Quantum electrodynamics in the squeezed vacuum state: Electron mass and anomalous magnetic moment

K. Svozil
Institut für Theoretische Physik
Technische Universität Wien
Wiedner Hauptstraß e 8-10/136
A-1040 Vienna, Austria
e1360dab@awiuni11.edvz.univie.ac.at

Abstract

Due to the nonvanishing average photon population of the squeezed vacuum state, finite corrections to the scattering matrix are obtained. The lowest order contribution to the electron mass shift for a one mode squeezed vacuum state is given by dm(W,s)/m = a(2/p)(W/m)2sinh2(s), where W and s stand for the mode frequency and the squeeze parameter and a for the fine structure constant, respectively. The correction to the anomalous magnetic moment of the electron is dae(s) = -(4a/p)sinh2(s).

The dependece of the scattering matrix on the vacuum state of the theory and on exterior parameters has been studied for the thermal equilibrium [1], in cavity-quantum electrodynamics [2], on fractal space-time support [3] and, to some extent, in the presence of strong electromagnetic fields [4,5]. Here, quantum electrodynamics is investigated in the presence of squeezed vacuum fluctuations [6]; i.e., fluctuations with reduced noise in amplitude or phase.

A caveat: the following derivation is heuristic. In particular, no attempt will be made to derive the photon propagator from first principles. Rathe it is assumed the squeezed vacuum state [7] exhibits a nonvanishing average photon density proportional to sinh2(s) per squeezed mode, where s is the squeeze parameter [8]. This can be accounted for in the perturbation series by the introduction of a causal photon propagator as follows [9]. Denote the squeezed vacuum by |svñ. The photon propagator in the Landau gauge is

Dmn(x-y)
=
-iásv|T[Am (x)An (y)]| svñ
=
igmn ó
õ 		d3k
(2p)3
 dk¢3
2(EkEk¢)1/2
 ásv|q(x0-y0)[e-i(k x -k¢y)akafk¢+ei(k x-k¢ y)afkak¢]+ x« y |svñ
=
igmn ì
í
î 		ó
õ 		d3k
(2p)3
 1
2Ek
 [q (x0-y0)e-ik (x-y)+q(y0-x0)eik (x-y)]+
              + ó
õ 		d3k
(2p)3
 1
2Ek
 n(k)[eik (x-y)+e-ik (x-y)] ü
ý
þ 		,
(1)
where the aaf terms generate the usual causal propagator while the af a terms count the particle density in the squeezed vacuum. Notice, however, that by defining the photon propagator, the squeezed vacuum state had to be assumed ``quasi-stationary,'' otherwise the final state of the vacuum cannot be identified with the initial state. This assumption can be justified only in the appropriate spacial and temporal ranges. The propagator can be rewritten using contour-integral techniques
Dmn(x-y)
=
ó
õ 		d4k
(2p)4
 e-ik(x-y)Dmn (k)
Dmn(k)
=
-gmn é
ê
ë 		1
k2+ie
 -2pid(k2)n(k) ù
ú
û 		    .
(2)
For the one mode squeezed state,

n(k;W,s) = Wsinh2(s)d(Ek -W), where Ek is the photon energy parameter and W and s stand for the frequency of the squeezed mode and the squeezing parameter, respectively. The electron propagator S(p) = 1/(\rlap/p -m+ie), as well as the bare vertex gm remain unchanged. Notice however that a preferred frame of reference has been introduced due to the noncovariant choice of the density n(k;W,s), i.e., the one at rest with respect to the squeezed vacuum. The resulting breakdown of Lorentz invariance necessitates a careful interpretation of the usual renormalisation prescriptions.

In what follows, the lowest order correction to the radiative mass of the electron will be calculated. This can be done by evaluating the second order contribution to the self energy of the electron

S(p;W,s) = -ie2 ó
õ 		d4k
(2p)4
 [iDmn(k;W,s)]gm i
\rlap/p-\rlap/k-m
 gn     .
(3)
The physical mass is interpreted as usual as the pole of the renormalized electron propagator. For dm(W,s) << m,

m(W,s)
»
m+dm+S(p;W ,s)|\rlap/p = m
       = m+dm+S(p;s = 0)|\rlap/p = m+dS(p;W,s)|\rlap/p = m
       = m+dm(W,s)     ,
(4)
where m stands for the renormalized unsqueezed mass of the electron.

The correction term dm(W,s) = dS(p;W,s)|\rlap/p = m due to squeezing adds up coherently to the renormalization contributions of m. Its explicit form is given by

dm(W,s)
=
- e2
(2p)3
 ó
õ 		d4k d(k2)n(k;W,s)gm \rlap/p-\rlap/k+m
(p-k)2-m2+ie
 gm \mid\rlap/p = m
=
a
2p2
 Im (p)pm
m
 |p2 = m2    ,
(5)
where Gordon's identity which reduces to gm = pm /m has been used,

a = e2/4p stands for the fine structure constant and

Im (p) = ó
õ 		d3 ®
k
 
 km
| ®
k
 
 |(pk)
 n(| ®
k
 
 |;W,s)    .
(6)
In the rest frame of the electron this expression can be evaluated, yielding
dm(W,s)/ m = a(2 / p)(W/ m)2sinh2(s)    .
(7)
For optical frequencies, dm(s)/m » 10-13sinh2(s).

The correction dae to the anomalous magnetic moment of the electron ae can be extracted from a decomposition of the vertex function Lm = gm+Gm on shell

_
u
 
 (p-q)Lm u(p) = _
u
 
 (p-q) é
ê
ë 		gm f1(q2)+ i
2m
 smnqn f2(q2) ù
ú
û 		u(p)
(8)
with ae = f2(q2 = 0). To lowest order one obtains
Gm (p,p-q;W,s)
=
Gm (p,p-q;s = 0)+d Gm (p,p-q;W,s)
=
(-ie)2 ó
õ 		d4k
(2p )4
 [iDab(k;W,s)]ga i
\rlap/p-\rlap/q-\rlap/k-m
 gm i
\rlap/p-\rlap/k-m
 gb .
(9)
The correction due to squeezing for p = (m,0,0,0) and q = (q0,0,0,0), q02 << m2 can be written as
d Gm (p,p;W,s)
=
-(ia/2p2)(m2Igm -Imngn )
(10)
I
=
ó
õ 		d4k d(k2)n(k;W,s)
(pk)2
(11)
Imn
=
ó
õ 		d4k km kn d(k2)n(k;W,s)
(pk)2
     .
(12)
For a definition of ae the term proportional to smnqn is chosen. Hence, only the first term proportional to Igm on the right hand side of (10) is relevant. Using Gordon's identity, which is gm = (1/2m)(2pm -qm -ismnqn) here, one obtains for the correction to the anomalous magnetic moment of the electron
dae(s) = -(4a/p)sinh2(s) .
(13)

This correction to the anomalous magnetic moment of the electron becomes comparable to the unsqueezed value ae = a/2p for s » 0.35 and increases rapidly as the population of the squeezed vacuum increases. However, despite the relatively ``huge'' magnitude of the effect when compared to corrections from other sources, one has to bear in mind that the above calculation did not take into account the spacial and temporal characteristics of the squeezed vacuum states. Therefore, a more careful calculation would have to take into account the nonstationary property of the squeezed vacuum.

References

[1]
G. Barton, Annals of Physics (N.Y.) 200, 271 (1990); A. Romero, J. Math. Phys. 34, 2206 (1993).

[2]
K. Svozil, Phys. Rev. Lett. 54, 742 (1985); M. Kreuzer and K. Svozil, Phys. Rev. D34, 1429 (1986); E. Fischbach and N. Nakagawa, Phys. Rev. D30, 3320 (1984); G. Barton and N. S. J. Fawcett, Phys. Rep. 170, 1 (1988).

[3]
A. Zeilinger and K. Svozil, Phys. Rev. Lett. 54, 2553 (1985); K. Svozil and A. Zeilinger, Journal of Modern Physics A1, 971-990 (1986); K. Svozil, J. Phys. A19, L1125 (1986); ibid. A20, 3861 (1987).

[4]
W. Greiner, B. Müller and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, Berlin 1985).

[5]
for instance Issues in Intense-Field Quantum Electrodynamics, ed. by V. L. Ginzburg (Nova Science Publishers, Commack, New York, 1987).

[6]
for a calculation of corrections to the Lamb shift which used different methods, see G. J. Milburn, Phys. Rev. A34, 4882 (1986).

[7]
R. Loudon and P. L. Knight, Journal of Modern Optics 34, 709 (1987).

[8]
other squeezing parameters associated with more general canonical transformations [isomorphic to elements of the real symplectic group Sp(2N,R) for N-mode squeezing] have been set zero, but a generalisation is straightforward.

[9]
In the following, units are used such that (h/2p) = c = 1 . The notation of J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964) is adopted.

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