Generalized Feynman Amplitudes. (AM-62), Volume 62 9781400881864

Table of contents :
TABLE OF CONTENTS
INTRODUCTION
CHAPTER I: Renormalization in Lagrangian Field Theory
Section 1. INTRODUCTION
Section 2. FIELD THEORY
Section 3. RENORMALIZATION
Section 4. THE RESULTS OF HEPP
CHAPTER II: Definition of Generalized Amplitudes
Section 1. INTRODUCTION
Section 2. GENERALIZATION OF THE PROPAGATOR
Section 3. PRELIMINARY DEFINITION OF THE GFA
Section 4. REDUCTION OF SPIN TERMS
Section 5. THE 2 → 0 LIMIT
CHAPTER III: Analytic Renormalization
Section 1. INTRODUCTION
Section 2. ANALYTIC PROPERTIES
Section 3. REMOVING THE λ - SINGULARITY
Section 4. VALIDITY OF ANALYTIC RENORMALIZATION
CHAPTER IV: Summation of Feynman Amplitudes
Section 1. INTRODUCTION
Section 2. Q DEPENDENCE OF THE GFA
Section 3. GENERALIZED FEYNMAN AMPLITUDES IN A FIELD THEORY
Section 4. SUMMATION OF AMPLITUDES
CONCLUSION
APPENDIX A: Graphs
APPENDIX B: Distributions
APPENDIX C: The Free Field
BIBLIOGRAPHY

Citation preview

Annals of Mathematics Studies Number 62

GENERALIZED FEYNMAN AMPLITUDES BY

Eugene R. Speer

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1969

Copyright ©

1969, by Princeton University Press AL L RIGHTS RESERV ED

L.C. Card: 72-77595

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

Acknowledgements I would like to thank my advisor, Prof. Arthur Wightman, for giving generously of his time during the past two years. His help and advice were invaluable in the preparation of this thesis. I would also like to thank Prof. Edward Nelson for reading the manuscript and for various suggestions, and Prof. Tullio Regge for several helpful discussions. I am grateful to the National Science Foundation for support during four years of grad­ uate school, to the Princeton University Mathematics Department for support during the summer of 1968, and to Dr. Carl Kaysen for his hospitality at the Institute for Advanced Study. Part of this work was sponsored by the Air F o rce Office of Scientific R esearch, Office of Aerospace R esearch, United States Air F o rce, under AFOSR Grant 68- 1365.

A BSTRAC T Renormalization in the context of Lagrangian quantum field theory is reviewed, with emphasis on two points:

(a) the Bogoliubov-Parasiuk definition of the renormalized amplitude of an arbitrary

Feynman graph, including some generalizations of the rigorous work of Hepp, and (b) a discussion of the implementation of this renormalization by counter terms in an arbitrary interaction Lagrangian. A new quantity called a generalized Feynman amplitude is then defined. It depends analytically on complex parameters A1, .. . ,

, and these analytic properties may be used to define renormalized

Feynman amplitudes in a new way; the method is shown to be equivalent to that of Bogoliubov, Parasiuk, and Hepp. The generalized Feynman amplitude depends on other parameters also; when these take on certain values, it is equal to the Feynman amplitudes for various graphs (aside from problems of renormalization, which are handled via the A dependence).

The generalized amplitude

thus interpolates the Feynman amplitude between different graphs. Some partial results are ob­ tained which exploit this interpolation to give an integral representation for a sum of Feynman amplitudes.

T A B LE OF CONTENTS

INTRODUCTION........................................................................................

1

C H A P T E R I: Renormalization in Lagrangian

F ie ld T h e o ry .................................................................... 5 Section 1.

I N T R O D U C T IO N ..........................................................5

Section 2.

F I E L D T H E O R Y ..........................................................5

Section 3.

R E N O R M A L I Z A T I O N ............................................. 15

Section 4.

T h e R e s u l t s OF He p p ................................... 38

Definition of G eneralized Amplitudes . . . 43

C H A P T E R II: Section 1.

I N T R O D U C T IO N .......................................................43

Section 2.

GENER ALIZ AT ION OF THE

Section 3.

PR E LIM IN A R Y DEFINITION OF

Section 4.

REDUCTION OF SPIN T E R M S .....................

50

Section 5.

THE

52

P r o p a g a t o r ....................................................

44

THE G F A ...................................................................... 47

C H A P T E R III: Section 1.

-> 0 L I M I T ............................................

Analytic R enorm alization ............................. 61 I N T R O D U C T IO N .................................................... 61

Section 2.

ANALY TIC P R O P E R T I E S . . . : ..................

62

Section 3.

REMOVING T HE A - SINGULARITY . . . .

72

Section 4.

VALID ITY OF AN ALYTIC

C H A P T E R IV:

R e n o r m a l i z a t i o n .......................................

75

Summation of Feynman Amplitudes . . . .

81

Section 1.

I N T R O D U C T I O N .................................................... 81

Section 2.

Q D E P E N D E N C E OF THE G F A ................

Section 3.

G E N E R A L IZ E D FEYNMAN

Section 4.

SUMMATION OF A M P L I T U D E S ..................

90

C O N C L U S I O N .........................................................................................

95

82

A m p l i t u d e s in a fi e l d t h e o r y . . . 86

A P P E N D I X A:

G ra p h s ....................................................................

97

A P P E N D I X B:

D istrib u tio n s ......................................................

103

A P P E N D I X C:

The F re e F i e l d .....................................................109

B I B L I O G R A P H Y ...................................................................................

119

INTRODUCTION The b asic subject matter of this thesis is the Feynman amplitude (sometimes called a Feynman integral) which is always associated with a Feynman graph. A Feynman graph G is a graph [see Definition A. l] together with an assignment of a propagator A ® to each line i of the graph; A®

is a distribution in c^'(R4 ) whose Fo urier transform has the form

a (E)(p) = z ?(P) - — \ — p2 - m2 + iO with Zj? a polynomial.

,

Suppose the graph has n vertices and L lines;

then the corresponding

Feynman amplitude T(G ) is the function of n 4-vectors x v ..., xn given by L I = 1 with e ® the incidence matrix of the graph (Definition A. 6). (We lean throughout on the mathei matical props of graph theory and the theory of distributions. The basic definitions we will use, and some simple results we will need, are given in Appendices A and B .) Feynman amplitudes arise when one studies quantum field theory by perturbation-theoretic techniques [see, for example, 2, 32]. In such a study the amplitudes (whose squares give the probabilities) for, say, a scattering process are given as an infinite sum of Feynman amplitudes for different graphs. Now the validity of these techniques is questionable, and the resulting series is not known to converge in any ca se and known to diverge in some [20]. Nevertheless, we are interested in studying the terms of the series for several reasons:

(a) we would like to

prove convergence (or even divergence) for series whose behavior is not yet known, (b) the su cce ss of the perturbation approach in quantum electrodynamics suggests that we may have an asymptotic series even if not a convergent one, (c) the analytic properties of the summands in the momentum variables suggest analytic properties which may be valid for a true scattering amplitude. Part of the work in this thesis bears on these problems, as we will discuss later. There is another problem connected with Feynman amplitudes which must be dealt with be­ fore any of the above: the product ( 1) is in fact not even well defined for many graphs, due to the coincidence of the singularities of the factors (or, in momentum sp ace, due to the divergence of the convolution integrals).

Dyson [7] has shown how to define these amplitudes appropriately.

In this process of renormalization the divergences are attributed to unobservable physical effects; they can then be “ subtracted out” to produce finite observable results.

To state this in a slight­

ly different way: for any graph G in which the product (1) contains divergences, we may define instead a renormalized Feynman am plitude; moreover, the renormalization of graphs in a field theory has an acceptable physical interpretation. 1

GENERALIZED FEYNMAN AMPLITUDES

2

We treat these matters in Some detail in Chapter I (see also Appendix C, a brief treatment of the theory of free fields). There we review the basic concepts in the Lagrangian theory of interacting fields, the Bogoliubov-Parasiuk definition of renormalization [ 2 ,3 , 27], and the mathe­ matically p recise work of Hepp [18] on the properties of this definition (referred to as BogoliubovParasiuk-Hepp renormalization, to distinguish it from the somewhat different approach of Dyson and Salam [ 7 ,2 9 ] ) . The chapter contains a thorough analysis of the implementation of renormali­ zation by counterterms in the Lagrangian, although we do not d iscu ss the physical interpretation. We also give a relatively simple extension of Hepp’s results which is needed in Chapter III. Chapter II defines a new quantity called a generalized Feynman amplitude, or GFA for short. This amplitude is not associated with a particular Feynman graph, but rather depends on certain parameters A, Q , e , Z , and m (whose specific nature does not concern us here).

When these

parameters take on appropriate values, the GFA becomes (at least formally) equal to the Feynman amplitudes for various graphs. The “ appropriate value” for A is a fixed point A0; if we take A = A0 , we vary the structure of the graph in question by varying Q and e , and vary the propa­ gators associated with the lines by varying Z e

and m . That is, considering only the Q and

dependence, we have gone from the Feynman amplitude, defined for a discrete set (the graphs),

to the GFA, which is defined for a continuous variable and is equal to various Feynman amplitudes at discrete points.

Stated another way, the GFA is an interpolation of the Feynman amplitude be­

tween different graphs. As implied above, the equality of the Feynman amplitude and the GFA is in some ca se s only formal. This occurs when the graph in question needs to be renormalized, in which ca se the Feynman amplitude (1) is not well defined. The parameter A , which is really an L-tuple ( A j,... , Aj^) of complex variables, plays an important role in these divergence problems; this is the subject of Chapter III. Suppose then that Q, e , Z ,

and m have been chosen to correspond

to some graph G. The resulting GFA is meromorphic as a function of A, and, as stated above, is formally equal to 3*(G) at A0 . Now it turns out that a divergence in the amplitude T (G ), necessitating a renormalization, corresponds precisely to a singularity in the GFA at the point A = A0 . Moreover, we can give a prescription for removing singularities of this type, in such a way that application of the prescription to the GFA produces a correctly renormalized Feyn­ man amplitude for the graph. It is important to note that this prescription does not depend on the structure of the graph (in contrast to the usual renormalization procedure, whose recursive subtractions depend explicitly on various subgraphs, e tc.).

Thus, for example, we may renor­

malize a sum of Feynman amplitudes by one operation, rather than treating each summand sepa­ rately. In Chapter IV we turn to applications of the dependence of the GFA on the parameters Q and e (the Z and m dependence plays only a minor role). We originally introduced these parameters in the hope that we could convert a sum of Feynman amplitudes, such as occurs in the perturbation expansion, into an integral over the parameters Q and e . Such an integral

INTRODUCTION

3

representation might be useful in estimating the size of the sum, for study of the convergence properties of theexpansion, or indetermining the

analytic properties of the

sum in the momenta.

This program has been onlypartially su ccessful: we have found such anintegralrepresentation, but it is valid only for A restricted to a region which does not contain the physical point A0 . Any applications depend on finding an explicit analytic continuation of the integral to a neigh­ borhood of A0. The following notation will be used throughout; it is all standard except possibly (b). Further notation is introduced where appropriate; see in particular Appendices A and B. (a) If x and y are 4-vectors, we write

x -y

=

% IL,V= 0

-

where

i j

if n = v = 0 ,

- 1, if [i = v = i > 0 , 0,

if otherwise.

(b). In general, we denote a k-tuple of variables ( x ^ - .^ x ^ ) by underlining:

( x 1, . . . , x ^ ) = x .

We will use this notation with different values of k simultaneously [e .g ., ( a^ ..., a (c1 ,

= a,

= c ]; the dimension of each variable will be clear from context.

‘We also write: (c). x 1 , ..., x - , ..., x R — the variable x^ is omitted; (d).

Qx^] = the greatest integer less than or equal to the real number x ;

(e).

# (K) = the number of elements in the finite set K ;

(f).

Sn = the complete permutation group on n elements;

(g).

det A

(h).

A (j) = the (signed) cofactor of the entry A - in the matrix A ;

=

the determinant of the matrix A;

( i) .

A(-

) = the (signed) cofactor of Aj^ in the matrix obtained by deleting the i ^ row and

column of A.

A( j ^ ) vanishes if i = j or i = k .

We comment finally on the organization of the thesis.

Each chapter is divided into sections

numbered 1, 2 , ... e tc.; these are sometimes further subdivided for clarity, using the sequence capital letter, small Roman numeral, small letter [e.g ., there is a Chapter I, Section 3 (c. iii. a)]. Theorems, definitions, remarks, e tc., are numbered in one continuous sequence for each chapter (e .g ., Definition 1. 1, e tc .) and appendix (Definition A. 1). in different sequences.

Equations are numbered similarly, but

All references are to the bibliography at the end of the thesis.

CHAPTER I Renormalization in Lagrangian F ie ld Theory Section 1. I N T R O D U C T I O N This chapter is devoted to a brief discussion of Lagrangian Quantum Field Theory, the subject which underlies the entire th esis but which will be mentioned explicitly only in this chapter. The review of the basic material (Section 2) is much too brief even to be called self-contained; we are interested mainly in motivating the study of the perturbation series for the time-ordered vacuum ex­ pectation values of a theory and of the Feynman integrals which are the terms of this series.

These

integrals are often divergent, and renormalization theory is the study of ways to modify them to con­ vergent integrals while maintaining an acceptable physical interpretation of the theory. The final formulas for these “ renormalized” integrals have a complicated combinatorial structure. In Section 3 we d iscu ss the relation of these formulas to the field theory, given by the idea of “ counterterms” in the Lagrangian of the theory.

Finally, Section 4 summarizes the mathematically rigorous work of

Hepp on the properties of these renormalized amplitudes, and gives some extensions of his results which we will need in Chapter III. In this chapter we will frequently d iscu ss products of non-commuting operators on Hilbert space. We therefore adopt the following convention:

if |A- |i = l,...,n | are such operators, then

n II A- = Ax A2 -- An . i=1 Similarly, n

Section 2.

m(j)

II

II

i = l

]= 1

An

"• A i m ( i ) A 2 i

An,m(n)

FIELD TH EO RY

Our discussion of field theory will follow the notation and spirit of PC T, Spin and Statistics, and All That, by Streater and Wightman [34], Thus the physical states of the theory form a Hilbert space H, which is equipped with a unitary representation, la, Aj - U(a, A) , of the group “ inhomogeneous S L(2,

C).” A field 0 is, technically speaking, an operator-valued

distribution on S (R 4), but we follow the usual procedure and write symbolically 0 ( x ) , where 0 (f) = / 7r 4

0 (x )f(x )d x

6

GENERALIZED FEYNMAN AMPLITUDES

for any f e S(R4). We refer to Streater and Wightman for a discussion of the finite-dimensional representations of SL(2, C); see also [11]. (A). The free field of mass m. (i). Transformation law. Our free field (f> actually co n sists of M components 1cj>v ..., 0 Mi =

They transform

under Lorentz transformations by the formula

(1 .1 )

M = % Sa g( A- 1) is either a fermion or boson field. Condition (b) implies the existen ce of an M xM invertible Hermitian matrix 7/ satisfying (1 .2 )

t/"~ 1 S(A)f 77 = S_ 1(A)

for any A € SL(2, C ) (here S(A)^ is the Hermitian adjoint of the matrix S(A)) [ l l ] . Equations ( 1. 1) and ( 1.2) imply that the quantity

^

transf° rms like a scalar field

under Lorentz transformations, so that (1 3 )

is a scalar.

/ /u d 4x a X

va;

We will use this matrix 77 in constructing a Lagrangian for the free field. Note

that if we consider to be a column vector,

and let

be the row vector [