Quantum Fluctuations 9780691218021

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Quantum Fluctuations

Princeton Series in Physics edited by Arthur S. Wightman and Philip W. Anderson Quantum Mechanics for Hamiltonians Defined as Quadratic Forms by Barry Simon Lectures on Current Algebra and Its Applications by Sam B. Treiman, Roman Jackiw, and David J. Gross Physical Cosmology by P.J.E. Peebles The Many-Worlds Interpretation of Quantum Mechanics edited by B. S. DeWitt and N. Graham The P(Q>)2 Euclidean (Quantum) Field Theory by Barry Simon Homogeneous Relativistic Cosmologies by Michael P. Ryan, Jr., and Lawrence C. Shepley Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann edited by Elliott H. Lieb, B. Simon, and A. S. Wightman Convexity in the Theory of Lattice Gases by Robert B. Israel Surprises in Theoretical Physics by Rudolf Peierls The Large-Scale Structure of the Universe by P.J.E. Peebles Quantum Theory and Measurement edited by John Archibald Wheeler and Wojciech Hubert Zurek Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager by Stephen G. Brush Quantum Fluctuations by Edward Nelson

Quantum Fluctuations by Edward Nelson

Princeton Series in Physics

Princeton University Press Princeton, New Jersey

Copyright © 1985 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 IN THE UNITED KINGDOM:

Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Cataloging in Publication Data will be found on the last printed page of this book ISBN 0-691 -08378-9 (cloth)

0-691 -08379-7 (paper)

Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America 8

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5

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CONTENTS PREFACE

vii

Chapter I. KINEMATICS OF DIFFUSION 1. Differentiable Manifolds 2. Affine Connections 3. Measures on Path Space 4. Martingales 5. Diffusion 6. Markovian Diffusion 7. Continuity of Paths 8. Stochastic Integrals 9. Stochastic Action 10. Stochastic Parallel Translation 11. Existence of Diffusions

3 3 11 15 18 22 34 37 41 44 47 51

Chapter 12. 13. 14. 15.

58 58 60 65 77

II. DYNAMICS OF CONSERVATIVE DIFFUSION Newtonian Dynamics Lagrangian Dynamics Stochastic Quantization Nodes

Chapter III. STOCHASTIC MECHANICS 16. Gaussian Processes 17. Interference 18. Momentum 19. Bound States 20. Statistics 21. Spin

83 83 90 95 98 100 102

Chapter IV. PHYSICS OR FORMALISM? 22. Measurements 23. Locality 24. Fields

112 112 119 130

A LIST OF OPEN PROBLEMS

133

vi

CONTENTS

BIBLIOGRAPHY

135

INDEX

141

LIST OF SYMBOLS

145

PREFACE These are the revised lecture notes of a course given in June, 1983 for the Troisieme cycle de la physique en Suisse romande. This course was given at a time when my thinking about stochastic mechanics was in a state of flux. There are many loose ends; this is not a treatise. I publish it in the attempt to express a viewpoint about the nature of quantum fluctuations, and in the hope that others will be encouraged to work on the many mathematical problems that are left unresolved here. I wish to express my deep thanks to P. Huguenin, W. Amrein, J.-P. Eckmann, and all of the Troisieme cycle for the invitation to give this course and for the many arrangements they made, to the audience for their lively participation, to R. Chacon for an invitation to talk about stochastic mechanics in Vancouver in the summer of 1982, to W. Faris and S. Goldstein for insisting that Markovian stochastic mechanics violates locality, to A. Rechel-Cohn and L. Smolin for sharing their ideas about gravity and quantum effects, to E. Carlen for invaluable help in writing this book (finding errors, suggesting ideas, and informing me about the literature —a role reversal of the student-teacher relationship), to J. Lafferty and M. Campanino for correcting computational and conceptual mistakes, to F . Guerra for generous help and criticisms, to K. Yasue for helpful correspondence and for locating our beautiful apartment, to J . S . Bell, S. Buss, and N. Gisin for stimulating conversations, and to J.-C. Zambrini for making our stay in Geneva extremely enjoyable both scientifically and personally. This work was partially supported by the National Science Foundation grant MCS-8101877 A 02.

VII

Vlll

PREFACE

The thesis of this investigation is that perhaps quantum fluctuations are real, and have physical causes. No physical system of finitely many degrees of freedom is truly isolated; it is always in interaction with a background field. Consider a system with Lagrangian

The inverse m1^ of the mass tensor m- is a symmetric positive definite contravariant tensor. So is the diffusion tensor a1^ , given by

of a diffusion process ^, where Ej. denotes the conditional expectation given the configuration f(t) at time t . We may set a1^ proportional to m1-' , but dimensional considerations require the proportionality constant to have the dimensions of action. The background field hypothesis is that interaction with the background field causes the system to undergo a diffusion process with CT1-' = lim^ satisfying a variational principle SE/Ldt = 0. The paths of a diffusion process are nondifferentiable. This can be guessed from (1), which shows that df 1 is of order \Jdt. This means that the kinematics of diffusion is very different from deterministic kinematics. What is meant by velocity and acceleration? How are tensors transported along paths? What is meant by an action integral involving the square of the velocity? How can one construct a diffusion process from an infinitesimal description of it? These questions are discussed in Chapter I. Next we take up stochastic quantization, basing it on a variational principle, and derive the Schrodinger equation. Then familiar topics from quantum mechanics — interference, momentum, bound states, statistics, spin—are discussed. Following this we investigate the physical interpretation of the theory, including problems of measurement and nonlocality. The notes conclude with a brief discussion of stochastic field theory.

Quantum Fluctuations

Chapter I KINEMATICS OF DIFFUSION This chapter is long, and much of the material in it is well known. But some readers may be familiar with differential geometry but not measures on path space, or vice versa, so I have included a lot of elementary material for reference. Most of the differential geometric material is physically relevant only to the discussion of spin, and it may be omitted by the reader who is willing to accept the result that one obtains the correct equation of motion even on a curved manifold. The main exposition of this chapter begins in §5 where I attempt to construct a stochastic calculus with a minimum of technicalities. This is done at the cost of a loss of generality and elegance, and for a better approach the reader is referred to the French and Japanese schools of probability theory, e.g. [4], [32], [36], [45]. §1. Differentiable Manifolds The first two sections of this chapter are a review of deterministic kinematics. The configuration space of a mechanical system is a differentiable manifold M . If the system has n degrees of freedom, then M is an n-dimensional manifold. An n-dimensional manifold is specified as follows. It is a pair consisting of a set M and a collection 11, called an atlas, of charts. A chart is a pair (U, ^TJ) consisting of a subset U of M, called a coordinate neighborhood, and a bijective mapping Oy from U to an open subset of R n . The components q ,---,q n of $ y are called local coordinates.

If U' is also a coordinate neighborhood,

4

1. DIFFERENTIABLE MANIFOLDS

then we have two systems of local coordinates, q , - - , q n and q 1 , - - - ^ 1 1 , on U H U". (It is convenient to use different index sets 1, ••-, n and 1', •••, n' for different local coordinates.) We require that the mapping ^U' O< ^U 1

1

dq /dq .

ke C°°. Its matrix of partial derivatives is denoted by We give M a topology by defining a subset G of M to be

open in case each ^ ( G f l U ) is open. We also require that M be a Hausdorff space; that is, for any two distinct points x^ and x 2 in M there are disjoint open sets Gj and G 2 containing them. Finally, we require that the coordinate neighborhoods cover M, and that M be connected. For example, Rn is a manifold if we take the atlas consisting of a single chart: R n and the identity mapping. If n = 3N, this is the configuration space of N spinless particles in 3-space. Let M and N be differentiable manifolds. We say that O:M-^N is C°°, or smooth, in case each $ v o0»y 1 is C°° (where U is a coordinate neighborhood in M and V is a coordinate neighborhood in N ). If is bijective and ~

is also smooth, then is called a diffeomor-

phism of M onto N, and M and N are diffeomorphic in case there exists such a R (1.1)

such that

iF$(t,x)=X(d>(t,x)), $(0,x) = x

for all t in R and x in R n . H we let $* be x t-> O(t,x), then t t-^ 0 (1.2)

is a one-parameter group of diffeomorphisms of Rn : $

t,+t9 l

=0

t,

°$

t

2

.

Then

1. DIFFERENTIABLE MANIFOLDS

Proof. Let K be a Lipschitz constant for X ; that is, |X(xj)-X(x 2 )|
R n , let

UII = sup |f(s)| . sel

The initial value problem (1.1) is equivalent to tt

(1.3)

O(t,x) = x x+ + jj X(

M . Then we may ask for a family Y(t) in T>- M such that

2. AFFINE CONNECTIONS V

(2.15)

x(t) Y W = °
T*. M. This is parallel transla-

tion. There is an induced linear mapping, again denoted by r,.(s,t), of T

P into T

P.

Using this, we can make precise the heuristic notion used to motivate the definition of an affine connection: VXY is (2.2) if the previously undefined T in it is taken to be 7>(t +dt, t) . Let £ be a smooth path. Each Y in T*. M is an equivalence class of mutually tangent smooth curves r H> T](T) with r)(0) = £(t). Let y = r](dr); this is well defined up to o(dr). That is, we may think of a tangent vector Y as being a neighboring point y. On an affinely connected manifold, parallel translation moves an infinitesimal neighborhood of f(t) in such a way that the points in it all have the same velocity as §3. Measures on Path Space You all know probability theory in depth, thanks to the labors among you of the apostle of probability to Swiss physics. Nevertheless, let me review the basics. A probability space is a measure space of total measure one —that is, a triple (S, S, /x) where S is a set, S is a oalgebra of subsets of S , and /x is a positive countably additive function on S such that /x(S) = 1. A random variable f is a measurable function on S . If it is numerical valued, we define its expectation (or mean) by

Ef = I fd/x

16

3. MEASURES ON PATH SPACE

and its variance by E(f-Ef) , provided the integrals exist. An event is an element of S. A stochastic process is a function from some index set I to the random variables on a probability space. One peculiarity of probabilists is that they take a-algebras seriously, rather than regard them as a technical nuisance. Let $ be a sub-a-algebra of S. Then (S, S , ^ f S ) is also a probability space. If f e L ^ S , S,/x) , then A t-> J fd/n for A in $ is a finite measure on (S, 55) that is A

absolutely continuous with respect to ^ , so by the Radon-Nikodym theorem there is a unique (up to equality a.e.) element f Q of L (S,S,/z| £) such that f fdu = f f-du for all A in S . Then fn is called the conditional expectation of f with respect to SB, and it is denoted by E{f|fB|. We may think of 8 as being the events accessible to observation; then E{f|Si is the best prediction of f that we can make on the basis of available knowledge. We denote by $ the set of ^-measurable random variables. Then E{f|$! is S-linear: E{gf|$} = gE{f|$! for g in % , provided the integrals exist. Suppose we want to study a stochastic process indexed by a set I and taking values in a locally compact Hausdorff space M. Form the onepoint compactification M = M U {oo}. This is a compact Hausdorff space. Now form the Cartesian product Q of M indexed by I :

An element of Q is a completely arbitrary function a converges in fl to a> if and only if 0, and as soon as this happens the investor sells the share. That is,

(

l,f(s)-£(a) A if successful, and are £(b)-f(a) otherwise. That is, (4.10)

£(b) > A X A + (f(b) -f(a))x A c ,

where x denotes the indicator function and c denotes the complement. Now suppose that, unfortunately for the investor, f is a supermartingale. Since r/ > 0 , C, is also a supermartingale, and since £(a) = 0 we have E£(b) < 0. By (4.10) this implies that PrA < ||£(b)-£(a)|| / A . We have proved the following theorem: THEOREM 4.1. Let £ be a supermartingale indexed by a Unite subset I of R , and let A > 0. Then Pr{max(f(t)-(a))>\l 0 .

be a supermartingale

Then

Prl sup (f(t)-f(a))>AS A } < i - ||&b)-£(a)||2 a