The retarded electron propagator S_{R}(p_{0},p) at high momentum p>>T was shown by Blaizot and Iancu to be an entire function of complex p_{0}. In this paper a specific form for S_{R}(p_{0},p) is obtained and checked by showing that its temporal Fourier tr

Bloch-NordsieckcalculationofBlaizotandIancu[7,8,9]

gives

s

R

(

t,

p)= iθ(t)e iptexpf(z)isgivenby

αTtf(ωpt)

,wheref(z)=C γ+

1 e z

s

.

Thisrepresentationshowsthatf(z)isanalyticevery-whereinthecomplexzplane.(SeealsoAppendixB.)

Thustheretardedpropagatorwithθ(t)omittedisana-lyticeverywhereinthecomplextplane.

Analyticityintisveryunusual.AppendixAshowsthatingeneraltheretardedpropagator(withouttheθ(t))isthedi erenceoftwofunctions:one,S>(t,p),isan-alyticintheopenstrip β<Im(t)<0;theotherS<(t,p)isanalyticintheopenstrip0<Im(t)<β.Thusgenerally,theretardedpropagatorisonlyde nedontherealtaxisandisnotanalytico -axis.AppendixAalsoshowsthattheanalyticityintofsR(t,p)doesnotimproveon,ordetractfrom,thevalidityoftheKubo-Martin-Schwingercondition[12],whichrequiresS>(t iβ,p)= S<(t,p).

III.

THEPROPAGATORFORp0≈p

Thissectionwillprovideevidencethatagoodapprox-imationtotheretardedelectronpropagatorintheregion

p

παT

2

,

(3.1)

isthefunction

sR(p0,p)= iNexp ip0 p

(3.2)

whereN=

√

αT

, ×cos

2αT

p0 p

αT

.

Ifaissmall,thisispositiveintherangegivenbyEq.

(3.1).For|p0 p| απTtheapproximatebehaviorρ(p0,p)≈2Nea

1

12

pis

0 p

2

2(γ0 γ·p )sR(p0,p)

1

ψλ(x′

)O(y1,...yn)

1

2π

e ip0tsR(p0,p)J

whereJ

(p (p0,p),(3.4)

0,p)istheFouriertransformofJ(t,p).Example:Fortheconventionalretardedpropagatorofthequarktype(witha niteγ)thenecessaryenergyintegrationis

Ψ(t,p)=

dp0

pJIfJ

(p (p0 p+iγ/2

0,p)0,p)isanalyticinp0inthelower-halfplaneandvanishessu cientlyrapidlyasImp0p0contourcanbeclosedinthelower-half→ ∞plane,thenandthetheresultevaluatedbyCauchy’stheorem.Thesimplestwaytoinsurethatthesourcehasthesepropertiesinp0istoinsistthatitvanishfortimeslaterthansometJ(t′,p)= f:

0t′>tf

arbitraryt′<tf.(3.5)

TheFouriertransformofthesourceis

J

(p0,p)= tf

dt′eip0t′

J(t′,p).(3.6)

∞

ThisisanalyticforImp0<0andsothep0integrationtobeperformedbyCauchy’st>tf:Ψ(t,p)=exp theoremtogive

Theremainderofthissection ip γ

2

,p).(3.7)

willshowthatwithacur-rentofthistype,thesimplefunctiongiveninEq.(3.2)producestheanalogousresultbutwithγ/2replacedbyαT[ln(ωpt)+C].