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Quantum electrodynamics in an external constant field
Nuclear Physics B258 (1985) 435-467 © N o r t h - H o l l a n d Publishing C o m p a n y
QUANTUM ELECTRODYNAMICS IN AN EXTERNAL CONSTANT FIELD I.A. B A T A L I N and E.S. F R A D K I N
P.N. Lebedev Physical Institute, Moscow, USSR SH.M. S H V A R T S M A N
Department of Mathematical Analysis, Tomsk Pedagogical Institute, Tomsk, USSR Received 3 December 1984
The generating functional of all q u a n t u m Green functions in an external field is found. The Green function of a scalar particle in a constant and uniform external field is calculated with the help of modified perturbation theory. Its infrared asymptotics is studied. It is shown that an e n h a n c e m e n t of the pole takes place, as in the vacuum case.
1. Introduction
There has been great interest in electromagnetic processes in external fields during the last few years. This is accounted for, on the one hand, by the creation of the intensive electromagnetic fields in lasers and the appearance of electrons and photons of high energies, and, on the other hand, by possible astrophysical applications. In the above processes one can verify quantum electrodynamics in the domain of high energies and large fields. The study of this phenomenon is of fundamental importance because a correct description of the phenomenon in intensive external fields is connected with the necessity of going beyond the framework of the usual perturbation theory. For this reason the processes in the e 2 approximation in different external fields [1-14] have been studied in detail lately with the external field taken into account exactly (in this connection the constant uniform field and the plane wave field or their combination are of a special interest because in these external fields one can exactly solve the Dirac and Klein-Gordon equations and find the one-particle Green functions). However, the contribution of radiative corrections at high energies and intensive fields becomes essential. For this reason it becomes of special interest to work out such methods of calculation of radiative corrections to the processes in an external field which, in their final result, go essentially beyond the scope of the usual perturbation theory with the external field kept exactly. This is due to the fact that the generalized (with the external field being taken into account exactly) Feynman diagram technique encounters essential difficulties caused 435
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LA. Batalin et aL / QED in external constant fieM
by the necessity of correct calculation of the vertex function in a self-consistent approximation, since otherwise the results obtained are gauge noninvariant, and the corresponding generalized Ward identities in the presence of an external field [15-17] are not correct. The most adequate method for solving this problem is the functional method developed in papers [17,19]. With the help of this method a closed expression for the generating functional of the Green function is obtained in [16,17,20] and the modified perturbation theory is constructed. The method of modified perturbation theory proposed in [16,17, 20] leads to a result which goes far beyond the scope of the usual perturbation theory. This method appeared to be especially fruitful in quantum electrodynamics, because in the first order of the modified perturbation theory the contribution of the soft photons, both virtual and real, is summed completely. In particular, the first order of the modified perturbation theory gave the possibility of obtaining the correct asymptotic behaviour of the Green function and the cross section of the processes in QED [16,20, 21] in the double logarithmic approximation. Not long ago [22] the method of the modified perturbation theory was also used in statistic quantum electrodynamics. These general results (which are beyond the scope of the usual perturbation theory) will allow us to essentially advance in solving the problem of finding the quantum Green function in the presence of a real external field. In the present paper we consider the calculation of the one-particle Green function of a scalar field in a constant uniform external field by means of the modified perturbation theory. The Green function is found in the representation of eigenfunctions of the Klein-Gordon equation in a constant field. Such functions were first obtained in papers [12,13]. It was shown there that Dirac's eigenfunctions (in the spinor case) in a constant field diagonalize not only the Green function without radiative correction, but also the Green function in the e: approximation (see also [8]). The use of eigenfunctions appears fruitful in the modified perturbation theory because they contain considerable information about the dynamics of the system in a constant field.
2. Generating functional in QED with external field The principal value of the functional method is the generating functional of all Green functions: Z[~*, ~, J] = Cv 1out(0lS [~*, ~, J]10)in, where S [ , * , , , J] =
Texp{ifEs(x)d4x),
= L(x)A.(x)
+
+
(2.1)
LA. Batalinet al. / QED in externalconstantfield
437
Cv = out(0i0)i n is the probability amplitude for the vacuum to remain the vacuum, 10) in, 10)out are initial and final vacuum states, Jr(x) is the source of the electromagnetic field At(x), ~(x) and ~*(x) are sources of the scalar fields ~ + ( x ) and ~ ( x ) respectively. Operators are taken in the usual Heisenberg representation (i.e. the interaction between the fields is taken into account in the absence of the external sources) and satisfy the equations
m2] [
[ngr,
= 0, +(x)=O,
god
=
- e a r ( x ) -ear°x"x'
J,
~ * ( x ) = ia r + eAr(x ) + eAe~Xt(x),
+(dl- 1)dtlc~rS,]A"(x) = Ir(X ) ,
[] = _ era
Ir(x)=e~+(x)°'2~(x)~(x)-e(P~'(x)~+(x))~(x).
(2.2)
A~xt is an external electromagnetic field, d t is a gauge parameter. Using the method proposed in [16,17, 20] one may show that the solution of the functional equations for the generating functional (2.1) resulting from (2.2) has the form
- i f d4xd4y,*(x)a(x, yiAext+i-~)~(y))
× exp{½iS d4xd'iYJr(x)Dr'(x-y)J~(y)},
(2.3)
where G (x, y la) is the causal Green function of the scalar particle in the "external" field ar(x ) = A~Xt(x)- iSIS Jr(x) satisfying the equation
[(iOr-ea~(x))(iOr-ear(x))-m2]G(x,
yla)=8(4)(x-y).
(2.4)
/ / ( A ext + 81i8J) is the polarization correction to the action of the classical electromagnetic field:
/-/'(Aextq-/~-~'=) Spin[G( 'Aext-[-/.~-~')/G(10)1.
(2.5)
C is a constant to be determined by the condition Z[0, 0, 0] = 1. Further, it is convenient to find a functional representation for the Green function in an arbitrary field a~(x). To do this let u s write G(x, yla) in the form of the inverse operator
G(x, yla) = - i fo~dvexp( i(er 2 - m2-t- ie)v } ~(4)(x - y ) , % =- i0 r - ea r •
(2.6)
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438
Following paper [20] let us write the Green function (2.6) in the form of a functional integral:
G(x, yla)= --i (
d4p
J (2~) 4
e - i p ( x - Y ) [ ° ° d l , e-iOn2-i~)~
Jo
× f D4texp(-ifo~[t~(u')t~(~,') - 2p.t~(v')]
dr'} q~(x, p,vlt), (2.7)
where the function ¢(x, p,
v It) satisfies
the equation
• Oq~=2%t~(v)q~, -- 1-0-~" p
,(v=0)=l
D4t is the normalized volume element in the functional space
d4t(v ')
D4t = ~ ~t = 0
[J ~
/ J0~t.(v')t"(v')dv' /1' .
d4t(~,')exp - i
~r = 0
Let us consider the subsidiary function ~(x, p, viA ext, u) satisfying the equation
.0~
--t-~--
~
(2%t,(v)+u,(v)t,(v))~,
~(v=0):l
Note that ~(u = 0) = ~. It is not difficult to find the solution of this equation:
Separating the external field a~ ×t and the "'radiative field" making some simple transformations we obtain [20, 23]
G(x,ylAe×t-l-i-~)~-if d4P
e-iP(X
8/i~J ~ from
y)f°°dpe-i(m2,~.)~,
"~ (2~)'
Jo
× exp( _ 2efo'p~(p,)
Y(x,p,~'la'X'.u)=fD4texp(-ifo"[e(v'lt.
6
dv'}r(x, p, vlAex', u)l.=o,
Aex')-u.(v')t"(v')]d~," ) ,
where
E(v'lt,
A'X') = [,~(r.')- 2p~, +
a. and
2eA:Xt(x-
2 ~ : t ( ~ ) d h ) ] tt~(v')
(2.8)
LA. Batalin et al. / QED in external constantfield
439
is the classical action of the particle in the external field A~xt, P~(v') =
-i ~ u ~~----~--x(p')=x-2f~P(~)d~. (~') '
Substituting (2.8) into (2.3) we obtain for the generating functional Z[~*, ¢, J] the expression
Z[~*'~'J]=Cexp{ f d4xf~dse-i("~-m s~Jo,
(
0+(xp(ej0
t/ t)
×exp(- f d'xd4y~*(X)fo°°dve-`(m2-i~)~
)) ~(y) )
8J~(x(~,)) d~' xexp{lif
xy
d4xdayj~(x)D~(x-y)J'(y)),
(2.9)
where
Y(x, y, plAeXt,u)=j/* ~d4p e-ip(x-Y)Y(x,p, viA ext, u) , D.o(x)= f a'~k,D.o(k)e'~x, (2~)
(2.10) Some remarks about expression (2.9) are in order. It is known that the expression for Z[~*, ~, J ] contains divergences before the programme of renormalization is carried out. It is convenient to carry out this programme after all the calculations have been done. To do so it is necessary to regularize Z[~*, ~, J] (2.9), so that the divergences are absent during the calculation. As in papers [16, 20] this can be done
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LA. Batalin et al. / Q E D in external constant field
by introducing the cut-off s o for the proper-time of the particle in the polarization term in (2.9) and the cut-off t o for the photon proper-time in (2.9). As long as s o and t o are not zero, the whole expression (2.9) is finite. After carrying out the programme of renormalization it is necessary to remove the regularization by making s o and t o zero. The resulting renormalized expressions should not depend on s o and t o. In (2.9) one may carry out the partial functional differentiation over the sources J~ of the electromagnetic field A~ by expanding (2.9) into a series in powers of the sources ~* and ~:
c-xz[~ *, t, J] = c-~z[o,o, J] +
(-1)" I-I f d4x,d4Y,~*(xk)~(Y~,)
~,
"
n!
n=l
Xexp(
k=l
fo ~
--
i(m'-ie)vk
dyke
An,,n-I-~ Bn,(J)}exp(-I-fd4zf°°d~Se -i(rn2-'Os
~ nl , n 2 = 1
nl = 1
× exp n
So
- 2e
°~(s')
S
8j~(z(s,))
ds'
x
× ~ Y(xt, Yl; viiAex',ut)l.=o l~l
× exp(½ifd4xd4yj'(x)D,.(x-y)JV(y)),
(2.11)
where
Pn2dv P~(,)D,.(x.~(v)-x.2(v A .....=2ieEfo%dV'fo ))P~2(v),(2.12) p!
=4ie
P~(v')
= -i
~.(s')
=
-i
2
I
!
ii
ip
II
dv"f d4xJ'(x)D~.(x-xn,(v'))P~(v'),
(2.13)
ds' "~dv'P~(s')D..(z(s')-x.~(v'))P"(v '~ I'
(2.14)
Bnl(J)= -Rlefo Rn
~
~1 ~
8
(~)' ~ ~ 8o~(s')
'
~(s') = z - 2f
'~(X)dX,
(2.15)
I.A. Batalin et al. / QED in external constant field
441
and
Z[0,0, J]=
Cexp(H(AeXt+ / -8~ ) ) e x p ( s t f d 1"
4
xd
4
yJ (x)D~.,(x-y)j,,(y)} I.~
is the generating functional for the photon Green functions. To carry out the complete functional differentiation over the sources J, one should expand H ( A ext + 8/i8J) into a series with respect to the functions G(lAext + 8/i8J). In this way we obtain the following expression for Z[f*, f. J]:
c-Xz[~*,~,J]=C-lZ[O,O,J]+ f n=l
fd4x,d4ykt*(x,)f(y,)
( -1)n fi n !•
k=l
×fo=d,ke-iC"2-iO".~exp{ f
A.1..2+ f B.,(J)
n 1, n 2 = 1
n 1= 1
- f d 'z C "So s
z,
slO,O)/ )
d s p e_i(mZ_ie)sp X l + f ~ .1 p(Ifdz, 4 fs= Sp =l o r=l
xexp
( "Y". f \n l=l
a=l
C~l , . ~,-
f BI,~(J) + f
AI,,,,~'
a,al~l
a=l
}
X af :i l ~ 7 ( Z a ' Z a ' S a l ~ 4 e x t ' V a ) l ° = O }
X fi I"(xt,y,, VllA~xt,ul)l,,=o /=i
x exp{lif
d4xd4yy'(x)D~,(x-y)J'(y)},
(2.16)
where £un~ Z ~(s*))P:(s u t), C.,,.=4te2 Jo dv'fosa ds'P~(n')D"(X'l(V')(2.17) •
Bla (J) =
-2iej:ads' fd4xJ~(x)D~,,(x-
z~(s'))@~(s'),
sa p!
P:(s')= -i 8 v a , 8O '~)
'
z.(s') =
zo -
2 f : a P . ( X ) d~. .
v
gp
(2.18)
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LA. Batalin et al. / QED in external constantfieM
The expression for Z[~*,~, J] will have the simplest form if we neglect the polarization effects. Then we obtain: /[~*'~5'J] = [ 1+ n=X ~'~ (--1)n ~' k=l fi
fd4xkd4yk~*(Xk)~(Yk)
Xfo°°dvkei(m2-i~>kexp{ ~
Anl,/.12-]- ~ Bnl('))
nl,n2=l
nl=l
X 1=1 f i ~r(x/' Yl'Ol[heXt'Ul)[u=O]
×exp{½if daxdgyJ~(x)D~(x-y)J~(y)}.
(2.19)
If we are interested only in the Green functions of the scalar particle, the expression for the generating functional becomes still more simplified: Z[~*,~,J]=I+
~ (-1)" fi n! k=l
n=l
exp{
nl,n2=l
fd4xkdayk~.(xk)~(yk) 1=1
Using (2.16) one can write the one-particle scalar Green function in the form
O(x, y ) =
-if0m dvexp{-i(m
1 x 1+ ~. 7y r=l
2- ie)v+ A1,1 }
Hr fd4zpf oo dSpe-i"2-i~'sP Sp p=l so
×exp{a=l~ CiaS- oe,a,=lkA1 .... } f = i l r ( za , Za , Sa [Aext, ua )[ v=O
×exp{- f d4z [~ dS e-i(m2-i~)s~'(z,z,slO,O)}5"(x,y, vlAeXt,u)[u=o" % -7 (2.21) When neglecting polarization effects this expression takes an especially simple form:
G(x, y) =
- ifo~°dvexp{
-i(m 2-ie)v} ×(expAl,1)x,y.
(2.22)
LA. Batalin et al. / QED in external constant field
443
Consider now expression (2.20) in detail. Suppose we know a complete set of solutions of the eigenvalue problem. X ~r~Xt(x)~reXt~(x)~(k}(x)=p 2~b(k}(),
%eXt(x) =
ion-- eAe~Xt(x),
(2.23)
where ( k } is a set of quantum numbers ({ k } contains the quantum number p2). Let us call such functions eigenfunctions. In the constant uniform field such functions were first obtained in [12,13]. Suppose, further, that the eigenfunctions form a complete orthonormalized system of functions with respect to the scalar product
(~p, qJ) - f d4xep*(x)~(x),
£ ~p{k}(X)+~k}(y)
= g(4)(X - - y ) .
(2.24)
(2.25)
{k}
(If the eigenvalues are discrete, then g{k},{k'} in (24) is Kronecker's symbol and if they are continuous, then g{k },{k'} is the &function.) With the help of the eigenfunctions one can construct the causal Green function of the Bose particle in the external field A~xt without radiative corrections:
G(°)(x, ylA ext) =
Y'. ~ P { k } ( x ) ~ } ( Y ) , {k}
pZ --
(2.26)
m 2 q- ie
which satisfies the equation
{rr~iext (x)Ir extp. (x) If we represent (p2 _ m E + can be written as
ie)-i
G(°)(x, ylA ~xt) = -
_ m 2
}G~°)(x, ylAeXt)=g(4)(x-y).
(2.27)
in an exponential form, the Green function (2.26)
i [ °° dl, e-i(-,2-i~> £ e 'p:~a, v{k}(X)q~k}(y). ~0 {k}
On the other hand, the Green function expression (2.8):
G(°)(x, ylA ~xt)
G(m(x, ylA ~xt) = -ifomdve-~("=-i')~I'(x, y,
(2.28)
can be obtained using
vlA~Xt,0).
(2.29)
Comparing (2.28) and (2.29) we have I"(x, y, vlheXt,0) = £ (k}
ei*'="~p{k}(x)tp'~t,}(y).
(2.30)
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LA. Batalin et al. / QED in external constant field
Using (2.24) and (2.30) we obtain an important property of eigenfunctions:
f d4x d4y ~tk} (X)I"(X, y, via ext,0) ~P{k')(Y) = e ip2v'~v(k},(k, } .
(2.31)
Let us represent the sources ~*(x) and ~(y) as a superposition of the eigenfunctions
I2
E
(k}
(k)
Then the expression for the generating functional Z [ a * , a] (2.20) has the form (--l)n
Z [ a * , a] = 1 +
- -nI n=l
Y'~ ( k l } .....
(k2)
pFI__lO~(kp}O~{kp} *
(kl) ..... (k;,)
X f0~ dpp e-i("~-m",
exp
~
A,~,,:
\ nl"n2~l
k 1..... k.; k{ ..... k"
(2.32) where
X Ht~tk,}(Xl)A
/=1
~U
yp,
p=l
Up
(2.33) Expression (2.32) allows one to find the representation of Green functions in the basis of eigenfunctions.
3. Green functions in constant field
Consider the constant and uniform external field A~Xt(x) =
-
1
5F~,~x
v.
(3.1)
As can be seen from above the calculation of the generating functional and the Green functions reduces to the calculation of the functional integral (2.8), i.e. the
LA. Batalin et al. / QED in external constant field
445
function ~'(x, y, viAext, u). In the constant field (1) the functional integral (2.8) reduces to the gaussian one. The function Y(x, p, viA ext, u) was found in ref. [23]:
Y(x, p, ~,[A ext, u) = [detch(eFp)]
-1/2
x exp{ i f f f d~,' dp" ~r~(x, p, p')G~o(v', 1,"; p)~r°(x, p, 1,")},
(3.2) where
%(x, p, v') =p~ + ½eF~x~+ ½u~(v') G(v', v"; v) = B(v'- v") + eFexp(2eF(v'- v") ) [ e ( v ' - v") + th(eFv)] e(v'- v")=
O, v'= v". - -
1,
(3.3)
p, < pt,
Thus, the integral over p, in (2.10) also reduces to the gaussian form and we find Y(x, y, 1,1Aext, u) =
i
(4~r) 2
exp(½iexFy - i(x - y ) L ( u ) ( x - y ) - ½Spin sh(eFv)
eF
)K(x - y , plu)
L (u) = lefcth(eFu),
(//o
K(x-y,~,lu)=exp ¼i
+½i(x-y) . eFexp( eFu ) f~ exp(-2ef~')u(v')dl,'} G(p', u"; 1,) = G(p', ~,"; ~)-
2er e x p ( 2 e r ( v ' - t,") sh(eFu) }.
'
(3.4)
Substituting expression (3.4) into (2.21) we find the quantum Green function of the
LA. Batalin et al. / QED in external constant field
446
scalar particle:
G ( x , y ) - ( 4 ; ) i 2 fo~ dv
exo( X (1 +
'
j 'zr
J'o
s
/)
(4,rrs) 2
r@l 1 fl fd4zpf c~dSp Sp =
p=
so
1
} i~
{ a.al=l
a=l
( 4 ~ ) 2r a = l
h( eFs a ) + ¼ i f f 0 d s ' d s " O a ( S )G,v(s ,s ,Sa)OVa(S '') eF . . . .
{
> exp { - ½ Sp In s__, - -
t
[
×exp{ liexFy - i ( x - y ) L ( v ) ( x t
- y ) - ½Spin
• ~ +1'ffo
d v ' d v " u ~ ( v' )G~r(v " . ,v ...
))v=o
sh( eFv ) eF eFeeF"
, v)ur(v '') +½i(x -y) sh(eFv)
X fo"e-2er"'u(v')dv '} .=o"
(3.5)
If we perform the substitution zp + x = z~ in the integral over zp in (3.5) it is not difficult to see that the Green function G(x, y) has the following structure:
G ( x , y ) = exp( ½iexFy }Gl(X - y ) ,
(3.6)
where G l ( x - y ) is a scalar function of the difference x - y and of the constant tensor F. Hence it follows that G x ( x - y) may depend only upon arguments of the form (x - y ) K ( F ) ( x - y ) , where K ( F ) is an even function of the field. The eigenfunctions of the problem (2.23) in the external field (3.1) can be found in a covariant (see appendix) form:
+(k} (x) = N{k}exp{ ieCp(x)
-- ipl(n -- x) -- ip2(m -- x) -- ½pZ }
XHn(o)D,[~ei~/4"r],
(3.7)
LA. Batalin et al. / QED in externalconstantfield
447
where { k } = { p l , p2, n, p 2 , ~ } , ~ = +1, n = 0 , 1 , 2 . . . . ; the numbers Pl, P2 are eigenvalues of the operators i a / 3 ( n _ x ) and i O / O ( m _ x ) respectively, p = ½(1 + iX), X = [p2 + l e l ~ ( 2 n + 1)]/lelE ' E, % = [(oy2 + ~32)1/2 -Y-~-]1/2; ~, oy are the invariants of the field (which we suppose are not zero), p = ([e ~)1/2((m + x ) + p2/e~C), "r = ( 2 [ e [ E ) x / 2 ( n + x ) + p x / e E ) , eg(x ) = - ½ E ( n + x ) ( n _ x ) - ½g(~(m+x)(m x), H , ( p ) are Hermite polynomials, D~[6oei(~/a)r] are the parabolic cylinder functions, and n _+, m_+ are vectors which are solutions of the equations -
Fn += En =;, m+-2--1,
Fro+= -Y-gCm ~ ,
n 2 = +_1,
( n + n _ ) = ( n + m + ) = ( n _ m + ) =_( m
N{k) = exp{ ~iqrp ) F( - l,)( ~C/2E)l/4(Z'n !vr~) - 1 / 2 Using the integral representation for the parabolic cylinder functions, one may show that the eigenfunctions (3.7) form a complete orthonormal system of functions in the sense of (2.24) and (2.25). Note that the following important property of eigenfunctions (3.7) is valid:
fdnxd4y+~k)(x)exp{½iexFy - i(x -y)T(F)(x
- y ) } ~b(k,) ( y ) - 8(k},(k'} •
(3.8) Relation (3.8) permits us to prove that eigenfunctions (3.7) diagonalize not only the Green function G(°)(x, ylAeXt), but also the quantum Green function G(x, y), eq. (3.5). Indeed, suppose that the function G ~ ( x - y ) can be expanded as a Fourier integral:
al(x-y)
=
f dkl . . d k t f ( k l
{
.... ki)ex p - i ( x - y )
,
E kjKj(F)(x-y)
}
.
j=l
(3.9) Here 1 is the number of the independent structure matrices K ( F ) on which the function G I ( X - y ) may depend. Then we have
f d'xd4yg~k)(x)G(x, y)ap(k,} ( y ) = f dkl...dktf(k 1.....
{
x f d4xd4y4,~,)(x)exp
½iexFy-i(x-y)
'
kt)
E kjKj(F)(x-y)
/
+{k')(Y)-
j=l
(3.10)
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LA. Batalin et al. / QED in external constant field
Choosing in (3.8)
T(F) = Y~=lkjKj(F) we obtain f d4xd'y~k}(x)G( x, Y)g'(k,)(Y) - ~(~,(i,,~.
(3.11)
Proceeding from this we can write
G(x,y)= ~_, ~ f dpxdp2dp2 ,o=_+1 n=0
(27r)4
* ~p{k}(x)G{k}~p{k}(y),
G(k~= f d'xd4yg,'(k~(x)G(x, Y)~P(I,}(Y). Neglecting the polarization effects we obtain for
(3.12)
(3.13)
G(x, y)
G(k} (4~) 2 f0~ du e-i(m2-i~lV({expAl,1))k,k"
(3.14)
Let us rewrite the definition of the average ( ~ ) ) k , k in a more convenient, for our subsequent work, form:
((h(--~u)))k,k= fd4xd4y~b~,~}(x)(A(~U))x,y~b(k}(y).
(3.15)
Relation (3.14) permits one to develop the modified perturbation theory for G(k}. To obtain the first order of the modified perturbation theory let us expand G(k / into a series with respect to the radiative interaction. To within e 2 we have G(k}= (4--~)2 fo°°dve-i'm2-i~)J(~}(v)(1 + ( ( Aa,1))k,k/J{k}(v)), J(k) (1,)= (~l))k,k.
(3.16)
According to [16,17, 20] the first order of the modified perturbation theory consists in the replacement 1 + ( ( A I , , ) ) k, k/J{k} (I') ~ exp( ( (AI,1) ) k , k / J { k } ( P ) ) . Thus in the first order of the modified perturbation theory one has
~(expAl,1))k,k=J(k)(~')exp(((AiA))k,k/J(k)(l,)),
(3.17)
LA. Batalinet al. / QED in externalconstantfield
449
and for the Green functions in this approximation we obtain
(
/
a~12}--- (4~r)2 fo dvJ{k}(v)exp -i(m2-ie)v+ ()k'kJ{k}(~'") (3.18) Substituting the solutions (3.7) into (3.16) and using expression (A.25) we find J{k i(v) = (4~r)2 exp(ip2v).
(3.19)
Finally we have m 2 + ie)v + ((Al'l))k'k I
G(x)_ { k ) - - _ifo °° dr exp{ i( p 2
(3.20)
4. Calculation of ((A 1,l) ) k, Using the definition of ((Al,1))k,~ from (3.15) we can write
((Al,1))k,k = f d4xd4yq,'~,}(x)(A~a)x,y@{k}(y),
x,y =
2e 2 exp{ XiexFy - i ( x - y ) L ( v ) ( x - y) (art)2
-
Aa,~(x-y,~)=
(4.1)
½Spin sh(eFV)eF } A',l(x - Y ' v) ,
fro a ~ ' d ~ " a ( x - y , ~ ' , ~ " , ~ ) ,
(4.2)
(4.3)
a(x - y, ~', ~", ~) = e"(~')V.~(x(~') - x ( ~ " ) ) e ~ ( ~ ' ) ~ : ( x - y, .lu)u=O, (4.4) where K ( x - y , vlu ) is determined by (3.4). To find (4.4) let us use the integral representation of the photon Green function
D~(x(v')-x(r"))=
j~--~
,~(k)exp 2i
t(X; v', v") = 0(X - v") - 0 ( X - v').
t(X;~',.")k.e°(X)dX
, (4.5)
450
LA. Batalin et al. / QED in external constant field
As a result we obtain
a ( x - y , v', v", v) = / " d4k D ~ ( k ) e x p ( i k ~ k J (2~r) 4
+ 2iqk}
× {- ~,e.~(.,, .,,; .)+ [ K,s,~,(.,, .,,, .)+i~(.,,, .)] × [~j::(is', is", is) +/:(is', is)] }, fl(V',
(4.6)
is", is) = f0VG (is '', ~k; is)t(~k; is', is") dh,
r tis, , is" , is)= ~2,
So"G(is',X;is)t(X'is',is")dX,
,
eF exp{ eF(is - 2v')} f3(is', is) = ½(x - y ) sh(eFv)
(4.7)
£= ff0"dXdX' t(X; is', is")d(X, X'; is)t(X'; is', is"),
q= So~f3(X,v)t(X;is',is")dX. The integrals over X and X' in (4.7) are not difficult to calculate.
~= s h ( e F s ) s h [ e F ( i s - s)] eFsh(eFis) , e eFl'
[e_2eFv,
s = Its'- is'l, e_2eFv. ]
q = ¼(x - y ) sh(eFis) fl(is',is",is) = ½e(is'- is")+ ½(e 2eV(~'-~")- l ) ( e ( i s ' - v") +cth(eFis)), fz(is', is", is) = ½e(is' - is") + ½(e 2ev(/-"'') - 1)(e(is' - is") - cth(eFis)).
(4.8)
To integrate expression (4.6) over K s it is convenient to use the representation
451
LA. Batalin et al. / QED in external constantfield (2.10), having rewritten it first in a more convenient form:
D~p(k)= Df.(k) + DXv(k ),
Df.(k)=g,.(k= + ie)-x= -ig,.S~
eik=tdt,
-- lto
1)
DX,(k) = - k ~ k p ( d t -
teik2tdt.
(4.9)
-- tl 0
In accordance with this procedure it is also convenient to represent expression (6) in the form of two terms: a(x--y,v',v",v)=ad(x--y,v',v",v)+aX(x--y,v',v",v).
(4.10)
Then for ad( x -- y, V', v", V) and aX( x - y, v', v", v) we obtain, respectively, a d ( x - - y, v', v " , v ) = - i [ - ½ i G ~ ( v ' , v " ; ×
f~ Sdt -
v)+f3~(v', v)f3~(v '', v)]
if~#(v ,, v-, v ) f ~ ( v , ,
v,,, v)
-- it o
I~l~dt-i(f~lt(v',v",v)f3~(v',v)
× - -
it o
"[-ftfl(1.'#,l)
aX(x - y , v', v", v) = - ( a , -
l*)f3/,(/-'##, ")) f_Ttoladt,
'',
(4.11)
1 ) { [ - ½iGU°(v ', v", v) + f3~(v ', v)f3°(v '', v)]
×
f~ I.odt +frl~(v ', .", v)ft~(v ', .', v) -- it o
x f_~,o/.oo~at + (fl~(~ ', ~", v)f,.(~', ~)
Ia' .... "
= ( d4k k ...ko, e x p { i k A k + 2 i q k ) , d (2,n.) 4 'h
A = ~ + t.
(4.13)
LA. Batalinet aL / QEDin externalconstantfield
452
All the integrals 1~, ..... involved in (4.11) and (4.12) are expressed by means of the integral I:
I= f
d4k exp(ikAk + 2iqk)= ,.--~(detA)-l/2exp(-iqA-lq), ~-~ (47r) (4.14)
I.= -(A-lq)¢,I,
I,,~= [½iA,7~ +(A-lq).(A-lq)fl]I,
I~¢= -[½i(A~2(A-lq)~+ A~A(A-lq). + AJ(A-'q).) + ( A-lq)~,( A-lq)a( A-lq)13]I,
+ ½i(a2¢(A-lq).(a-lq)¢ + A72(A-lq).(A
lq)t~
+ a~¢(A-lq).(a-lq). + a#(aql).(a-lq)B+
a3(a-lq).(a
lq)o
+aJ(a-aq).(a-aq).)+(A-lq).(a-lq)~(a-lq).(a-lq)l~}I. (4.15) Substituting (4.15) into (4.11) and (4.12) we obtain after cumbersome calculations
_ (detA)-l/2exp a~(x - y, ,,', ,,", ,,) = (4.)-~f~,o .
aX(x -
{-i(x-y) 8sh2(eFp) ,AI ,x 1 ~A -2
× - 2 i 8 ( s ) + 2-S(x
-y)shT(eFv) (x - y )
+~iSpA-I( l+4eFt
ch[eF(v- 2s)] sh(eFv))}dt,
y, v', v", v) = -i(4~r ) - 2 ( d , _ 1)!i~ot (det a )-1/2
X exp
{ -i(x -y) 8sh2(eFv) "qA-1 (x-Y)}
(4.16)
453
LA. Batalin et al. / QED in externalconstantfield [ × (4t)-l~(s)+i2-4(x-y)
A -2
e F ' q c h [ e F ( v - 2s)] sh3(eFv)
]
+ A - X 2eZF2ch(2eFs) • -1 ~4 sh[eF(v - 2s)] sh2(eFv) -li s--~eF-v) ](x-y)
+2-'
"r4 (x - y ) - i S p A ¼(x - y ) A -2 sh2(eFv)
[
2eFch[eF(v- 2s)]
+-~Sp A -1
sh(eFv)
-1 sh[eF(v - 2s)] ]2 sh(eFv)
_A_2Sh2[eF(v - 2s)l sh~e~)
]}
dt, (4.17)
"q = ch(2eFs) - 1,
~2 = 2eFt + sh(2eFs),
"r3 = sh(2eFs) + 2eFtch(2eFs), *1= "q + 4eFt'r3,
~'4= ~', + 2eFtsh(2eFs),
A = ( 2 e F ) - l ( r 2 - rlcth(eFv)).
(4.18)
Note that a a and a x depend on v' and v" only in the combination s = Iv' - v" I. Therefore, when integrating over v' and v" in (4.3) one has
ffo d v' d r ' ¢ ( s ) = 2 fo~(V - s ) ¢ ( s ) d s ,
(4.19)
where ¢ ( s ) is an arbitrary function. Thus Al,l(x-y,v)=2
-
)a(x-y,v,s)ds
(4.20)
If in accordance with the breaking up of the photon function D~,(k) into the parts (9) one represents analogously A l , l ( x - y , v) in the form of the sum of two terms, then A ~ , I ( X - y, v) can be considerably simplified. For doing this note the identity
. OI {
2t~s =
"r4
¼(x-y)A-2shZ(eFv) (x-y)-iSpA-1
sh[eF(v - 2s)] s--~F-v)
}
I. (4.21)
Using this relation and integrating over s by parts, the expression for AXl(x - y , v)
454
LA. Batalin et al. / QED in external constant field
may be represented in the form
AX,l(x - y , ~,)=
½i(4~r)-2(dt
-1)~o-~-[exp(-i(x-T~y)21-1 ] .
(4.22)
Finally we obtain
Al,l(x -y,
£
r) = ip(8to) -1 + ½i(4~r) -2 ~(p - s ) d s
I - -
(detA) -1/2 ito
T1A-1 (X--y))
× exp - i ( x - y) 8 sh2(eFv)
×{(x-y)A :
n
sh2( eFp)
(x-y)
ch[eF(v - 2s)] +4iSpA-a( l +4eFt sh(eF~,)
)}
dt +½i(4~r)-2(d,-1)f£toV(e-'(~-':/4'-I ).
(4.23)
Substitute now (4.23) into (4.1). The result of this substitution is written in the form ((Ax'D)k'k=
.Ol -1 ia :'( --t-~ vt° Jk'k(L)+ 2~r.lo"V-s)ds f:
X
KaBJk,k;aB(L+
~'IA-1 )
,.1
ch[ e F ( p - 2s)] sh(eF,) )
×Jkk L+ , 8 sh2 (eFt,) _.o- }-
-1/2
8 shz ( eFt,)
+4iSpA-I( l +4eFt
(
(detA)
ito
)]o
~-~-( d, - 1)
Jk,k(L+ 1)-Jk,k(L )
det sh(eFp) -1/2
eF
'
if2
a = 4rr'
(4.24)
LA. Batalin et al. / QED in external constant fieM
455
where
Jk,k,( ~2) = f d'xd4y~'~k~(x)exp( ½iexFy - i(x - y)~2( F ) ( x - y ) } ~b(k,i(y), (4.25) J~,k,;,a(~2) = f d4xd'y (x - y ) , , ( x -y)#~p~ki(x)
Xexp( liexFy - i(x - y ) ~ ( F ) ( x - y ) } tp(k,i( y ) .
(4.26)
Differentiating (4.25) over I2~t~ one can find the connection between these two integrals:
Jk k'. ,~l~(12)= i 3Jk'k'(12) ,
,
(4.27)
012,,~
Making use of the integral representations for the parabolic cylinder functions and Hermite polynomials [24], the integrals (4.25) can be calculated to give Jk,k,(~2) = (41r)Z[det(16~22 - eZF2)] -'/4exp{ - i p x ( F ) p } 8{k},(*'),
- eF x ( F ) = ( 2 e F ) - l l n 412 412 + e ~ '
(4.28)
and p~, is a four-vector:
p~,p. = p 2 ,
p~.p.=p2. +p2,
p2=ppp = p 2 + [ e l ~ ( 2 n + 1),
p 2 = p t p = - l e l ~ ( 2 n + 1),
Pt~v=(~122-k-E 2 ) -1 ( F ~2+ g , ~ X 2 ) ,
L/x, = (~C 2 -t-- E 2 ) - I ( g . ~ , E 2 - F~/~).
The integral
Jk,k,;,a(12) enters into (4.24) together with the matrix K = A-2~l/sh2(eFp). Differentiating (4.27)over 12"a we obtain
K'~OJk,k';al~(~2)=4Jk,k'(~)
K
P16~22~e2F2P-
2K~2 } iSp 16$22 _ eZV 2 . (4.29)
456
LA. Batalin et al. / QED in external constant field
From (4.27) we have
Jk,k,(L) = (4¢r)2eiP2~3{k},{k,}, (
rlA-1
Jk,k' L + 8sh2(eFv )
) = Jk,,,(L)(detA)l/Z(det
"rff-T12/-1/4 4e2F 2 ]
×exp( -ip~(s, t)p }, ~b(s, t) = (2eF)-11n z2 + rl r2 - r 1
(
)
Jk,k' L + ~ - I
(
~2-T2 \-1/4 exp{-ip~p(v,t)p}.
=jk,k(L) det~Js=~
(4.30) Note that Jk,k(L)= J{k}(g). Substituting (4.29), (4.30) into (4.24) we get ((Al'l))k'k =J{k}(V)
--i~ v ` ° l - " at-~fOv(v-s)ds ~itoM(S, oo t; p, F)dt + - ~ ( d, -
t-l[t2g(,,
1)f - -
t ; p,
F) - 1]dr
ito
(4.31)
=bl(V;p,F), M(s, t; p, F) = g(s, t; p, F) p ~------~--p ~2z _ "1z + ½iSp -r22-_ rl2 , R = eF[2eFt(3 - 2ch(2eFs)) + (1 - 8eZFZtZ)sh(2eFs)],
]- [ ,1:2 ,1.2~]-1/4
g(s,t;p,F)=[dett~)]
exp{-ip~p(s,t)p).
(4.32)
This expression diverges when t o ~ 0. Before proceeding to the renormalization consider the case F = 0. In this limit
(
M(s, t; p, 0) = g(s, t; p, 0) { p2
(s + 2t) 2 + 2 ( s + t) 2
g(s, t; p,0) = exp{ -ip2s2/(s + t)} (s + 0 2
2 ( s + t)
LA. Batalin et al. / QED in external constant field
457
Thus ((Al,1))k
3a
'
-1
. Ot
oo
J(k)(v ) k = - i - ~ v t ° _ t2"-~ fo~(p-s)ds fito - e-ip:':/("+t) o0 e - i p 2 s 2 / ( s + t )
s+ XP2(t+S)3
~
+ 4--~(dt-1)f°°
folksy-ito
(t+s) 2
t-l( t2 e_ip~,~At+.)_l)dt
"-,,o
~ (t + ~)2
~bl(p;p),
(4.33)
which coincides with the results of [20]. Consider the Green function G[1) I (k}lvzo
=
_ifo°°dvexp(i(p2_m2+ie)~,+bl(,;p))
(4.34)
Let us separate the terms linear in v from bl(~; p) and rewrite (4.33) in the following form:
bx(~', P) = -ivMo(p) + No(l,, p),
(4.35)
where Mo(P)=-4--~t
°/5 ds f_,oMs't; p,O)dt,
o +~
a fosds ~ f ~ M(s,t;p,O)dt+iv~-~j~f No(~, p)= i~__~ a o~dsfo°°M(s,t;p,O)dt - -
a
4qr
ito
£ ds I:,0"(s,t;p,O)dt
+ ~_~ _ (dr a
1) L~itotdt[t2g(p't;p'O)-l]
.
(4.37)
Note that if we calculate the mass operator M0(p 2, m 2) of a scalar particle in the e 2 approximation and make the substitution m 2-~ p2 then we get just M0(p):
Mo(P ) =
_~0(p2, m E ~ p 2 ) .
(4.38)
Therefore, the renormalization of Mo(p) may be performed in a standard way by choosing the normalization point as p Z = m 2 where m 2 is the experimental mass
458
LA. Batalin et al. / QED in external constant field
squared [20]. To do this let us represent Mo(p) in the form
Mo(p)
= M o ( p 2 = me2 ) + ( p 2 - - m e 2
) 3M°(P)pZ=m+MoR(P)
,
(4.39)
where MoR(P ) is the renormalized mass operator. Let us make in (4.34) the substitution g = z2u',
(4.40)
where z 2 is the renormalization constant of the wave function, and 3m 2 is the mass renormalization
3m 2 = m 2 - m 2.
(4.41)
Then G(x) I _-- _ i Z 2 f o ~ d v e x p { i ( p 2 _ m Z ) ( z 2 _ l ) v + i ~ m 2 p (k}lF=O
_iuMo( p 2 = m2) _ iu OM°(P--------~ 2 p2=m2( p 2 - m2e)
+ i( p 2 -
m 2 )lp -
iuMog(p ) +N0(u; p ) } .
(4.42)
i
The constants z 2 and 3m 2 are chosen from the conditions
8m 2 = Mo ( pz = m2 ) ,
(4.43)
z 2 - 1 = aM°(p)op 2 p~=,~"
(nAn)
The integrals entering into (4.43) and (4.44) are easily calculated: 34 2[z 2 \-1 + ~ + ln(Tm2to)-l], 8m2= T-~me[[moto)
z2-1=
(4.45)
34~r[2 a l l + l n ( y m e 2t 0 ) -1 ]
(4.46)
where y is Euler's constant. The function No(v, p), eq. (4.37), is equal to
No(~, , p) =
-- -4--~dt Ol l n ( y m /20
a(p, p ) ~ 0,
u ~ oo.
a (3 - d , ) l n ( m 2 u ) + a ( u , p ) -x + ~-~
) (4.47)
LA. Batalinet aL/ QEDin externalconstantfield The diverging term - ( a / 4 ~ r ) d
t ln('/m~t0)-1
459
should also be related to the z 2 factor:
22=z2exp(-~dtln('ym:to)-l). In the e 2 approximation we have z2- 1=
~-~[(3-d,)ln(~tm2to)-i+
Let us return now to expression (4.31) and rewrite form:
½].
(4.48)
bl(v, p, F)
in the following
bl(v,p,F)= -iavtol-i~-~v a fo~ ds i~itoM(S,t; p,F)dt
° f f ds
-v +N°(v'p)+N(v'p'F)+ i -2~r
M(s,t; p,F)dt, (4.49)
O~
1~
O0
N(v,p,F)=i-~--~ fosds fo M,R(S,t;p,F)dt +__4_~(dt 1) fo~t[g(v,t;p,F)_g(v,t;p,O)]dt, MaR(s, t; p, F ) =
M(s, t; p, F) - M(s, t; p , 0 ) ,
(4.50)
where No(v, p) is determined by (4.37). In the last term in (4.49) and (4.50) we have put the lower proper-time-integration limit equal to zero everywhere; on removing the regularization they are finite. One can verify this fact by expanding the integrals into power series with respect to the field and calculating the corresponding integrals. Consider the frst two terms in (4.49) linear in v. Let us write M(s, t; p, F) in the form
M(s,t; p,F)= M(s,t;
p 2 = m¢2,0)
+( p2_ m2) OM(s,t;Op2p , 0 ) p2=m2
+ M R ( S , t ; p, F ) ,
MR(s, t; p, F) = M(s, t; p, F) -
(
t
exp - i m ~ t +----~
×(t+s)-a(m 2(s+2t)2 + i ) - e x p ( - t m2" s2 e 2(s ~ f i
×[
(s+t)2+t2t+s
Then the integrals over s and t in
t +----~I
im2s2(s+2t)Z(s+t)-2] "
MR(S, t; p, F)
(4.51'
are finite when t o ~ 0 and we
LA. Batalin et al. / QED in external constant field
460
obtain for bl(p; p, F )
61( I,; p, V ) = - ip6m 2 - i( p2 _ m 2)( z2 _ 1)v - ipMg ( p, F) Ol
t.O~
~00
+i~--~l'J~ dSJo M ( s , t ; p , r ) d t +1~---~ sds
R(S,t; p , F ) d t
+N0(~,p)+ ~ ( d , - 1 ) MR(p, e ) = ~a
~ds
tgR(~,t;p,F)dt,
(4.52)
MR(S,t;p,F)dt ,
gR(s, t; p, F) = e,(s, t; p, r ) -g(s, t; p,0),
(4.53)
where 8m 2 and z 2 - 1 are determined by (4.43) and (4.44). Thus we see that the procedure of renormalization in the presence of a constant field is the same as in the vacuum ( F = 0) and the constants of renormalization do not depend upon the external field. If we write the Green f u n c t i o n G~lk)} in the form
"~(k)~(l)_----i
f0 d p e x p ( i ( p Z - m 2 + i e ) 7 , + b x ( ~ , p , F ) }
,
(4.54)
and make the change of variables u = z2~,' and the mass renormalization m E = m e2 - - 8m 2, we obtain a ( 1{ k) } - __ ~,2"-' F;,(1) {k }R,
(4.55)
where G {o~ k } R is the renormalized Green function G~)}R= - i f o ° ° d v e x p ( i ( p 2 -
2
,
(4.56)
bin(g, p , F ) = - i ~ , M R ( p , F) + NR(~, , p , F )
+i_---p[ 2 rr .I ds
M(s,t;p,F)dt,
NR(v, p, F ) = N(u, p , F ) + NoR (~, , p ) , O~
2
NOR(V, p ) = No(v, p) + - ~ d t ln(3,met0)
-1
.
(4.57)
N o t e that if we calculate the mass operator in a constant field in the e 2 approximation and write it in the form
M(p,F)=
fo°°dsf~ ~(~,,;p,F,
mZ)dt,
(4.58)
-- it o
then the function M(s, t; p, F), eq. (4.32), is connected with M(s, t; p, F, m 2) in the
I.A. Batalin et al. / QED in external constant field
461
M(s,t; p, F) = M(s,t; p, F,m 2 ~ p 2 ) .
(4.59)
following way:
Then our result for M(s, t; p, F) coincides with that of ref. [8], i.e. M(s, t; p, F) is the kernel of the mass operator in the e z approximation. To find the infrared asymptotics of the Green function one should calculate the asymptotic behaviour of baR(V; p, F ) when v ---, ~ . The terms linear in v contribute to the position of the pole of the Green function. From (4.57) and (4.54) we see that the pole of the Green function is at the point
p2 - m e2- M R ( p , F ) = O .
(4.60)
In the vacuum there occurs, besides, the enhancement of the pole near the point p2 = m e2 [16,17, 20] owing to the fact that the function NoR(v; p) behaves logarithmically when v ~ ~ , i.e. ~a (3 - dr) In mZ~v
: i n a NOR(v, p ) -
Let us find the asymptotic behaviour of NR(V; p, F ) when v --, o¢. To do so we shall act in the following way: when v ~ ~ we write NR(V; p, F ) ln(m~v) lnrn~v
NR(v;p,F)
f(p,F)ln(m2eV),
=
NR(v, p, F) = N(v, p, F) +NoR(v , p ) ,
(4.61)
where
f ( p , F ) = lim ~ ~,---* OO
ONR(V; p, F)
(4.62)
0/)
Substituting expressions (4.57) and (4.50) into (4.62) we obtain
f(p,F)=
lim [ i - - v 2 (
M(v,t;p,F)dt
+-~(dt-1)Vfo°~tOg(v,t;p,F)/avdt).
(4.63)
Thus, the Green function ,--o) ~r{k~ near the point p2 -- m e2 -- MR(p, F ) = 0 has the form G(1) 2 MR(p,F)]-(X+f(P,F)). (k} - [ p 2 --me--
(4.64)
462
LA. Batalin et al. / QED in external constant fieM
If F - - 0 then Ot f ( p , O ) = ~-~ (3 - dl),
(4.65)
and we obtain the well-known results [16,20]. If F ¢ 0 , the function f ( p , F ) depends upon the parameters p2, X = - ( e F p ) 2m-6 and the invariants 6)-and 9. As the function f ( p , F ) is linear in the constant a it is necessary to substitute p2 ~ me2 near the pole (4.60), so as to remain within the accuracy. Consider the crossed field ( ~ = ~ = 0). In this case f ( p , F ) depends upon one parameter X and we have
f(p, F) =
~---~(3 - dr), O/ 1-~(29 + d/),
X << 1
(4.66)
x>>l
For constant magnetic field (9 = 0, oy< 0) if - ~3/m6e 2 << 1, X >> - ~/m4e2, the results of the calculations for the function coincide with those for the crossed field.
Appendix Here we find the eigenfunctions of Klein-Gordon's equation in a constant and uniform external field A~Xt(x)= - ~F~,~x. 1 These functions satisfy the equations
;Xt(x)
(x) =
(x).
(A.1)
Consider isotropic orthogonal eigenvectors of the tensor F,~:
Fl,~n ~ = End,, F~=-i~C~,
F ~ " = - E~,,
F~m ~ = i~)Cm~,,
E, ~3(~= [(6~2 + ~2)1/2 -T- 6S] 1/2 .
(A.2)
The vectors (A.2) are normalized by the conditions ( n ~ ) = - ( m ~ ) = 2 . If we introduce the vectors n_+= ½(n _+ ~), m+= ½(m + ~ ) , m = ½i(~ - m) satisfying the equations
Fn += En ~: , m±-2--1,
Fro±= T g(~m :~ ,
n2±= +-1,
(n+n_)=(n±m+)=(n+m_)=(m_m+)=O
the tensor F~, and the field A~Xt(x) can be represented in the form
F~ = E(n_~,n+, - n+~,n_,) + ~(~(m_~,m+~ - m+~,m_~), A~Xt(x) = - ½ E [ n _ ~ , ( n + x ) - n + , ( n _ x ) ]
(A.3)
- ½~[m_~,(m+x)-m+~,(m_x)]. (A.4)
463
LA. Batalin et aL / Q E D in external constant field
Let us perform the gauge transformation of the potential _ ext (x) + a~+(x), A~text (x)-A~
, ( x ) = - ½E(n+x)(n_x) - ½[}C(m+x)(m_x).
(A.5)
This transformation is due to the fact that in terms of the potential A~,ext(x) it is possible to separate the variables in eq. (A.1). If we transform the functions in accord with the gauge transformation (4.5) +(k}(x) = exp(ieeo(x))•'{k}(x),
(A.6)
then the functions ~k'(~(x) satisfy the equations
~;text•~)~'ext~(x)~'~(x)--,~ \ -- n21~t~,~(~), text [~x)\ = ion,- eA;×t(x). ~r/,
(A.7)
Substitute (A.5) into (A.7). Then we have 82
(
a(n+x) 2 i O(m x
8
)2
ia(n_x) +eE(n+x) + e%(m+x)
02 + a(m+x) 2
+'{k}(x) =p2*'fk}(X ) .
(A.8)
Carrying out the separation of the variables in (A.8) we obtain ~P'{k)(X) -- N{k)exp { -- ip,(n _x) -- iP2(m_x ) -- ½02 } H,,(p)D,[~o eif"/4)r ],
t3 = ( l e l ~ ) l / Z [ ( m + x ) + p 2 / e % ] ,
. = -½(a +ix),
~"= (21elE)l/2[(n+x) + p l / e $ ] ,
X= [ p 2 + l e l ~ ( 2 n + l )
lelE
]
n = 0,1,2,...,
0)-~- --[-1.
(A.9) H,(O) are the Hermite polynomials, Dv[oa ei(~/4)~-] are the parabolic cylinder functions, Pl and P2 are eigenvalues of the operator integrals of motion iO/O(n_x) and iO/O(m_x), respectively, N~k ~ is a normalizing factor. Let us verify that for the functions (A.9) the orthogonality condition (2.24) does take place and find in this way the normalizing factor. Using the orthogonality condition of the Hermite
464
LA. Batalin et aL / QED in external constant field
functions we have (~b(k} ,~b(k,}) = N{.k}N(k,}(2cr)2(2eZE~3f)-1/2
×2"n!v/~8.,.,8(p1 --p~) 3 ( p 2 --p~)V,
(A.IO)
T= f_+~ d'rD~.ttoe-i("/4)'r]D,,[to'ei('~/4)'r].
(A.11)
To calculate the integral T let us make use of the integral representation for the parabolic cylinder functions [24]: D~[ o~ ei(*r/4)'r ] ----F - 1 ( _ V) exp( - ¼i¢rv -
l i'r2 }
f°°t-~- I exp( - ½it 2 - it'r } dt. "0
(A.12) As a result we obtain T=
21e15(2~)2[/-(_~)r(_ u.)]-lexp{ ¼i~r(v. _
v)} 8~,~,3(p2 _ p 2 , ) . (A.13)
If the normalizing factor N(k ~ is equal to N(k}=(2~r)
-2
~ \1/4
(~-~)
(2nn!~/-~)-l/2I"(--v) ei(~'/4, ,
(A.14)
the functions (A.9) are normalized by condition (2.24). Let us show that the functions (A.9) form a complete set in the sense of (2.25):
E°° ~, :dpadp2dp2t~ n=0 to=+1
:
(2¢r)4
(x)~b~k}(Y) (k}
]1/2
= (~-~]
fdPldp2exp{-ipl(n-z)-ip2(m-z))
X ~
°=o.°(o)..(o t )j_ Sp2dp2,
(A.15)
z= - x + y ,
j.2 = e../.,<.-:,r(-.)r(-.*)
E
D. E~ e '<~/4~ ] D.. [ o~e -'<~/4)~'],
~o=_+1
v*= -v-l,
":'=(2[e[$)l/2[(n+y)+pl/eE],
p' = [lei'X, ]l/2[ ( m +y ) + p2/e * ] ,
(A.16)
I.A. Batalin et al. / QED in external constant field
465
where v , ( o ) = (2"n !v~-)-1/2exp(- ~p2)Hn(O) are the Hermite functions. Let us find J +_~~ J 2p d p 2. To do this let us use the identity F ( - u ) F ( - v * ) = -rr/sin(rrv) and introduce the new variable v = - ½ - ½i[p2 + l e [ ~ ( 2 n + 1)]/lel$" Then
f+=42de2=+ i21e[t9 )
f -1/2+i~
-~
J,=tr
exp(-~i~r(v + ~)} sin (cry)
E
-1/2-ic~
J, dp,
(A.17)
D~[°~ei(~r/4)'r]O-v-l[°ae-iOr/4)'r'] • (A.18)
~o=+1
The integral over v is calculated with the help of the Cherry formula [24] fcC+i~ e i(~r/2)(v+l/2) ic~
~
sin(~r u)
-
D~[a~ei("/4)r]D-~-t[°ae-'('~/4)z']
w=_+l
=
-4~riS(r-r'),
-1
(A.19)
Thus, choosing c = - ½ in (17) we obtain f_+ °~Jpzdp 2 = - ( 2 le[$ )t/2(2 ~r)28( n +z ).
(A.20)
We see that this expression does not depend on the quantum numbers. Therefore one may integrate over P2 and sum over n in (A.15) using the condition of completeness of the Hermite functions: oc
E v,(O)v,(o') = 8(0 - O') = (lel%)-X/ES(m+z).
(A.21)
n=O
After integrating over Pl we obtain the condition of completeness (2.25):
E ~{k}(x)+'~k}(Y)
= 8(4)( x - Y ) -
(A.22)
{k} To calculate the integral
Jk,k,(~) = f d4xd4y+Tk}(x)exp{~iexFy- iz52(r)z} ~,{~,}(y),
(A.23)
which occurs when calculating ((Ax,x))k,k , let us use the method of refs. [12,13]. Using this method we obtain
Jk,k'(52) = (4rr)2[det( 16522 - e2F2)] - X/4exp( - i p x ( r ) p } 8{k},{k,} , 452 -er x ( F ) = (2eV)-I In 452 +~"
(A.24)
LA. Batalin et al. / QED in external constant field
466
Proceeding from this expression we shall calculate the integral occurring in (2.24):
Jk,k'; ,1~( fa) = i OJk'k'( af~, #I2) X
iJk,k'( ~2)
0
-- ( -ipx(F)p 0$2~
- ¼ Sp ln(16122 - eEF2)}
(A.25)
Let ~(I2) be an arbitrary function of ~2 which can be expanded as a series in ~2:
n~0
•
Let us find the derivative of ~,(Ia) with respect to ~2~B: '(A.26) Here we make use of the relation n
n--i
O~l~'
E
012"t~
k=o
k
n-k-1
Expression (A.25) contains derivatives of two types: OSp °Y(~2)/OIa~a. Using (A.26) we have
(A.27)
OpC3(fa)p/O~2 "B and
O .-1
0 pOy(fa)p= O~aJ8
n! n=l
•
(pgak),(la"-k-lp)/~.
(A.28)
k=0
But the expressions (A.28) are included in (3.24) together with the factor K = A-2Tl/sh2(eFp). Then we obtain finally from (A.28) K~/~ O S p ~ ( I a ) = S p [ K ~ ' ( I 2 ) ] 0~2~
K'~ o - ~ p ~ ( ~ ) p = p ~ ' ( ~2) Kp.
(A.29)
Using relations (A.29) we have
[
1
K"/~Jk,k,;,~t~(/a) = 4Jk,k,(fa) P 16122--e2F 2p - iSp(2KJ2)/(16122 - e2F 2) . (A.30)
LA. Batalin et al. / QED in external constant field
467
Substituting the expressions
~2(F)=L(p),
~2(F)=L(v)+
~.IA - 1 8shZ(eFu) ,
12(F)=L(p)+
1 4--t '
i n s t e a d o f ~ 2 ( F ) i n t o (A.24) w e o b t a i n t h e r e l a t i o n s (3.30).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
V.I. Ritus, Proc. P.N. Lebedev Physical Institute, vol. 111 (1979) A.I. Nikishov, Proc. P.N. Lebedev Physical Institute, vol. 111 (1979) I.A. Batalin and A.E. Shabad, ZhETF 60 (1971) 894 W.-Y. Tsai, Phys. Rev. D8 (1973) 3446 W.-Y. Tsai, Phys. Rev. D8 (1973) 3460 A.A. Sokolov and I.M. Ternov, Relativistic electron (Nauka, Moscow, 1974) V.N. Baler, V.M. Katkov and V.S. Fadin, Radiation of relativistic electron (Atomizdat, Moscow, 1973) V.N. Baler, V.M. Katkov and V.M. Strakhovenko, ZhETF 67 (1974) 453 V.N. Baler, V.M. Katkov and V.M. Strakhovenko, ZhETF 68 (1975) 405 V.I. Ritus, Ann. Phys. 69 (1972) 555 V.I. Ritus, Problems of theoretical physics (Nauka, Moscow, 1972) V.I. Ritus, Pisma ZhETF 12 (1970) 416 V.I. Ritus, ZhETF 75 (1978) 1560 J.J. Volfengaut, S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Yad. Fiz. 33 (1981) 743 E.S. Fradkin ZhETF 29 (1955) 258 E.S. Fradkin Proc. P.N. Lebedev Physical Institute, vol. 29 (1965) E.S. Fradkin Nucl. Phys. 76 (1966) 588 E.S. Fradkin Dokl. Akad. Nauk SSSR, 98 (1954) 47 E.S. Fradkin Dokl. Akad. Nauk SSSR, 100 (1954) 897 E.S. Fradkin Acta Phys. Hung. XIX (1965) 1975 G.A. Milekhin and E.S. Fradkin, ZhETF 46 (1963) 1926 A.S. Babichenco and A.E. Shabad, preprint PhlAN N 127 (1984) I.A. Batalin and E.S. Fradkin, Teor. Mat. Fiz. 5 (1970) 190 H. Bateman and A. Erdelyi, Higher transcendental functions, vol. 2 (1953)
QUANTUM ELECTRODYNAMICS IN AN EXTERNAL CONSTANT FIELD I.A. B A T A L I N and E.S. F R A D K I N
P.N. Lebedev Physical Institute, Moscow, USSR SH.M. S H V A R T S M A N
Department of Mathematical Analysis, Tomsk Pedagogical Institute, Tomsk, USSR Received 3 December 1984
The generating functional of all q u a n t u m Green functions in an external field is found. The Green function of a scalar particle in a constant and uniform external field is calculated with the help of modified perturbation theory. Its infrared asymptotics is studied. It is shown that an e n h a n c e m e n t of the pole takes place, as in the vacuum case.
1. Introduction
There has been great interest in electromagnetic processes in external fields during the last few years. This is accounted for, on the one hand, by the creation of the intensive electromagnetic fields in lasers and the appearance of electrons and photons of high energies, and, on the other hand, by possible astrophysical applications. In the above processes one can verify quantum electrodynamics in the domain of high energies and large fields. The study of this phenomenon is of fundamental importance because a correct description of the phenomenon in intensive external fields is connected with the necessity of going beyond the framework of the usual perturbation theory. For this reason the processes in the e 2 approximation in different external fields [1-14] have been studied in detail lately with the external field taken into account exactly (in this connection the constant uniform field and the plane wave field or their combination are of a special interest because in these external fields one can exactly solve the Dirac and Klein-Gordon equations and find the one-particle Green functions). However, the contribution of radiative corrections at high energies and intensive fields becomes essential. For this reason it becomes of special interest to work out such methods of calculation of radiative corrections to the processes in an external field which, in their final result, go essentially beyond the scope of the usual perturbation theory with the external field kept exactly. This is due to the fact that the generalized (with the external field being taken into account exactly) Feynman diagram technique encounters essential difficulties caused 435
436
LA. Batalin et aL / QED in external constant fieM
by the necessity of correct calculation of the vertex function in a self-consistent approximation, since otherwise the results obtained are gauge noninvariant, and the corresponding generalized Ward identities in the presence of an external field [15-17] are not correct. The most adequate method for solving this problem is the functional method developed in papers [17,19]. With the help of this method a closed expression for the generating functional of the Green function is obtained in [16,17,20] and the modified perturbation theory is constructed. The method of modified perturbation theory proposed in [16,17, 20] leads to a result which goes far beyond the scope of the usual perturbation theory. This method appeared to be especially fruitful in quantum electrodynamics, because in the first order of the modified perturbation theory the contribution of the soft photons, both virtual and real, is summed completely. In particular, the first order of the modified perturbation theory gave the possibility of obtaining the correct asymptotic behaviour of the Green function and the cross section of the processes in QED [16,20, 21] in the double logarithmic approximation. Not long ago [22] the method of the modified perturbation theory was also used in statistic quantum electrodynamics. These general results (which are beyond the scope of the usual perturbation theory) will allow us to essentially advance in solving the problem of finding the quantum Green function in the presence of a real external field. In the present paper we consider the calculation of the one-particle Green function of a scalar field in a constant uniform external field by means of the modified perturbation theory. The Green function is found in the representation of eigenfunctions of the Klein-Gordon equation in a constant field. Such functions were first obtained in papers [12,13]. It was shown there that Dirac's eigenfunctions (in the spinor case) in a constant field diagonalize not only the Green function without radiative correction, but also the Green function in the e: approximation (see also [8]). The use of eigenfunctions appears fruitful in the modified perturbation theory because they contain considerable information about the dynamics of the system in a constant field.
2. Generating functional in QED with external field The principal value of the functional method is the generating functional of all Green functions: Z[~*, ~, J] = Cv 1out(0lS [~*, ~, J]10)in, where S [ , * , , , J] =
Texp{ifEs(x)d4x),
= L(x)A.(x)
+
+
(2.1)
LA. Batalinet al. / QED in externalconstantfield
437
Cv = out(0i0)i n is the probability amplitude for the vacuum to remain the vacuum, 10) in, 10)out are initial and final vacuum states, Jr(x) is the source of the electromagnetic field At(x), ~(x) and ~*(x) are sources of the scalar fields ~ + ( x ) and ~ ( x ) respectively. Operators are taken in the usual Heisenberg representation (i.e. the interaction between the fields is taken into account in the absence of the external sources) and satisfy the equations
m2] [
[ngr,
= 0, +(x)=O,
god
=
- e a r ( x ) -ear°x"x'
J,
~ * ( x ) = ia r + eAr(x ) + eAe~Xt(x),
+(dl- 1)dtlc~rS,]A"(x) = Ir(X ) ,
[] = _ era
Ir(x)=e~+(x)°'2~(x)~(x)-e(P~'(x)~+(x))~(x).
(2.2)
A~xt is an external electromagnetic field, d t is a gauge parameter. Using the method proposed in [16,17, 20] one may show that the solution of the functional equations for the generating functional (2.1) resulting from (2.2) has the form
- i f d4xd4y,*(x)a(x, yiAext+i-~)~(y))
× exp{½iS d4xd'iYJr(x)Dr'(x-y)J~(y)},
(2.3)
where G (x, y la) is the causal Green function of the scalar particle in the "external" field ar(x ) = A~Xt(x)- iSIS Jr(x) satisfying the equation
[(iOr-ea~(x))(iOr-ear(x))-m2]G(x,
yla)=8(4)(x-y).
(2.4)
/ / ( A ext + 81i8J) is the polarization correction to the action of the classical electromagnetic field:
/-/'(Aextq-/~-~'=) Spin[G( 'Aext-[-/.~-~')/G(10)1.
(2.5)
C is a constant to be determined by the condition Z[0, 0, 0] = 1. Further, it is convenient to find a functional representation for the Green function in an arbitrary field a~(x). To do this let u s write G(x, yla) in the form of the inverse operator
G(x, yla) = - i fo~dvexp( i(er 2 - m2-t- ie)v } ~(4)(x - y ) , % =- i0 r - ea r •
(2.6)
LA. Batalin et al. / QED in external constant field
438
Following paper [20] let us write the Green function (2.6) in the form of a functional integral:
G(x, yla)= --i (
d4p
J (2~) 4
e - i p ( x - Y ) [ ° ° d l , e-iOn2-i~)~
Jo
× f D4texp(-ifo~[t~(u')t~(~,') - 2p.t~(v')]
dr'} q~(x, p,vlt), (2.7)
where the function ¢(x, p,
v It) satisfies
the equation
• Oq~=2%t~(v)q~, -- 1-0-~" p
,(v=0)=l
D4t is the normalized volume element in the functional space
d4t(v ')
D4t = ~ ~t = 0
[J ~
/ J0~t.(v')t"(v')dv' /1' .
d4t(~,')exp - i
~r = 0
Let us consider the subsidiary function ~(x, p, viA ext, u) satisfying the equation
.0~
--t-~--
~
(2%t,(v)+u,(v)t,(v))~,
~(v=0):l
Note that ~(u = 0) = ~. It is not difficult to find the solution of this equation:
Separating the external field a~ ×t and the "'radiative field" making some simple transformations we obtain [20, 23]
G(x,ylAe×t-l-i-~)~-if d4P
e-iP(X
8/i~J ~ from
y)f°°dpe-i(m2,~.)~,
"~ (2~)'
Jo
× exp( _ 2efo'p~(p,)
Y(x,p,~'la'X'.u)=fD4texp(-ifo"[e(v'lt.
6
dv'}r(x, p, vlAex', u)l.=o,
Aex')-u.(v')t"(v')]d~," ) ,
where
E(v'lt,
A'X') = [,~(r.')- 2p~, +
a. and
2eA:Xt(x-
2 ~ : t ( ~ ) d h ) ] tt~(v')
(2.8)
LA. Batalin et al. / QED in external constantfield
439
is the classical action of the particle in the external field A~xt, P~(v') =
-i ~ u ~~----~--x(p')=x-2f~P(~)d~. (~') '
Substituting (2.8) into (2.3) we obtain for the generating functional Z[~*, ¢, J] the expression
Z[~*'~'J]=Cexp{ f d4xf~dse-i("~-m s~Jo,
(
0+(xp(ej0
t/ t)
×exp(- f d'xd4y~*(X)fo°°dve-`(m2-i~)~
)) ~(y) )
8J~(x(~,)) d~' xexp{lif
xy
d4xdayj~(x)D~(x-y)J'(y)),
(2.9)
where
Y(x, y, plAeXt,u)=j/* ~d4p e-ip(x-Y)Y(x,p, viA ext, u) , D.o(x)= f a'~k,D.o(k)e'~x, (2~)
(2.10) Some remarks about expression (2.9) are in order. It is known that the expression for Z[~*, ~, J ] contains divergences before the programme of renormalization is carried out. It is convenient to carry out this programme after all the calculations have been done. To do so it is necessary to regularize Z[~*, ~, J] (2.9), so that the divergences are absent during the calculation. As in papers [16, 20] this can be done
440
LA. Batalin et al. / Q E D in external constant field
by introducing the cut-off s o for the proper-time of the particle in the polarization term in (2.9) and the cut-off t o for the photon proper-time in (2.9). As long as s o and t o are not zero, the whole expression (2.9) is finite. After carrying out the programme of renormalization it is necessary to remove the regularization by making s o and t o zero. The resulting renormalized expressions should not depend on s o and t o. In (2.9) one may carry out the partial functional differentiation over the sources J~ of the electromagnetic field A~ by expanding (2.9) into a series in powers of the sources ~* and ~:
c-xz[~ *, t, J] = c-~z[o,o, J] +
(-1)" I-I f d4x,d4Y,~*(xk)~(Y~,)
~,
"
n!
n=l
Xexp(
k=l
fo ~
--
i(m'-ie)vk
dyke
An,,n-I-~ Bn,(J)}exp(-I-fd4zf°°d~Se -i(rn2-'Os
~ nl , n 2 = 1
nl = 1
× exp n
So
- 2e
°~(s')
S
8j~(z(s,))
ds'
x
× ~ Y(xt, Yl; viiAex',ut)l.=o l~l
× exp(½ifd4xd4yj'(x)D,.(x-y)JV(y)),
(2.11)
where
Pn2dv P~(,)D,.(x.~(v)-x.2(v A .....=2ieEfo%dV'fo ))P~2(v),(2.12) p!
=4ie
P~(v')
= -i
~.(s')
=
-i
2
I
!
ii
ip
II
dv"f d4xJ'(x)D~.(x-xn,(v'))P~(v'),
(2.13)
ds' "~dv'P~(s')D..(z(s')-x.~(v'))P"(v '~ I'
(2.14)
Bnl(J)= -Rlefo Rn
~
~1 ~
8
(~)' ~ ~ 8o~(s')
'
~(s') = z - 2f
'~(X)dX,
(2.15)
I.A. Batalin et al. / QED in external constant field
441
and
Z[0,0, J]=
Cexp(H(AeXt+ / -8~ ) ) e x p ( s t f d 1"
4
xd
4
yJ (x)D~.,(x-y)j,,(y)} I.~
is the generating functional for the photon Green functions. To carry out the complete functional differentiation over the sources J, one should expand H ( A ext + 8/i8J) into a series with respect to the functions G(lAext + 8/i8J). In this way we obtain the following expression for Z[f*, f. J]:
c-Xz[~*,~,J]=C-lZ[O,O,J]+ f n=l
fd4x,d4ykt*(x,)f(y,)
( -1)n fi n !•
k=l
×fo=d,ke-iC"2-iO".~exp{ f
A.1..2+ f B.,(J)
n 1, n 2 = 1
n 1= 1
- f d 'z C "So s
z,
slO,O)/ )
d s p e_i(mZ_ie)sp X l + f ~ .1 p(Ifdz, 4 fs= Sp =l o r=l
xexp
( "Y". f \n l=l
a=l
C~l , . ~,-
f BI,~(J) + f
AI,,,,~'
a,al~l
a=l
}
X af :i l ~ 7 ( Z a ' Z a ' S a l ~ 4 e x t ' V a ) l ° = O }
X fi I"(xt,y,, VllA~xt,ul)l,,=o /=i
x exp{lif
d4xd4yy'(x)D~,(x-y)J'(y)},
(2.16)
where £un~ Z ~(s*))P:(s u t), C.,,.=4te2 Jo dv'fosa ds'P~(n')D"(X'l(V')(2.17) •
Bla (J) =
-2iej:ads' fd4xJ~(x)D~,,(x-
z~(s'))@~(s'),
sa p!
P:(s')= -i 8 v a , 8O '~)
'
z.(s') =
zo -
2 f : a P . ( X ) d~. .
v
gp
(2.18)
442
LA. Batalin et al. / QED in external constantfieM
The expression for Z[~*,~, J] will have the simplest form if we neglect the polarization effects. Then we obtain: /[~*'~5'J] = [ 1+ n=X ~'~ (--1)n ~' k=l fi
fd4xkd4yk~*(Xk)~(Yk)
Xfo°°dvkei(m2-i~>kexp{ ~
Anl,/.12-]- ~ Bnl('))
nl,n2=l
nl=l
X 1=1 f i ~r(x/' Yl'Ol[heXt'Ul)[u=O]
×exp{½if daxdgyJ~(x)D~(x-y)J~(y)}.
(2.19)
If we are interested only in the Green functions of the scalar particle, the expression for the generating functional becomes still more simplified: Z[~*,~,J]=I+
~ (-1)" fi n! k=l
n=l
exp{
nl,n2=l
fd4xkdayk~.(xk)~(yk) 1=1
Using (2.16) one can write the one-particle scalar Green function in the form
O(x, y ) =
-if0m dvexp{-i(m
1 x 1+ ~. 7y r=l
2- ie)v+ A1,1 }
Hr fd4zpf oo dSpe-i"2-i~'sP Sp p=l so
×exp{a=l~ CiaS- oe,a,=lkA1 .... } f = i l r ( za , Za , Sa [Aext, ua )[ v=O
×exp{- f d4z [~ dS e-i(m2-i~)s~'(z,z,slO,O)}5"(x,y, vlAeXt,u)[u=o" % -7 (2.21) When neglecting polarization effects this expression takes an especially simple form:
G(x, y) =
- ifo~°dvexp{
-i(m 2-ie)v} ×(expAl,1)x,y.
(2.22)
LA. Batalin et al. / QED in external constant field
443
Consider now expression (2.20) in detail. Suppose we know a complete set of solutions of the eigenvalue problem. X ~r~Xt(x)~reXt~(x)~(k}(x)=p 2~b(k}(),
%eXt(x) =
ion-- eAe~Xt(x),
(2.23)
where ( k } is a set of quantum numbers ({ k } contains the quantum number p2). Let us call such functions eigenfunctions. In the constant uniform field such functions were first obtained in [12,13]. Suppose, further, that the eigenfunctions form a complete orthonormalized system of functions with respect to the scalar product
(~p, qJ) - f d4xep*(x)~(x),
£ ~p{k}(X)+~k}(y)
= g(4)(X - - y ) .
(2.24)
(2.25)
{k}
(If the eigenvalues are discrete, then g{k},{k'} in (24) is Kronecker's symbol and if they are continuous, then g{k },{k'} is the &function.) With the help of the eigenfunctions one can construct the causal Green function of the Bose particle in the external field A~xt without radiative corrections:
G(°)(x, ylA ext) =
Y'. ~ P { k } ( x ) ~ } ( Y ) , {k}
pZ --
(2.26)
m 2 q- ie
which satisfies the equation
{rr~iext (x)Ir extp. (x) If we represent (p2 _ m E + can be written as
ie)-i
G(°)(x, ylA ~xt) = -
_ m 2
}G~°)(x, ylAeXt)=g(4)(x-y).
(2.27)
in an exponential form, the Green function (2.26)
i [ °° dl, e-i(-,2-i~> £ e 'p:~a, v{k}(X)q~k}(y). ~0 {k}
On the other hand, the Green function expression (2.8):
G(°)(x, ylA ~xt)
G(m(x, ylA ~xt) = -ifomdve-~("=-i')~I'(x, y,
(2.28)
can be obtained using
vlA~Xt,0).
(2.29)
Comparing (2.28) and (2.29) we have I"(x, y, vlheXt,0) = £ (k}
ei*'="~p{k}(x)tp'~t,}(y).
(2.30)
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LA. Batalin et al. / QED in external constant field
Using (2.24) and (2.30) we obtain an important property of eigenfunctions:
f d4x d4y ~tk} (X)I"(X, y, via ext,0) ~P{k')(Y) = e ip2v'~v(k},(k, } .
(2.31)
Let us represent the sources ~*(x) and ~(y) as a superposition of the eigenfunctions
I2
E
(k}
(k)
Then the expression for the generating functional Z [ a * , a] (2.20) has the form (--l)n
Z [ a * , a] = 1 +
- -nI n=l
Y'~ ( k l } .....
(k2)
pFI__lO~(kp}O~{kp} *
(kl) ..... (k;,)
X f0~ dpp e-i("~-m",
exp
~
A,~,,:
\ nl"n2~l
k 1..... k.; k{ ..... k"
(2.32) where
X Ht~tk,}(Xl)A
/=1
~U
yp,
p=l
Up
(2.33) Expression (2.32) allows one to find the representation of Green functions in the basis of eigenfunctions.
3. Green functions in constant field
Consider the constant and uniform external field A~Xt(x) =
-
1
5F~,~x
v.
(3.1)
As can be seen from above the calculation of the generating functional and the Green functions reduces to the calculation of the functional integral (2.8), i.e. the
LA. Batalin et al. / QED in external constant field
445
function ~'(x, y, viAext, u). In the constant field (1) the functional integral (2.8) reduces to the gaussian one. The function Y(x, p, viA ext, u) was found in ref. [23]:
Y(x, p, ~,[A ext, u) = [detch(eFp)]
-1/2
x exp{ i f f f d~,' dp" ~r~(x, p, p')G~o(v', 1,"; p)~r°(x, p, 1,")},
(3.2) where
%(x, p, v') =p~ + ½eF~x~+ ½u~(v') G(v', v"; v) = B(v'- v") + eFexp(2eF(v'- v") ) [ e ( v ' - v") + th(eFv)] e(v'- v")=
O, v'= v". - -
1,
(3.3)
p, < pt,
Thus, the integral over p, in (2.10) also reduces to the gaussian form and we find Y(x, y, 1,1Aext, u) =
i
(4~r) 2
exp(½iexFy - i(x - y ) L ( u ) ( x - y ) - ½Spin sh(eFv)
eF
)K(x - y , plu)
L (u) = lefcth(eFu),
(//o
K(x-y,~,lu)=exp ¼i
+½i(x-y) . eFexp( eFu ) f~ exp(-2ef~')u(v')dl,'} G(p', u"; 1,) = G(p', ~,"; ~)-
2er e x p ( 2 e r ( v ' - t,") sh(eFu) }.
'
(3.4)
Substituting expression (3.4) into (2.21) we find the quantum Green function of the
LA. Batalin et al. / QED in external constant field
446
scalar particle:
G ( x , y ) - ( 4 ; ) i 2 fo~ dv
exo( X (1 +
'
j 'zr
J'o
s
/)
(4,rrs) 2
r@l 1 fl fd4zpf c~dSp Sp =
p=
so
1
} i~
{ a.al=l
a=l
( 4 ~ ) 2r a = l
h( eFs a ) + ¼ i f f 0 d s ' d s " O a ( S )G,v(s ,s ,Sa)OVa(S '') eF . . . .
{
> exp { - ½ Sp In s__, - -
t
[
×exp{ liexFy - i ( x - y ) L ( v ) ( x t
- y ) - ½Spin
• ~ +1'ffo
d v ' d v " u ~ ( v' )G~r(v " . ,v ...
))v=o
sh( eFv ) eF eFeeF"
, v)ur(v '') +½i(x -y) sh(eFv)
X fo"e-2er"'u(v')dv '} .=o"
(3.5)
If we perform the substitution zp + x = z~ in the integral over zp in (3.5) it is not difficult to see that the Green function G(x, y) has the following structure:
G ( x , y ) = exp( ½iexFy }Gl(X - y ) ,
(3.6)
where G l ( x - y ) is a scalar function of the difference x - y and of the constant tensor F. Hence it follows that G x ( x - y) may depend only upon arguments of the form (x - y ) K ( F ) ( x - y ) , where K ( F ) is an even function of the field. The eigenfunctions of the problem (2.23) in the external field (3.1) can be found in a covariant (see appendix) form:
+(k} (x) = N{k}exp{ ieCp(x)
-- ipl(n -- x) -- ip2(m -- x) -- ½pZ }
XHn(o)D,[~ei~/4"r],
(3.7)
LA. Batalin et al. / QED in externalconstantfield
447
where { k } = { p l , p2, n, p 2 , ~ } , ~ = +1, n = 0 , 1 , 2 . . . . ; the numbers Pl, P2 are eigenvalues of the operators i a / 3 ( n _ x ) and i O / O ( m _ x ) respectively, p = ½(1 + iX), X = [p2 + l e l ~ ( 2 n + 1)]/lelE ' E, % = [(oy2 + ~32)1/2 -Y-~-]1/2; ~, oy are the invariants of the field (which we suppose are not zero), p = ([e ~)1/2((m + x ) + p2/e~C), "r = ( 2 [ e [ E ) x / 2 ( n + x ) + p x / e E ) , eg(x ) = - ½ E ( n + x ) ( n _ x ) - ½g(~(m+x)(m x), H , ( p ) are Hermite polynomials, D~[6oei(~/a)r] are the parabolic cylinder functions, and n _+, m_+ are vectors which are solutions of the equations -
Fn += En =;, m+-2--1,
Fro+= -Y-gCm ~ ,
n 2 = +_1,
( n + n _ ) = ( n + m + ) = ( n _ m + ) =_( m
N{k) = exp{ ~iqrp ) F( - l,)( ~C/2E)l/4(Z'n !vr~) - 1 / 2 Using the integral representation for the parabolic cylinder functions, one may show that the eigenfunctions (3.7) form a complete orthonormal system of functions in the sense of (2.24) and (2.25). Note that the following important property of eigenfunctions (3.7) is valid:
fdnxd4y+~k)(x)exp{½iexFy - i(x -y)T(F)(x
- y ) } ~b(k,) ( y ) - 8(k},(k'} •
(3.8) Relation (3.8) permits us to prove that eigenfunctions (3.7) diagonalize not only the Green function G(°)(x, ylAeXt), but also the quantum Green function G(x, y), eq. (3.5). Indeed, suppose that the function G ~ ( x - y ) can be expanded as a Fourier integral:
al(x-y)
=
f dkl . . d k t f ( k l
{
.... ki)ex p - i ( x - y )
,
E kjKj(F)(x-y)
}
.
j=l
(3.9) Here 1 is the number of the independent structure matrices K ( F ) on which the function G I ( X - y ) may depend. Then we have
f d'xd4yg~k)(x)G(x, y)ap(k,} ( y ) = f dkl...dktf(k 1.....
{
x f d4xd4y4,~,)(x)exp
½iexFy-i(x-y)
'
kt)
E kjKj(F)(x-y)
/
+{k')(Y)-
j=l
(3.10)
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LA. Batalin et al. / QED in external constant field
Choosing in (3.8)
T(F) = Y~=lkjKj(F) we obtain f d4xd'y~k}(x)G( x, Y)g'(k,)(Y) - ~(~,(i,,~.
(3.11)
Proceeding from this we can write
G(x,y)= ~_, ~ f dpxdp2dp2 ,o=_+1 n=0
(27r)4
* ~p{k}(x)G{k}~p{k}(y),
G(k~= f d'xd4yg,'(k~(x)G(x, Y)~P(I,}(Y). Neglecting the polarization effects we obtain for
(3.12)
(3.13)
G(x, y)
G(k} (4~) 2 f0~ du e-i(m2-i~lV({expAl,1))k,k"
(3.14)
Let us rewrite the definition of the average ( ~ ) ) k , k in a more convenient, for our subsequent work, form:
((h(--~u)))k,k= fd4xd4y~b~,~}(x)(A(~U))x,y~b(k}(y).
(3.15)
Relation (3.14) permits one to develop the modified perturbation theory for G(k}. To obtain the first order of the modified perturbation theory let us expand G(k / into a series with respect to the radiative interaction. To within e 2 we have G(k}= (4--~)2 fo°°dve-i'm2-i~)J(~}(v)(1 + ( ( Aa,1))k,k/J{k}(v)), J(k) (1,)= (~l))k,k.
(3.16)
According to [16,17, 20] the first order of the modified perturbation theory consists in the replacement 1 + ( ( A I , , ) ) k, k/J{k} (I') ~ exp( ( (AI,1) ) k , k / J { k } ( P ) ) . Thus in the first order of the modified perturbation theory one has
~(expAl,1))k,k=J(k)(~')exp(((AiA))k,k/J(k)(l,)),
(3.17)
LA. Batalinet al. / QED in externalconstantfield
449
and for the Green functions in this approximation we obtain
(
/
a~12}--- (4~r)2 fo dvJ{k}(v)exp -i(m2-ie)v+ (
(3.19)
Finally we have m 2 + ie)v + ((Al'l))k'k I
G(x)_ { k ) - - _ifo °° dr exp{ i( p 2
(3.20)
4. Calculation of ((A 1,l) ) k, Using the definition of ((Al,1))k,~ from (3.15) we can write
((Al,1))k,k = f d4xd4yq,'~,}(x)(A~a)x,y@{k}(y),
2e 2 exp{ XiexFy - i ( x - y ) L ( v ) ( x - y) (art)2
-
Aa,~(x-y,~)=
(4.1)
½Spin sh(eFV)eF } A',l(x - Y ' v) ,
fro a ~ ' d ~ " a ( x - y , ~ ' , ~ " , ~ ) ,
(4.2)
(4.3)
a(x - y, ~', ~", ~) = e"(~')V.~(x(~') - x ( ~ " ) ) e ~ ( ~ ' ) ~ : ( x - y, .lu)u=O, (4.4) where K ( x - y , vlu ) is determined by (3.4). To find (4.4) let us use the integral representation of the photon Green function
D~(x(v')-x(r"))=
j~--~
,~(k)exp 2i
t(X; v', v") = 0(X - v") - 0 ( X - v').
t(X;~',.")k.e°(X)dX
, (4.5)
450
LA. Batalin et al. / QED in external constant field
As a result we obtain
a ( x - y , v', v", v) = / " d4k D ~ ( k ) e x p ( i k ~ k J (2~r) 4
+ 2iqk}
× {- ~,e.~(.,, .,,; .)+ [ K,s,~,(.,, .,,, .)+i~(.,,, .)] × [~j::(is', is", is) +/:(is', is)] }, fl(V',
(4.6)
is", is) = f0VG (is '', ~k; is)t(~k; is', is") dh,
r tis, , is" , is)= ~2,
So"G(is',X;is)t(X'is',is")dX,
,
eF exp{ eF(is - 2v')} f3(is', is) = ½(x - y ) sh(eFv)
(4.7)
£= ff0"dXdX' t(X; is', is")d(X, X'; is)t(X'; is', is"),
q= So~f3(X,v)t(X;is',is")dX. The integrals over X and X' in (4.7) are not difficult to calculate.
~= s h ( e F s ) s h [ e F ( i s - s)] eFsh(eFis) , e eFl'
[e_2eFv,
s = Its'- is'l, e_2eFv. ]
q = ¼(x - y ) sh(eFis) fl(is',is",is) = ½e(is'- is")+ ½(e 2eV(~'-~")- l ) ( e ( i s ' - v") +cth(eFis)), fz(is', is", is) = ½e(is' - is") + ½(e 2ev(/-"'') - 1)(e(is' - is") - cth(eFis)).
(4.8)
To integrate expression (4.6) over K s it is convenient to use the representation
451
LA. Batalin et al. / QED in external constantfield (2.10), having rewritten it first in a more convenient form:
D~p(k)= Df.(k) + DXv(k ),
Df.(k)=g,.(k= + ie)-x= -ig,.S~
eik=tdt,
-- lto
1)
DX,(k) = - k ~ k p ( d t -
teik2tdt.
(4.9)
-- tl 0
In accordance with this procedure it is also convenient to represent expression (6) in the form of two terms: a(x--y,v',v",v)=ad(x--y,v',v",v)+aX(x--y,v',v",v).
(4.10)
Then for ad( x -- y, V', v", V) and aX( x - y, v', v", v) we obtain, respectively, a d ( x - - y, v', v " , v ) = - i [ - ½ i G ~ ( v ' , v " ; ×
f~ Sdt -
v)+f3~(v', v)f3~(v '', v)]
if~#(v ,, v-, v ) f ~ ( v , ,
v,,, v)
-- it o
I~l~dt-i(f~lt(v',v",v)f3~(v',v)
× - -
it o
"[-ftfl(1.'#,l)
aX(x - y , v', v", v) = - ( a , -
l*)f3/,(/-'##, ")) f_Ttoladt,
'',
(4.11)
1 ) { [ - ½iGU°(v ', v", v) + f3~(v ', v)f3°(v '', v)]
×
f~ I.odt +frl~(v ', .", v)ft~(v ', .', v) -- it o
x f_~,o/.oo~at + (fl~(~ ', ~", v)f,.(~', ~)
Ia' .... "
= ( d4k k ...ko, e x p { i k A k + 2 i q k ) , d (2,n.) 4 'h
A = ~ + t.
(4.13)
LA. Batalinet aL / QEDin externalconstantfield
452
All the integrals 1~, ..... involved in (4.11) and (4.12) are expressed by means of the integral I:
I= f
d4k exp(ikAk + 2iqk)= ,.--~(detA)-l/2exp(-iqA-lq), ~-~ (47r) (4.14)
I.= -(A-lq)¢,I,
I,,~= [½iA,7~ +(A-lq).(A-lq)fl]I,
I~¢= -[½i(A~2(A-lq)~+ A~A(A-lq). + AJ(A-'q).) + ( A-lq)~,( A-lq)a( A-lq)13]I,
+ ½i(a2¢(A-lq).(a-lq)¢ + A72(A-lq).(A
lq)t~
+ a~¢(A-lq).(a-lq). + a#(aql).(a-lq)B+
a3(a-lq).(a
lq)o
+aJ(a-aq).(a-aq).)+(A-lq).(a-lq)~(a-lq).(a-lq)l~}I. (4.15) Substituting (4.15) into (4.11) and (4.12) we obtain after cumbersome calculations
_ (detA)-l/2exp a~(x - y, ,,', ,,", ,,) = (4.)-~f~,o .
aX(x -
{-i(x-y) 8sh2(eFp) ,AI ,x 1 ~A -2
× - 2 i 8 ( s ) + 2-S(x
-y)shT(eFv) (x - y )
+~iSpA-I( l+4eFt
ch[eF(v- 2s)] sh(eFv))}dt,
y, v', v", v) = -i(4~r ) - 2 ( d , _ 1)!i~ot (det a )-1/2
X exp
{ -i(x -y) 8sh2(eFv) "qA-1 (x-Y)}
(4.16)
453
LA. Batalin et al. / QED in externalconstantfield [ × (4t)-l~(s)+i2-4(x-y)
A -2
e F ' q c h [ e F ( v - 2s)] sh3(eFv)
]
+ A - X 2eZF2ch(2eFs) • -1 ~4 sh[eF(v - 2s)] sh2(eFv) -li s--~eF-v) ](x-y)
+2-'
"r4 (x - y ) - i S p A ¼(x - y ) A -2 sh2(eFv)
[
2eFch[eF(v- 2s)]
+-~Sp A -1
sh(eFv)
-1 sh[eF(v - 2s)] ]2 sh(eFv)
_A_2Sh2[eF(v - 2s)l sh~e~)
]}
dt, (4.17)
"q = ch(2eFs) - 1,
~2 = 2eFt + sh(2eFs),
"r3 = sh(2eFs) + 2eFtch(2eFs), *1= "q + 4eFt'r3,
~'4= ~', + 2eFtsh(2eFs),
A = ( 2 e F ) - l ( r 2 - rlcth(eFv)).
(4.18)
Note that a a and a x depend on v' and v" only in the combination s = Iv' - v" I. Therefore, when integrating over v' and v" in (4.3) one has
ffo d v' d r ' ¢ ( s ) = 2 fo~(V - s ) ¢ ( s ) d s ,
(4.19)
where ¢ ( s ) is an arbitrary function. Thus Al,l(x-y,v)=2
-
)a(x-y,v,s)ds
(4.20)
If in accordance with the breaking up of the photon function D~,(k) into the parts (9) one represents analogously A l , l ( x - y , v) in the form of the sum of two terms, then A ~ , I ( X - y, v) can be considerably simplified. For doing this note the identity
. OI {
2t~s =
"r4
¼(x-y)A-2shZ(eFv) (x-y)-iSpA-1
sh[eF(v - 2s)] s--~F-v)
}
I. (4.21)
Using this relation and integrating over s by parts, the expression for AXl(x - y , v)
454
LA. Batalin et al. / QED in external constant field
may be represented in the form
AX,l(x - y , ~,)=
½i(4~r)-2(dt
-1)~o-~-[exp(-i(x-T~y)21-1 ] .
(4.22)
Finally we obtain
Al,l(x -y,
£
r) = ip(8to) -1 + ½i(4~r) -2 ~(p - s ) d s
I - -
(detA) -1/2 ito
T1A-1 (X--y))
× exp - i ( x - y) 8 sh2(eFv)
×{(x-y)A :
n
sh2( eFp)
(x-y)
ch[eF(v - 2s)] +4iSpA-a( l +4eFt sh(eF~,)
)}
dt +½i(4~r)-2(d,-1)f£toV(e-'(~-':/4'-I ).
(4.23)
Substitute now (4.23) into (4.1). The result of this substitution is written in the form ((Ax'D)k'k=
.Ol -1 ia :'( --t-~ vt° Jk'k(L)+ 2~r.lo"V-s)ds f:
X
KaBJk,k;aB(L+
~'IA-1 )
,.1
ch[ e F ( p - 2s)] sh(eF,) )
×Jkk L+ , 8 sh2 (eFt,) _.o- }-
-1/2
8 shz ( eFt,)
+4iSpA-I( l +4eFt
(
(detA)
ito
)]o
~-~-( d, - 1)
Jk,k(L+ 1)-Jk,k(L )
det sh(eFp) -1/2
eF
'
if2
a = 4rr'
(4.24)
LA. Batalin et al. / QED in external constant fieM
455
where
Jk,k,( ~2) = f d'xd4y~'~k~(x)exp( ½iexFy - i(x - y)~2( F ) ( x - y ) } ~b(k,i(y), (4.25) J~,k,;,a(~2) = f d4xd'y (x - y ) , , ( x -y)#~p~ki(x)
Xexp( liexFy - i(x - y ) ~ ( F ) ( x - y ) } tp(k,i( y ) .
(4.26)
Differentiating (4.25) over I2~t~ one can find the connection between these two integrals:
Jk k'. ,~l~(12)= i 3Jk'k'(12) ,
,
(4.27)
012,,~
Making use of the integral representations for the parabolic cylinder functions and Hermite polynomials [24], the integrals (4.25) can be calculated to give Jk,k,(~2) = (41r)Z[det(16~22 - eZF2)] -'/4exp{ - i p x ( F ) p } 8{k},(*'),
- eF x ( F ) = ( 2 e F ) - l l n 412 412 + e ~ '
(4.28)
and p~, is a four-vector:
p~,p. = p 2 ,
p~.p.=p2. +p2,
p2=ppp = p 2 + [ e l ~ ( 2 n + 1),
p 2 = p t p = - l e l ~ ( 2 n + 1),
Pt~v=(~122-k-E 2 ) -1 ( F ~2+ g , ~ X 2 ) ,
L/x, = (~C 2 -t-- E 2 ) - I ( g . ~ , E 2 - F~/~).
The integral
Jk,k,;,a(12) enters into (4.24) together with the matrix K = A-2~l/sh2(eFp). Differentiating (4.27)over 12"a we obtain
K'~OJk,k';al~(~2)=4Jk,k'(~)
K
P16~22~e2F2P-
2K~2 } iSp 16$22 _ eZV 2 . (4.29)
456
LA. Batalin et al. / QED in external constant field
From (4.27) we have
Jk,k,(L) = (4¢r)2eiP2~3{k},{k,}, (
rlA-1
Jk,k' L + 8sh2(eFv )
) = Jk,,,(L)(detA)l/Z(det
"rff-T12/-1/4 4e2F 2 ]
×exp( -ip~(s, t)p }, ~b(s, t) = (2eF)-11n z2 + rl r2 - r 1
(
)
Jk,k' L + ~ - I
(
~2-T2 \-1/4 exp{-ip~p(v,t)p}.
=jk,k(L) det~Js=~
(4.30) Note that Jk,k(L)= J{k}(g). Substituting (4.29), (4.30) into (4.24) we get ((Al'l))k'k =J{k}(V)
--i~ v ` ° l - " at-~fOv(v-s)ds ~itoM(S, oo t; p, F)dt + - ~ ( d, -
t-l[t2g(,,
1)f - -
t ; p,
F) - 1]dr
ito
(4.31)
=bl(V;p,F), M(s, t; p, F) = g(s, t; p, F) p ~------~--p ~2z _ "1z + ½iSp -r22-_ rl2 , R = eF[2eFt(3 - 2ch(2eFs)) + (1 - 8eZFZtZ)sh(2eFs)],
]- [ ,1:2 ,1.2~]-1/4
g(s,t;p,F)=[dett~)]
exp{-ip~p(s,t)p).
(4.32)
This expression diverges when t o ~ 0. Before proceeding to the renormalization consider the case F = 0. In this limit
(
M(s, t; p, 0) = g(s, t; p, 0) { p2
(s + 2t) 2 + 2 ( s + t) 2
g(s, t; p,0) = exp{ -ip2s2/(s + t)} (s + 0 2
2 ( s + t)
LA. Batalin et al. / QED in external constant field
457
Thus ((Al,1))k
3a
'
-1
. Ot
oo
J(k)(v ) k = - i - ~ v t ° _ t2"-~ fo~(p-s)ds fito - e-ip:':/("+t) o0 e - i p 2 s 2 / ( s + t )
s+ XP2(t+S)3
~
+ 4--~(dt-1)f°°
folksy-ito
(t+s) 2
t-l( t2 e_ip~,~At+.)_l)dt
"-,,o
~ (t + ~)2
~bl(p;p),
(4.33)
which coincides with the results of [20]. Consider the Green function G[1) I (k}lvzo
=
_ifo°°dvexp(i(p2_m2+ie)~,+bl(,;p))
(4.34)
Let us separate the terms linear in v from bl(~; p) and rewrite (4.33) in the following form:
bx(~', P) = -ivMo(p) + No(l,, p),
(4.35)
where Mo(P)=-4--~t
°/5 ds f_,oMs't; p,O)dt,
o +~
a fosds ~ f ~ M(s,t;p,O)dt+iv~-~j~f No(~, p)= i~__~ a o~dsfo°°M(s,t;p,O)dt - -
a
4qr
ito
£ ds I:,0"(s,t;p,O)dt
+ ~_~ _ (dr a
1) L~itotdt[t2g(p't;p'O)-l]
.
(4.37)
Note that if we calculate the mass operator M0(p 2, m 2) of a scalar particle in the e 2 approximation and make the substitution m 2-~ p2 then we get just M0(p):
Mo(P ) =
_~0(p2, m E ~ p 2 ) .
(4.38)
Therefore, the renormalization of Mo(p) may be performed in a standard way by choosing the normalization point as p Z = m 2 where m 2 is the experimental mass
458
LA. Batalin et al. / QED in external constant field
squared [20]. To do this let us represent Mo(p) in the form
Mo(p)
= M o ( p 2 = me2 ) + ( p 2 - - m e 2
) 3M°(P)pZ=m+MoR(P)
,
(4.39)
where MoR(P ) is the renormalized mass operator. Let us make in (4.34) the substitution g = z2u',
(4.40)
where z 2 is the renormalization constant of the wave function, and 3m 2 is the mass renormalization
3m 2 = m 2 - m 2.
(4.41)
Then G(x) I _-- _ i Z 2 f o ~ d v e x p { i ( p 2 _ m Z ) ( z 2 _ l ) v + i ~ m 2 p (k}lF=O
_iuMo( p 2 = m2) _ iu OM°(P--------~ 2 p2=m2( p 2 - m2e)
+ i( p 2 -
m 2 )lp -
iuMog(p ) +N0(u; p ) } .
(4.42)
i
The constants z 2 and 3m 2 are chosen from the conditions
8m 2 = Mo ( pz = m2 ) ,
(4.43)
z 2 - 1 = aM°(p)op 2 p~=,~"
(nAn)
The integrals entering into (4.43) and (4.44) are easily calculated: 34 2[z 2 \-1 + ~ + ln(Tm2to)-l], 8m2= T-~me[[moto)
z2-1=
(4.45)
34~r[2 a l l + l n ( y m e 2t 0 ) -1 ]
(4.46)
where y is Euler's constant. The function No(v, p), eq. (4.37), is equal to
No(~, , p) =
-- -4--~dt Ol l n ( y m /20
a(p, p ) ~ 0,
u ~ oo.
a (3 - d , ) l n ( m 2 u ) + a ( u , p ) -x + ~-~
) (4.47)
LA. Batalinet aL/ QEDin externalconstantfield The diverging term - ( a / 4 ~ r ) d
t ln('/m~t0)-1
459
should also be related to the z 2 factor:
22=z2exp(-~dtln('ym:to)-l). In the e 2 approximation we have z2- 1=
~-~[(3-d,)ln(~tm2to)-i+
Let us return now to expression (4.31) and rewrite form:
½].
(4.48)
bl(v, p, F)
in the following
bl(v,p,F)= -iavtol-i~-~v a fo~ ds i~itoM(S,t; p,F)dt
° f f ds
-v +N°(v'p)+N(v'p'F)+ i -2~r
M(s,t; p,F)dt, (4.49)
O~
1~
O0
N(v,p,F)=i-~--~ fosds fo M,R(S,t;p,F)dt +__4_~(dt 1) fo~t[g(v,t;p,F)_g(v,t;p,O)]dt, MaR(s, t; p, F ) =
M(s, t; p, F) - M(s, t; p , 0 ) ,
(4.50)
where No(v, p) is determined by (4.37). In the last term in (4.49) and (4.50) we have put the lower proper-time-integration limit equal to zero everywhere; on removing the regularization they are finite. One can verify this fact by expanding the integrals into power series with respect to the field and calculating the corresponding integrals. Consider the frst two terms in (4.49) linear in v. Let us write M(s, t; p, F) in the form
M(s,t; p,F)= M(s,t;
p 2 = m¢2,0)
+( p2_ m2) OM(s,t;Op2p , 0 ) p2=m2
+ M R ( S , t ; p, F ) ,
MR(s, t; p, F) = M(s, t; p, F) -
(
t
exp - i m ~ t +----~
×(t+s)-a(m 2(s+2t)2 + i ) - e x p ( - t m2" s2 e 2(s ~ f i
×[
(s+t)2+t2t+s
Then the integrals over s and t in
t +----~I
im2s2(s+2t)Z(s+t)-2] "
MR(S, t; p, F)
(4.51'
are finite when t o ~ 0 and we
LA. Batalin et al. / QED in external constant field
460
obtain for bl(p; p, F )
61( I,; p, V ) = - ip6m 2 - i( p2 _ m 2)( z2 _ 1)v - ipMg ( p, F) Ol
t.O~
~00
+i~--~l'J~ dSJo M ( s , t ; p , r ) d t +1~---~ sds
R(S,t; p , F ) d t
+N0(~,p)+ ~ ( d , - 1 ) MR(p, e ) = ~a
~ds
tgR(~,t;p,F)dt,
(4.52)
MR(S,t;p,F)dt ,
gR(s, t; p, F) = e,(s, t; p, r ) -g(s, t; p,0),
(4.53)
where 8m 2 and z 2 - 1 are determined by (4.43) and (4.44). Thus we see that the procedure of renormalization in the presence of a constant field is the same as in the vacuum ( F = 0) and the constants of renormalization do not depend upon the external field. If we write the Green f u n c t i o n G~lk)} in the form
"~(k)~(l)_----i
f0 d p e x p ( i ( p Z - m 2 + i e ) 7 , + b x ( ~ , p , F ) }
,
(4.54)
and make the change of variables u = z2~,' and the mass renormalization m E = m e2 - - 8m 2, we obtain a ( 1{ k) } - __ ~,2"-' F;,(1) {k }R,
(4.55)
where G {o~ k } R is the renormalized Green function G~)}R= - i f o ° ° d v e x p ( i ( p 2 -
2
,
(4.56)
bin(g, p , F ) = - i ~ , M R ( p , F) + NR(~, , p , F )
+i_---p[ 2 rr .I ds
M(s,t;p,F)dt,
NR(v, p, F ) = N(u, p , F ) + NoR (~, , p ) , O~
2
NOR(V, p ) = No(v, p) + - ~ d t ln(3,met0)
-1
.
(4.57)
N o t e that if we calculate the mass operator in a constant field in the e 2 approximation and write it in the form
M(p,F)=
fo°°dsf~ ~(~,,;p,F,
mZ)dt,
(4.58)
-- it o
then the function M(s, t; p, F), eq. (4.32), is connected with M(s, t; p, F, m 2) in the
I.A. Batalin et al. / QED in external constant field
461
M(s,t; p, F) = M(s,t; p, F,m 2 ~ p 2 ) .
(4.59)
following way:
Then our result for M(s, t; p, F) coincides with that of ref. [8], i.e. M(s, t; p, F) is the kernel of the mass operator in the e z approximation. To find the infrared asymptotics of the Green function one should calculate the asymptotic behaviour of baR(V; p, F ) when v ---, ~ . The terms linear in v contribute to the position of the pole of the Green function. From (4.57) and (4.54) we see that the pole of the Green function is at the point
p2 - m e2- M R ( p , F ) = O .
(4.60)
In the vacuum there occurs, besides, the enhancement of the pole near the point p2 = m e2 [16,17, 20] owing to the fact that the function NoR(v; p) behaves logarithmically when v ~ ~ , i.e. ~a (3 - dr) In mZ~v
: i n a NOR(v, p ) -
Let us find the asymptotic behaviour of NR(V; p, F ) when v --, o¢. To do so we shall act in the following way: when v ~ ~ we write NR(V; p, F ) ln(m~v) lnrn~v
NR(v;p,F)
f(p,F)ln(m2eV),
=
NR(v, p, F) = N(v, p, F) +NoR(v , p ) ,
(4.61)
where
f ( p , F ) = lim ~ ~,---* OO
ONR(V; p, F)
(4.62)
0/)
Substituting expressions (4.57) and (4.50) into (4.62) we obtain
f(p,F)=
lim [ i - - v 2 (
M(v,t;p,F)dt
+-~(dt-1)Vfo°~tOg(v,t;p,F)/avdt).
(4.63)
Thus, the Green function ,--o) ~r{k~ near the point p2 -- m e2 -- MR(p, F ) = 0 has the form G(1) 2 MR(p,F)]-(X+f(P,F)). (k} - [ p 2 --me--
(4.64)
462
LA. Batalin et al. / QED in external constant fieM
If F - - 0 then Ot f ( p , O ) = ~-~ (3 - dl),
(4.65)
and we obtain the well-known results [16,20]. If F ¢ 0 , the function f ( p , F ) depends upon the parameters p2, X = - ( e F p ) 2m-6 and the invariants 6)-and 9. As the function f ( p , F ) is linear in the constant a it is necessary to substitute p2 ~ me2 near the pole (4.60), so as to remain within the accuracy. Consider the crossed field ( ~ = ~ = 0). In this case f ( p , F ) depends upon one parameter X and we have
f(p, F) =
~---~(3 - dr), O/ 1-~(29 + d/),
X << 1
(4.66)
x>>l
For constant magnetic field (9 = 0, oy< 0) if - ~3/m6e 2 << 1, X >> - ~/m4e2, the results of the calculations for the function coincide with those for the crossed field.
Appendix Here we find the eigenfunctions of Klein-Gordon's equation in a constant and uniform external field A~Xt(x)= - ~F~,~x. 1 These functions satisfy the equations
;Xt(x)
(x) =
(x).
(A.1)
Consider isotropic orthogonal eigenvectors of the tensor F,~:
Fl,~n ~ = End,, F~=-i~C~,
F ~ " = - E~,,
F~m ~ = i~)Cm~,,
E, ~3(~= [(6~2 + ~2)1/2 -T- 6S] 1/2 .
(A.2)
The vectors (A.2) are normalized by the conditions ( n ~ ) = - ( m ~ ) = 2 . If we introduce the vectors n_+= ½(n _+ ~), m+= ½(m + ~ ) , m = ½i(~ - m) satisfying the equations
Fn += En ~: , m±-2--1,
Fro±= T g(~m :~ ,
n2±= +-1,
(n+n_)=(n±m+)=(n+m_)=(m_m+)=O
the tensor F~, and the field A~Xt(x) can be represented in the form
F~ = E(n_~,n+, - n+~,n_,) + ~(~(m_~,m+~ - m+~,m_~), A~Xt(x) = - ½ E [ n _ ~ , ( n + x ) - n + , ( n _ x ) ]
(A.3)
- ½~[m_~,(m+x)-m+~,(m_x)]. (A.4)
463
LA. Batalin et aL / Q E D in external constant field
Let us perform the gauge transformation of the potential _ ext (x) + a~+(x), A~text (x)-A~
, ( x ) = - ½E(n+x)(n_x) - ½[}C(m+x)(m_x).
(A.5)
This transformation is due to the fact that in terms of the potential A~,ext(x) it is possible to separate the variables in eq. (A.1). If we transform the functions in accord with the gauge transformation (4.5) +(k}(x) = exp(ieeo(x))•'{k}(x),
(A.6)
then the functions ~k'(~(x) satisfy the equations
~;text•~)~'ext~(x)~'~(x)--,~ \ -- n21~t~,~(~), text [~x)\ = ion,- eA;×t(x). ~r/,
(A.7)
Substitute (A.5) into (A.7). Then we have 82
(
a(n+x) 2 i O(m x
8
)2
ia(n_x) +eE(n+x) + e%(m+x)
02 + a(m+x) 2
+'{k}(x) =p2*'fk}(X ) .
(A.8)
Carrying out the separation of the variables in (A.8) we obtain ~P'{k)(X) -- N{k)exp { -- ip,(n _x) -- iP2(m_x ) -- ½02 } H,,(p)D,[~o eif"/4)r ],
t3 = ( l e l ~ ) l / Z [ ( m + x ) + p 2 / e % ] ,
. = -½(a +ix),
~"= (21elE)l/2[(n+x) + p l / e $ ] ,
X= [ p 2 + l e l ~ ( 2 n + l )
lelE
]
n = 0,1,2,...,
0)-~- --[-1.
(A.9) H,(O) are the Hermite polynomials, Dv[oa ei(~/4)~-] are the parabolic cylinder functions, Pl and P2 are eigenvalues of the operator integrals of motion iO/O(n_x) and iO/O(m_x), respectively, N~k ~ is a normalizing factor. Let us verify that for the functions (A.9) the orthogonality condition (2.24) does take place and find in this way the normalizing factor. Using the orthogonality condition of the Hermite
464
LA. Batalin et aL / QED in external constant field
functions we have (~b(k} ,~b(k,}) = N{.k}N(k,}(2cr)2(2eZE~3f)-1/2
×2"n!v/~8.,.,8(p1 --p~) 3 ( p 2 --p~)V,
(A.IO)
T= f_+~ d'rD~.ttoe-i("/4)'r]D,,[to'ei('~/4)'r].
(A.11)
To calculate the integral T let us make use of the integral representation for the parabolic cylinder functions [24]: D~[ o~ ei(*r/4)'r ] ----F - 1 ( _ V) exp( - ¼i¢rv -
l i'r2 }
f°°t-~- I exp( - ½it 2 - it'r } dt. "0
(A.12) As a result we obtain T=
21e15(2~)2[/-(_~)r(_ u.)]-lexp{ ¼i~r(v. _
v)} 8~,~,3(p2 _ p 2 , ) . (A.13)
If the normalizing factor N(k ~ is equal to N(k}=(2~r)
-2
~ \1/4
(~-~)
(2nn!~/-~)-l/2I"(--v) ei(~'/4, ,
(A.14)
the functions (A.9) are normalized by condition (2.24). Let us show that the functions (A.9) form a complete set in the sense of (2.25):
E°° ~, :dpadp2dp2t~ n=0 to=+1
:
(2¢r)4
(x)~b~k}(Y) (k}
]1/2
= (~-~]
fdPldp2exp{-ipl(n-z)-ip2(m-z))
X ~
°=o.°(o)..(o t )j_ Sp2dp2,
(A.15)
z= - x + y ,
j.2 = e../.,<.-:,r(-.)r(-.*)
E
D. E~ e '<~/4~ ] D.. [ o~e -'<~/4)~'],
~o=_+1
v*= -v-l,
":'=(2[e[$)l/2[(n+y)+pl/eE],
p' = [lei'X, ]l/2[ ( m +y ) + p2/e * ] ,
(A.16)
I.A. Batalin et al. / QED in external constant field
465
where v , ( o ) = (2"n !v~-)-1/2exp(- ~p2)Hn(O) are the Hermite functions. Let us find J +_~~ J 2p d p 2. To do this let us use the identity F ( - u ) F ( - v * ) = -rr/sin(rrv) and introduce the new variable v = - ½ - ½i[p2 + l e [ ~ ( 2 n + 1)]/lel$" Then
f+=42de2=+ i21e[t9 )
f -1/2+i~
-~
J,=tr
exp(-~i~r(v + ~)} sin (cry)
E
-1/2-ic~
J, dp,
(A.17)
D~[°~ei(~r/4)'r]O-v-l[°ae-iOr/4)'r'] • (A.18)
~o=+1
The integral over v is calculated with the help of the Cherry formula [24] fcC+i~ e i(~r/2)(v+l/2) ic~
~
sin(~r u)
-
D~[a~ei("/4)r]D-~-t[°ae-'('~/4)z']
w=_+l
=
-4~riS(r-r'),
-1
(A.19)
Thus, choosing c = - ½ in (17) we obtain f_+ °~Jpzdp 2 = - ( 2 le[$ )t/2(2 ~r)28( n +z ).
(A.20)
We see that this expression does not depend on the quantum numbers. Therefore one may integrate over P2 and sum over n in (A.15) using the condition of completeness of the Hermite functions: oc
E v,(O)v,(o') = 8(0 - O') = (lel%)-X/ES(m+z).
(A.21)
n=O
After integrating over Pl we obtain the condition of completeness (2.25):
E ~{k}(x)+'~k}(Y)
= 8(4)( x - Y ) -
(A.22)
{k} To calculate the integral
Jk,k,(~) = f d4xd4y+Tk}(x)exp{~iexFy- iz52(r)z} ~,{~,}(y),
(A.23)
which occurs when calculating ((Ax,x))k,k , let us use the method of refs. [12,13]. Using this method we obtain
Jk,k'(52) = (4rr)2[det( 16522 - e2F2)] - X/4exp( - i p x ( r ) p } 8{k},{k,} , 452 -er x ( F ) = (2eV)-I In 452 +~"
(A.24)
LA. Batalin et al. / QED in external constant field
466
Proceeding from this expression we shall calculate the integral occurring in (2.24):
Jk,k'; ,1~( fa) = i OJk'k'( af~, #I2) X
iJk,k'( ~2)
0
-- ( -ipx(F)p 0$2~
- ¼ Sp ln(16122 - eEF2)}
(A.25)
Let ~(I2) be an arbitrary function of ~2 which can be expanded as a series in ~2:
n~0
•
Let us find the derivative of ~,(Ia) with respect to ~2~B: '(A.26) Here we make use of the relation n
n--i
O~l~'
E
012"t~
k=o
k
n-k-1
Expression (A.25) contains derivatives of two types: OSp °Y(~2)/OIa~a. Using (A.26) we have
(A.27)
OpC3(fa)p/O~2 "B and
O .-1
0 pOy(fa)p= O~aJ8
n! n=l
•
(pgak),(la"-k-lp)/~.
(A.28)
k=0
But the expressions (A.28) are included in (3.24) together with the factor K = A-2Tl/sh2(eFp). Then we obtain finally from (A.28) K~/~ O S p ~ ( I a ) = S p [ K ~ ' ( I 2 ) ] 0~2~
K'~ o - ~ p ~ ( ~ ) p = p ~ ' ( ~2) Kp.
(A.29)
Using relations (A.29) we have
[
1
K"/~Jk,k,;,~t~(/a) = 4Jk,k,(fa) P 16122--e2F 2p - iSp(2KJ2)/(16122 - e2F 2) . (A.30)
LA. Batalin et al. / QED in external constant field
467
Substituting the expressions
~2(F)=L(p),
~2(F)=L(v)+
~.IA - 1 8shZ(eFu) ,
12(F)=L(p)+
1 4--t '
i n s t e a d o f ~ 2 ( F ) i n t o (A.24) w e o b t a i n t h e r e l a t i o n s (3.30).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
V.I. Ritus, Proc. P.N. Lebedev Physical Institute, vol. 111 (1979) A.I. Nikishov, Proc. P.N. Lebedev Physical Institute, vol. 111 (1979) I.A. Batalin and A.E. Shabad, ZhETF 60 (1971) 894 W.-Y. Tsai, Phys. Rev. D8 (1973) 3446 W.-Y. Tsai, Phys. Rev. D8 (1973) 3460 A.A. Sokolov and I.M. Ternov, Relativistic electron (Nauka, Moscow, 1974) V.N. Baler, V.M. Katkov and V.S. Fadin, Radiation of relativistic electron (Atomizdat, Moscow, 1973) V.N. Baler, V.M. Katkov and V.M. Strakhovenko, ZhETF 67 (1974) 453 V.N. Baler, V.M. Katkov and V.M. Strakhovenko, ZhETF 68 (1975) 405 V.I. Ritus, Ann. Phys. 69 (1972) 555 V.I. Ritus, Problems of theoretical physics (Nauka, Moscow, 1972) V.I. Ritus, Pisma ZhETF 12 (1970) 416 V.I. Ritus, ZhETF 75 (1978) 1560 J.J. Volfengaut, S.P. Gavrilov, D.M. Gitman and Sh.M. Shvartsman, Yad. Fiz. 33 (1981) 743 E.S. Fradkin ZhETF 29 (1955) 258 E.S. Fradkin Proc. P.N. Lebedev Physical Institute, vol. 29 (1965) E.S. Fradkin Nucl. Phys. 76 (1966) 588 E.S. Fradkin Dokl. Akad. Nauk SSSR, 98 (1954) 47 E.S. Fradkin Dokl. Akad. Nauk SSSR, 100 (1954) 897 E.S. Fradkin Acta Phys. Hung. XIX (1965) 1975 G.A. Milekhin and E.S. Fradkin, ZhETF 46 (1963) 1926 A.S. Babichenco and A.E. Shabad, preprint PhlAN N 127 (1984) I.A. Batalin and E.S. Fradkin, Teor. Mat. Fiz. 5 (1970) 190 H. Bateman and A. Erdelyi, Higher transcendental functions, vol. 2 (1953)