Fortschritte der Physik / Progress of Physics: Band 26, Heft 5 1978 [Reprint 2021 ed.] 9783112519103, 9783112519097

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ISSN 0015 - 8208

FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 5 • 1978 • B A N D 26

A K A D E M I E

- V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung f ü r fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST U N D WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortachritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschlubn, P r o f . Dr. Artur Lasche, Prof. D r . Rudolf Ritsch], Prof. D r . Robert Rompe t im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2 23 6221 und 2 2 3 6 2 29; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Strafle 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik'* erscheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/26/5. © 1978 by Akademie-Verlag Berlin. Printed in the German Democratio Republio. AN (EDV) 57618

ISSN 0015 - 8208 Fortschritte der Physik 26, 2 8 9 - 3 5 6 (1978)

Source Methods in Quantum Field Theory WALTEE

DITTRICH

Institut für Theoretische Physik der Universität Tübingen, Tübingen, Federal Republic of Germany Abstract Schwinger's source theory is used to present a pedagogical review of various methods and models in quantum field theory. In the interest of simplicity we start out with a charged scalar model. Later on we also incorporate more complicated concepts in order to treat quantum electrodynamics, effective Lagrangians and model approximations.

Introduction We advocate, following Schwinger, the fundamental description of a particle to be its creation. There are no speculative assumptions about the inner structure of the observed particles. The particle is defined by the collisions that create it. Although we do not know the detailed aspects of how a particle is created, we recognize the production of a particle in a collision process as the result of a transfer of certain properties (energy, charge, spin, etc.) to the particle of interest. All the other particles are present to supply the net balance, i.e., all the others act simply as a "source". The active creation of a particle is a collision; the conversion into a final state is a collision act, too. The collision process used to detect particles may be called a sink. Here the particle's properties are handed over to the colliding partners. Hence, the same mechanism that creates particles can be used to detect them. Consequently we can unite source and sink under the general heading of source. Source can be defined indirectly through its effectiveness to create a particle (as the electric field is defined by how it acts on a charge). If we use S(x) as source symbol, then S(x) is a numerical measure that describes where the collision act occurs distributed in space-time. The same degree of control in momentum space is written as S(p); this is the numerical representation of the fact that in some part in momentum space a particle is created. The localizability in configuration and momentum space is not independent. S(x) and S(p) are complementary aspects of the same source; they are related by a Fourier transform, S(p) = J S(x) e~ipx d*x, thereby we retain the uncertainty of position and momentum. 1. Source Theory of Neutral Scalar Particles 1.1. Preliminaries This section is meant to be an introduction to various notions used in source theory without, however, going into all the details as displayed, e.g., in Schwinger's Volume I on the matter [1, 2, 3]. The concepts which are sufficient for our discussion are the ideas 22

Zeitschrift „Fortschritte der Physik", Heft 5

290

W A L T E R DITTRICH

of a simple source, extended source and causal diagrams. We use the following conventions: "circles represent sources, thin lines represent the causal propagation of real particles, heavy lines indicate non-causal propagation of virtual particles; time is read vertically. In symbols:

O

detection time.

•creation Fig. 1:1

We start with the vacuum which describes the region where nothing is present. The common notation |0) will be used for this state. Then in a small space-time region a particle is created. After it has been detected it disappears, and we are left with the vacuum again. The simplest situation is conveyed by an undeflected beam:

—

Q_

Fig. 1.2

The introduction of a causal sequence is indicated by the two distinct vacua: |0_) indicates the state before any act of creation while |0+) denotes the state after the particle has been annihilated. The quantity of general interest is then expressed by the vacuum persistence amplitude (0 + | (L) s in presence of the source S. 1.2. Spinless particles Let us consider a non-interacting spinless neutral particle of mass m. There are two basic facts which can be derived systematically: spin 0 particles can be described by scalar sources K(x) and furthermore, any number of such freely propagating particles (between production and annihilation) can be described by the transition amplitude (0+ | 0_)K =

e x p j y f (dx) (dx') K{x)

A+{x

-

x)

K{x'^,

(1.1)

where A+{x — x) propagates the source effect from x' to x. Consequently A+(x — x') is called a propagation function. It is symmetrical in x and x', A+(x — x') — A+(x'

—

x).

We want to analyze our basic formula for a causal arrangement in which particles are created by K 2 , propagate in space-time, and then are detected by K y , which is localized later in time than the emission source K 2 Before proceeding with our causal analysis we have to state another basic fact. We regard the disjoint sources K 1 and K 2 as two different realizations of the same source mechanism. Only the total source K{x) = Kx(x) + K2(x) is of physical significance. This fact is also known as principle of space-time uniformity.

Source Methods in Quantum Field Theory

291

If we now restrict ourselves to weak sources, eq. (1.1) becomes approximately * = £ a;0' the Vacuum amplitude (0+ | 0_)Ki has the form (cf. eq. (1.2)) * « = 1 + / ( * . ) ,

with

| / ( Z , ) K l , / ( 0 ) = 0.

(1.4)

Hence, we can continue to write eq. (1.3) in the form * ^

1 + f(K{) + f(K2)

Kt (-p)i

K2(p),

(1.5)

p

where we introduced the effectiveness of the source in terms of the one-particle production and detection amplitudes * = %tfdaTpK(p)

(x) ='/

which satisfies (—da +

(dx') A+(x + x') K(x')

(1.13)

m2) (x) = K(x).

(1-14)

There are now various alternative expressions for W: (0 + | 0_) K =

where

eimK),

(1.15)

W(K) = j J KA+K =jfX =

T /

+

m2)

*

=

j

f

(1-16)

^

+

m22]

(L17)

after having performed a partial integration. (Here and in the following we will frequently omit integration variables as well as inte-. gral signs.) The action expression W can also be rearranged according to ,

I

W = 2W, -

W = J

- y [ ( ^ ) 2 + m24>2]j

(1.18)

Source Methods in Quantum Field Theory

293

or W = f (dx) [(*(*)

+ 1(4>)]

K(x)

(1.19)

where Jf() denotes the well-known free Lagrangian •W) = — v [ ( W d»4> + W ] •

(1-20)

If we take eq. (1.18) assuming K and to be independent and W stationary with respect to (/>, i.e., insist that, W depends only on K, as expressed in eq. (1.12), we get d0W

= f {dK -

[8U(I> 8" d + m24> 6])

(m2 -

= / d[K -

82) 4>] = O.J

(1.21)

The introduction of charge follows the conventional treatment, namely by doubling the corresponding Klein-Gordon fields and sources. An equivalent way uses complex quantities, so that our basic formula (1.1) is modified according to + Hf*

+

I]

with * =

-

j

K W

+ ^

~ tW*)

W

+

m2v*4>]

+

# * # •

The principle of stationary action then implies the following set of coupled field equations (_£)2

+

W 2)

=

H(X)

+

8 2 + to2) ip*(x) = # * ( z ) . + gij>*(x) 4>(x), (-8*

+ ^

=

K { x )

+

#

*

( x )

(1.24)

294

W A L T E R DITTRICH

For the coupling constant a;0', we get

(A™(x - x'))2 = (/i dupe^-^Y = -f dcop daye^+P'«*-*'» -j dcop da)p-eik^-x">{2nf 6(p(2(dk) +n)3p' - k) ' (dk) / (2n)3 d{p + p' (¿ny — k) 2 and 2 2 = dM , —k = M > (2m)2,'2k°dk° ,•, • W= ^ (die) 7in F(eZ£) 1 , „ , J71„ wh h ves ,dk° = -L-i = d*». 1C

gl

Here we introduced the mass variable M . Therefore

x° > x0': [A+m(x — x'))2 = i f dMH dwkeik^-x"> J dmp dmp.(2nf = »•/dM2A+(x - x'\ M2) f dcop dmp-(2nf b(p + The last integral is a scalar, f(M2). In order to evaluate it, we go to the rest frame of k,

i.e.,

(p + p' =) Ic = 0 ,

p° = p0' = j/|p|a + m2 =

Then we obtain f(M2)

=

/ da>p

(2*)» «5(2, + V

-

*) =

~k2

¿i

-¿ji ]/

1

-

= M2.

(if)' •

(L32)

0+ 0

aT Fig. 1.6

Carrying out the space-time extrapolation, i.e., leaving the causal analysis of the graph 2 plotted in fig. 1.6, we first have to replace + r ikl x x ) by

A (x — x'; M ) = i J da>i e - ~ ' r (dJA • eikix-x')

A+(x - x'; M2)'=

j

—

k2

and so obtain for the square in (1.31) {x dM2

^

-

=

(ibfS

+

M

^

i e

] l l ~ (iw)

+ contact terms A+{x

-

;

•

Finally then we get the two-particle exchange contribution to the vacuum amplitude (1.31) in momentum space

(1.33)

296

Waltee Dittkich

The original field-source relation had the form

If we remove the fields in terms of the sources, i.e., insert = K>

^

= k2 + l

» +

2

-ie

in eq. (1.33), we create a double pole, which is unacceptable. This can be seen as follows: rewriting eq. (1.33) in terms of sources gives, up to contact terms,

(47t)2 J

]/

\M ) (F + !*2)2 k2 + M1 2 — ie

K2(k).

Together with the free propagating particle «» = i f K , { - k ) A+m(A) K2(k) we can extract the modified ^-particle propagation function from

l K2{k) fi2)2 k2 + M2 — ie

x namely:

M*) = M \ — ,is. + Mk2 +, .1 Je2 , fj? ju? —„,ieM(k*) • k2 + /¿2 — is where 00'

M{k2)

(4tt): (2m)

However, M(k2) does not vanish for k2 = —/j,2 and so changes the behavior of the propagation function for k2 + fi2 ~ 0. Consequently we will choose the contact terms in such a way as to suppress the singularities of A+(k) in the neighborhood of k2 + ¡u.2 = 0. This requirement is stated in d m e ) 2

M m , . - , - « ,

dk

= 0,

which can be achieved by s the following choice for c.t.: 1 A + ilf2 — ie 2

1 M - ft2! 2

1

k2 + fi2 _ 2 2 2 I(M7IT9 - fi,.2\2 )

k2 + P? 1 2 2 7.2 I MS75 - fiÄ2 k? + M2-

k2 + fj2 T T "T" ie ' (M2 - fi2)2

_ lk2-\-[n2 \2 1 ~ I i i 2 - ¿t2/ k2 + M2 — is'

297

Source Methods in Quantum Field Theory

With these contact modifications we obtain , M(k2) = {k2 + ft2)2

2

dM

(4 n f j (2m) 2

(2 mV

(M2-ft2)2

k2 +

M2-ie'

which suffices to meet our requirements. Finally we arrive at the modified /¿-particle propagator

1 = p +

ie

g2 (4jt)2

+

4 m2 Id2 1 2 (M - fi2)2 k2 + M21 -

dM2

ie

(2m)2

or in configuration space

A+"(x — x') =

— x') +

4m2 H2 A+{x (M2 - fi2)2

dM2

(4jt)2

x';M2).

(2m) 2

Removing the causal indices 1,2 and thereby picking up a factor 1/2, we can now rewrite (1.33) as h 2 _L ,.2 \2 1 = i 2 2(4:71) M J {2nf (1.34)

(

But we are still with (k) near the source, i.e., the 's are the fields of the virtual particles. If we add the original action

/

(dk)

*(-*)

K(k) - - 4>(-k) (k2 + v2)

m

to the V.A., we obtain from the action principle a modified relation between field and source and thereby the modified propagation function for the particle of mass ¡u:

A+"(k) =

k2 + ¡u? - (k2 + ft2)2 (4n) 2

dM2

(2m) 2

4m2 M2 1 2 (M - fl2)2 k2 + M2-

(1.35)

ie

In general then, when computing the modified /«-particle propagator, we obtain a single spectral form A+"(k) = k2 + ju,2 - ie - (k2 + ft2)2 I dM2

a(M2) {M2 - fi2)2 k2 + M2-

ie

, (1.36)

where for the simple two-particle exchange process the spectral measure a(M2) reduces to a(M2) =

/j2 (4n)

2

1 -

4m2 H2'

(1.37)

298

W A L T E R DITTBICH

Clearly we have reproduced a Lehmann-Kallen-type representation [4] for a pure scalar field theory. In fact, it is quite easy to identify the new spectral measure 2J {M2-) in A+"(k)

=

i . + k2 + [i2 — ie

r

•f

„„ dM2

Jc2 + M2-

(1.38)

ie '

We just have to compare the imaginary parts of the two different versions of A+*(k) at = -if2. Introducing a slightly different spectral measure 5{M2), where 5(M2)

a{M2) (if2-/*2)2'

=

we can write (1.36) as -M2 + p2 -

{-M2

+ fi2 J

1 + M'2 — ie

dM'25{M'2

(1.39) where we dropped the «-instruction in the propagator associated with the free particle, which is allowed since M2 > /A The imaginary part is only contained in 1

M'2 -M2

1

— ie

M'2 -

M2

+ in 8{M'2 -

M2).

+ fi2)2 P dM'2

a(M'2 —2 M' - M2

Thus, eq. (1.39) can be written in the form -M2

+ ¡x2 -

- Ì7t(-M2 In an obvious notation we have Im A+f(k)\ k i = ^ M i =

+

1 x — iy

(-M2

fi2)25(M2) x + iy and therefore x2 + y2

y x2 + y2 n(-M2

(~M2+ju2)

1 - {-M2

+ fi2)T

- fi2)2 a(M2) a{M'2 dM'2 —2 M' - M2

+ [n{-M

+ p2)2

a(M2)]2

na(M2)

(1.40) + [n{-M2

+ fjt2) a{M2)}2

Since the imaginary part of (1.38) evaluated at k2 — —M2 is simply Im

=7i£(M2),

we find the relation ' ZW2)

a{M2)

= 1 + (M 2

fi ) P I dM' M'2JL- ' M2 i ' 2

2

(M 2)

(1.41) + [:m(M2 - p.2)

a{M2)f

299

Source Methods in Quantum Field Theory

1.4. Formfactor calculation, scattering So far we were concerned with a rather limited description of physical processes in which the particles involved crossed the region between emission and absorption without interaction. We now want to extend this idealized description allowing the propagation particles to interact (fig. 1.7). There are two types of improved interaction processes possible.

In the first diagram (fig. 1.8) the emitted ^-particles are permitted to interact via spacelike (^-particle exchange while in the second picture, the two outgoing ¡/»-particles recombine to produce the modified propagation function A+'*. The upper box of this dia-

Fig. 1.8

gram shows the exchange of a time-like ^-particle. Both pictures of fig. 1.8 contain in the lower box the primitive interaction where

r

=

j =

(1.42)

=

The different upper boxes which contain the ^-particle scattering are extracted in fig. 1.9. There is, however, a crucial difference between the diagrams in fig. 1.8. While the interaction which is mediated through the modified ^-particle propagator (b) is still a local

Fig. 1.9

300

WALTEK DITTRICH

coupling, i.e., the original local coupling has been replaced by another one, namely i fj(x) (x) by i f j{x) (x), where (x) = f A+"{x — x') K(x'), we will find that the scattering process (a) induces a non-local coupling. The vacuum amplitude of interest is now given by =

L I / F J

+

M

7

-

(1.43)

with j given by eq. ( 1 . 4 2 ) , j(x) = gf*(x) 4>{x). The field ip(x) is required in the interaction region which is causally intermediate between the emission source H2(x) and the detection source Hx(x) (fig. 1.9). The total field is the superposition =

Substituting

in the V.A.

(1.44)

(1.43)

i*(x) f2(x) + i>i{x) tp2*(x)), and jn = gipHence, becomes =

j

f

(/„ +

fa

+

fa)*

(x) A+"(x

-

x')

(fa +

the V.A.

+

fa

fa)

(1.43)

(1.45)

(x').

The process with single time-like /¿-particle exchange is contained in the V.A. < 0 + I 0_> =

i ffaA+»

fa,

while the scattering process is given by w = - 1 f fa*(x) A+"(x - x') fa(x') ' = ±.gHf

(dx) (dx') {MS

+

(x) A+"(x - x')

+ M**)

(*')• (1.46)

All the other terms in ( 1 . 4 5 ) have either too many or too few emission or detection indices. Let us first analyze the scattering process (fig. 1.8). The two boxes shown in the first diagram act as effective emission and detection source. Previously we found for the effective source in emission ( 1 . 2 9 ) iH2{x) H2*{x')\ett.em. = g Ô(x — x') 2(x). The effective source in absorption is deduced by comparing exp { i / H^A^H,} with expression

(1.46).

iH*{x)

exp { i / H,A+mH2*)

= - / M*) H*(x)

H¿x') Pl-e-i^'x'Hi(—px') I

ziPl'x' {x') and the density ip{x) ip(x). The result (1.51) corresponds to an excitation-exchange of mass M which propagates

x

u

Hex t

Fig. 1.10

from x' to x (fig. 1.10). Defining h{x -x')=

JdM2I{M2)

Z, M2 -

fi2

M*

~

; M2),

we obtain in momentum space h{k2)

-{k2 + fi2)

dM2

I{M2) M2 - /¿2k2 + M2-

is'

h { k % * = ^ = 0,

with I{M2) given by (1.49) and all the integrals exist. To complete our calculation we still have to compute the second diagram of fig. 1.8. The comments given in the beginning of this section make it clear that we need to compute JT.o» = i J j{x) [${x) _ ^(a;)],

$

=

f

a+"K

= i f j{x) [Z/(a; — x') — A+"{x - a;')] K{x') i f g>p*(x) {x') dx) (dx')

j(x) cj)(x) (dx) — i f j(x) (x) (dx)

= i f j(x) h(x -

x') (x') + i f j(x) 4>(z).

(1.52)

Let us substitute (x) for (x) in the first term of (1.52). This means that the (/»-field tied to the extended source in the first diagram of fig. 1.8 becomes replaced by the iterated version = f A+^K. Equation (1.52) can then be written in the form W = i f (dx) (dx') j(x) [d(x - x') + h(x — x'j\ $(x').

If we define F(x -

x') = d(x -

x')

+

f

dM2

J^M2)

2

(82

-

fi2) A+(x -

x';

F(-fS)

=

M2),

we obtain in momentum space F(k2)

= 1-

(*» + ¡i2) I dM2 -

1,

and

= 1+

n

j

—

M>2

_ ^

16w

M

,2 _

4w2

, / M ' 2 - 4m 2 - w X logl — /x2 J M'2 — 'M2 — is ' 1.5. External field problem, non-causal method In this section we want to investigate processes with no external ^-particles. An example is given in fig. 1.11 which shows a two-particle exchange of (/»-particles between sources. The effective source in emission (lower box) for the two ^-particle is seen to be a — scattering process with a virtual if> exchange. We will indicate this by writing the argument in the action In fig. 1.11 the fields act twice in the emission act. An arbitrary number of interactions is pictorially given by fig. 1.12 which we finally want to extrapolate down to fig. 1.13. The latter process, with an arbitrary number of external (^-particle lines, is analytically given by (0+ I 0_> =

ëWW

304

W A I T E R DITTRICH

where

iW[4>] = O

C ^ Q - '

1

^

grcM+T 1

Triog í é r r n

=

Q

Trlog (1 -

\

—

O

S

(1.53)

QQ

•

N N\

O

/

V c

/]

=

04>{x)

A+[]

= i

(1.57)

gA+ m(x,x\).

There is still another useful presentation for presentation for A+[],

W[].

iW[],

If we employ the proper-time re-

s+m s—g4>) i! dse~is(-0

f

Ö

and use the differential version of (1.57), dW[]

we find

=

- i g f

A

+

( x , x | ) =

—i

oo

SW[4>]

Using

{ d x ) d{x)

gd(j>

=

8(d 2

+

g].

Tr

[gdA+[]]

(1.58) (1.59)

we can continue to write oo

SW[4>]

=

f

Tr [d{8

dse~ ismt

2

+

g$)

rh

/

(1.60)

S

o

Hence we obtain within,an additive constant oo

iW[4>]

=

J

j

e~ ism'

T r

[e

i s { e l +

sV]

(1.61)

o

or

oo /Jo

/

o

— e-fcc»'-»'-»*).

(1.62)

Let us subtract the free theory (g — 0) and unwanted divergences in (1.61). We then obtain iW[cj>]

o ds

/

—

Tr

{[U{s)

-

I70(«)] + c.t.}.

(1.63)

o Following SCHWINGER [6], we call

U{s) = (1.64) a unitary operator given in terms of the proper-time parameters s. If we define U0(s) = e isd* and introduce F(s) = e " i 5 5 V s ( 8 ' + ^ ) = U 0 ( - s ) U { s ) = U 0 H s ) U ( s ) = U ^ s ) U { s ) 23

Zeitschrift „ F o r t s c h r i t t e der P h y s i k " , H e f t 5

306

W A I T E R DITTBICH

which satisfies the differential equation 1

^

%

V(8)

OS

=

U

( - s )

0

gU0(s)

V{8)

with boundary condition F(0) = 1. The related integral equation is then given by V{a)

=

1

+

i j d s ' U

( - s ' )

0

gU0(s')

V(s').

o Multiplying from theJeft by U0(s) and recalling U(s) = U0(s) F(s), we get U(s)

=

U0(s)

+

ijds'U0(s)

U

0

( — s ' )

gU(s').

o Upon using the group property, Ua(s) Ua(t) = U0{$ U(s)

=

U0(s)

+

i j

ds'U0(s

o

-

Being interested in the lowest order dependence of U(s)

U0(s)

=

i j

+

ds1

o

U0(s

-

Sl)

i2

(1.65)

s') gU{s').

U

on

we may iterate

—

s2)

gU0(s2)

(1.65):

gU0(Sl)

Si

S +

t), we can continue to write

J

dsJJ0(a

—

g4>

s,)

0

J

ds2

U0(,s,

+

0

Originally we were interested in (1.63), i.e., in Tr

[U(s)

-

U0(s)]

i f

Tr [*70(a)

ds,

0 S +

having used Tr

/

Si dst

f

ds2

[U0{s

T r

+

s2

-

sj

A and again the group property of Si 0 «1 then, calling t = sx — s2, j ds2 -> — J dt = J dt, A

B

= Tr

i2

g]

B

0

Tr [17(a) -

SI U0(s)]

«

is

+

¿2

/

f

Tr

[U(s)

-

ds,

dt

0

U0(s)]

is

+

«2)

9]

To second order

/

g4>]

[ d t

j

dSi^r

dt(s

Tr

[U0(s

=

f

-

t) gU0(-h

-

g4>]

t)

Tr [F 0 ( a -

t) g Ua(t)

g#\.

(1.66)

Source Methods in Quantum Field Theory

307

So far the traces were understood to be performed in configuration space. However, it is more convenient to go to momentum space. In momentum space we find for the first term on the right-hand side of (1.91) isg Tr [U0(s) ] = isgp £ (j>\ e^" \p) = isg £ e-W 0, we obtain

isgTr[U0(s)]

= -g4>(0).

(1.67)

S

The second term in (1.66) yields the contribution i2 / dt (s - t) Tr o

[e^-'^g^'g^]

s

e-«5-»^!

= (igf jdt(s-t)2J 0

$ \p') e~ilf"{p'\

|p)

p,p'

s

= (igf J dt(s - t ) f (dp) (dp') e-^-W-Wfap o

— p') 4>(p' — p).

Calling p — p' = k, we find 5

-H = (igf f dt(s - t ) f (dp) e~ i(s-i)p2 J (dk) 4>(k) 4>(-k) o k/ Another change of variables, — q = p — — , i.e., p — k = \p k V I . \ •w = -I(-k) J(dq) exp j - t ( « - i) s

o s(l — u) =

k\ k k —I — = —q — 2 / 2 / 2

- - I ) ' - * (« + I " ) }

1

with new variables t = us, so t h a t J dt -> s J du and o o l i r u = — (1 -f- v), so that I du

and furthermore (s — t)

er^v-W.

A

+i i r — I dv -1

s(l — v), we obtain

+1 2

= (ig)

dv( 1 - v)J

(dk) 4>(k) 4>(-k)

-l X

23*

J (dp) exp

(1 - v) (p - - | j 2 - ¿ - 1 ( 1 + » ) ^ + A j j .

308

Waiter Dittrich

k2 | The exponent of this term is —is p2 + vpk + — I and we need to evaluate e-iS*V4

j ^p) e-is(p*+vpk)

= e-iskVi J

^p) expj—is ( p +-J

kJ

~

|•

712 A change of variables and J (dp) e~i3p* = — i— then gives s e-isk2li

(-v)

_ —i— e-isk'na-t")_

e i s

So far we have found that +i -f-=-¿02 i - j j 2 j"dv(l-v)

f

(dtyMtyM-tye-™"*!^1-^,

Here it is convenient to perform an integration by parts : risr''4a4 f dv{ 1 - ») e-(-k). while the remainder of rem.

(1.68)

is given by

= (ig2 ^Çj

/ (dk) 4>(k) 4>(-k) k2 f

dw^"'^1^.

(1.69)

The corresponding contributions of iW[(f>], eq. (1.63), are therefore given by oo iW™[] = f j < +

8

S2

2

2

— g4>{0) + ig2 S

i*

Z J

/ (di)