂S˜F ∂Σ . = γμ − μ ∂p μ ∂p

(4.62)

Splitting the three-point vertex function into leading and sub-leading parts via

Γμ(p , q , p + q ) = γμ + Λμ(p , q , p + q ),

(4.63)

we see that the Ward identity implies

Λμ(p , 0, p ) = −

∂Σ . ∂p μ

(4.64)

We would now like to verify this explicitly. The leading-order electron self-energy is given by a diagram of the form

where it is understood that the ‘legs’ on the diagram are just place holders. Using the Feynman rules developed in volume 1, in Feynman gauge ξ = 1, we obtain

Σ(p ) = −ie 2

∫

d 4k γνSF (p − k )γ ν . (2π )4 k2

(4.66)

The other quantity of interest is the leading correction to the QED vertex, which appears at O(e 3)

This can be written as

Λμ(p , q , p + q ) = −ie 2

∫

ν d 4k γνSF (p − k )γμSF (p − k + q )γ . (2π )4 k2

4-13

(4.68)

Relativistic Quantum Field Theory, Volume 2

To verify the Ward identity, we need to evaluate ∂Σ(p )/∂p μ. For this purpose, it follows from SF−1SF = 1, that

∂SF−1(p ) ∂SF (p ) = − S ( p ) SF (p ) = −SF (p )γμSF (p ). F ∂p μ ∂p μ

(4.69)

Using equation (4.66), this allows us to find

∂Σ(p ) = ie 2 ∂p μ

∫

ν d 4k γνSF (p − k )γμSF (p − k )γ . (2π )4 k2

(4.70)

Comparing this to equation (4.68) we see that

∂Σ(p ) = −Λμ(p , 0, p ), ∂p μ

(4.71)

which shows that the Ward identity is satisfied by the one-loop order electron selfenergy and one-loop QED vertex correction. Exercise 4.6 Show that equation (4.68) is correct.

4.7.3 Extension to higher orders Using equation (4.69) we can come up with a graphical rule for relating the electron self-energy and the electron-photon vertex function. The electron self-energy to nextto-leading order can be written graphically as

Equation (4.69) tells us that if we take a derivative of this result with respect to p μ then this is equivalent to inserting a zero-momentum photon vertex midway in every propagator. Doing this graphically, we obtain

As can we see from this figure, the form of the O(e 3) vertex corrections can be automatically generated from derivatives of the self-energy. This construction can be extended to arbitrary order in e .

4-14

Relativistic Quantum Field Theory, Volume 2

4.7.4 Implications for renormalization As we saw in volume 1, the self-energy and vertex corrections presented above are divergent and one must absorb these infinities through a rescaling of the bare n -point functions of the theory. For the propagator, the one has

S˜F → Z2SF ,

(4.74)

and, for the vertex function, the convention is

Γμ(p , 0, p ) →

1 γμ. Z1

(4.75)

In order to satisfy the Ward identity, it must be true that Z1 = Z2 . We saw this emerge from the explicit calculations of Z1 and Z2 we made last semester, but the Ward identity allows us to prove that this is something that must occur at all loop orders in the renormalization procedure.

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 5 Groups and Lie groups

Having covered the basics of abelian gauge theory, we turn to non-abelian gauge theories so that we can write down the generating functional of quantum chromodynamics (QCD). To begin, we need to understand some basics of group theory, in particular, Lie groups that correspond to continuous symmetries. For additional reading materials on this topic, I refer the reader to refs. [1, 2].

5.1 Group theory basics A group is a set of algebraic objects G = {M1, …, Mn} plus an operation, indicated here by ∗, which combines two objects in the set into another object. To be a proper group, four conditions must be satisfied: 1. Closure: operation on members of the set always produces a member of the same set, that is G ∗ G → G or Mi ∗ Mj = Mk . 2. Associativity: Mi ∗ Mj ∗ Mk = (Mi ∗ Mj ) ∗ Mk = Mi ∗ (Mj ∗ Mk ). 3. Identity: there exists an identity element I such that any other object Mi combined with it returns Mi back without modification: I ∗ Mi = Mi ∗ I = Mi . 4. Invertibility: each element Mi has an inverse that satisfies Mi−1 ∗ Mi = Mi ∗ Mi−1 = I . For example, if the operator ∗ is scalar multiplication, then Mi−1 = 1/Mi and I = 1, and, if the operator is addition, then Mi−1 = −Mi and I = 0. If Mi ∗ Mj = Mj ∗ Mi , the group is called abelian and, if Mi ∗ Mj ≠ Mj ∗ Mi , the group is called nonabelian. 5.1.1 Subgroups A subgroup H of G is a subset of the elements of G that form a proper group under the ∗ operation of G . For example, the cyclic group  8 has two subgroups: one

doi:10.1088/2053-2571/ab3108ch5

5-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

generated by ϕ4 and another by ϕ2 . In the first case, the subgroup’s set is {I , ϕ2 , ϕ4 , ϕ6} and in the second case it is {I , ϕ2 , ϕ4}. 5.1.2 The group center For a group G , the center Z (G ) is the set of all elements that commute with all elements of G . Z (G ) is an abelian subgroup by definition.

5.2 Examples Let us consider some examples [2] in order to make things more clear. 5.2.1 Finite groups 1. The nth roots of unity form an abelian group of order n under multiplication. Defining ϕ = exp(2πi /n ), then G = {ϕ m} with m = 1, …, n . The identity is ϕ n = 1. The inverse of simply ϕ m = ϕ−m . Since ϕ can be used to generate all elements in the set, it is called a generating element. This group is closed under multiplication and, as one can easily see, multiplication of group elements just moves us to a different root of unity in the complex plane. This group is an example of a cyclic group, which is defined as a group generated by a single element ϕ that satisfies ϕ n = I . One denotes the cyclic group of order n as  n . 2. The integers under addition modulo n form another finite group, which is isomorphic to the previous example. A group isomorphism is a map between two groups that establishes a one-to-one correspondence between the elements of the groups. Additionally and most importantly, an isomorphism respects the given group operation. For mapping the integers under addition modulo n to the example above, the isomorphism would be provided by the function f (k ) = exp(2πik /n ) where k ∈  . 5.2.2 Infinite discrete groups 1. The set of integers under addition, ( , +). The identity is I = 0 for this group and inverse is M → −M . This group is like the cyclic group, since it can be generated by a single element, 1, by repeated addition, however, it is an infinite discrete group. 2. The ‘modular group’ SL(2,  ), which is the group of integer-valued 2 × 2 matrices with unit determinant. The group operation in this case is simply 2 × 2 matrix multiplication. Exercise 5.1 Show that SL(2,  ) forms a proper group and that it is infinite and discrete.

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Relativistic Quantum Field Theory, Volume 2

5.2.3 Continuous compact groups 1. The orthogonal group, O(n ), is the group of real n × n matrices O satisfying OTO = 1. This maps to continuous rotations and reflections in n-dimensional Euclidean space n . O(3) has a discrete subgroup given by {I , −I }, with the second member of this subgroup corresponding to reflections. 2. The unitary group, U (n ), which is the group of complex n × n matrices U satisfying the condition U †U = 1. In the connection to subgroups (properly defined below), we note that U(2) has two one-dimensional subgroups that are not related by a similarity transformation.

⎧⎛ iϕ 0 ⎞⎫ ⎟⎬ , U (2)A = ⎨⎜ e ⎩⎝ 0 eiϕ ⎠⎭

(5.1)

⎧⎛ iθ 0 ⎞⎫ ⎟⎬ . U (2)B = ⎨⎜ e ⎩⎝ 0 e−iθ ⎠⎭

(5.2)

and

3. In general, orthogonal matrices have determinant equal to ±1 and unitary matrices have a determinant given by an arbitrary phase. There is a special version of each of these groups, called SO(n ) and SU (n ), respectively. For the ‘special’ version of these groups one adds an extra restriction that the matrices must have determinant equal to one. Exercise 5.2 Show that SO(2) forms a proper group and that it is continuous and compact.

5.3 Representations of groups Let us focus on 3d rotations in quantum mechanics as our canonical example. As we know, we can represent rotations through a 3d rotation matrix, R . The canonical set of matrices R , forms the defining representation of the group. It is possible to have other representations of the group as we shall see. Let us assume that we have a complete set Ψ = {ψ1, …, ψn} of wave functions that transform into one another upon rotation by R . After rotation, ψ1 transforms into a linear combination of ψ1, …, ψn

ψ ′1(x ⃗ ) ≡ ψ1(Rx ⃗ ) = d11ψ1(x ⃗ ) + d12ψ2(x ⃗ ) + ⋯ + d1nψn(x ⃗ ),

(5.3)

and likewise for ψ2, …, ψn so that, in general, we can write

ψ ′A =

∑dABψB

(A , B = 1, … , n ).

B

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(5.4)

Relativistic Quantum Field Theory, Volume 2

The coefficients dAB depend on the rotation R and can be collected into a matrix D(R ), allowing us to write compactly

(5.5)

Ψ′(x ⃗ ) = Ψ(Rx ⃗ ) = D(R )Ψ(x ⃗ ),

where we have collected all of the wave functions making up the set into a column vector Ψ(x ). One can show that, given two rotations R1 and R2 , one has D(R1R2 ) = D(R1)D(R2 ). As a result, the matrices D(R ) have the same multiplication rules, commutation rules, etc. as the original matrices R . A mapping of R onto explicit matrices D(R ) is called a representation of the group. Note also, that since overlaps between states should be the same before and after rotations, one can infer that D(R ) must be a unitary matrix. To see this, consider two states ∣ψ 〉 and ∣φ〉 that have an overlap 〈φ∣ψ 〉. Under rotation by R one has ∣ψ ′〉 ≡ D(R )∣ψ 〉 and ∣φ′〉 ≡ D(R )∣φ〉. Since 〈φ′∣ψ ′〉 = 〈φ∣ψ 〉, it follows that D is hermitian D†(R )D(R ) = 1. A general 3d rotation can be described by picking a direction to rotate around the direction specified by α⃗ and exponentiating the generators, which are the angular momentum matrices,

⎛0 0 0 ⎞ ⎛ 0 0 i⎞ L1 = ⎜⎜ 0 0 − i ⎟⎟ L 2 = ⎜⎜ 0 0 0 ⎟⎟ L 3 = ⎝0 i 0 ⎠ ⎝− i 0 0 ⎠

⎛0 − i 0⎞ ⎜ i 0 0⎟, ⎜ ⎟ ⎝0 0 0⎠

(5.6)

and

⎛ ⎞ R(α⃗ ) = exp ⎜⎜i∑αkLk⎟⎟ . ⎝ k ⎠

(5.7)

The generators of the group are the angular momentum matrices and the corresponding group is SO(3). In general, the commutator of two generators for a group defines the structure constants fabc . The structure constants are antisymmetric under the interchange of any two indices. For rotations, one has

[La , Lb ] = ifabc Lc = iεabcLc ,

(5.8)

where we used the fact that for SO(3) the structure constants are fabc = εabc . Since the generators do not commute, this group is non-abelian. We also note that the group is compact and our first example of a Lie group. A Lie group is a group where the transformations can be arbitrarily small (continuous symmetry). Together with the commutation relations (5.8), the generators Lk define a continuous algebra, called the Lie algebra. Any set of matrices Ak with the same commutation relations can be used to define an alternative representation of the Lie algebra. The structure constants of a group obey certain relations. For example, they should satisfy the Jacobi identity, which holds for any three matrices, A, B , and C

[[A , B ], C ] + [[B, C ], A] + [[C , A], B ] = 0.

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(5.9)

Relativistic Quantum Field Theory, Volume 2

Taking A = L i , B = Lj , and C = Lk , one finds

fabd fdce + fbcd fdae + fcad fdbe = 0.

(5.10)

For the case of SO(3), this translates to εijmεmkn + εjkmεmin + εkimεmjn = 0, which can be shown to be true using the identity εijmεmkl = δikδjl − δilδjk . We can define n, n × n matrices Aa

(Aa )bc ≡ −ifabc ,

(5.11)

which allow us to write equation (5.10) as

ifabd Adce + (Ab Aa )ce − (Aa Ab )ce = 0,

(5.12)

Aa Ab − Ab Aa = [Aa , Ab ] = ifabd Ad .

(5.13)

or, equivalently

This demonstrates that the n, n × n matrices Aa obey the same commutation relations as the original matrices and, therefore, define an alternative representation of the group called the adjoint representation. In the case of rotations (SO(3)), the adjoint matrices are the same as the original matrices, so it is a bit boring; however, in general, the adjoint matrices can be different from the original or ‘fundamental’ representation. 5.3.1 Equivalent representations We can easily obtain different representations by making a unitary transformation of the fields. In quantum mechanics, for example, we can transform the wave function ψ˜ = Uψ . Requiring that expectation values are unaffected, we find that under a rotation D˜ = UDU −1. The original matrices D and D˜ both define representations of SO(3), but these trivial alternative representations are not considered to be really different because they are related by a unitary transformation. Representations related by such a unitary transformation are called equivalent representations. These type of transformations also allow us to make a general statement: All finite dimensional representations of finite or compact groups can be made to be unitary. In practice, if we are given a representation that is not unitary, then we can apply a unitary transformation such that D˜ † = D˜ −1. 5.3.2 Reducible and irreducible representations So far we have considered three-dimensional rotations that act on a 3d vector x ⃗ = (x1, x2, x3). We can trivially construct higher-dimensional representations by constructing matrices that act on n vectors, for example for n = 2, x ⃗ and y ⃗ , which are grouped together to form a 3n-dimensional vector z ⃗ , for example for n = 2 we can define z ⃗ = (x1, x2, x3, y1, y2 , y3). Under rotations z ⃗ transforms as

z ⃗ → z ′⃗ = Dz ⃗ .

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(5.14)

Relativistic Quantum Field Theory, Volume 2

The matrix D above has been decomposed into 3 × 3 matrices. For example, for n = 2 we would have

D=

⎛R 0 ⎞ ⎜ ⎟. ⎝ 0 R⎠

(5.15)

If it possible to factorize the group as above, it is called reducible. In this case, the reducible six-dimensional representation can be regarded as a direct sum of two three-dimensional representations

6 = 3 ⊕ 3.

(5.16)

In quantum mechanics such a direct sum representation can occur when a particle is in a superposition of two different kinds of quantum states. Representations that cannot be cast into a block diagonal form similar to equation (5.15), are called irreducible representations. 5.3.3 Product representations In the context of the addition of angular momentum, we saw another way construct representations called product representations. If we have two particles with wave functions ψ1(x ⃗ ) and ψ2(y ⃗ ), the wave function of the combined system, Ψ(x ⃗, y ⃗ ) then consists of all possible products of wave functions ψ1 and ψ2 . We call this a tensor product and it is denoted as following

Ψ = ψ1 ⊗ ψ2.

(5.17)

Frequently, such a product representation is reducible, but can be decomposed into a number of distinct irreducible representations. For example, the product of two vectors can be decomposed into three irreducible elements

3 ⊗ 3 = 1 ⊕ 3 ⊕ 5,

(5.18)

which map to a scalar, a pseudovector, and a symmetric traceless tensor. For the rotation group the Clebsch–Gordon coefficients tell us how to construct states with fixed total angular momentum in each set on the right and the Wigner–Eckart theorem can be used to reduce calculations to evaluation of the reduced matrix element [3].

5.4 The group U(1) The U(1) Lie group emerges in quantum mechanics from the invariance of the wave function under multiplication by an arbitrary local phase factor

ψ → e iθ(x ) ψ .

⏟ ≡U

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(5.19)

Relativistic Quantum Field Theory, Volume 2

Since ∣ψ ∣2 is unmodified by the phase factor, physics is the same. We can think of this as a group with one one-dimensional matrix as the generator (scalar). This group is abelian ([U1, U2 ] = 0), unitary (U † = U −1), and special (∣U ∣ = 1).

5.5 The group SU(2) Let us consider next the angular momentum generators for a spin-1/2 particle1. As we know, the states can be represented as two-dimensional vectors called spinors. The generators in this case can be written in terms of Pauli matrices. A spin-1/2 rotation operator can be written, in general, as X = exp(i ∑i αiSi ), where the generators are Si ≡ σi /2 with

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ σ1 = ⎜ 0 1 ⎟ , σ2 = ⎜ 0 − i ⎟ , σ3 = ⎜1 0 ⎟ . ⎝1 0 ⎠ ⎝i 0 ⎠ ⎝ 0 − 1⎠

(5.20)

These matrices obey σiσj = δij  + iεijkσk and

[Si , Sj ] = iεijkSk .

(5.21)

As a result, an arbitrary rotation of a spin-1/2 state can be written as

⎛α ⎞ ⎛ α ⎞ αj σj X (α⃗ ) = cos ⎜ ⎟  + i sin ⎜ ⎟ , ⎝2⎠ ⎝2⎠ α

(5.22)

where α ≡ α12 + α22 + α32 . The generators Si form elements of the group SU(2), which is the group of unitary 2 × 2 complex matrices with determinate equal to 1. For SU(2), product representations can be easily constructed using standard Clebsch–Gordon coefficients. A Casimir operator is an operator that commutes with all other generators. For SU(2), there is just one Casimir, which is SaSa → J 2 = J12 + J22 + J32 . Since [J 2, J3] = 0, we can label states by J3 and J 2 , resulting in the usual basis states ∣jm〉 with j = 1/2. The eigenvalues of the Casimir operator label the irreducible representations. We can connect different states in each multiplet using the ladder operators J± = J1 ± iJ2 . Hence, for SU(2), the ladder operators are constructed from σ1 and σ2 . The Cartan sub-algebra associated with each multiplet is one-dimensional and can be spanned using the ladder operators. Since the commutation relation (5.21) is the same as that of the rotation group SO (3), we expect that there is a relation between the two groups. For the rotation group we saw that the direction of α⃗ is the axis of rotation and the length α is the angle of rotation. The range for α can be taken to be 0 to π and rotations by a negative angle −π to 0 can be obtained by inverting the vector. All rotations can be parameterized by points that live inside a sphere in α -space with radius π . Points on the surface of this sphere correspond to rotations by π . Since rotations by +π ane −π are equal, one has that R(α⃗ ) = R( −α⃗ ) if α = π . This means that points on the sphere in α -space related

1

This can also be used to describe isospin, which is an internal rotational symmetry.

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Relativistic Quantum Field Theory, Volume 2

Figure 5.1. Two-dimensional cartoon showing the sphere in α -space that can be used to describe the rotation group SO(3). In the figure, we show the case that α = π which corresponds to a rotation by π . We also show the mirror point −α⃗ which is equivalent to a rotation by −π and hence maps to the same point when α = π , that is P = P′.

by α⃗ → −α⃗ correspond to the same rotation and can be identified. This is sketched in figure (5.1). Elements of SU(2) can be parameterized by the same vector α⃗ and a similar visualization; however, to parametrize all elements of X (α⃗ ), the radius must be taken to be 2π instead of π . This is related to the fact that if we rotate our 2d spinors by 2π , instead of getting the same result back, we get −1 times the result back, that is X (α′⃗ ) = −X (α⃗ ), where α + α′ = 2π . We have to rotate our spinors by 4π in order to get the original result back. These observations follow directly from equation (5.22) and tell us that the elements of SU(2) are not a representation of the threedimensional rotation group. They are instead called a projective representation. The group SU(2) does have the same structure constants, and thus the same group product structure as the rotation group, but, due to the fact that we can pick up a sign flip in SU(2), the equivalence of the group product only holds true in a small domain surrounding the unit element, and not for the entire group. We can introduce a contravariant spinor φ α that transforms as

ϕ α → ϕ α ′ = X α βϕ β ,

(5.23)

and a complex conjugate spinor (covariant spinor) that transforms as

ϕ α⁎ = ϕα → ϕ′α = (X α β )⁎ϕβ = (X †) β α ϕβ .

(5.24)

As a result, one has

ϕ′α ψ ′α = (X α β )∗X α γϕβ ψ γ = (X †X ) β γ ϕβ ψ γ = δ β γϕβ ψ γ = ϕβ ψ β ,

(5.25)

where I have used the fact that X †X =  . This shows that the product of covariant and contravariant spinors is invariant under SU(2) transformations. We can construct two invariant tensors, ε αβ and εαβ , which satisfy ε αβ = εαβ = −εβα and ε12 = ε12 = 1. We observe that

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Relativistic Quantum Field Theory, Volume 2

X α α′X β β ′ε α′ β ′ = det(X )ε αβ = ε αβ ,

(5.26)

and, similarly, εαβ is invariant. This tells us that the representations X ∗ and X are probably equivalent. We can demonstrate this explicitly by using the general form for an element of SU(2)

X=

⎛ a b⎞ ⎜ ⎟, ⎝−b∗ a∗⎠

(5.27)

with ∣a∣2 + ∣b∣2 = 1, to show that

ε · X · ε −1 =

⎛ 0 1 ⎞⎛ a b ⎞⎛ 0 1 ⎞ ⎛ a∗ b∗⎞ ⎜ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟ = X ∗. ⎝−1 0 ⎠⎝−b∗ a∗⎠⎝−1 0 ⎠ ⎝−b a ⎠

(5.28)

So we see that X ∗ and X are equivalent representations in SU(2) (related by a unitary transformation). This does not hold in general for other SU (n ).

5.6 The group SU(3) The group SU(3) describes the set of unitary complex 33 matrices having determinate 1. It can be used to describe, for example, color SU(3) symmetry or flavor SU(3) symmetry. In the former case, it becomes the fundamental symmetry of QCD and in the latter case it can be used to understand the symmetry patterns of the hadronic states. Being complex we start with 2∗3∗3 = 18 degrees of freedom. Unitarity restricts this to nine degrees of freedom than can be described by nine real-valued numbers. The requirement that the determinate of the matrices is equal to one further reduces this to eight real-valued numbers. As a result, the dimension of the group is d (G ) = 8 and there are eight generators ta . These can be written in terms of the Gell-Mann matrices with ta = λa /2 and

⎛ λi = ⎜ ⎜ ⎝0 ⎛0 λ 4 = ⎜⎜ 0 ⎝1 ⎛0 λ6 = ⎜⎜ 0 ⎝0

0⎞ σi 0 ⎟ for i = 1, 2, 3 ⎟ 0 0⎠ ⎛0 0 −i ⎞ 0 1⎞ 0 0 ⎟⎟ , λ5 = ⎜⎜ 0 0 0 ⎟⎟ , ⎝i 0 0 ⎠ 0 0⎠ ⎛0 0 0 ⎞ 0 0⎞ 0 1 ⎟⎟ , λ7 = ⎜⎜ 0 0 −i ⎟⎟ , ⎝0 i 0 ⎠ 1 0⎠ ⎛ ⎞ 1 ⎜1 0 0 ⎟ λ8 = 0 1 0 . 3 ⎜⎝ 0 0 −2 ⎟⎠

(5.29)

These matrices satisfy

[ta , tb ] = ifabc tc ,

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(5.30)

Relativistic Quantum Field Theory, Volume 2

Table 5.1. The antisymmetric structure constants fabc and symmetric symbols dabc for SU(3). Other coefficients can be obtained by permutation (symmetric and anti-symmetric, respectively). If a particular abc cannot be generated by permutations of the ones listed here, then it is zero.

fabc

abc

dabc

abc

1

123 147, 246, 257, 345

1 3 1 2 1 −2

118, 228, 338

1 2

1

156, 367

3 2

458, 678

−2

−

146, 157, 256, 344, 355 247, 366, 377 1

2 3 1 − 3

448, 558, 668, 778 888

and

{ta , tb} =

1 δab + d abctc . 3

(5.31)

Like SU(2), since the generators of the group do not necessarily commute, SU(3) is non-abelian. The antisymmetric structure constants fabc and symmetric symbols dabc can be obtained by taking traces of the generators in the fundamental representation

fabc = − 2i Tr([ta , tb ]tc ), d abc = 2Tr({ta , tb}tc ).

(5.32)

We summarize the non-vanishing elements in table 5.1. Note also that the antisymmetric structure constants fabc obey the Jacobi identity derived earlier

fabc fcde + fadc fceb + faec fcbd = 0.

(5.33)

The matrices ta above provide the defining representation of SU(3), which we refer to as the fundamental representation. The elements t1, t2 , and t3 generate an SU(2) subalgebra, which acts on the first two elements of a vector, leaving the third element unchanged. There are also SU(2) subgroups formed by {λ 4 , λ5, ( 3 λ8 + λ3)/2} and {λ 6 , λ7 , ( 3 λ8 − λ3)/2}. The matrices are orthonormal

Trtatb =

1 δab. 2

(5.34)

The normalization of the matrices is a convention. In general, one has Trtatb = C (R )δab , where C (R ) is called the Dynkin index. As was the case in SU(2), we can construct the adjoint representation of the group. For SU(3), the adjoint can be constructed using the general relation equation (5.11). The result is

(Ta )bc ≡ −ifabc .

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(5.35)

Relativistic Quantum Field Theory, Volume 2

Since a , b, and c all take values 1, …, 8, we see that the adjoint representation corresponds to eight 8 × 8 matrices. As we saw previously, they satisfy the group commutation relation, [Ta, Tb ] = ifabc Tc . The corresponding states are eight-dimensional column vectors. The adjoint matrices are normalized as

TrTaTb = 3δab.

(5.36)

For SU (N ) this becomes TrTaTb = Nδab. Exercise 5.3 Show that equation (5.36) is correct.

5.6.1 Quadratic Casimir invariants SU(3) has two Casimir operators, tata and dabctatbtc 2. Both Casimir operators commute with every generator in the group. Let us focus here on the quadratic Casimir invariants. For a representation R , let the generators be denoted G . The quadratic operator is

G 2 ≡ GaGa .

(5.37)

Computing the commutator of G 2 with the generators we find the expected result

[GaGa , G b ] = Ga[Ga , G b ] + [Ga , G b ]Ga = ifabc {Ga , Gc} = 0,

(5.38)

where we have used the antisymmetry of fabc in the last step. As a result, G 2 must be proportional to the identity matrix R of the representation

G 2 = C2(R )R ,

(5.39)

where C2(R ) is a constant that depends on the representation. Taking the trace of equation (5.39) we obtain TrG 2 = C2(R )d (R ), where d (R ) is the dimension of the matrices in the representation. Based on equations (5.34) and (5.36), for a general representation we can write

Tr(GaG b) = C (R )δab.

(5.40)

Taking a = b in this expression and summing over a = 1, …, 8, we obtain Tr(GaGa ) = C (R )d (G ). As a result,

C 2 (R ) =

C ( R ) d (G ) . d (R )

(5.41)

For the fundamental representation, we obtain

CF ≡ C2(F ) =

2

In both cases, the sum over repeated indices is implied.

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4 . 3

(5.42)

Relativistic Quantum Field Theory, Volume 2

For SU (N ) this becomes CF = (N 2 − 1)/2N . For the adjoint representation, we obtain

CA ≡ C2(A) = 3.

(5.43)

For SU (N ) this becomes CA = N . Exercise 5.4 Show that equation (5.39) is correct. 5.6.2 Cartan sub-algebra and ladder operators The Cartan sub-algebra is generated by the two diagonal generators t3 and t8 which maps to H = 〈t3, t8〉. In the diagrams, that follow we will define our axes as H1 = t3 and H2 = t8. The positions of the points in this space are called the weights. In the language of the flavor SU(3) quark model, the corresponding eigenvalues are called I3 and Y , which are the third component of isospin and the hypercharge. In this state, these two quantum numbers label the states within each multiplet and, as a result, each multiplet can be drawn as a planar object. We will now construct some twodimensional weight diagrams for SU(3). We can define six ‘raising/lowering’ matrices t± = t1 ± it2 , u ± = t6 ± it7, and v± = t4 ± it5

⎛0 t+ = ⎜⎜ 0 ⎝0 ⎛0 u + = ⎜⎜ 0 ⎝0 ⎛0 v+ = ⎜⎜ 0 ⎝0

1 0⎞ 0 0 ⎟⎟ , 0 0⎠ 0 0⎞ 0 1 ⎟⎟ , 0 0⎠ 0 1⎞ 0 0 ⎟⎟ , 0 0⎠

⎛0 t− = ⎜⎜1 ⎝0 ⎛0 u − = ⎜⎜ 0 ⎝0 ⎛0 v− = ⎜⎜ 0 ⎝1

0 0⎞ 0 0 ⎟⎟ , 0 0⎠ 0 0⎞ 0 0 ⎟⎟ , 1 0⎠ 0 0⎞ 0 0 ⎟⎟ . 0 0⎠

(5.44)

I will refer to these as the ladder operators in what follows. 5.6.3 Irreducible representations We now review how to construct the irreducible representations of SU(3), with an eye towards the flavor SU(3) quark model and SU(3) color. In general, the representations can be labeled as D pq and they depend on two quantum numbers p, q = 0, 1, 2, …. In general, the dimension of an irreducible representation of SU(3) is

dim D pq =

1 ( p + 1)(q + 1)( p + q + 2), 2

with, for example,

5-12

(5.45)

Relativistic Quantum Field Theory, Volume 2

dim D 00 = 1, dim D10 = dim D 01 = 3, dim D11 = 8, ⋮.

(5.46)

We can denote the lowest-dimensional irreducible representations by their dimension and distinguish the complex conjugate representation with a bar over the symbol

D 00 = 1 , D10 = 3 , D 20 = 6 , D11 = 8 , D 30 = 10 , ⋯ D 01 = 3, D 02 = 6 , D 30 = 10 .

(5.47)

5.6.4 The singlet –D00 The simplest representation is the singlet, which has only one state and maps to a point in weight space corresponding to (w1, w2 ) = (0, 0). This representation is denoted as 1. 5.6.5 The triplet and anti-triplet –D10 and D01 The natural representation for the triplet state vectors is a three-dimensional column vector. The weights in this case are combinations (H1ii , H2ii ), where there is no sum implied. As a result one has

⎛1 1 ⎞ (w1, w2 )3 = ⎜ , ⎟, ⎝2 2 3 ⎠  ⎛ 1⎞ u ≡ ⎜⎜ 0 ⎟⎟ ⎝0⎠

⎛ 1 1 ⎞ ⎜− , ⎟, ⎝ 2 2 3⎠    ⎛0⎞ d ≡ ⎜⎜ 1⎟⎟ ⎝0⎠

⎛ 1 ⎞ ⎜0, − ⎟, ⎝ 3⎠  ⎛0⎞ s ≡ ⎜⎜ 0 ⎟⎟ ⎝ 1⎠

(5.48)

where below each point, I have indicated the state associated with it and a label appropriate for applications in the quark model. I could have equally well have made the labels r , g , and b instead of u , d , and s if I were interested in color SU(3). We can make transitions between the weights using the ladder operators. For example, u = t+d , d = t−u , u = v+s , s = v−u , d = u +s , and s = u−d . Since this representation has three states, we call it the triplet representation and label it as 3. We have drawn a weight diagram along with the transitions that can be made using the ladder operators in the left panel of figure 5.2. Unlike SU(2), in SU(3), the complex-conjugate representation is not equivalent to the original representation, so we need to keep track of the difference. The complex conjugate of the 3 representation is called the 3 representation and the weights are obtained by multiplying the weights of the 3 representation by −1. We have drawn the corresponding weight diagram for 3 in the right panel of figure 5.2.

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Relativistic Quantum Field Theory, Volume 2

Figure 5.2. Weight diagram for the triplet (D10 ) and anti-triplet (D01) representations of SU(3). The axes are generically H1 and H2 . For the quark model, the axes map to I3 and Y .

Figure 5.3. The form of a general SU(3) weight diagram is shown on the far left. Some examples are indicated as well.

5.6.6 Higher dimensional representations A general SU(3) representation D pq can be represented by a weight diagram of the form shown on the left in figure 5.3. We have shown the case that p = 2 and q = 3. The states on the outermost ring have multiplicity 1, which is indicated by a filled circle. Proceeding in rings inward, we encounter states marked with two circles indicating that each of these states has multiplicity 2. Finally, in the innermost triangle we have states with multiplicity 3. The rule for multiplicities is that the multiplicity should increase as one proceeds in rings from the outermost states and, if a triangle is generated, then all nested triangles have the same multiplicity. With this general rule in place, we can easily draw the other important irreducible representations of SU(3). These are depicted in figure 5.3. 5.6.7 Product representations In the quark model, the multiplets shown in figures 5.2 and 5.3 map to statements about the possible quantum numbers of hadrons. In addition to simply visualizing each of the irreducible representations, the weight diagrams introduced so far also help us to construct direct sum decompositions if we combine two elements from two irreducible representations. For example, if we want to combine one triplet with another triplet, 5-14

Relativistic Quantum Field Theory, Volume 2

that is 3 ⊗ 3. We take the first triplet and then place the copies of the second triplet’s weight diagram at each of the vertices of the first tripet’s weight diagram. We then try to decompose the resulting points in standard irreducible representations taking into account the multiplicities of the overlapping points, for example

and likewise one finds 3 ⊗ 3 = 6 + 3. Next consider the states that can be constructed from a triplet and an anti-triplet

Physics-wise, in the context of color SU(3) this last relation tells us that if we combine a color triplet and color anti-triplet then we can make either an octet (colored) or a singlet (colorless). In the context of the SU(3) quark model, the spin1/2 8 ⊕ 1 maps to the nonet of light mesons, π 0, π ±, η , η′, K 0, K ±, and K . If the flavor symmetry were true, then all of these mesons would have the same mass. As one final example, let us consider 3 ⊗ 3 ⊗ 3

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Relativistic Quantum Field Theory, Volume 2

This last relation is directly applicable to the determination of the possible quantum numbers of, for example the hadrons in a three-flavor quark model.

5.7 The group SU(N) Here I would like to summarize some basic facts about SU (N ). For SU (N ) there are N 2 − 1 generators in the fundamental representation, each of which is an N × N matrix. They obey the same commutation relation as SU(3)

[ta , tb ] = ifabc tc ,

(5.52)

but

{ta , tb} =

1 δab + d abctc . N

(5.53)

The normalization convention for the fundamental representation is the same, that is

Trtatb =

1 δab, 2

(5.54)

however, the normalization of the adjoint representation depends on N

TrTaTb = Nδab.

(5.55)

As a result, we have

CF = C2(F ) =

N2 − 1 , 2N

(5.56)

and

CA = C2(A) = N .

(5.57)

In preparation for our discussion of the Faddeev–Popov method for gauge-fixing in non-abelian gauge field theory, we need to understand some formalities about the integration over elements over a group. In the context of abelian gauge field theories, we did not pay this detail much attention because, as we will see, for abelian gauge fields the final answer was the simple one I already presented.

5.8 The Haar measure We would like to consider a general gauge theory in which the fields belong to a representation of a group G . Since physics is invariant under group (gauge) transformations, integrations over group elements should also be invariant. The invariant integration measure on a group is called the Haar measure. It is invariant in the sense that a change of variables (angles) caused by group multiplication, does not change the measure. At first this seems a bit abstract, but we already know one simple case, which is integration of three-dimensional space with measure dV = dxdydz under translations. If we act on our coordinate system using the 5-16

Relativistic Quantum Field Theory, Volume 2

translation operator (translation group T ) to translate our coordinates by a constant vector (a, b, c ), we have that dV = d (x + a )d (y + b )d (z + b ) = dxdydz . This demonstrates that the measure on Euclidean space is a invariant with respect to translations (isometries). Now we want to be able to do this in terms of some continuous variables that parametrize the group, for example α , β , and γ which parameterize a group element U , such that

dα dβ dγ f (α , β, γ ) = dα′ dβ′dγ′ f (α′ , β′ , γ′),

(5.58)

where α′, β′, and γ′ parameterize U ′ = VU , with V being an arbitrary group element, or U ′ = UV , or both. Symbolically, the properly constructed left- and rightinvariant Haar measure guarantees that for a compact group

∫ dU = ∫ d (UV ) = ∫ d (VU ).

(5.59)

This will come in handy later. The existence of the Haar measure is enough, but I would like to make this a bit more concrete by describing how to construct the Haar measure and then do it for a particular parameterization of SU(2). 5.8.1 General method for constructing invariant Haar measures Let A, B , and C belong to a Lie group G . The elements A, B , and C can be parameterized by vectors of real numbers a ⃗ , b ⃗ , and c ⃗ , where the dimension d of the vectors is set by the dimension of the group, for example N 2 − 1 for SU (N ). The quantities A′ = BA and A″ = AC are also elements of G and are parametrized by a ⃗′ and a ⃗″, respectively. Based on this, we can construct left and right one forms, L⃗ and R⃗:

L⃗ ≡ iA−1

∂(BA) ∂A′ ∂A′ ∂a ⃗′ ∂A · = L′⃗ · = i (BA)−1 = i (A′)−1 = i (A′)−1 ∂a ⃗ ∂a ⃗ ∂a ⃗′ ∂a ⃗ ∂a ⃗

∂a ⃗′ , ∂a ⃗

(5.60)

where, in the first step, we used the fact that B does not depend on a ⃗ . Translating to ∂a ′j index notation, this can be stated as L i = L′ j ∂a , where it is understood that each i

component of L⃗ is a matrix with dimensions appropriate for the representation of the group. Similarly, one finds

∂A −1 ∂(AC ) ∂A″ (AC )−1 = i (A″)−1 A =i ∂a ⃗ ∂a ⃗ ∂a ⃗ ∂a ⃗″ ∂a ⃗″ ∂A″ (A″)−1 · = R ⃗″ · =i ∂a ⃗ ∂a ⃗ ∂a ⃗″

R⃗ ≡ i

(5.61)

where, once again, we can express this in terms of the components as Ri = R″ j (∂a″ j /∂ai ).

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Relativistic Quantum Field Theory, Volume 2

Having determined the one-forms, we can construct a d × d metric in the parameter space

⎛ ∂A ∂A ⎞ gij = N Tr(L iL j ) = −N Tr⎜A−1 A−1 ⎟ = N Tr(Ri Rj ), ∂ai ∂aj ⎠ ⎝

(5.62)

where N is an arbitrary normalization and, in establishing the last equivalence, I have used the cyclicity of trace. Using equation (5.60), one finds

gij = N Tr(LiLj ) = N Tr(L′k L′ℓ )

∂a′ ∂a′ℓ ∂a′k ∂a′ℓ . = g′kℓ k ∂ai ∂aj ∂ai ∂aj

(5.63)

Taking the determinate of both sides, we obtain

det g = J 2(a ⃗′ , a ⃗ )det g′ ,

(5.64)

where J(a ′⃗ , a ⃗ ) = det(∂a ′⃗ /∂a ⃗ ) stands for Jacobian. As a result, the left- and rightinvariant measure is

dU = da1 … dad det g = da′1… da′d det g′ = d (VU ) = d (UV ),

(5.65)

where we have used da1 … dad J(a ⃗′, a ⃗ ) = da′1… da′d . Equation (5.65) is our final result. 5.8.2 Haar measure example - SU(2) Let us work out the details for SU(2) to see how this works in practice. For SU(2), there are three parameters and there are different ways that one might consider parameterizing a general SU(2) matrix each of which can be used to construct the appropriate invariant integration measure. Let us consider first the following parameterization

⎛ e−iα cos ϕ eiβ sin ϕ ⎞ U = ⎜ −iβ ⎟, ⎝− e sin ϕ eiα cos ϕ ⎠

(5.66)

where 0 ⩽ α ⩽ 2π , 0 ⩽ β ⩽ 2π , and 0 ⩽ ϕ ⩽ π /2 are real parameters that span the entire group space. As we can see, det U = 1 and U †U = 1. We compute the metric

⎛ ∂U † ∂U ⎞ ⎛ ∂U ∂U ⎞ ⎟, gij = −Tr⎜U † U † ⎟ = Tr⎜ ∂aj ⎠ ⎝ ∂ai ∂aj ⎠ ⎝ ∂ai

(5.67)

where, in this case, a ⃗ = (α , β, ϕ ), I have used the fact that U −1 = U † for SU(2), and I made use of d (U †U ) = 0 = U †dU + d (U †)U to simplify the expression. Using this we obtain

⎛ cos2 ϕ 0 0⎞ ⎜ ⎟ g = 2N ⎜ 0 sin2 ϕ 0 ⎟ , ⎜ ⎟ ⎝ 0 0 1⎠

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(5.68)

Relativistic Quantum Field Theory, Volume 2

and, hence,

2 N 3/2 sin(2ϕ).

det g =

(5.69)

We can fix the constant N by normalizing the group volume to one, that is

2 N 3/2

∫0

2π

dα

∫0

2π

dβ

∫0

π /2

dϕ sin(2ϕ) = 1,

(5.70)

which gives

dU =

1 dα dβ dϕ sin(2ϕ). 4π 2

(5.71)

Alternatively, one could consider the ‘standard’ parameterization αj σj U = cos(α /2) + i sin(α /2) , α

(5.72)

where α ≡ α12 + α22 + α32 . In this case, if we use spherical coordinates α1 = α cos ϕ sin(θ ), α2 = α sin ϕ sin θ , and α3 = α cos θ with 0 ⩽ α ⩽ 2π , 0 ⩽ θ ⩽ π , and 0 ⩽ ϕ ⩽ 2π one finds

dU =

1 dα dθ dϕ sin θ sin2(α /2). 4π 2

(5.73)

Using either parameterization, one now obtains the same answer for the integral of an arbitrary smooth function f (U ) over the group space, for example

(∫ dU f (U )) = (∫ dU f (U )) , 1

(5.74)

2

where the subscripts indicate two different parameterizations of U . With this understanding under our belts, we can now abstractly write the integration over variables that are elements of a Lie group. Exercise 5.5 Show that equation (5.73) is correct.

References [1] Veltman M, de Wit B and ’t Hooft G Lie groups in physics http://staff.science.uu.nl/hooft101/ lectures/lieg07.pdf [2] Lüdeling C Group theory (for physicists) http://th.physik.uni-bonn.de/nilles/people/luedeling/ grouptheory/data/grouptheorynotes.pdf [3] Sakurai J J 2017 Modern Quantum Mechanics (Cambridge: Cambridge University Press)

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 6 Path integral formulation of quantum chromodynamics

In the original SU (3) quark model, which was independently proposed by GellMann [1] and Zwieg in 1964 [2, 3], quarks were combined into SU (3) flavor multiplets. The model was quite successful in describing the observed hadron spectra; however, it was not compatible with the spin-statistics theorem. For example, the observed Δ++ baryon with S = 3/2 required three up quarks in the ∣↑ ↑ ↑〉 state and L = 0 which has a symmetric wave function with respect to parity. However, since the Δ++ is fermionic (half-integer spin), it has to have a wave function that is antisymmetric under the interchange of any two quarks. The naive model clearly violated this. In 1964 and 1965, Oscar Greenberg [4] followed quickly by Han and Nambu [5] proposed a solution to the problem by positing that quarks possessed an additional SU (3) gauge degree of freedom, which was later called color charge. Han and Nambu also suggested that quarks interacted via an octet of vector gauge bosons1. This became the SU (3)flavor ⊗ SU (3)color model for hadrons. This fixed the spin-statistics problem. At the level of the quarks, the idea was that, in analogy to QED, where one has an invariance of the action under local U (1) gauge transformations of the electron field

ψ (x ) → eiλ(x )ψ (x ),

(6.1)

one should require that the action for the quantum field theory of quarks should be invariant under arbitrary local non-abelian SU (3) transformations of the quark fields

1

The works of Greenberg, Han, and Nambu were preceded by a decade by the seminal work of Yang and Mills who formulated pure SU (N ) gauge theory [6]. Their groundbreaking contribution should not be forgotten and I kind of hate to relegate it to a footnote.

doi:10.1088/2053-2571/ab3108ch6

6-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

eiαa(x )ta ψ (x ), ψ (x ) → 

(6.2)

≡ Ω(x )

where αa(x ) are sight independent functions of spacetime, ta are the SU (3) generators in the fundamental representation, and the sum over a is implied. In this picture (fundamental representation), the quark fields are represented as three-dimensional column vectors in color space, with each row mapping to a particular color. As we learned from QED, it does not suffice to have only a fermion field. One needs a gauge field coupled to the electron field in order to maintain local gauge invariance. In QED, the gauge field is the photon field and one ‘gauges’ the theory by taking the normal partial derivative appearing in the kinetic part of the QED Lagrangian and promote it to be a covariant derivative via

∂μ → Dμ ≡ ∂μ − i eAμ ,

(6.3)

and then adding the action for the gauge fields themselves to obtain

LQED = ψ (i D − m)ψ −

1 FμνF μν. 4

(6.4)

As we have learned, the QED Lagrangian is invariant under the simultaneous gauge transformation of the gauge and electron fields

Aμ → Aμ +

1 ∂μλ(x ), e

ψ → eiλ(x )ψ .

(6.5) (6.6)

In addition, in QED we know that the field strength tensor is F μν = ∂ μAν − ∂ νAμ. For our extension to quantum chromodynamics (QCDs), it is useful to note that this can formally be written as

i F μν = [D μ, D ν ] e = [∂ μ − i eAμ , ∂ ν − i eAν ] = ∂ μAν − ∂ νAμ ,

(6.7)

where I have used the fact that in QED Aμ commutes with itself. Therefore, −i eF μν = [D μ, D ν ].2 Turning to QCD, since the quark fields are three-dimensional column vectors, when we gauge the field, this means that the gauge field itself must be a 3 × 3 matrix, since it will eventually be sandwiched between a ψ and ψ in the Lagrangian density. In QCD, one has

Dμ ≡ ∂μ − igAμ ,

2

(6.8)

This relationship should be considered as an operator relationship acting on an arbitrary function f .

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Relativistic Quantum Field Theory, Volume 2

where Aμ now represents the matrix-valued gluon field, g has been introduced as the coupling constant for the interaction of quarks and gluons, and there is an implicit 3 × 3 identity matrix multiplying the first term on the right-hand side. The covariantized quark contribution to the QCD Lagrangian density will therefore take the form

LQCD,quark = ψ (iγ μDμ − m)ψ .

(6.9)

In order for this contribution to be invariant under an arbitrary SU (3) gauge transformation Ω(x ) ≡ exp(iαa(x )ta ), it is obvious from the above expression that Dμψ has to transform in the same way as the quark field itself, that is

Dμψ → ΩDμψ ,

(6.10)

since ψ → ψ Ω† = ψ Ω−1 under a gauge transformation. As a result, this allows us to determine the gauge transformation rule for our non-abelian gauge field. To determine the rule, we evaluate

→ → Ωψ Dμ(Ωψ ) Dμψ ⎯ψ⎯⎯⎯⎯⎯⎯ = ∂μ(Ωψ ) − i eAμ Ωψ = Ω∂μψ + (∂μΩ)ψ − i eAμ Ωψ .

(6.11)

The second term on the right is the same as one would have obtained in QED. In QED, Ω = exp(iλ(x )) and this term would be cancelled by the gauge transformation of the gauge field Aμ → Aμ − ei (∂μΩ)Ω−1 = Aμ + 1e ∂μλ(x ). In order for the Ω in the third term to be moved to left, the full non-abelian gauge transformation required is

Aμ → ΩAμ Ω−1 −

i (∂μΩ)Ω−1. g

(6.12)

As in QED, the second term cancels the second term on the right-hand side of equation (6.11). And the form of the first term guarantees that equation (6.10) is satisfied. In the abelian case, this trivially reduces to the old rule for gauge transformations of the photon and electron fields since the gauge field commutes with Ω. With these two transformations, we are now in a position to add the contribution of the gluons to the QCD Lagrangian density. To do this, we use −igF μν = [D μ, D ν ] to obtain

F μν = ∂ μAν − ∂ νAμ − ig[Aμ , Aν ],

(6.13)

where now, like the vector potential itself, the field-strength tensor F μν is a 3 × 3 matrix. To make this explicit, we can expand the vector potential in terms of the SU (3) generators Aμ = A μa ta .3 This maps to an expansion in terms of eight different

3 The position of color indices does not indicate anything important and can be chosen at will in order to satisfy your aesthetic whims.

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Relativistic Quantum Field Theory, Volume 2

scalars, which can be thought of as eight different gluon fields. Likewise, we can expand F μν = Faμνta to obtain the field strength for each of the eight gluon fields

(

)

Fcμνtc = ∂ μAcν − ∂ νAcμ tc − igA μa Aνb [ta , tb ].

(6.14)

Using the commutation relation [ta, tb ] = ifabc tc , we obtain

(

)

Fcμνtc = ∂ μAcν − ∂ νAcμ tc + g fabcA μa Aνb tc .

(6.15)

To extract the coefficient Fcμν we can use the orthonormality relation Tr tatb = 12 δab to obtain

Fdμν = ∂ μAdν − ∂ νAdμ + g fabd A μa Aνb ,

(6.16)

and using the cyclicity of fabc and relabeling d → a , a → b, and b → c , giving finally

Faμν = ∂ μAaν − ∂ νAaμ + g fabc A μb Aνc .

(6.17)

To obtain the gluonic contribution to the Lagrangian density, we simply have to sum up the eight ‘copies’ of the electromagnetic field

1 a . LQCD,gluon = LYang−Mills = − FaμνF μν 4

(6.18)

Using the orthonormality relation for the generators in the fundamental representation, this can be compactly expressed as

1 LQCD,gluon = − Tr[F μνFμν ]. 2

(6.19)

One can verify that this term by itself is invariant under equation (6.12) (homework). Putting the pieces together and summarizing, we obtain the total Lagrangian density for QCD Nf

LQCD =

∑ψi (i D

− mi )ψi −

i=1

1 Tr[F μνFμν ] 2

(6.20)

where I have introduced Nf flavors of quarks with masses mi and Dμ ≡ ∂μ − igAμ. The QCD Lagrangian is invariant under local SU (3)-color gauge transformations of the quark and gluon fields

ψ (x ) → Ωψ (x ), Aμ → ΩAμ Ω−1 −

i (∂μΩ)Ω−1 ≡ g

Ω

Aμ ,

(6.21)

with Ω(x ) ≡ exp(iαa(x )ta ). Note that the form written obtain holds for an arbitrary integer number of colors; we would simply have to replace Ω by an SU (Nc ) transformation and the treatment

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Relativistic Quantum Field Theory, Volume 2

presented above would go through unchanged. In this case, we would have Nc2 − 1 different gluon fields, but the mathematical structure would remain the same. Exercise 6.1 Show that equation (6.19) is invariant under equation (6.12).

6.1 The Fadeev–Popov method The quantization of non-abelian gauge theories was made possible by Fadeev and Popov in 1967 who showed how to incorporate gauge fixing into the path integral [7]. As mentioned previously, one runs into a problem when you attempt to write down a path integral formulation of a gauge theory because the path integral naively includes an integration over an infinite number of gauge-equivalent copies of the fields. These copies result in an overall multiplicative infinity if one is not careful. The idea of Fadeev and Popov was to factorize the path-integral over the gauge field into a part that involved integration over gauge transformations and a part in the ‘fundamental sector’. By doing so in a well-defined manner, the integration over the ‘gauge transformation component’ cancels between the numerator and denominator in the generating functional. For this purpose, we can focus on the gauge sector which has

S=−

1 4

∫x FaμνF μνa ,

(6.22)

where, for generality, we consider a = 1, …, Nc2 − 1. To proceed, we introduce a general gauge condition

f a (A μb ) = 0,

(6.23)

where a and b are color indices. For compactness, below we will drop the color and Lorentz indices on f and A unless they are explicitly needed. As discussed in the last lecture, for all compact Lie groups G , there exists an invariant Haar measure dU with U ∈ G such that, for fixed V ∈ G , dU = d (VU ) = d (UV ). Based on this observation, we introduce the quantity

Δ−1[A] ≡

∫ DUδ[ f (UA)],

(6.24)

where the square brackets in the δ function indicate that it is a delta functional which is a product of Dirac delta functions at each point in space-time. This quantity represents a path integral over all possible gauge-transformed versions of A with the gauge condition enforced at all space-time points on the gauge-transformed A. Due to the invariance of the Haar measure, Δ−1[A] is gauge invariant

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Relativistic Quantum Field Theory, Volume 2

∫ DUδ[ f (UVA)] = ∫ D(UV )δ[ f (UVA)] = ∫ DWδ[ f (WA)]

Δ−1[VA] =

(6.25)

= Δ−1[A], where, in going from the second to third lines, I simply relabeled UV → W . Based on this, we can define a gauge-invariant decomposition of unity

∫ DUδ[ f (UA)].

1 = Δ[A]

(6.26)

To proceed, we consider infinitesimal gauge transformations. Rescaling A → A/g in equation (6.21) in order to make the notation more compact, we find that the gauge transformed field subject to U is U

Aμ = UAμ U −1 − i (∂μU )U −1,

(6.27)

with U ≡ exp(iαa(x )ta ). For an infinitesimal transformation, one has

U (α⃗ ) = 1 + iαata + O(α 2 ).

(6.28)

As a result, for U ≈ 1, the group measure dU can be expressed as N

dU =

∏

dαa = dα⃗ .

(6.29)

a=1

Additionally, for an infinitesimal transformation, equation (6.27) becomes

( αA)aμ ≡ A μa + ∂μα a + fabc A μb αc

(

)

≡ A μa + δac ∂μα a + fabc A μb αc .

(6.30)

This can be simplified further by noting that, in the adjoint representation, one has A μac = A μb (T b )ac and from equation (6.8)

(Dμϕ)a = ∂μϕa − igA μb (T b)ac ϕc = ∂μϕa − g fbac A μb ϕc

(

)

= δac ∂μ + g fabc A μb ϕc ,  

(6.31)

ac

≡ D˜μ

where the tilde indicates that the derivative is computed in the adjoint representaac tion. This can be summarized as (Dμϕ )a = D˜μ ϕc = (D˜μϕ )a . Rescaling A → A/g as ac before we have D˜μ = δac ∂μ + fabc A μb , which allows us to write equation (6.30) compactly as

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Relativistic Quantum Field Theory, Volume 2

ac ( αA)aμ = A μa + D˜ μ αc .

(6.32)

As a result, in the infinitesimal limit, the decomposition of unity becomes

∫ Dαδ⃗ [f ( αA)].

1 = Δ[A]

(6.33)

We can now insert this decomposition of unity into the path integral over gauge fields

∫ DAeiS = ∫ DAΔ[A]∫ Dαδ⃗ [( αA)]eiS .

(6.34)

Exploiting the gauge invariance of Δ[A] and S , we can make a gauge transformation from αA to A

∫ DAeiS = ∫ DAΔ[A]∫ Dαδ⃗ [f (A)]eiS = ∫ DAΔ[A]δ[f (A)]eiS ∫ Dα⃗ ,

(6.35)

where, in going from the first to second lines, we made it explicit that the quantities in front do not depend on α⃗ . The resulting overall (infinite) multiplicative factor of ∫ Dα⃗ is now factorized in a well-defined manner. Since such a factor will cancel in the generating functional, we can redefine Z = Z[0] to be

Z=

∫ DAΔ[A]δ[f (A)]eiS .

(6.36)

Next we turn to the evaluation of Δ[A]. For infinitesimal transformations, using equation (6.32) one finds

f a ( αA) = f a (A) +

∂f a δA μb ∂A μb

∂f a ˜ bc c Dμ α , f (A) + =  ∂A μb

(6.37)

a

=0

where the first term is zero due to the gauge constraint (6.23). As a result, from equation (6.33) we have

Δ−1[A] =

⎡

a

⎤

∫ Dα aδ⎢⎢⎣ ∂∂Af b D˜μbcαc(y )⎥⎥⎦.

(6.38)

μ

In order to evaluate the path integral of the delta functional, we define a matrix operator M

Mac(x ) ≡

∂f a ˜ bc Dμ , ∂A μb

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(6.39)

Relativistic Quantum Field Theory, Volume 2

which satisfies (homework)

δf a [A(x )] . δα c(y )

Mac(x )δ 4(x − y ) ∝

(6.40)

If M has eigenfunctions ηac with associated eigenvalues λc , then

∑Mab(x)ηbc(x) = λcηac(x),

(6.41)

b

where there is no summation over c on the right-hand side. We now expand ωc(y ) in terms of the eigenfunctions of M

α c (y ) =

∑α d fcd (y ),

(6.42)

d

which can be used to find

Mac(y )α c(y ) =

∑α d λd f ad (y ).

(6.43)

d

As a result, Δ−1[A] can be written as

Δ−1[A] =

⎡

⎤

∫ Dα aδ⎢⎢⎣∑α d λd fad (y )⎥⎥⎦.

(6.44)

d

Defining ua(y ) = Σd α d λd f ad (y ), we can evaluate the path integral over the delta function to obtain

Δ−1[A] =

…) ∏ δ (u a ) ∫ Dua ∂∂((αu11,, αu22,, … ) a

=

∂(α1, α2 , …) ∂(u1, u2, …)

(6.45) u=0

∝ (λ1λ2⋯)−1. The product of the eigenvalues can be written as det M , giving us our final result

⎛ δf a [A(x )] ⎞ Δ[A] ∝ det M = det ⎜ ⎟, ⎝ δα c(y ) ⎠

(6.46)

where the proportionality constant can be discarded since it will cancel in the generating functional. Summarizing, we have found that for a gauge theory the generating functional with J = 0 can be written as

Z=

∫ DA det(M )δ[f (A)] eiS .

6-8

(6.47)

Relativistic Quantum Field Theory, Volume 2

Note that the remaining delta function simply enforces the gauge fixing constraint. Exercise 6.2 Show that equation (6.30) is correct. Exercise 6.3 Show that equations (6.39) and (6.40) are consistent. 6.1.1 General Lorenz gauge To make this concrete, let us consider a general LG with

f a [A] = ∂ μA μa − g a(x ),

(6.48)

where g a(x ) is arbitrary smooth function does not depend on A. With this choice, Z becomes

Z=

∫ DA det(M )δ⎡⎣∂ μAμa − g a(x)⎤⎦ eiS .

(6.49)

Since Z is gauge invariant, it cannot depend on the choice of g . As a result, the equation above holds even for a linear combination of different choices for gi a(x ), for example

∫ DA det(M )δ⎡⎣∂ μAμa − g1a(x)⎤⎦eiS + a2∫ DA det(M )δ⎡⎣∂ μA μa − g2a(x )⎤⎦eiS + ⋯ ,

Z = a1

(6.50)

where ai are constants. Since each of the terms is independent of the choice of, for example g1a(x ), each term above is a constant times the same result. As a result, this just reduces to the result we had before (6.49) times an overall constant N = a1 + a2 + ⋯. We can use this freedom to choose the weights and functionally integrate (continuous sum of an infinite number of possibilities) over all different possibilities by multiplying equation (6.49) by the path-integral operator

⎛

∫ Dg exp⎜⎝− 2iξ ∫x

⎞ ga2(x )⎟ , ⎠

(6.51)

where ξ is an arbitrary constant. Doing this and using the functional delta function to perform the integration over g one finds

Z=

⎛

⎡

⎤⎞

∫ DA det(M ) exp ⎜⎝i∫x ⎢⎣LYang−Mills − 21ξ (∂ μAμ)2⎥⎦⎟⎠.

(6.52)

Next, we use something we learned in our discussion of fermionic fields. In equation (3.30) we learned that

⎛

⎞

∫ DυDυ exp ⎝i∫x υ aMabυb⎠ = det(−iM ) ∝ det M , ⎜

⎟

6-9

(6.53)

Relativistic Quantum Field Theory, Volume 2

where υ and υ are independent Grassmann variables and I have adjusted the notation a bit to conform to our case. We can use this to write the determinant appearing in equation (6.52) as an integral over Grassmann-valued ghost fields υ and υ . The result is

Z=

⎛

⎞

∫ DADυDυ exp ⎝i∫x Leff ⎠, ⎜

⎟

(6.54)

with

1 μ 2 (∂ Aμ ) 2ξ + Lgaugefixing .

Leff = LYang−Mills + υ a Mabυ b − = LYang−Mills + Lghost

(6.55)

The only thing left to do is to evaluate Mab . Using equation (6.39) we see that we must evaluate

δf a [A(x )] = δα c(y )

δA b ( z )

a

∫z δfδA[Ab((zx))] δανc(y ) ,

(6.56)

ν

where no sum over ν is implied. From equation (6.48) we have

δf a [A(x )] = δ ab∂ νδ 4(x − z ), δAνb (z )

(6.57)

and from equation (6.32) we have

δAνb (z ) bc = D˜ν δ 4(z − y ). δα c(y )

(6.58)

Plugging these into equation (6.56) and integrating by parts, we obtain

δf a [A(x )] ac = −∂ νD˜ν δ 4(x − y ). c δα (y )

(6.59)

This allows us to identify ab

Mab = − ∂ νD˜ν

= − ∂ ν(δab∂ν − fabc Aνc ) = − δab□ +

fabc ∂ νAνc

+

(6.60) fabc Aνc ∂ ν.

From this, we see that Mab will generate a kinetic energy term for the ghost fields and coupling with the gluon field. Inserting this into the general form, we obtain finally

1 1 μ 2 a Leff = − FaμνF μν − υ a □υ a + g fabc Aνc υ a ∂ νυ b − (∂ Aμ ) , 4 2ξ

(6.61)

where I have integrated the second term coming from Mab by parts and rescaled A → gA in the Mab contribution to undo the rescaling we made before.

6-10

Relativistic Quantum Field Theory, Volume 2

Finally, I mention that, if we were to repeat the same exercise in QED which is abelian, then only the term −υ □ υ would have been generated and, since this does not couple to the other fields, we could integrate the ghost fields out by performing the trivial Gaussian path-integral and absorb the result into the normalization of Z . For this reason, QED in general LG is ghost free. This is not necessarily the case in other gauges. With that under our belts, we can now write down the QCD Lagrangian in general LG including the ghost fields and gauge fixing term. This will allow us to proceed in establishing the Feynman rules necessary. 6.1.2 Summary We learned in the last lecture that, after implementing general LG fixing, some new effective degrees of freedom appear in the Lagrangian density called ghosts. These degrees of freedom are not physical and will not be coupled to sources4. What we learned in the end was that, in general LG, one has Nf

LQCD = ∑ψi (i D − mi )ψi − i=1 a

a

− υ □υ + g

1 Tr[F μνFμν ] 2

fabc Aνc υ a ∂ νυ b

(6.62)

1 μ 2 (∂ Aμ ) . − 2ξ

The first term contains a sum over Nf flavors of quarks with masses mi and Dμ = ∂μ − igAμ is the matrix-valued covariant derivative which includes the coupling between the quarks and gluons. The next term corresponds to eight ‘copies’ of the field strength tensor, expressed compactly through the trace; however, these ‘copies’ are complicated because of the non-commutative nature of the matrix-valued gluon vector potential. This results in a non-abelian field strength tensor

F μν = ∂ μAν − ∂ νAμ − ig[Aμ , Aν ].

(6.63)

The next two terms in equation (6.62) map to a propagator for the Grassmannvalued (fermionic) ghost fields υ and υ and a Yukawa-like interaction between the ghost fields and the gluon field. The last term is the gauge-fixing term appropriate for general LG, with ξ being a free parameter upon which physically measurable quantities should not depend. As in QED, the gauge-fixing term ensures that the propagator is well-defined (invertible inverse propagator). With this in hand, we can obtain the action for QCD, SQCD = ∫ LQCD and, x hence, the generating functional for QCD

Z ⎡⎣ η , η , J μ⎤⎦ =

∫ D⎡⎣ψ , ψ , Aμ , υ , υ ] exp (iSQCD + ∫x ⎡⎣ηψ + ψ η + JμAμ ⎤⎦) ∫ D⎡⎣ψ , ψ , Aμ , υ , υ ⎤⎦ exp (iSQCD)

. (6.64)

4 An exception will be during the derivation of the Slavnov–Taylor identities, in which case we introduce ghost sources as a formal device.

6-11

Relativistic Quantum Field Theory, Volume 2

6.2 QCD Feynman rules Before stating the Feynman rules, I would like to demonstrate how to derive the momentum-space Feynman rules directly from the action. I do so explicitly only for some simple cases and leave the details for the others as homework. 6.2.1 Bare quark propagator The quadratic quark contribution to the QCD action for a single flavor is of the form (2) SQCD, ψψ =

∫x ψa(x)(i ∂ − m)ψa(x),

(6.65)

where a = 1, …, Nc . We first write ψ (x ) and ψ (x ) as

ψa(x ) =

∫p ψa(p)e−ip·x ,

ψa(x ) =

∫k

(6.66)

ψa(k )eik·x ,

where ∫ =∫ d 4p/(2π )4 and a · b ≡ a μbμ. This gives p

(2) SQCD, ψψ =

∫p ∫k ∫x ψa(k )eik·x(i ∂ − m)e−ip·xψa( p) ⎛

⎞

=

∫p ∫k ⎝∫x e−ix·(p−k )⎠ψa(k )( p − m)ψa( p)

=

∫p ψa(p)( p − m)ψa( p),

⎜

⎟

where, in going from the second to third lines, we used

∫x

(6.67)

exp( −ix · ( p − k )) =

(2π )4δ 4( p − k ) and performed the k integration using the delta function. In doing so, we have obtained the momentum-space Lagrangian density from which we can obtain

(

SFab( p1,

p2 )

−1

)

=

(2) δ 2SQCD, ψψ

δψa(p1 )δψb(p2 )

= ( p 2 − m)δ abδ 4(p1 − p2 ),

(6.68)

which gives us our first Feynman rule

where we have suppressed the delta-function corresponding to energy-momentum conservation. It is now implicit. We also simplified the notation to one argument since p1 = p2 in practice.

6-12

Relativistic Quantum Field Theory, Volume 2

6.2.2 Bare gluon propagator An exactly analogous procedure can be followed for the bare gluon propagator, giving the rule

Exercise 6.4 Derive the Feynman rule for the bare gluon propagator (6.70).

6.2.3 Bare ghost propagator Again, the same procedure can be used here to obtain

Exercise 6.5 Derive the Feynman rule for the bare ghost propagator (6.71).

6.2.4 Quark-gluon vertex The relevant contribution to the action is now (3) SQCD, ψ Aψ = g

∫x ψ (x) A (x)ψ (x).

(6.72)

Expanding the gauge field in the fundamental representation A (x ) = A a (x )t a = Aaμ(x )γμt a and making the Fourier decomposition, one obtains

∫p ∫p ψ (p1 + p2 )Aaμ(p2 )ψ (p1).

(3) a SQCD, ψ Aψ = gt γμ

1

(6.73)

2

Evaluating the necessary functional derivatives gives

Γ 0b,ν(p1 ,

p2 , p3 ) ≡

(3) δSQCD, ψ Aψ

δψc(p3 )δAbν (p2 )δψa(p1 )

This maps to

6-13

= gt bγνδ 4(p1 + p2 − p3 ).

(6.74)

Relativistic Quantum Field Theory, Volume 2

Exercise 6.6 Derive the Feynman rule for the bare quark-gluon vertex (6.75). 6.2.5 Gluon self-interactions The gluon self-interactions come from the commutator terms in

−

1 2

∫x Tr[F μνFμν ] = − 14 ∫x FaμνF μνa = − 14 ∫x (∂ μAaν − ∂ νAaμ + gfabc AbμAcν )2 .

(6.76)

From this, two types of non-abelian terms will be generated, for example

1 μ ν ∂ Aa − ∂ νAaμ gfabc A μb Aνc = −gfabc ∂ μAaν A μb Aνc , 2

(6.77)

1 1 gfabc A μb Aνc gfab ′ c ′ Abμ′ Acν′ = − g 2feab fecd A μa Aνb Acμ Adν . 4 4

(6.78)

−

(

)(

)

(

)

and

−

(

)(

)

The first type will generate the three-gluon coupling and the second type with generate the four-gluon coupling. Skipping the intermediate steps, the result for the three- and four-gluon vertices are

Exercise 6.7 Derive the Feynman rule for the three- (6.79) and four-gluon vertices (6.80). 6.2.6 Ghost-gluon vertex Again, following a similar procedure, we can extract the ghost-gluon vertex, which comes from the Yukawa-like interaction term between the ghost fields and the gluon field. The result is (homework) 6-14

Relativistic Quantum Field Theory, Volume 2

Exercise 6.8 Derive the Feynman rule for the ghost-gluon vertex (6.81). 6.2.7 Closed ghost and quark loops As in QED, if there are closed fermionic loops (quark or ghost), one has to multiply by a factor of −1 for every closed loop5.

6.3 Simple example application of the QCD Feynman rules Since there are now complications involving color structure constants etc, let us do a quick example of how to compute the invariant amplitude for a simple graph from these rules. As our example, let us consider the gluon polarization graph stemming from a loop involving only one quark flavor

Note that, in this graph, the external gluon legs are just there visually since the selfenergy (polarization function in this case) is defined without legs. Putting the pieces together, we obtain ⎡ ⎤ i i μν b 2 a ⎥ i Π ab Tr⎢γ μ γν ,quark–loop(p ) = ( − 1)(ig ) (t )dc (t )cd k ⎣ k − m k − p − m⎦ ⎡ ⎤ 1 1 ⎥ (6.83) Tr⎢γ μ γν = − g 2 Tr[t at b ] k ⎣ k − m k − p − m⎦ ⎡ ⎤ 1 1 g2 ⎥, Tr⎢γ μ γν = − δ ab 2 k ⎣ k − m k − p − m⎦

∫

∫

∫

where we have used Tr[t at b ] = δ ab /2 in color space and the remaining trace above is in spinor space. Apart from the overall factor of δ ab /2, this is exactly the same result we obtained in QED in volume 1. 5

This was derived in chapter 3.

6-15

Relativistic Quantum Field Theory, Volume 2

6.4 Becchi, Rouet, Stora, and Tyutin symmetry In QED, we derived the Ward–Takahashi identities (WTIs) by requiring that the generating functional Z was gauge-invariant even though, subject to the gauge fixing term, the Lagrangian density is not. For QED in the LG, there are no ghost fields required for the gauge fixing to be well-defined, but in Lorenz-gauge QCD things are more complicated and we cannot avoid them. Knowing how to deal with the ghosts formally, we will now try to generalize the WTIs to QCD and obtain an analogous set of identities called the Slavnov–Taylor identities (STIs). First, however, we need to understand a symmetry of the QCD effective Lagrangian found by Becchi, Rouet, and Stora and, independently, Tyutin, which will be integral in obtaining the STI [8–10]. For this purpose, let us again focus on the pure gauge sector with

⎛

⎞

∫ D(Aμ , υ, υ )exp ⎝i∫x L⎠,

Z=N

⎜

⎟

(6.84)

and L = LYang–Mills + Lghost + Lgauge fixing . BRST found that, if one makes a gauge transformation parameterized by

Ωa = −υ a λ ,

(6.85)

where both υ and λ are Grassmann-valued quantities with (υa )2 = λ2 = 0 and λ is a constant and requiring additionally that

1 δυ a = − fabc υ bυc λ , 2

(6.86)

1 (∂μAaμ )λ , gξ

(6.87)

δυ a = −

(6.88)

δλ = 0,

one finds that the QCD effective Lagrangian density (6.62) is invariant. This transformation is called a BRST transformation and the invariance under such transformations is called BRST symmetry. We will now prove that L is invariant under BRST transformations. To begin, we note that the pure-gauge term −(1/2)Tr[F μνFμν ] is invariant since equation (6.85) simply represents a particular type of gauge transformation and this term is invariant under all types of SU (N ) gauge transformations. Considering the gauge fixing term Lgauge fixing = −(1/2ξ )(∂ μAμ )2 , one finds

δ Lgauge fixing = −

1 μ a ν ∂ A μ )∂ δAνa . ( ξ

(

)

(6.89)

Using equation (6.32), for an infinitesimal gauge transformation, one has

δA μa =

1 ˜ ac c 1 ac Dμ Ω = − (D˜ μ υc )λ , g g

6-16

(6.90)

Relativistic Quantum Field Theory, Volume 2

where I have undone the scaling of A used to obtain equation (6.32) in order to reinstate g . Using this and equation (6.85), one has

δ Lgauge fixing =

1 μ a ν ˜ ac c (∂ Aμ )(∂ Dμ υ )λ. gξ

(6.91)

Next, we turn to the ghost field contribution to L. From the first line of equation (6.60), we can express Lghost compactly as ab

Lghost = −υ a ∂ νD˜ν υ b ,

(6.92)

which gives

(

)

ab ab δ Lghost = −(δυ a )∂ νD˜ν υ b − υ a ∂ νδ D˜ν υ b .

(6.93)

Using equation (6.87), the first term becomes

1 1 ab ab ∂μAaμ λ∂ νD˜ν υ b = − ∂μAaμ ∂ νD˜ν υ b λ , gξ gξ

(

)

(

)(

)

(6.94)

where we have used the fact that υa and λ anti-commute. As a result we find that

(

)

ab δ L = δ Lghost + δ Lgauge fixing = −υ a ∂ νδ D˜ν υ b .

(6.95)

ab Next, we focus on δ (D˜ν υ b ). Explicitly this is

(

) (

ab δ D˜ν υ b = δ ∂μυ a + g fabc Aνb υc

)

fabc δAνb υc

= ∂μ(δυ a ) + g + g fabc Aνb δυc (6.96) 1 1 bd = − fabc ∂μ υ bυc λ − fabc D˜ν υd λυc − g fabc fcde Aνb υd υ e λ 2 2 1 1 = − fabc ∂μ υ bυc λ − fabc ∂νυ b + fbde Aνd υ e λυc − g fabc fcde Aνb υd υ e λ , 2 2

( (

(

) )

)

(

)

bd where in going from the third to fourth lines, we used D˜ν = δ˜bd ∂ν + fbed Aνe and then relabeled the indices. Looking at the first term, we note that

fabc ∂μ υ bυc = fabc ⎡⎣(∂μυ b)υc + υ b(∂μυc )⎤⎦

(

)

= fabc ⎡⎣(∂μυ b)υc − (∂μυc )υ b⎤⎦ = 2fabc (∂μυ b)υc ,

6-17

(6.97)

Relativistic Quantum Field Theory, Volume 2

which gives,

1 ab δ D˜ν υ b = − fabc (∂μυ b)υc λ − fabc ∂νυ b + fbde Aνd υ e λυc − g fabc fcde Aνb υd υ e λ 2 1 d e c b c c = − fabc (∂μυ )  + λ λυ) + fabc fbde Aν υ υ λ − g fabc fcde Aνb υd υ e λ (6.98) (υ  2

(

)

(

)

=0

= fabc fbde Aνd υ e υc λ −

1 g f f Aνb υd υ e λ . 2 abc cde

Next we make use of the Jacobi identity (5.33) obeyed by the structure constants fabc fcde + fadc fceb + faec fcbd = 0 in the second term on the last line, to obtain

1 1 ab δ D˜ν υ b = fabc fbde Aνd υ e υc λ + g fadc fceb Aνb υd υ e λ + g faec fcbd Aνb υd υ e λ . 2 2

(

)

(6.99)

Relabeling indices here and there, using the anti-commuting nature of υ, and the anti-symmetry of fabc under exchanges of any two indices, one can show that the second two terms cancel the first, giving

(

)

ab δ D˜ν υ b = 0,

(6.100)

δ L = 0,

(6.101)

and, therefore, subject to a BRST transformation. Exercise 6.9 Show that equation (6.100) is correct.

6.5 Slavnov–Taylor identities As mentioned in a footnote a few pages ago, for the derivation of these identities, it is convenient to introduce a generating functional with sources that couple to the ghost fields as a formal device. We will call these two new sources x and y . In addition, it proves to be convenient to add two new sources χ and ς that couple to non-linear terms. For the Yang–Mills theory, this translates into

Z [J μ, x , y ; χ , ς ] =

⎛

⎞

∫ D(Aμ , υ, υ )exp ⎝i∫x Leff ⎠, ⎜

⎟

(6.102)

where

⎛ 1 ab ⎞ ⎛ 1 ⎞ Leff = LQCD + JμaAaμ + x aυ a + y a υ a + χμa ⎜ D˜ υ b⎟ + ς a⎜ − fabc υ bυc ⎟ , ⎝ 2 ⎠ ⎝g ⎠

(6.103)

with LQCD defined in equation (6.62). The coefficients of χ and ς are constructed in such a way as to be invariant under a BRST transformation. The invariance of the first follows from equation (6.100) and the invariance of the second δ ( fabc υ bυc ) = 0 6-18

Relativistic Quantum Field Theory, Volume 2

can be proven straightforwardly using the Jacobi identity. From this invariance and equation (6.86), it follows that δ 2υa = 0. Since LQCD and the two new source terms are invariant under BRST transformations, any change in the generating functional term will come from the other source terms and a possible Jacobian J induced by the transformation, that is, after a BRST transformation one would have

Z [J μ, x , y ; χ , ς ] =

∫ D(Aμ , υ, υ )J ⎛⎡ × exp ⎜i ⎢S + ⎝⎣

∫x (

JμaδAaμ

⎤⎞ + x δυ + y δυ )⎥⎟ , ⎦⎠ a

a

a

(6.104)

a

where

⎡ ⎛ A μa (x ) + δA μa (x ), υ a (x ) + δυ a (x ), υ a (x ) + δυ a (x ) ⎞⎤ det ⎢∂⎜ ⎟⎥ . ⎢⎣ ⎝ Aνb (y ), υ b(y ), υ b(y ) ⎠⎥⎦

J=

(6.105)

ac From equation (6.90) we see that δA μa = −(1/g )(D˜μ υc )λ = −(1/g )(∂μυa + g fab ′ c A μb ′υc )λ depends on A and υ, giving two non-vanishing contributions to the Jacobian

δ⎡⎣A μa + δA μa ⎤⎦

= δ μνδ 4(x − y )(δ ab − fabc υc λ) ,

(6.106)

1 = − δ 4(x − y ) δ ab∂μ + g fab ′ b Ab ′ λ = δ 4(x − y )fabc Ac λ , g

(6.107)

δAνb and

δ⎡⎣A μa + δA μa ⎤⎦ δυ

b

(

)

where, in obtaining the second relation, we used the right Grassmann derivative. Next, from equation (6.86) we have δυa = −(1/2)fade υd υeλ . As a result, there is another non-vanishing contribution of the form

δ[υ a + δυ a ] ⎛ 4 = ⎜δ (x − y )δ ab − ⎝ δυ b ⎛ = δ 4(x − y )⎜δ ab + ⎝

⎞ 1 δ f υ d υ e )λ ⎟ b ( ade ⎠ 2 δυ ⎞ 1 fabe υ e − fadb υd λ⎟ ⎠ 2

(

)

(6.108)

= δ 4(x − y )(δ ab + fabc υc λ) , where we have again used the right Grassmann derivative. Finally, from equation (6.87) we have

δ[υ a + δυ a ] = 0, δAνb

6-19

(6.109)

Relativistic Quantum Field Theory, Volume 2

and

δ[υ a + δυ a ] = δ 4(x − y )δ ab. δυ b

(6.110)

As a result, the Jacobian has the form

δ μν(δ ab − fabc υc λ)

fabc Ac λ

0

0

(δ ab + fabc υc λ)

0

0

0

δ ab

J = [δ (x − y )] 4

3

(6.111)

= δ abδ μν[δ 4(x − y )]3 , where we have used the fact that λ2 = 0. On a discrete spacetime grid, this translates into the Jacobian being unity. As a result,

Z [J μ, x , y ; χ , ς ] =

∫ D(Aμ , υ, υ ) ⎛⎡ × exp ⎜i ⎢S + ⎝⎣

=

⎤⎞

∫x (JμaδAaμ + x aδυa + y a δυ a )⎥⎦⎟⎠ ⎡

(6.112) ⎤

∫ D(Aμ , υ, υ )exp (iS )⎢⎣1 + ∫x (JμaδAaμ + x aδυa + y a δυ a )⎥⎦.

Requiring that Z is invariant implies that

∫ D(Aμ , υ, υ )exp (iS )∫x (JμaδAaμ + x aδυa + y a δυ a ) = 0.

(6.113)

Since the coefficients of each of these three terms map to the three variations given in equations (6.85)–(6.87), translating this into a functional differential equation gives

⎡

δZ ∫x ⎢⎢Jμa δχ a ⎣

+ xa

μ

⎤ 1 a ⎛ δZ ⎞⎥ δZ ⎜⎜∂μ a ⎟⎟ = 0, − y δζ a ξ ⎝ δJμ ⎠⎥⎦

(6.114)

and rewriting in terms of the generating functional for connected diagrams W = −i log Z , one obtains

⎡

∫x ⎢⎢Jμa δδχWa ⎣

μ

+ xa

⎤ 1 a ⎛ δW ⎞⎥ δW ⎜⎜∂μ a ⎟⎟ = 0, − y δζ a ξ ⎝ δJμ ⎠⎥⎦

(6.115)

since the relationship involves only first functional derivatives and we assume zero vacuum expectation value for the fields. Finally, we make the Legendre transform to the effective action in the three physical variables using the relationships

W [J μ, x , y ; χ , ς ] = Γ[Aμ, ϒ, ϒ; χ , ς ] +

6-20

∫x ⎡⎣A aμJaμ + x a ϒa + y a ϒa⎤⎦

(6.116)

Relativistic Quantum Field Theory, Volume 2

with

δW = ϒ = 〈υ 〉J ,x,y , δx δW = ϒ = 〈υ〉J ,x,y , δy δW = Aμ = 〈Aμ 〉J ,x,y , δJμ

δΓ = −y, δϒ δΓ = −x, δϒ δΓ = − J μ, δAμ

(6.117)

and

δW δΓ = , δχ δχ δW δΓ = . δζ δζ

(6.118)

As a result, equation (6.115) becomes

⎡

∫x ⎢⎢⎣ δδAΓaμ δχδ Γμ a

+

⎞⎤ δΓ δΓ δΓ ⎛ 1 μ ∂ − A ⎜ ⎟⎥ = 0. μ a ⎠⎥⎦ δ ϒa δζ a δ ϒa ⎝ ξ

(6.119)

Since this is independent of the spacetime support of the fields, it can be turned into a local relationship

⎞ δΓ δΓ δΓ δΓ δΓ ⎛ 1 μ ∂ − A ⎜ ⎟ = 0. μ a a μ + a ⎠ δ A μ δχa δ ϒa δζ a δϒ ⎝ ξ

(6.120)

This relation can be used to generate an entire hierarchy of relationships between the gauge field and ghost field vertex functions in QCD called the STIs for the ghostgluon sector. It is the non-abelian analogue of equation (4.52), which could be used to generate the hierarchy of relationships between the photon and electron vertex functions in QED. If we restrict our attention to the quark-gluon sector, then following the discussion in section (4.7), there is a non-abelian extension of the quark-gluon WTIs of the form (homework)

−

⎡ δΓ δΓ ⎤ 1⎡ μ μ ν ⎥ (tc )ij Ψj − Ψi (tc )ij ⎣ □∂μAc − g fabc (∂μAa )(∂νAb )⎤⎦ + ig⎢ ξ δ Ψj ⎦ ⎣ δ Ψi ⎛ δΓ ⎞ δΓ b + ∂μ⎜⎜ c ⎟⎟ − g fabc A μ = 0, δ A aμ ⎝ δA μ ⎠

(6.121)

where we have added the quark sources and fields to the generating functional and effective action, respectively. Following a similar procedure as we used to obtain equation (4.56), we obtain the QCD version of the WTI relating the exact quarkgluon vertex function with the exact inverse quark propagator (homework)

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Relativistic Quantum Field Theory, Volume 2

−1 −1 μ qμΓ aij (p , q , p + q ) = ⎡⎣S˜F , ik(p + q ) − S˜F , ik(p )⎤⎦(ta )kj ,

(6.122)

where a = 1, …, Nc2 − 1 is a color index in the adjoint representation and i , j = 1, …, Nc is a color index in the fundamental representation. It can be verified μ straightforwardly that the bare vertex function Γ aij (p, q, p + q ) = γ μ(ta )ij and bare inverse propagator SF−,1ij (p ) = δij ( p − m ) satisfy this relationship. This can be μ simplified further by defining Γ aij (p, q, p + q ) = Γμ(p, q, p + q )(ta )ij to obtain −1 −1 Γμ(p, q, p + q ) = γ μ and S˜F , ij = S˜F δij , which results in −1 −1 qμΓμ(p , q , p + q ) = S˜F (p + q ) − S˜F (p ).

(6.123)

Exercise 6.10 Show that the non-linear terms proportional to χ and ς in equation (6.103) are invariant under BRST transformation. Exercise 6.11 Derive equation (6.121). Exercise 6.12 Derive equation (6.122).

References [1] Gell-Mann M 1964 A schematic model of baryons and mesons Phys. Lett. 8 214–5 [2] Zweig G 1964 An SU(3) model for strong interaction symmetry and its breaking Version 1 CERN-TH−401 [3] Zweig G 1964 An SU(3) model for strong interaction symmetry and its breaking Version 2 Developments in the Quark Theory of Hadronsvol. 1 1964−1978 ed D Lichtenberg and S P Rosen (Cambridge: Cambridge University Press) pp 22–101 [4] Greenberg O W 1964 Spin and unitary spin independence in a paraquark model of baryons and mesons Phys. Rev. Lett. 13 598–602 [5] Han M Y and Nambu Y 1965 Three triplet model with double SU(3) symmetry Phys. Rev. 139 B1006–10 [6] Yang C N and Mills 1954 Conservation of isotopic spin and isotopic gauge invariance Phys. Rev. 96 191–5 [7] Faddeev L and Popov V 1967 Feynman diagrams for the Yang-Mills field Phys. Lett. B 25 29–30 [8] Becchi C, Rouet A and Stora R 1974 The abelian Higgs-Kibble model. Unitarity of the S-operator Phys. Lett. 52B 344–6 [9] Becchi C, Rouet A and Stora R 1976 Renormalization of gauge theories Annals Phys. 98 287–321 [10] Tyutin I V 1975 Gauge invariance in field theory and statistical physics in operator formalism LEBEDEV-75-39 (Preprint 0812.0580)

6-22

IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 7 Renormalization of QCD

As we learned in volume 1, when one evaluates the one-loop electron self-energy or photon polarization function, one finds that they are divergent. By regulating the divergences and introducing a finite number of counterterms in the bare Lagrangian, we were able to remove the divergences and obtain a result that was well defined as the regulator was removed. At one-loop order for QED we found

α( μ ) =

α( μ 0 ) , α( μ 0 ) μ2 log 2 1− 3π μ0

(7.1)

where α ≡ e2 /(4π ) is the electromagnetic fine structure constant. The consequence of this result is that the effective fine structure constant is scale dependent, with α being a monotonically increasing function of the renormalization scale μ. We would now like to repeat this exercise for QCD, however, before embarking upon the explicit calculation, I will discuss the formalism of perturbative renormalization using our new understanding of the effective action and the central role played by the one-particle irreducible vertex functions. Then we will perform the detailed one-loop renormalization of QCD reproducing the results of the original seminal works of Gross, Wilczek, and Politzer [1, 2]. Before doing that, however, let us start with a simple scalar field theory in order to introduce the basic concepts.

7.1 Divergences in scalar field theories As we learned in volume 1, the leading correction to the propagator in λϕ4 theory is a divergent quantity. In momentum-space, we found

Since there are four powers of momentum in the numerator of the integrand and two

doi:10.1088/2053-2571/ab3108ch7

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ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

in the denominator, the integral will diverge quadratically in the ultraviolet. Similarly, at O(λ2 ) one obtains a correction to the four-point function of the form

where p1 and p2 are the incoming four-momenta. In this case, in the integrand we have four powers of the momentum in both the numerator and the denominator. This will result in a logarithm ultraviolet divergence. Rather than analyzing each graph, we would like to determine the degree of divergence of a given graph without having to write down the explicit integral expression. As we can see from the two examples above, each propagator contributes a power k −2 , each vertex contributions a power of k 4 , and then there is a delta function that contributes k −4 . If we were working in d -dimensions instead of four, then the last two would change to k d and k −d . The number of independent momenta which remain after the delta function is used is given by the number of loops L . In both examples above, the number of loops was one. Next consider a general graph that contributes at nth order in perturbation theory, such that there are n vertices. If the diagram has E external lines, I internal lines, and L loops, then the superficial degree of divergence D is given by

D = dL − 2I .

(7.4)

To proceed, we would like eliminate I and L in favor of n and E . If there are I internal momenta, there is momentum conservation at every vertex plus overall four-momentum conservation, implying n − 1 constraints. As a result there are I − n + 1 independent momenta involved at nth order in perturbation theory. Since the number of independent momenta is equal to the number of loops, we have

L = I − n + 1.

(7.5)

For λϕ4 theory each vertex has four legs, so there are 4n legs if we count both internal and external legs. If we are counting the total number legs, internal lines count twice since they are always connect to two vertices. As a result, one has

4n = E + 2I .

(7.6)

⎛d ⎞ D = d − ⎜ − 1⎟E + n(d − 4). ⎝2 ⎠

(7.7)

Combining these results, we find

If D < 0 then the integral is convergent. When d = 4, we obtain

D = 4 − E,

(7.8)

which indicates that all graphs with more than four external legs are convergent.

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Relativistic Quantum Field Theory, Volume 2

If d > 4 then the last term in equation (7.7) would be particularly worrisome since it would imply that, as we go to higher orders in perturbation theory, the degree of divergence would increase without bound so that when summed over n we would obtain an infinite number of divergences, each worse than the prior one. For ϕ4 theory in d = 4 we are lucky because this factor is zero and for d < 4 the degree of divergence would be a decreasing function of n. In both cases there would be only a finite number of divergent terms so there would be some hope that we could eliminate these divergences using a finite number of renormalization counterterms. If only a finite number of counterterms are necessary, the theory is said to be renormalizable. However, there is an additional requirement for renormalizability, namely that the counterterms necessary should also have the same algebraic structure as terms contained in the original bare Lagrangian. The analysis of the degree of divergence does not prove that the theory is renormalizable, but it is an indication that the situation is not hopeless. If we instead were to consider a theory with a ϕ ℓ interaction then equation (7.7) would become (homework)

⎛ℓ ⎞ ⎛d ⎞ D = d − ⎜ − 1⎟E + n⎜ (d − 2) − d ⎟ , ⎝2 ⎠ ⎝2 ⎠

(7.9)

which, for d = 4 becomes

D = 4 − E + n(ℓ − 4).

(7.10)

If ℓ = 6, that is ϕ6 theory, then the degree of divergence increases without bound as n increases. Such as theory is unrenormalizable. If ℓ = 3, that is ϕ3 theory, the degree of divergence decreases with increasing n and the theory is said to be super-renormalizable. Finally, we note that for d = 2, then D = 2 − 2n and so all types of scalar field theories are super-renormalizable in this case. Returning to the case of λϕ4 theory, equation (7.7) implies that all graphs with six external legs (E = 6) should be convergent, however, this is misleading and why D is called the superficial degree of divergence. For example, consider the three graphs shown below.

The first graph (a) is convergent, however, (b) and (c) are divergent because they both contain subgraphs that are divergent. The correct statement, which is called Weinberg’s theorem, says that a graph will be finite if its degree of divergence D and the degree of divergence of all its subgraphs is negative [3]. In (7.11) (b) and (c) we have indicated the two divergent subgraphs by outlining them with a dashed box. These divergences are called primitive divergences and correspond to divergences in the one-particle irreducible n -point functions. In ϕ4 theory, the boxes shown are the

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Relativistic Quantum Field Theory, Volume 2

only two primitive divergences. We can therefore focus our attention on them. These map to the 1PI two- and four-point vertex functions. Exercise 7.1 Show that equation (7.9) is correct.

7.2 Divergences in Yang–Mills theory Next, let us repeat this kind of analysis for pure-glue Yang–Mills theory to assess the possibility that this theory is renormalizable. By identifying the superficial degree of divergence, we will be able to at least find out if it is possible that the theory is renormalizable. For the Yang–Mills sector, there are three types of vertices corresponding to the three-gluon interaction, the four-gluon interaction, and the gluon-ghost interaction. Defining

n3g = number of three−gluon vertices, n 4g = number of four−gluon vertices, ngg = number of ghost−gluon vertices, Eg = number of external gluonlines, Ig = number of internal gluonlines, Igh = number of (internal) ghost lines,

(7.12)

we can determine the degree of divergence similarly to the scalar cases

D = 4L − 2(Ig + Igh ) + n3g + ngg ,

(7.13)

where the last two terms account for the fact that both the three-gluon and gluonghost vertices are linear in the external momentum. As in the case of the scalar theory, the number of loops can be determined by counting the number of independent integration variables. The result is

L = Ig + Igh − n3g − n 4g − ngg + 1 = Ig − n3g − n 4g + 1,

(7.14)

where we have used the fact that ngg = Igh . Since each three-gluon vertex has three legs, each four-gluon vertex four legs, and each gluon-ghost vertex has one gluon leg and two ghost legs we have

4n 4g + 3n3g + 3ngg = 2Ig + Eg + 2Igh.

(7.15)

Combining these results, one obtains

D = 4 − Eg ,

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(7.16)

Relativistic Quantum Field Theory, Volume 2

suggesting that Yang–Mills theory should be renormalizable. In the next section, we will evaluate the counterterms necessary to renormalize QCD at one-loop order. I will not attempt to make a general proof of the renormalizability of QCD1. Exercise 7.2 Show that equations (7.13) and (7.16) are correct.

7.3 Dimensional regularization refresher Since we will use dimensional regularization with d = 4 − 2ε space-time dimensions, we first need to remind ourselves about the dimensions of the fields and the coupling constant g . By examining the QCD Lagrangian, one finds

d−1 , 2 d [Aμ ] = − 1. 2 [ψ ] =

(7.17)

This means that we must multiply the coupling constant g by a factor of μ2−d /2 , where μ is the renormalization scale in dimensional regularization. So, when applying the Feynman rules, one must take g → gμ2−d /2 = gμε at every vertex. We will also need the rules for γ -matrices in d dimensions

{γ μ, γ ν} = 2η μν ,

(7.18)

γ μγμ = d,

(7.19)

γμγ νγ μ = (2 − d )γν,

(7.20)

Tr[odd # of γ matrices] = 0,

(7.21)

Tr[I ] = f (d )

f (4) = 4,

(7.22)

Tr[γ μγ ν ] = f (d )η μν ,

(7.23)

Tr[γ μγ νγ λγ σ ] = f (d )(η μνη λσ − η μλη νσ − η μσ η νλ ),

(7.24)

where f (d ) is an arbitrary function that satisfies f (4) = 4. Also note that in d -dimensions, Tr[η] = η μ μ = d . In what follows we will use the shorthand

1

For the gory details see for example refs. [4, 5].

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Relativistic Quantum Field Theory, Volume 2

⎛ e γE μ2 ⎞ε ≡⎜ ⎟ ⎝ 4π ⎠ k  ≡μ 2ε

∫

d

∫ (2dπk)d

d

= μ 2ε

∫ (2dπk)d .

(7.25)

Finally, I also note that, in dimensional regularization, an integral of the following form (scaleless power-law divergence) vanishes

∫k (k 2)n = 0,

(7.26)

with n = −1, 0, 1, …, ∞.

7.4 One-loop renormalization of QCD We will now embark on the calculation of the one-loop beta function for QCD. For this purpose, we will specialize to the Feynman gauge ξ = 1. 7.4.1 Quark wave function and mass renormalization We will start with the quark self-energy graph shown below

The factor of tctc is summed over c and reduces to the quadratic Casimir in the fundamental representation (5.56) times δ ab . For SU (Nc ), one has CF = (Nc2 − 1)/(2Nc ), giving μν

∫k γμ p − ki − m γν −kiη2

−i∑ (p ) = (ig )2 CF δ ab ab

.

(7.28)

Up to the overall factor of CF δ ab , this is exactly the same result that we obtained in QED in volume 1 chapter 5, so we can take the QED result multiply it by the overall Casimir. The final result is

∑ab (p) =

g 2CF δ ab ( − p + 4m) + O(ε 0), 16π 2ε

(7.29)

where m is the mass of one of the quarks. We will renormalize each quark flavor independently. From this, we learn that, to one-loop order the quark wave function renormalization is

Z2 = 1 −

g 2CF , 16π 2ε

and that the quark mass renormalization is

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(7.30)

Relativistic Quantum Field Theory, Volume 2

Zm = 1 −

g 2CF . 4π 2ε

(7.31)

7.4.2 Gluon wave function renormalization Next we turn to the gluon polarization graphs. There are four graphs in this class

Graph (a) is one that we already evaluated in section 6.3 and, for a single quark flavor with mass m, we found

where, once again, the only difference from the corresponding graph in QED is the overall factor of δ ab /2. Taking the result from volume 1 chapter 5 on the renormalization of QED, and summing over the Nf flavors of the quarks, we obtain μν

∏ab,(a ) (p) =

g 2Nf δ ab μ ν (p p − η μνp 2 ) + O(ε 0). 24π 2ε

Next, we consider to the ghost loop. Using the Feynman rules, it is

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(7.33)

Relativistic Quantum Field Theory, Volume 2

where the −1 takes into account that the ghost loop is a Grassmann loop and we have used the fact that fabc = (T a )bc , where T a is an SU (Nc ) generator in the adjoint representation. Using equation (5.55), one has Tr[T aT b ] = CAδ ab = Ncδ ab giving

i∏

μ

μν ab,(b )

ν

∫k kk 2((kk −− pp))2 .

(p ) = g 2Ncδ ab

(7.35)

Evaluating the integral and extracting the divergent part gives (homework) μν

∏ab,(b) (p) =

⎞ g 2Ncδ ab ⎛ 1 μ ν 1 ⎜ p p + η μνp 2 ⎟ + O(ε 0). 2 ⎝ ⎠ 32π ε 3 6

(7.36)

Turning to graph (c) involving the gluon three-point self-interaction we obtain

where the 1/2 is a symmetry factor and we have used equation (6.79) to obtain

G μν ≡ [η μγ (2p − k )α + η γα(2k − p ) μ − η αμ(p + k )γ ] × [ −η ν α(p + k )γ + ηαγ (2k − p )ν + ηδ ν(2p − k )α ].

(7.38)

Simplifying using the fact that Tr[η] = η μ μ = d and introducing some Feynman parameters, we can perform the integration and extract the divergent terms, with the result being (homework) μν

∏ab,(c ) (p) = −

g 2Ncδ ab ⎛ 11 μ ν 19 μν 2 ⎞ ⎜ p p − η p ⎟ + O(ε 0). 2 ⎝ ⎠ 32π ε 3 6

(7.39)

Turning, finally, to graph (d) we find (homework)

where the fact that this is zero follows from equation (7.26). Combining the three non-vanishing results, we obtain μν

μν

μν

g 2δ ab

∏ab,(a ) + ∏ab,(b) + ∏ab,(c ) = 16π 2ε (η μνp2

⎛ 5N 2Nf ⎞ − p μ p ν )⎜ c − ⎟. ⎝ 3 3 ⎠

(7.41)

This implies that the gluon wavefunction renormalization is

Z3 = 1 +

2Nf ⎞ g 2 ⎛ 5Nc − ⎜ ⎟. 2 ⎝ 16π ε 3 3 ⎠

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(7.42)

Relativistic Quantum Field Theory, Volume 2

Exercise 7.3 Show that equations (7.36), (7.39), and (7.40) are correct.

7.4.3 Vertex renormalization Finally, we must evaluate the vertex renormalization. There are two leading-order corrections

The vertex correction (a) is similar to the leading-order modification of the QED vertex. Using the Feynman rules one obtains a f 3 f ig Λ abc μ,(a )(p , q − p , q ) = (ig ) (t )bd (t )de (t )ec λν

×

∫k γν q − ki − m γμ p − ki − m γλ −kiη2

(7.44)

(p , q − p , q ). = (t f t at f )bc Λ QED μ The group theory factor can be written as

t f t at f = t f [t a , t f ] + t f t f t a = ifafg t f t g + CF t a ,

(7.45)

or, equivalently,

t f t at f = [t f , t a ]t f + t at f t f = −ifafg t gt f + CF t a ,

(7.46)

where CF = C2(F ) = (Nc2 − 1)/(2Nc ). Adding these two and dividing by two, we obtain

i t f t at f = fafg [t g, t f ] + CF t a 2 1 = − fafg fgfh t h + CF t a 2  Tr[T aT h ] ⎛ N ⎞ = ⎜ − c + CF ⎟t a . ⎝ 2 ⎠ For the QED-like vertex, in chapter 5 of volume 1, we obtained

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(7.47)

Relativistic Quantum Field Theory, Volume 2

Λ QED (p , q − p , q ) = μ

e2 γμ + O(ε 0), 16π 2ε

(7.48)

which gives us, for vertex graph (a),

⎛ Nc ⎞ g2 ⎜ Λ abc + CF ⎟ (t a )bc γμ + O(ε 0). μ,(a )(p , q − p , q ) = − ⎝ 2 ⎠ 16π 2ε

(7.49)

Turning, finally, to the vertex graph (b) d e 3 ig Λ abc μ,(b )(p , q − p , q ) = − i (ig ) fade (t t )bc i −i −i γ γ V μνρ × 2 2 ρ k −m ν k (p − k ) (q − k )

∫

(7.50) = − (ig )3fade (t d t e )bc V μνργρ( k + m)γν

∫k (p − k )2(q − k )2(k 2 − m2) , where,

V μνρ ≡ η μν(k + q − 2p ) ρ + η νρ(p + q − 2k ) μ + η μρ(p + k − 2q )ν .

(7.51)

The color factor can be evaluated using

fade t d t e = fade [t d , t e ] + fade t et d = ifade fdeg t g + fade t et d .

(7.52)

Alternatively, the last line can be written as

ifade fdeg t g − fade t d t e .

(7.53)

Adding these two results, we obtain

2fade t d t e = 2ifade fdeg t g + ifade fedg t g = ifade fdeg t g.

(7.54)

As a result,

fade t d t e =

i iN Tr[T aT g ]t g = c t a . 2 2

(7.55)

Evaluating the integral and extracting the divergent contribution gives (homework)

Λ abc μ,(b )(p , q − p , q ) =

3g 2Nc γ (t a )bc + O(ε 0). 32π 2ε μ

Combining the two vertex contributions, we obtain

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(7.56)

Relativistic Quantum Field Theory, Volume 2

Λ aμ,(b)(p , q − p , q ) + Λ aμ,(b)(p , q − p , q ) =

g2 (Nc + CF )γμt a . 16π 2ε

(7.57)

This tells us that the vertex renormalization is

Z1 = 1 −

g2 (Nc + CF ). 16π 2ε

(7.58)

Exercise 7.4 Show that equation (7.56) is correct.

7.4.4 Collection of results From our analysis of the one-loop quark self-energies, gluon polarization tensors, and vertex corrections, we have determined all of the necessary renormalization constants

g2 (Nc + CF ) 16π 2ε g 2CF Z2 = 1 − 16π 2ε 2Nf ⎞ g 2 ⎛ 5Nc Z3 = 1 + − ⎜ ⎟ 2 ⎝ 16π ε 3 3 ⎠ Z1 = 1 −

Zm = 1 −

vertexre normalization, quark wave function renormalization, (7.59) gluon wave function renormalization,

g 2CF 4π 2ε

quark mass renormalization.

7.4.5 The renormalized QCD Lagrangian Combining the three pieces determined above, we obtain the full bare (B) Lagrangian for QCD a L QCD,B = Z2ψ p ψ − mZmψ ψ + gZ1ψ γ μψAμ − (Z3/4)FaμνF μν

+ ghost and gauge fixing terms.

(7.60)

In order to absorb these infinities in the bare Lagrangian, we introduce rescaled fields

ψB =

Z2 ψ ,

(7.61)

ABμ =

Z3 Aμ ,

(7.62)

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Relativistic Quantum Field Theory, Volume 2

and write the Lagrangian in terms of these

⎛Z m⎞ LQCD,B = ψB p ψB − ⎜ m ⎟ ψBψB ⎝ Z ⎠ 2 ≡mB ⎛ gμ ε Z ⎞ 1 1 a ,B ⎟ ψBγ μψBA μB − Faμν,BF μν +⎜ 4 ⎝Z Z ⎠ 23 ≡gB 1 a ,B = ψB p ψB − mB ψBψB + gBψBγ μψBA μB − Faμν,BF μν , 4

(7.63)

where we have not listed the ghost and gauge fixing terms and we have introduced the scaled coupling and mass

gB ≡

gμ ε Z1 , Z2 Z3

(7.64)

Zmm . Z2

(7.65)

mB ≡

The fact that we were able to use these multiplicative rescalings and the Lagrangian ended up in the same form as the original QCD Lagrangian, proves the renormalizability of QCD. Note that, it may seem strange that we can rescale the fields in this way but, as we will see in the next semester, all physical quantities are expectation values of quantum fields defined in terms of a normalized path integral and the field rescaling cancels exactly between the numerator and denominator (normalization term) and, as a result, physical quantities are independent of this field rescaling.

7.5 The one-loop QCD running coupling Using equation (7.64), we can determine the scale dependence of the physical coupling g . We start from

gB =

⎤ ⎡ g2 gμ ε Z1 4 O (2 N 11 N ) ( g ) = gμ ε ⎢1 + − + ⎥, f c ⎦ ⎣ 96π 2ε Z2 Z3

(7.66)

with μ ≡ g γμ /(4π ) where γ = 0.57721… is the Euler–Mascheroni constant. To proceed, we express this in compact form as

⎛ a⎞ gB = μ ε ⎜g + 1 ⎟ + O(g 4), ⎝ ε⎠

(7.67)

with a1 ≡ g 3(2Nf − 11Nc )/96π 2 . Taking a derivative with respect to the scale μ on the left and right and using the fact that the bare coupling is independent of the scale (∂ μ eB = 0), we obtain

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Relativistic Quantum Field Theory, Volume 2

⎛ ∂g ⎛ a⎞ 1 ∂a1 ∂g ⎞ + 0 = εμ ε−1⎜g + 1 ⎟ + μ ε ⎜ ⎟ + O(g 4). ⎝ ε⎠ ε ∂g ∂μ ⎠ ⎝ ∂μ

(7.68)

Simplifying this a bit, we obtain

μ

⎛ 1 ∂a1 ⎞ a1 ⎞ ∂g ⎛ ⎜1 + ⎟ = −ε⎜g + ⎟ + O(g 4). ⎝ ∂μ ⎝ ε ∂g ⎠ ε⎠

(7.69)

∂g

Solving for μ ∂μ to leading-order in g , we obtain

μ

⎛ ⎛ 1 ∂a1 ⎞ a ⎞⎛ ∂a1 ⎞ ∂g = −ε⎜g + 1 ⎟⎜1 − ⎟ = − ⎜a1 − g ⎟ + O(g 4). ⎝ ∂g ⎠ ∂μ ε ⎠⎝ ε ∂g ⎠ ⎝    −2a1

(7.70)

From this, we arrive at our final expression for the one-loop running of the QCD coupling constant

μ

g3 ∂g = (2Nf − 11Nc ) + O(g 4), ∂μ 48π 2

(7.71)

which shows that the running of the coupling constant is independent of the renormalization scheme, which is defined by the overall constant scale in front of μ . This differential equation tells us how the coupling constant changes with the scale. It can be solved analytically subject to a boundary condition at a reference scale μ0. The result is

g 2 (μ 0 )

g 2 (μ ) = 1+

g 2 (μ 0 ) 48π 2

(11Nc − 2Nf ) log

μ2

.

(7.72)

μ 02

Translating this into αs ≡ g 2 /4π and taking Nc = 3, we obtain

αs(μ0)

αs(μ) = 1+

αs(μ 0 ) 4π

(

11 −

2N f 3

)

log

μ2

.

(7.73)

μ 02

This is the solution for the one-loop running of QCD. Note that, for Nf < 33/2, the sign of the running is opposite to that of QED running coupling (7.1) implying that the QCD coupling constant decreases with increasing μ. As a result, at very high energies, quarks and gluons will behave as weakly interacting quanta. This property is called asymptotic freedom. Since there are six observed quark flavors, QCD is an asymptotically free theory. Note also that, in the limit Nc → ∞, SU (Nc ) theories are asymptotically free for an arbitrary number of quark flavors. What we presented above, was a one-loop calculation of the running coupling. If one computes the loop corrections to higher orders in g , one can express the n-loop running in terms of the QCD β -function

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Relativistic Quantum Field Theory, Volume 2

Figure 7.1. One-, two-, three-, and four-loop running of αs . The differential equation is solved at each loop order with the boundary condition αs(1.5 GeV) = 0.326 which was determined by a lattice calculation.

μ2

dαs = β(αs ) = −(b0α s2 + b1α s3 + b2α s4 + ⋯), dμ2

(7.74)

where, for Nc = 3, one has [6],

33 − 2Nf , 12π

(7.75)

153 − 19Nf , 24π 2

(7.76)

b0 = b1 =

2857 − b2 =

5033 325 2 Nf + Nf 9 27 , 128π 3

(7.77)

which map to the one-, two-, and three-loop running of the coupling in the MS scheme2. Note that the number of contributing flavors is a function of the scale itself since, at low-energies, heavy quarks cannot be excited in loops. As a result, one should treat Nf as a scale-dependent quantity when applying to cases where quark thresholds are crossed. The differential equation (7.74) requires a boundary condition specifying αs at some scale. Using numerical evaluation on the lattice it has been found that αs(1.5 GeV) ≃ 0.326 [9]. The results of solving this differential equation at various loop orders are shown in figure 7.1. We have ignored the quark mass thresholds and simply taken Nc = Nf = 3 to generate this figure.

2

The four-loop running is known and the corresponding b3 can be found in refs. [7, 8].

7-14

Relativistic Quantum Field Theory, Volume 2

References [1] Gross D J and Wilczek F 1973 Phys. Rev. Lett. 30 1343–6 [2] Politzer H D 1973 Phys. Rev. Lett. 30 1346–9 [3] Weinberg S 2005 The Quantum Theory of Fields, Volume 1: Foundations (Cambridge: Cambridge University Press) [4] Eimerl D 1975 Phys. Rev. D12 427 [5] Joglekar S D and Lee B W 1976 Annals Phys. 97 160 [6] Tanabashi M et al 2018 Particle data group Phys. Rev. D 98 030001 [7] van Ritbergen T, Vermaseren J A M and Larin S A 1997 Phys. Lett. B400 379–84 [8] Czakon M 2005 Nucl. Phys. B710 485–98 [9] Bazavov A, Brambilla N, Garcia i Tormo X, Petreczky P, Soto J and Vairo A 2012 Phys. Rev. D86 114031

7-15

IOP Concise Physics

Relativistic Quantum Field Theory, Volume 2 Path integral formalism Michael Strickland

Chapter 8 Topological objects in field theory

In non-linear classical field theories there can exist stable configurations with welldefined energy, which are solutions to the classical equations of motion, but which possess some special properties such as propagation without dissipation. For example, the sine-Gordon model possesses solitonic solutions. Since non-abelian gauge theory is non-linear, there may exist important topological solutions such as vortices, monopoles, and ‘instantons’ that are soliton-like solutions.

8.1 The kinky sine-Gordon model The 1+1d ‘sine-Gordon’ equation of motion is

( ∂t2 − ∂ x2)ϕ + α sin βϕ = 0,

(8.1)

which describes a scalar field in one space and time dimension in a periodic potential. The corresponding Lagrangian is

1 ∂μϕ∂ μϕ − V (ϕ), 2

(8.2)

1 ⎛ ∂ϕ ⎞2 1 ⎛ ∂ϕ ⎞2 ⎜ ⎟ + ⎜ ⎟ + V (ϕ), 2 ⎝ ∂t ⎠ 2 ⎝ ∂x ⎠

(8.3)

L= and the Hamiltonian density is

H= with

V (ϕ ) =

α (1 − cos βϕ). β

(8.4)

We have plotted this potential in figure 8.1. As we can see from this figure, there are an infinite number of stable degenerate ‘vacuum’ solutions given by

doi:10.1088/2053-2571/ab3108ch8

8-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 2

Figure 8.1. Sine-Gordon potential.

ϕn =

2πn β

with

n ∈ .

(8.5)

Such trivial static solutions have zero energy (H = 0) and all non-trivial solutions we obtain below will only be defined up to a shift by ϕn . As we will see, there are solutions to these equations that allow us to connect the different degenerate minima. Note that, if we were to treat this problem perturbatively, we would expand equation (8.4) ∞

V (ϕ ) = −

α ( −1)n(βϕ)2n 1 1 ≃ αβ ϕ 2 − αβ 3 ϕ4 + ⋯ , ∑ β n=1 (2n )! 2 2 24

⏟

≡m

(8.6)

⏟ ≡λ

which looks like a potential for a particle with mass m2 = αβ and coupling λ = αβ 3. We begin by looking for static solutions with ∂tϕ = 0, and hence need to solve

∂ 2xϕ = α sin βϕ .

(8.7)

Multiplying left and right by ∂xϕ and simplifying gives

1 ∂x(∂xϕ)2 = α(∂xϕ)sin βϕ . 2

(8.8)

1 α (∂xϕ)2 = − cos βϕ + c1. 2 β

(8.9)

Integrating, gives

We will search for solutions that have ∂xϕ → 0 at both x → ±∞ and ϕ → 0 and ϕ → 0 mod 2π /β for x → −∞ and x → +∞, respectively. This implies that c1 = α /β , giving

⎛ βϕ ⎞ 2α 1 α (∂xϕ)2 = (1 − cos βϕ) = V (ϕ) = sin2 ⎜ ⎟ , ⎝ 2 ⎠ 2 β β

8-2

(8.10)

Relativistic Quantum Field Theory, Volume 2

and hence

⎛ βϕ ⎞ dϕ α sin ⎜ ⎟ . = ±2 ⎝ 2 ⎠ dx β

(8.11)

Separating and integrating, we obtain

∫ϕ

ϕf

0

dϕ α = ±2 β sin(βϕ /2)

∫x

xf

dx .

(8.12)

0

This gives

⎡ ⎛ βϕ(x ) ⎞⎤ 2 α ⎟⎥ = ±2 (x − x0), log ⎢tan ⎜ ⎝ 4 ⎠⎦ ⎣ β β

(8.13)

with x0 being an integration constant. Solving for ϕ, we obtain

ϕ(x ) =

4 arctan (e± β

αβ (x−x 0 )

).

(8.14)

The kink (+) and anti-kink (−) solutions are plotted figure 8.2 left panel. In the right panel of the same figure we show a visualization of field rotation associated with the kink solution. As can be seen from this figure, the kink solution moves from a vacuum state with ϕ = 0 to one with ϕ = 2π /β . The kink is topological in the sense that one cannot continuously deform the right panel of figure 8.2 while keeping the final vectors the same and get rid of the kink (rotation in the phase vector). The antikink is similar but instead moves one from ϕ = 2π /β to ϕ = 0. Classically, these solutions are ones in which the system starts asymptotically at one minimum and reaches the other minimum at large times. In the non-stationary case, we search for solutions of the form ϕ(t , x ) = f (x − vt ) with v < 1. Plugging this ansatz into the sine-Gordon equation gives

f ″(u ) = γ 2α sin(βf (u )),

(8.15)

Figure 8.2. (Left) Kink and anti-kink sine-Gordon solutions for α = β = 1 and x0 = 0 . (Right) Visualization of the phase rotation associated with the kink solution.

8-3

Relativistic Quantum Field Theory, Volume 2

with γ = (1 − v 2 )−1/2 and u = x − vt . This looks exactly like equation (8.7), so we can take the solution directly over to obtain

4 arctan (e±γ β

ϕ(t , x ) =

αβ (x−vt −x 0 )

).

(8.16)

This solution represents a kink (or anti-kink) moving at finite velocity. It has a finite energy, which can be computed from the integral of the Hamiltonian density. Since adding the time dependence just gives a moving kink, it suffices to consider the energy of a static kink

E=

=

⎡ ⎛

⎞2

dϕ 2V (ϕ) =

2α β

∫ dxH = ∫ dx⎢⎣ 12 ⎝⎜ ∂∂ϕx ⎠⎟ ∫0

=8

2π β

⎤ + V (ϕ )⎥ = ⎦

∫0

2π β

∫ dx2V (ϕ)

dϕ(1 − cos βϕ)1/2

(8.17)

8m3 α = , 3 β λ

where we used equations (8.10) and (8.11) and, in the last line, we re-expressed the energy in terms of the mass and coupling constant introduced in equation (8.6). This shows that the energy of the kink solution is finite and, importantly, that the energy goes inversely with the coupling λ meaning that, in the strong coupling limit, the energy associated with such configurations can become small. We note that similar non-trivial solutions can be obtained by assuming that ϕ(t , x ) can be expressed in the form [1]

ϕ(t , x ) =

⎛ F (x ) ⎞ 4 arctan ⎜ ⎟. β ⎝ G (t ) ⎠

(8.18)

For example, for α = β = 1, there is a two-kink solution of the form

⎛ v sinh(γx ) ⎞ ϕ(t , x ) = 4 arctan ⎜ ⎟. ⎝ cosh(vγt ) ⎠

(8.19)

This solution is plotted and visualized in figure 8.3 for the case v = 0.6 at t = 0. In fact, a family of solutions with an arbitrary number of kinks + anti-kinks can be constructed in this manner. Each of these solutions has a conserved number associated with the number of kinks minus anti-kinks. One can construct a conserved current of the form

Jμ =

β μν ε ∂νϕ , 2π

(8.20)

with μ, ν ∈ {0,1} and ε μν being the 1+1 anti-symmetric tensor with ε 01 = 1. This current is conserved, that is ∂μJ μ = 0, and the conserved charge associated with it is

8-4

Relativistic Quantum Field Theory, Volume 2

Figure 8.3. (Left) Two-kink sine-Gordon solution for α = β = 1. (Right) Visualization of the phase rotation associated with the two-kink solution. For both panels we took v = 0.6 and t = 0 .

Q=

∫ dx J 0 = 2βπ ∫ dx∂xϕ = 2βπ [ϕ(∞) − ϕ(−∞)] = N ,

(8.21)

where N can be identified as the difference of the number of kinks minus anti-kinks. From this we see the important role played by the boundary conditions at infinity. The (anti)-kink and two-kink solutions presented earlier in this section have charge ±1 and 2, respectively. Note that the solitonic current is not a Noether current associated with a global symmetry, it instead comes from the conservation of ‘winding number’, which is a topological invariant.

8.2 Two-dimensional vortex lines Next we consider a complex scalar field (charged scalar) in two spatial dimensions. We take the boundary of space to be the circle at infinity, S1. We will impose a boundary condition at infinity

ψ = a einθ

(8.22)

(r → ∞),

where r and θ are polar coordinates in the plane, a is an arbitrary amplitude, and n is an integer in order to guarantee that ψ is single valued. The Lagrangian and Hamiltonian densities are

1 1 L = ∂tψ †∂tψ − ∇ψ † · ∇ψ − V (∣ψ ∣2 ), 2 2 1 † 1 H = ∂tψ ∂tψ + ∇ψ † · ∇ψ + V (∣ψ ∣2 ). 2 2

(8.23)

As our example, let us take V (∣ψ ∣2 ) = (a 2 − ∣ψ ∣2 )2 such that V → 0 at the boundary. To start with, let us consider a static configuration. In this case, the Hamiltonian density at the boundary is

lim H =

r →∞

1 1 1 ∣∇ψ ∣2 = ∇(a einθ ) 2 = 2 2 2

8-5

ina inθ e r

2

=

n 2a 2 . 2r 2

(8.24)

Relativistic Quantum Field Theory, Volume 2

Since the Hamiltonian density only falls off like r −2 , this implies that the energy of such a configuration is infinite. This means that there is no finite-energy generalization of the 1d kink solution for a 2d complex scalar. Next, let us consider what happens if we couple our complex scalar field to an abelian gauge field by taking ∂μ → Dμ with

Dμψ = ∂μψ + i eAμ ψ ,

(8.25)

and require that the vector potential has the boundary condition

A=

1 ∇(nθ ) e

(8.26)

(r → ∞).

As we demonstrate below, by adding a gauge field with these non-trivial boundary conditions, one can construct a discrete set of finite-energy configurations of a complex scalar coupled to an abelian gauge field. Including the gauge-field contribution to the Lagrangian density, we have

L=

1 1 μ † D ψ Dμψ − V (∣ψ ∣2 ) − F μνFμν. 4 2

(8.27)

To see that the energy of a configuration satisfying equations (8.22) and (8.26) is finite, we first note that, given the boundary condition for Aμ above, one has

Ar → 0, Aθ → −

n , er

(8.28)

which implies that, at asymptotically large distances, one has

Drψ → 0, 1 ⎛ ∂ψ ⎞ Dθψ → ⎜ ⎟ + i eAθ ψ = 0. r ⎝ ∂θ ⎠

(8.29)

This means Dθ ψ falls faster than 1/r at infinity and the kinetic energy contribution will be UV finite. Additionally, since equation (8.26) can be expressed in the form Aμ = ∂μχ with χ = nθ /e, this gauge field is a pure gauge transform and carries zero energy. As a result, all terms in equation (8.27) result in a finite contribution to the total energy. The gauge field configuration possesses a quantized magnetic flux. To see this, consider the magnetic flux Φ generated by equation (8.28) through a disk of radius R with boundary C

Φ=

∫ B · d S = ∮C A · d l = ∫0

2π

RdθAθ (R ) = −

2πn , e

(8.30)

which shows that the flux is quantized. We have demonstrated that it is possible to construct a 2d configuration consisting of a charged scalar field plus a gauge field that carries a quantized magnetic flux.

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Relativistic Quantum Field Theory, Volume 2

This 2d solution can be extended to 3d by simplying requiring cylindrical symmetry of the 3d system. In this case, the quantized flux lines are the ‘Abrikosov flux lines’ which appear in the theory of type II superconductors. The model we analyzed (8.23) corresponds to scalar electrodynamics with spontaneous symmetry breaking (Higgs model). This is a relativistic generalization of the condensed matter field theory, with the field ψ corresponding to the BardeenCooper-Schrieffer condensate. In a type II superconducting medium, the magnetic field normally cannot penetrate the material and if it does, it can only do so through quantized flux lines called Abrikosov flux lines. For a more extensive discussion of topological solutions in the context of condensed matter see ref. [2].

8.3 Topological solutions in Yang–Mills The next question one naturally asks is whether it is possible to construct analogous classical Yang–Mills topological solutions. To determine whether such non-abelian topological solutions are possible, we ask if it is possible to construct finite-energy classical solutions with non-trivial boundary conditions. We specialize to the case of pure-gauge fields, in which case the canonical energy momentum tensor is expressible solely in terms of the field-strength tensor (see volume 1)

T μν = FaμλFaλ ν +

1 μν αβ η Fa Faαβ , 4

(8.31)

which obeys

∂μT μν = 0.

(8.32)

The energy-momentum tensor is gauge-invariant. From equation (8.31) we have

T μ μ = FaμλFaλμ +

1 1 (d + 1)FaαβFaαβ = (d − 3)FaαβFaαβ , 4 4

(8.33)

where d is the number of spatial dimensions. 8.3.1 Static solutions Topological solutions can be time-dependent or time-independent (static). A static solution is one where Aμ can be made time-independent using a continuous gauge transformation. For such static solutions, the general time evolution of Aμ can be obtained from a continuous gauge transformation

Aμ̇ (x ) =

1 μ ̇ ∂ Ω(x ) − i [Ω̇(x ), Aμ (x , t )], g

(8.34)

where Ω(x ) is an arbitrary continuous function. We can make the right-hand side vanish if

1 μ ̇ ∂ Ω(x ) = i [Ω̇(x ), Aμ (x , t )], g

8-7

(8.35)

Relativistic Quantum Field Theory, Volume 2

which is solved by

Ω̇(x ) = L(P ) Ω (x0)L−1(P ),

(8.36)

with L(P ) being the gauge-link, which is the exponential of the integral of the gauge potential along the path P

⎛ L(P ) ≡ P exp ⎜ −ig ⎝

⎞

∫P dx μAμ(x)⎠. ⎟

(8.37)

In the definition of L , P implies the path-ordering operator, which orders A similar to the time-ordering operator, but now for a path in space-time. With Ω determined in this way, we have Ȧμ(x ) = 0, that is the field configuration is time-independent. The gauge in which our static field is time-independent is called the static gauge. Static solutions with finite-energy are either the vacuum or a static solution. For pure Yang–Mills theory it turns out that there are no static topological solutions unless the number of spatial dimensions is four [3]. We will now review the proof of this statement by considering Yang–Mills in (d + 1)− dimensional Minkowski space. The requirement of finite-energy implies that

⎛

⎞

∫ d nxT 00 = ∫ d nx⎜⎝Fa0λFa0λ + 14 FaαβFaαβ⎟⎠ < ∞.

(8.38)

One can show that, for a static solution, Fai0 = 0 with i = 1, 2, 3 (homework), and hence one has

∫ d nx 14 FaijFaij < ∞.

(8.39)

As a result, limr→∞r n−1∣Faij∣2 < Cr −1−ε with ε > 0 and r 2 ≡ Σin=1xi2 and one must have limr→∞Faij ∼ O(r −n/2−ε ) for convergence in the UV. Next, consider

∂i(x j T ij ) = T i i + x j ∂iT ij .

(8.40)

Since ∂μT μν = 0, one has ∂iT ij = −∂tT 0j , giving

∂i(x j T ij ) = T i i − x j ∂tT 0j .

(8.41)

For a static solution, the second term vanishes, leaving

∂i(x j T ij ) = T i i .

(8.42)

Integrating on the left and right we see that, since T ij vanishes at infinity, the integral of the left-hand side has to vanish and, therefore, so does the integral over the righthand side

∫ d nxTii = 0.

8-8

(8.43)

Relativistic Quantum Field Theory, Volume 2

Using Tii = Tμμ − T 00, equation (8.33), and T 00 = (1/4)FaαβFaαβ we obtain

∫

(n − 4) d nxFaijFaij = 0.

(8.44)

Therefore, unless d = 4 one must have Faij = 0 everywhere for this integral to be zero (integrand is positive-definite). We thus conclude that for d ≠ 4 there cannot be static pure-gauge topological solutions. For d = 4 we can find a solution, which is called the instanton solution. We will construct such a solution next.

8.4 The instanton A static topological solution in 4 + 1 Minkowski space is a time-independent solution that only depends on the four-dimensional Euclidean space coordinates. Since we can formulate the path integral in four-dimensional Euclidean or Minkowski spaces, a four-dimensional Euclidean solution might be of some interest. As an example, let us search for a solution that is invariant under SU (2) gauge transformations with f abc = ε abc . What we are looking for is a mapping from the SU (2) group space S 3 to the boundary of the physical space, which is also S 3. We will label Euclidean space with spatial coordinates (x1, x2, x3, x4 ). The a Euclidean field tensor F μν is defined in the same way as the Minkowski tensor a F μν = ∂μAνa − ∂νA μa + gε abcA μb Aνc .

(8.45)

a Defining Aμ = t aA μa and Fμν = t aF μν we can write this compactly as Fμν = ∂μAν − . We can introduce the dual of Fμν as F˜μν which is defined by ∂νAμ − ig[Aμ , Aν ]

1 F˜μν ≡ εμνρσFρσ , 2

(8.46)

where, since we are in Euclidean space, we do not have to distinguish up and down indices. We can express

1 TrF˜μνFμν = ∂μK μ, 4

(8.47)

⎛ ⎞ g 1 K μ ≡ εμνκλ⎜Aνa ∂κA λa + εabcAνa Aκb A λc ⎟ ⎝ ⎠ 4 3 ⎛1 ⎞ ig = εμνκλ Tr⎜ Aν ∂κAλ − Aν Aκ Aλ ⎟ , ⎝2 ⎠ 3

(8.48)

with

being the Chern–Simons current. Since (1/4)TrF˜μνFμν can be expressed as a total derivative of the Chern–Simons current, its integral can only depend on the boundary conditions for the current. Considering a four-dimensional Euclidean volume V 4 with boundary ∂V 4 = S 3. Suppose that on the boundary we have a pure

8-9

Relativistic Quantum Field Theory, Volume 2

vacuum solution with Aμ = 0, Fμν = 0, and, hence, Kμ = 0. In the absence of matter, the equation of motion for the field in the entire volume V 4 is

DμF μν = 0.

(8.49)

μν DμF˜ = 0,

(8.50)

Additionally, the dual satisfies

which follows from the Bianchi identity. Using the current Kμ, we can write

∫V

4

d 4x TrF˜μνFμν = 4

∫V

4

d 4x∂μK μ = 4

∫S d 3xK⊥,

(8.51)

3

where K⊥ is the projection of K that is perpendicular to the surface S 3. Note that this is trivially satisfied if the solution is pure vacuum everywhere in V 4 . To proceed, we imagine performing a space-time dependent gauge transformation on the boundary S 3

i Aμ → − (∂μΩ)Ω−1. g

(8.52)

Since the result is a pure-gauge field, one still has F μν = 0 on the boundary, but it may be possible to have Kμ ≠ 0. Choosing

1 (x4 + ix · σ ), x2

(8.53)

i [xi − σi (x · σ + ix4)], gx 2

(8.54)

Ω= with x 2 ≡ Σi4=1xi2 gives

Ai =

A4 = −

1 x · σ, gx 2

(8.55)

and

Kμ =

2xμ . g 2x 4

(8.56)

Using this, we obtain

∫V

4

d 4x TrF˜μνFμν = 4

∫S d 3xK⊥ = g 28x3 ∫S d 3x = 16gπ2   

3

3

2

.

(8.57)

2 3

2π x

We see from this expression that although Fμν vanishes on the boundary at infinity, it cannot be zero everywhere within V 4 since the integral above is finite. This means that the solution cannot be a pure gauge everywhere in V 4 . We will return to this issue shortly.

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Relativistic Quantum Field Theory, Volume 2

Exercise 8.1 Show that equation (8.48) is correct. Exercise 8.2 Show that equation (8.50) is correct. Exercise 8.3 Show that equations (8.54) and (8.55) are correct.

8.5 The Potryagin index The quantity introduced above ∫ d 4x TrF˜μνFμν is related to the Potryagin or topological index, which is denoted as q ,

q≡

g2 16π 2

∫ d 4xTrF˜μνFμν.

(8.58)

The vacuum has q = 0. Equation (8.53) gives q = 1 according to equation (8.57). We will now show that q is the degree of the mapping of the group space from the spatial coordinate boundary (number of times it covers the group space). In this case we have a mapping of S 3, which is the group space of SU (2), onto a S 3 in coordinate space. Computing Kμ using equations (8.48) and (8.52) one obtains

Kμ =

1 μνκλ ε Tr[(Ω−1∂νΩ)(Ω−1∂λΩ)(Ω−1∂κΩ)]. 6g 2

(8.59)

This allows us to express q as

1 24π 2 1 = 24π 2

q=

∫S d 3xnˆ με μνκλTr[(Ω−1∂νΩ)(Ω−1∂λΩ)(Ω−1∂κΩ)] 3

∫S d xε 3

3

ijk

−1

−1

(8.60)

−1

Tr[(Ω ∂iΩ)(Ω ∂jΩ)(Ω ∂kΩ)],

where nˆ μ is an outward pointing unit vector. To proceed, let us consider a parameterization of the Ω using parameters ai . With this we have

⎡⎛

⎞⎛

⎞⎛

⎞⎤

⎡⎛

⎞⎛

⎞⎛

⎞⎤

q=

1 24π 2

∂Ω ∂a ℓ ∂am ∂an ⎟⎥ε ijk ⎟⎜Ω−1 ∫S d 3xTr⎢⎣⎜⎝Ω−1 ∂∂Ωaℓ ⎟⎠⎜⎝Ω−1 ∂∂Ω ∂an ⎠⎦ ∂xi ∂xj ∂xk am ⎠⎝

=

1 24π 2

∂Ω ∂(a ℓ , am , an) ⎟⎜Ω−1 ⎟⎥ ∫S d 3xTr⎢⎣⎜⎝Ω−1 ∂∂Ωaℓ ⎟⎠⎜⎝Ω−1 ∂∂Ω ∂an ⎠⎦ ∂(x1, x2 , x3) am ⎠⎝

=

1 4π 2

∂Ω ∂(a1, a2 , a3) ∂Ω ⎟⎜Ω−1 ⎟⎜Ω−1 ⎟⎥ ∫S d 3xTr⎢⎣⎜⎝Ω−1 ∂Ω ∂a3 ⎠⎦ ∂(x1, x2 , x3) ∂a2 ⎠⎝ ∂a1 ⎠⎝

=

1 4π 2

∂Ω ∂Ω ⎟⎜Ω−1 ⎟⎜Ω−1 ⎟⎥ ∫x∈S da1da2da3Tr⎢⎣⎜⎝Ω−1 ∂Ω ∂a3 ⎠⎦ ∂a2 ⎠⎝ ∂a1 ⎠⎝

=

∫x∈S dU .

3

3

⎡⎛

⎞⎛

⎞⎛

⎞⎤

3

⎡⎛

⎞⎛

3

3

8-11

⎞⎛

⎞⎤

(8.61)

Relativistic Quantum Field Theory, Volume 2

From the last line, we learn that q is the number of times that the group space is covered by the map the coordinate S 3 to the group S 3. This is sometimes called the ‘winding number’ of the map. As a simpler example of this, let us look at a map from S1 to S1 using a U (1) configuration of the form

Ω(x ) = e i 2πα(x ),

(8.62)

where x = S1 with 0 ⩽ x < 2π covering the circle. Requiring that Ω be singledvalued for x ∈ S1 we see that α (x + 2π ) = α (x ) + n where n ∈  . The corresponding pure-gauge vector potential constructed from this is

Ω−1∂μΩ = 2πi ∂μα(x ).

(8.63)

One, therefore, has

1 2πi

∫0

2π

dαΩ−1∂μΩ =

∫0

2π

dα∂μα = α(2π ) − α(0) = n ,

(8.64)

where n is the winding number, which counts the number of times Ω goes around the U (1) circle when x runs around the S1 circle once (x = 0 → 2π ). This is similar to the phase we accumulated in the sine-Gordon model when we made a transition from one vacuum to another. The situation is similar in pure-gauge QCD, the mapping of S 3 → S 3 generates a topologically conserved number, which is the Potryagin index of the field configuration. Exercise 8.4 Derive equation (8.59).

8.6 Explicit solution for a q = 1 instanton As mentioned previously, the field cannot be a pure-gauge configuration everywhere if q is non-zero. The instanton solution in all of V 4 is1

Aμ (x ) = −

i x2 (∂μΩ)Ω−1, 2 g x + ρ2

(8.65)

where x 2 = x2 + x42 and ρ is an arbitrary real constant which sets the size of the instanton solution. As x42 → ±∞ this solution reduces to the pure-gauge form given by equation (8.52). Generally, one has

Ai =

1 i [xi − σi (x · σ + ix4)], 2 g x + ρ2 A4 = −

1

1 1 x · σ. 2 g x + ρ2

See ref. [4], chapter 5.

8-12

(8.66)

(8.67)

Relativistic Quantum Field Theory, Volume 2

The asymptotic form of this solution can be obtained from gauge transformations of the type

gn = (g1)n ,

(8.68)

with

⎛ −iπ x · σ g1 ≡ exp ⎜⎜ ⎝ x 2 + ρ2

⎞ ⎟, ⎟ ⎠

(8.69)

and

lim Ai = i (gn)−1(∂ign),

x 4 →∞

lim Ai = i (gn−1)−1(∂ign−1).

(8.70)

x 4 →−∞

The gauge transform gn is an element of SU (2) but gn and gm for n ≠ m are not homotopic, that is they have a different topology and cannot be continuously deformed into one another. The q = 1 instanton configuration describes a solution of the gauge-field equations in which, as x4 goes from −∞ to ∞, a vacuum belonging to homotopy class n − 1 evolves into another vacuum with homotopy class n . The energy density of the pure-gauge field at the end-point is vanishing, however, the full configuration has a positive field energy. As a result, we see that the Yang–Mills vacuum is infinitely degenerate with an infinite number of homotopically nonequivalent vacua. The instanton represents a transition from one vacuum class to another. Due to the finite energy of the instanton, classically there can be no transition between the degenerate vacua, however, quantum mechanically we have the possibility of quantum tunneling.

8.7 Quantum tunneling, θ-vacua, and symmetry breaking The barrier potential amplitude is given exp( −SE ) where SE is the Euclidean action. To see this, consider the motion of a particle with energy E in a one-dimensional potential V in the WKB approximation. If V > E , the transition is classically forbidden, and the quantum tunneling amplitude is

⎛ exp ⎜ − ⎝

∫a

b

⎞ dx 2m(V − E ) ⎟ ≡ exp( −SE ), ⎠

(8.71)

where we have identified the Euclidean action SE with the integral appearing on the left. Let us see why this is true. If E > V , the transition is classically allowed and the wave function oscillates with the number of oscillations given by

∫a

b

dxp =

∫a

b

dx 2m(E − V ) .

8-13

(8.72)

Relativistic Quantum Field Theory, Volume 2

Alternatively, we can express the integral of p as

∫ dxp = ∫ dtpx ̇ = ∫ dt(H + L) = ∫ dt(E + L).

(8.73)

If the total energy is normalized to zero which it always can be, then

∫a

b

dxp =

∫t

tf

dtL = S.

(8.74)

0

This is the total action induced transversing from a to b. The only difference between the classically forbidden case and the allowed case just considered, is the sign of E − V and, since we have normalized the energy such that E = 0, we see that we simply pick up a relative factor of i from the square root. The sign of V in the classical equation of motion

mx ̈ =

∂V ∂x

(8.75)

is reversed if we take t → it . As a result, SE is the action for imaginary times. This justifies calling exp( −SE ) the tunneling amplitude. Now we need to determine the action for our instanton. For this we need to be able to evaluate

SE =

1 4

∫x F μνa F μνa .

(8.76)

The q = 1 solution given in equation (8.65) is special because it is self-dual meaning that

F˜μν = Fμν.

(8.77)

As a result,

SE =

1 4

2

∫x F˜μνa F μνa = 8gπ2 .

(8.78)

For the q = 1 instanton constructed herein, this gives a transition probability of exp( −8π 2 /g 2 ) and in general, one finds, exp( −8π 2∣q∣/g 2 ). For small g , such transitions are extremely strongly suppressed. Regardless of the magnitude of g , however, the fact that these tunneling solutions exist means that all of the degenerate vacua of Yang–Mills are coupled by instantons. We can label each of the generate vacuum by the topological index which must be an integer, so we have a basis of states of the form n〉 with n ∈  . Positive n map to multi-instanton solutions with more instanton than anti-instanton solutions and vice-verse for negative n and n = 0 is the instanton-free vacuum. The true wave function of the QCD vacuum is therefore a linear superposition of states, however, since instantons couple the various vacuum states, we must diagonalize the Hamiltonian to obtain the true ground state. The end result is that the QCD vacuum will have the form

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Relativistic Quantum Field Theory, Volume 2

∑einθ

θ ≡

n ,

(8.79)

n

where θ is an arbitrary constant. This is vacuum is called the theta vacuum. The form of the coefficients in this expansion guarantee that the θ -vacuum is invariant under gauge transformations of the type g1. Acting with g1 we find g1

n → n+1 ,

(8.80)

and hence the theta vacuum simply picks up a phase g1

θ → e−iθ θ .

(8.81)

Although one can construct stationary states for any value of θ , they are not excitations of the θ = 0 vacuum, because in QCD the value of θ cannot be changed. As far as the strong interaction is concerned, different values of θ correspond to different worlds. We can fix the value of θ by adding an additional term of the form

L=−

g 2θ a ˜ a F μνF μν, 32π 2

(8.82)

to the QCD Lagrangian. Does physics depend on the value of θ ? The interaction above violates both T and CP invariance. On the other hand, it is a surface term and it might be possible that confinement somehow screens the effects of the θ -term. A similar phenomenon is known to occur in three-dimensional compact electrodynamics. In QCD, however, one can show that, if the U (1)A problem is solved (there is no massless η′ state in the chiral limit) and none of the quarks are massless, a non-zero value of θ implies that CP symmetry is broken [5–8]. The most severe limits on CP violation in the strong interaction come from the electric dipole of the neutron. Current experiments imply that θ ≲ 10−10 [9, 10]. The question of why θ is so small is known as the strong CP problem. Exercise 8.5 Show that equation (8.77) is obeyed by a q = 1 instanton solution.

8.8 Quantum anomalies It is possible that classical symmetries of a system do not survive quantization, in which case it is said that the theory possesses a quantum anomaly. If this is case, the Noether current associated with the classical symmetry is no longer conserved after quantization and the current conservation law is said to receive an anomalous contribution. This is relevant for this course because QCD has a chiral anomaly, which is associated with the non-conservation of the chiral current in the limit of vanishing light quark masses. This will be the focus of the remainder of the chapter. Historically, the reaction π 0 → γγ is the best example of a process that proceeds primarily via the chiral anomaly. The original calculation of this anomalous decay was performed in 1969 by Bell and Jackiw and, independently, Adler. As a result, it is sometimes referred to as the Adler-Bell-Jackiw (ABJ) anomaly [11–13]. Earlier 8-15

Relativistic Quantum Field Theory, Volume 2

calculations of the π 0 → γγ decay width, which did not take into chiral anomaly, resulted in a decay lifetime on the order of 10−33 s, which was approximately three orders of magnitude longer than the experimentally observed pion lifetime. As of the 2015 PDG listings, the pion lifetime is τ = ℏ/Γtot(π 0 ) = (8.52 ± 0.18) × 10−17 s. The branching ratio for the 2γ decay channel is BR(π 0 → γγ ) ≃ 0.9882 and hence it dominates the total lifetime calculation for the pion. Taking into account the chiral anomaly, ABJ obtained a decay width of Γ(π 0 → γγ ) = 7.76 eV . This maps to a total pion lifetime of approximately 8.38 × 10−17 s, which is in the right ballpark and, within the modern experimental error bars2. In perturbation theory, one can understand the emergence of the chiral anomaly through the consideration of triangle graphs of the form

where A stands for ‘axial’ and V stands for ‘vector’. Such graphs naturally arise in the calculation of the pion decay rate. There are also VVV graphs and other configurations that occur at higher orders, however, it turns out that once we understand the anomaly in the AVV graph it is automatically handled in all of the other graphs. Before proceeding to this technical calculation, however, I would first like to discuss the physics of the anomaly. 8.8.1 The chiral anomaly in the Schwinger model The Schwinger model is simply 1+1d massless QED. The Lagrangian density is

L = ψ iDψ −

1 FμνF μν, 4

(8.84)

where, as usual, Dμ ≡ ∂μ − i eAμ is the covariant derivative and the 2 × 2 Dirac matrices can be written in terms of the Pauli matrices

γ 0 = σ1, γ1 = iσ2.

(8.85)

The resulting classical equations of motion are

iDψ = 0,

(8.86)

2 Since the ABJ result was obtained in the chiral limit (massless light quarks), it was not expected to be in full agreement with the data. Subsequent calculations using chiral perturbation theory have shown that, taking into account explicit chiral symmetry breaking, one obtains a lifetime, which is approximately 4% higher than the original ABJ calculation. See, for example, refs. [14–16].

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Relativistic Quantum Field Theory, Volume 2

□Aμ = −ejμ ,

(8.87)

jμ ≡ ψ γμψ ,

(8.88)

where

is the conserved vector current, ∂μ j μ = 0. This conserved current results from the local gauge invariance of the Schwinger model and QED in general. There is also an axial current or chiral current

jμ5 ≡ ψ γμγ5ψ ,

(8.89)

where γ5 ≡ −γ 0γ 1 = σ3. This chiral current is classically conserved ∂ μjμ5 = 0 and this conservation law is associated with classical invariance of QED under global chiral transformations of the fermionic fields

ψ → eiαγ5ψ = eiασ3ψ .

(8.90)

We can identify the two components of the spinor as ‘left’- and ‘right’-handed

⎛ψ ⎞ ψ = ⎜ ψL ⎟ , ⎝ R⎠

(8.91)

where ψL,R = (1/2)(1 ± γ 5)ψ ≡ PL,Rψ with PL,R being a left/right projector. Under a chiral transformation the left and right fields pick up opposite phases

ψL → eiαψL, ψR → e−iαψR.

(8.92)

Expanding out the free fermionic contribution to the Lagrangian in terms of ψL, R , we obtain

Lfree fermionic = iψ †γ 0(γ 0∂t + γ1∂x)ψ = iψ †(∂t − γ5∂x)ψ =

iψL†(∂t

− ∂x)ψL +

iψR†(∂t

(8.93) + ∂x)ψR,

and the general solution to the equations of motion in this case will be of the form

ψL(t , x ) = ψL(x + t ) left mover, ψR(t , x ) = ψR(x − t ) right mover. This demonstrates that, in the Schwinger model, left- and right-handed particles are quite literally left- and right-moving particles. This is different than the 3+1d case where chirality is related to the alignment of the particle’s spin with its momentum (helicity). In 1+1d, there is no spin and the handedness is related to particles propagating either to the left and right. This only makes sense in the massless case where particles move at the speed of light and the direction of propagation is the same in all reference frames. In addition, in this case we see that a parity transformation x → −x transforms left-movers into right-movers. This will be 8-17

Relativistic Quantum Field Theory, Volume 2

become important since, if the symmetry between left- and right-movers is broken, then we might break parity symmetry. We also see that the free part of the Lagrangian is invariant under chiral transformations (8.92) since left- and right-handed fields only couple to themselves. The same holds for the interaction part, where one finds

Linteraction = eψ Aψ = eψL†(A0 + A1)ψL + eψR†(A0 − A1)ψR.

(8.94)

Note that, in the classical theory, the number of left- and right-handed fermions, NL and NR , are independent constants of the motion. This will remain true if the fermions are coupled to a gauge field. Exercise 8.6 Derive equations (8.86) and (8.87). 8.8.2 Understanding the anomaly Before proceeding more formally, let us consider what would happen if we apply an external electric field E = E xˆ to a system of positively charged particles for a short amount of time Δt . In this case, the right-movers will gain energy

ΔE = eE Δt ,

(8.95)

and left-movers will lose the same amount of energy. If the initial state before the electric field was turned on is the Dirac vacuum, with all negative energy levels filled, then after the time interval Δt , the right-handed ‘Fermi-level’ has increased by ΔE and the left-handed Fermi level has decreased by ΔE . As a result, right-handed fermions are created along with left-handed anti-fermions (holes). This is sketched in figure 8.4. Since the one-dimensional density of states is dp /(2π ) = d E/(2π ), the number density per unit length of left- and right-handed fermions become

eE Δt , 2π eE ρR = Δt . 2π ρL = −

(8.96)

Figure 8.4. Action of an external electric field on left- and right-movers. Grey lines represent the filled fermion levels.

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Relativistic Quantum Field Theory, Volume 2

Therefore, the ‘vector’ fermion number associated with charge conservation is conserved during this process

ρ ̇fV = ρṘ + ρL̇ = 0,

(8.97)

however, the ‘axial’ fermion number, which is also conserved at the classical level, changes according to

ρȦ ≡ ρṘ − ρL̇ =

eE . π

(8.98)

The fact that the right hand side is non-vanishing is the chiral anomaly. The chiral anomaly corresponds to a kind of dielectric breakdown of the vacuum where we create particle-hole pairs with a chiral imbalance. At this point, you should be scratching your head in confusion since everything we just did was classical; however, the subtlety in the argument is the existence of an infinitely occupied Dirac sea in the first place. Imagine that instead of an infinite number of negative energy states, there were a finite number of them, regulated by a cutoff Λ on the lowest possible negative energy state before turning on the electric field. In that case, figure 8.4 would look instead like figure 8.5 and we would always have the same number of left- and right-moving states even in the presence of an external electric field. As a result, we see that the physics of the anomaly will be tied intimately with the regularization of the theory. If we regularize and do not remove the regulator, we might even miss it! Note that, if you hear condensed matter theorists discussing the chiral anomaly, the figure they use to illustrate the concept will look more like figure 8.6. 8.8.3 The chiral anomaly in 3+1d The natural followup question is, of course, if this kind of logic can be extended to 3+1d. As it turns out, the precise discussion we just had applies in 3+1d to massless fermions in a magnetic field. This is because fermions in a background magnetic field are restricted to Landau levels labeled by an integer n [17]. For massless fermions, the Landau levels are

Figure 8.5. In this case, we imagine that the Dirac sea is not an infinite reservoir, but is instead finite.

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Relativistic Quantum Field Theory, Volume 2

Figure 8.6. An alternative visualization of the chiral anomaly. The left panel shows the vacuum before the electric field is applied and the right panel shows after. Closed circles indicate filled states and open unfilled states. Diagonal lines are the light cones along which the massless fermions propagate.

En2, sz = k z2 + (2n + 1)∣q∣B − 2qBsz ,

(8.99)

where n ⩾ 0 is an integer, B = B zˆ , and sz = ±1/2. The fermionic motion in the x − y plane is quantized circular motion and the fermions effectively propagate along the z-axis like 1+1d fermions with mass 2 m eff = (2n + 1)∣q∣B − 2qBsz .

(8.100)

For a positively charged particle, q = e, with sz = 1/2, we see that the lowest Landau level n = 0 the effective mass vanishes and, hence, these particles will behave like massless 1+1d fermions. For a negatively charged particle, we have the same behavior for the sz = −1/2. For massless 3+1d fermions, chirality is identified with the helicity of the state. The right-handed fermion with sz = 1/2 is a right-mover along the z-axis and the lefthanded fermion is a left-mover. So, by adding a background field, at least some subset of the allowed states are effectively 1+1 and our previously setup can be applied. We now imagine applying an electric field along the z direction, E = E zˆ , and our earlier discussion applies. The density of states per unit area in a Landau level is

gn =

eB , 2π

(8.101)

therefore, axial charge is created at a rate of

ρȦ =

eB eE e2 = E · B, 2π π 2π 2

(8.102)

where now the right-hand side indicates the presence of a 3+1d axial anomaly. Stated succinctly, an electric field applied to the vacuum causes pair production and,

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Relativistic Quantum Field Theory, Volume 2

if a parallel magnetic field is applied, the pairs created are chiral. Positively-charged fermions will align their spins with B, while anti-fermions (negative charge in this case), anti-align their spins with B3. We would like to express ρA in terms of the axial (chiral) charge Q5

Q5 ≡

∫ dxj05 = ∫ dxψ γ0γ5ψ = ∫ dxψ †σ3ψ = ∫ dx(ψL†ψL − ψR†ψR ) ≡ ρL − ρR ,

(8.103)

and the contraction of the field strength tensor and the dual field strength tensor

FμνF˜

μν

= −4E · B,

(8.104)

where we remind you that F˜μν ≡ (1/2)εμνρσ Fρσ . This gives

dQ5 e2 FμνF˜ μν. = 8π 2 dt

(8.105)

This is the standard way to present the anomalous contribution, which breaks chiral current conservation. Exercise 8.7 Derive equation (8.99). Exercise 8.8 Derive equation (8.104).

8.9 An effective Lagrangian for the anomaly Now that we have a kind of intuitive understanding of the phenomenon, let us return to the mathematical development in the context of the 1+1d Schwinger model. To begin, we note that the axial current (8.89) can be expressed as

jμ5 = εμνj ν ,

(8.106)

where ε μν is the two-dimensional Levi-Civita tensor. Upon quantization, it can be shown that the Lagrangian can be written as

L = ψ i ∂ψ −

1 e2 FμνF μν − Aμ Aμ . 4 2π

(8.107)

This is the Lagrangian of a system of free massless spin-1/2 particles and free ‘photons’ having mass mγ2 = e2 /π . Also, in the quantized theory the axial current is no longer conserved. Instead, we have e ∂ μjμ5 = − ε μνFμν. (8.108) 2π 3 This effect is relevant to charge separation heavy-ion physics and has been dubbed the chiral magnetic effect. See for example refs. [18–20, 21] for a recent review.

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As a result, quantization breaks axial current conservation and there exists an anomaly. To see how this arises physically, returning to our consideration of an electric field applied to the chiral Schwinger model, we see that the applied electric field induces a current density along the x-direction j ≡ j 1 which grows in time with

dj = dt

∞

, ∫−∞ 2dpπ dv dt

(8.109)

where, using the Lorentz force law, dp /dt = eE , we have

eEm 2 dv d d . = m2 + p2 = 2 (p + m 2 )5/2 dt dt dp

(8.110)

Integrating equation (8.109), we then obtain

dj eE = , dt π

(8.111)

which is independent of the mass. Since the vacuum charge density ρvac is independent of the position, we have dj 0 /dx = 0. Defining j μ = ( j 0 , j ) and using equation (8.106), we have

∂ μjμ5 = ∂ μεμνj ν = e∂t j =

eE e = − ε μνFμν, 2π π

(8.112)

where, in the last step, we have used ε μνFμν = F01 − F10 = 2F01 = −2E . To see how we obtain a theory with massive photons, we note that the last equation can be written equivalently as

⎛ e ⎞ εμν∂ μ⎜j ν + Aν ⎟ = 0. ⎝ π ⎠ In Lorenz gauge, ∂μAμ = 0, we have (homework) e jμ = − Aμ . π

(8.113)

(8.114)

Plugging this into the equation of motion (8.87) gives

⎛ e2 ⎞ ⎜□ + ⎟Aμ = 0, ⎝ π⎠

(8.115)

which shows that the photon has developed an effective mass mγ2 = e2 /π . The picture is as before: the anomaly arises from an alteration of the vacuum state of a quantized system in the presence of an applied electric field. Exercise 8.9 Derive equation (8.107). Exercise 8.10 Derive equation (8.114).

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8.10 Instantons and the chiral anomaly Perhaps while you were reading the previous section you noticed that the rate of change of the axial (chiral) current (8.105) is proportional to the local topological density FμνF˜ μν . Although, the argumentation in the previous section relied on consideration of abelian electric and magnetic fields, the same phenomena occurs in the presence of color electric and magnetic fields Ea and Ba . Hence, if there is region where there is a non-vanishing topological density, there will also be breaking of chiral symmetry. Since instantons are ‘topological lumps’ that realized precisely this situation, one concludes that the presence of instantons in the QCD vacuum will result in local breaking of chiral symmetry. This leads to, for example, a nonzero amplitude for chirality breaking processes such as

uL + dL → uR + dR.

(8.116)

Our previous discussion of instantons was restricted to pure gauge theory (Yang– Mills); however, in relation to chiral symmetry breaking instantons are important because the Dirac operator has a chiral zero model in the instanton background. These zero modes correspond to localized quark states that can become collective if many instantons and anti-instantons interact. The delocalized state that results corresponds to the wave function of the quark condensate and the instanton zero modes generate an effective four-quark interaction as indicated above. That is all we will say on this point and instead refer you to the literature: see for example refs. [22–25] and references therein.

8.11 Perturbation theory for the chiral anomaly We will now discuss how to see the emergence of the chiral anomaly in the context of perturbation theory. Unfortunately, since there is no well-defined generalization of γ5 to non-integer dimensions, typically different regularization methods such as Pauli– Villars or Schwinger regularization are used. Here we will think in terms of momentum-space cutoffs with the understanding that, if done using a proper gauge-invariant regulator the same result emerges. The analysis begins by considering the perturbative corrections to the axial-vector vertex between gauge and matter fields. The corresponding diagrams through 2 (e5) are

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where a dashed line is a particle that couples via γμγ5 to quarks such as a pion field, but in general QFT it could also be the Weinberg–Salam theory of weak interactions as well. All that require is that there is an axial vector coupling between the matter and gauge fields of the form gAjμ5 W μ, where W μ is the particle that couples to the axial current and jμ5 is the axial current (8.89). At the classical level, we have from the Dirac equation

∂ μjμ5 = 2imψ γ5ψ ≡ 2mn5,

(8.118)

where n5 is the chiral density. For m ≠ 0 this current is not conserved since axial symmetry is explicitly broken, however, even in this case the relation results in an axial Ward identity. In the case that m = 0, this current is conserved at the classical level, however, as we have discussed previously, in the massless case this current is no longer conserved when the theory is quantized. The problem remains when m ≠ 0, so it suffices to consider the chiral limit to understand the problem.

In quantum field theory, conservation laws result from analysis of the vertex functions. Analyzing the graphs in (8.117), one finds that the last graph, which contains a triangle-shaped fermionic closed loop, fails to satisfy the axial Ward identity and gives rise to the chiral anomaly. Let us now focus our attention on the triangle subgraph that couples AVV. As noted in the last lecture, there are other configurations such as AAA-triangle, squares, pentagons, etc. but to understand the basic mechanism of the chiral anomaly it suffices to consider the AVV graph. Since the photons are indistinguishable, there are two graphs that enter. They can be expressed in terms of two permutations of the diagram shown to the right. The resulting Feynman diagrams can be expressed in terms of the three-tensor

T κλμ(q1, q2 ) = U κλμ(q1, q2 ) + U λκμ(q2 , q1),

(8.119)

where

U κλμ(q1, q2 ) ≡ −

i e 2KF 2

⎡

∫k Tr⎢⎢⎣ k +1 q 8-24

γκ 1

⎤ 1 λ 1 γ γ μγ5⎥ , ⎥⎦ k k − q2

(8.120)

Relativistic Quantum Field Theory, Volume 2

with ∫ = ∫ d 4k/(2π )4 . The factor of KF arises from a sum over colored light quark k loops and is

⎡⎛ 2 ⎞2 ⎛ 1 ⎞2 ⎤ KF = CA∑Q q2σ3q = 3⎢⎜ ⎟ − ⎜ ⎟ ⎥ = 1, ⎝3⎠ ⎦ ⎣⎝ 3 ⎠

(8.121)

u, d

where σ3u = (σ3)11 and σ3d = (σ3)22 . Generally, the quantity T κλμ is the Fourier transform of the AVV current amplitude

T κλμ(q1, q2 ) = −i e 2

∫x ∫y e−iq ·x−iq ·y〈0∣T ⎡⎣ Jemκ (x)Jemλ (y )J5μ(x)⎤⎦∣0〉. 1

2

(8.122)

The diagrams result from the perturbative expansion of this quantity μ We can now check the various conservation laws ∂μJem = 0 and ∂μJ5μ = 0. Based on the last equation, current conservation for the vector and axial-vector currents gives the following conditions

q1κTκλμ(q1, q2 ) = q2λTκλμ(q1, q2 ) = (q1 + q2 ) μTκλμ(q1, q2 ).

(8.123)

Looking at the first one, which expresses current conservation at one of the electromagnetic vertices, we find

q1κTκλμ(q1, q2 ) = − i e 2

⎡⎛ 1 1 ⎞ 1 ⎟⎟γλ γμγ5 Tr⎢⎜⎜ − ⎢ k k + q k − q2 k ⎣⎝ 1⎠

∫

⎤ ⎛1 1 1 ⎞ ⎥ ⎟⎟γμγ5 . + γλ⎜⎜ − k + q2 ⎝ k k − q 1 ⎠ ⎥⎦

(8.124)

The integrals involving a single factor of photon momentum q1 or q2 vanish since the Levi-Civita tensor associated with the trace

Tr[γκγλγαγβ ] = 4iεκλαβ ,

(8.125)

requires contraction with two independent momenta in order to be non-vanishing. Defining

⎡1 ⎤ 1 Wλμ(p ) = Tr⎢ γλ γμγ5⎥ , ⎢⎣ p p − q 1 − q 2 ⎥⎦

(8.126)

one finds

q1κTκλμ(q1, q2 ) = −i e 2

∫k ⎡⎣Wλμ(k + q1) − Wλμ(k + q2)⎤⎦.

(8.127)

This integral linearly divergent in the ultraviolet. If the integrals above were convergent, or diverged at worst logarithmically, then we could shift the integration variables by −q1 in the first term and by −q2 in the second term and they would

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Relativistic Quantum Field Theory, Volume 2

exactly cancel. The linear divergence, however, means that this shift would appear in the upper limit of the integration due to the need for regulation, and would break the symmetry between the two terms. In order to carry out the shift more carefully, we Taylor expand

∫k F (k + a) = ∫k [F (k ) + a ν∂νF (k ) + ⋯],

(8.128)

to obtain

q1κTκλμ(q1, q2 ) = −i e 2(q2 − q1)ν

∫k [∂νWλμ + ⋯].

(8.129)

The omitted higher-order terms in the Taylor series vanish when we remove the cutoff since, in the case at hand, F (k ) ∼ k −3 and the integration measure contains a factor of k 3. One can use Gauss’ law to evaluate the resulting integral, which gives (homework)

q1κTκλμ(q1, q2 ) = −

i e2 εκλμδq1κq2δ , 4π 2

(8.130)

where we have used

(

Tr[γ5γργλγσγκγτγμ ] = 4iεκτμα δ ραηλσ − δ λαηρσ + δσαηρλ −

(

4iερλσα δκαητμ

−

δταηκμ

+

)

δ μαηκτ

),

(8.131)

which gives

Tr[γ5 k γλ k γκ k γμ ] = 4iεκλμαk αk 2.

(8.132)

i e2 εκλμδq1κq2δ , 4π 2

(8.133)

Similarly, one finds

q2κTκλμ(q1, q2 ) = and

(q1 − q2 )κ Tκλμ(q1, q2 ) = 0.

(8.134)

This implies that current is not conserved and that electromagnetic gauge invariance is broken. Since gauge symmetry is special we can redefine the amplitude for the triangle graph by adding a polynomial in the external momentum, which can be be done without affecting the absorptive component of the amplitude. Thus, defining the physical decay amplitude via

i e2 εκλμδ(q1 − q2 )δ , T˜κλμ(q1, q2 ) ≡ Tκλμ(q1, q2 ) − 4π 2

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(8.135)

Relativistic Quantum Field Theory, Volume 2

we can enforce electromagnetic gauge invariance

q1κT˜κλμ(q1, q2 ) = q2κT˜κλμ(q1, q2 ) = 0,

(8.136)

at the expense of a non-vanishing axial divergence

i e2 εκλμδq1μq2δ , (q1 − q2 ) μT˜κλμ(q1, q2 ) = 2π 2

(8.137)

and the axial current is no longer conserved. Using the last expression we have

T˜κλμ(q1, q2 ) =

q1μ + q2μ i e 2 εκλμδq1μq2δ . (q1 + q2 )2 2π 2

(8.138)

Using

1 F μνF˜μν = εμνρσ(∂ μAν − ∂ νAμ )(∂ ρAσ − ∂ σAρ ) 2 = 2εμνρσ ∂ μAν ∂ ρAσ = ∂ μ(2εμνρσAν ∂ ρAσ ),

(8.139)

this maps to the condition

∂μJ5μ =

e 2 μν ˜ F Fμν. 8π 2

(8.140)

This is precisely the same form, we obtain based on more physical argumentation (8.105). If we were to repeat this exercise, without taking the fermion masses to zero in the beginning, we would have found instead

∂μJ5μ = 2mn5 +

e 2 μν ˜ F Fμν. 8π 2

Exercise 8.11 Derive equation (8.119). Exercise 8.12 Derive equation (8.123). Exercise 8.13 Derive equation (8.130).

References [1] [2] [3] [4] [5] [6] [7]

Dodd R K 1984 Solitons and Nonlinear Wave Equations (New York: Academic) Kleinert H 1989 Gauge Fields in Condensed Matter (Singapore: World Scientific) Deser S 1976 Phys. Lett. 64B 463–4 Huang K 1981 Quarks, Leptons and Gauge Fields (Singapore: World Scientific) Weinberg S 1975 Phys. Rev. D11 3583–93 ’t Hooft G 1976 Phys. Rev. Lett. 37 8–11 ’t Hooft G 1976 Phys. Rev. D14 3432–50

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(8.141)

Relativistic Quantum Field Theory, Volume 2

[8] Shifman M A, Vainshtein A I and Zakharov V I 1980 Nucl. Phys. B166 493–506 [9] Baluni V 1979 Phys. Rev. D19 2227–30 [10] Crewther R J, Di Vecchia P, Veneziano G and Witten E 1979 Phys. Lett. 88B 123 [Erratum: Phys. Lett. 91B, (1980)] [11] Adler S L 1969 Phys. Rev. 177 2426–38 [12] Bell J S and Jackiw R 1969 Nuovo Cim. A60 47–61 [13] Bardeen W A 1969 Phys. Rev. 184 1848–57 [14] Ananthanarayan B and Moussallam B 2002 J. High Energy Phys. 05 052 [15] Goity J L, Bernstein A M and Holstein B R 2002 Phys. Rev. D66 076014 [16] Kampf K and Moussallam B 2009 Phys. Rev. D79 076005 [17] Landau L D 1981 Quantum Mechanics: Non-Relativistic Theory (Oxford: ButterworthHeinemann) [18] Kharzeev D E, McLerran L D and Warringa H J 2008 Nucl. Phys. A803 227–53 [19] Kharzeev D 2006 Phys. Lett. B633 260–4 [20] Kharzeev D and Zhitnitsky A 2007 Nucl. Phys. A797 67–79 [21] Kharzeev D E 2014 Prog. Part. Nucl. Phys. 75 133–51 [22] Callan C G Jr, Dashen R F and Gross D J 1978 Phys. Rev. D17 2717 [23] ’t Hooft G 1986 Phys. Rept. 142 357–87 [24] Diakonov D 1996 Proc. Int. Sch. Phys. Fermi 130 397–432 (Preprint hep-ph/9602375) [25] Schäfer T and Shuryak E V 1998 Rev. Mod. Phys. 70 323–426

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