Quantum electrodynamics in the squeezed vacuum state: Electron mass shift

Volkmar Putz and Karl Svozil,
Institut für Theoretische Physik
Technische Universität Wien
Wiedner Hauptstraß e 8-10/136
A-1040 Vienna, Austria
svozil@tuwien.ac.at

Abstract

The squeezed vacuum is a fascinating nonclassical state of the quantized electromagnetic field [1]. Just as for the finite temperature case, the squeezed vacuum is populated by photons. Therefore, the scattering matrix, and in particular renormalization, has to be re-evaluated with these finite ground state photons in mind.

The dependence of the scattering matrix on the vacuum state of the theory and on exterior parameters has been studied previously for the thermal equilibrium [2], in cavity-quantum electrodynamics [3], on fractal space-time support [4] and, to some extent, in the presence of strong electromagnetic fields [5,6]. Here, quantum electrodynamics is investigated in the presence of squeezed vacuum fluctuations [7]; i.e., fluctuations with reduced noise in amplitude or phase.

At first we shall calculate the scattering matrix by Taylor-expansion up to second order of e². Let |iñ = a_e^(r)f([q\vec])|svñ be the initial state, r the incoming electron's spin, [q\vec] its momentum and |svñ the squeezed vacuum state. |svñ is a pure photonic state and behaves like an ordinary Fock-vacuum regarding the electron creation and annihilation operators. The final state is áf| = ásv|a_e^(r¢)([q\vec]¢). It is important to remark that the initial squeezed vacuum state will be assumed to be the same as the final one. Hence, in this approximation, |svñ is time independent.

The scattering matrix is given by

áf|S|iñ = áf|T eiòd4xLW|iñ,

where L_W stands for the interaction-term of the Lagrange-density, which, in QED is is given by L_W = -e:[`(y)]\rlap/Ay:. The expansion of S with respect to e is

S » 1 - ie ó õ d4x: y   (x) \rlap/A(x)y(x): +        (-ie)2 (2!) ó õ ó õ d4xd4yT[: y   (x) \rlap/A(x)y(x): : y   (y)(\rlap/A(y)y(y):]

We shall discuss the first three terms in the series expansion in e next. We find

O(e0): áf|1|iñ = d3( ® q   - ® q   ¢)drr¢ O(e1): áf|(-ie ó õ d4x: y   (x)\rlap/A(x)y(x):) |iñ = 0

A_m(x) contains a term with exactly one annihilation operator and a term with exactly one creation operator, so that áf|a^(f)|iñ = 0. The well known relations ásv|a|svñ = 0 and ásv|a^f|svñ = 0 hold.

O(e²): The electron- and photon-operators do not act on each other. Hence they commute and |svñ is a normal Fock-vacuum for the electron operators. Therefore it is possible to completely separate the electron and photon terms

áf | (-ie)2 (2!) ó õ ó õ d4xd4y T [: y   (x)\rlap/A(x)y(x):: y   (y)\rlap/A(y)y(y):] |i ñ = -e2 2 ó õ ó õ d4xd4yTásv |Am(x)An(y)|svñTá0 |ae(r¢)( ® q   ¢): y   (x)gm y(x):: y   (y)gn y(y):ae(r)f( ® q   )|0ñ

The electron term is given by

d( ® q   - ® q   ¢)drr¢á0|: y   (x)gm y(x):: y   (y)gn y(y):|0ñ disconnected  + + eiq¢x (2p)3/2 Ö 2[`q]¢0 u   (r¢)   gmiSc(x-y)gn e-iqy (2p)3/2 Ö 2[`q]0 u(r) +(x« y,m « n) connected

As usual, the disconnected-term is regarded as nonphysical.

The calculation demonstrated that effectively it would have been possible to build up the whole 2nd-order term by just replacing the usual photon propagator in the Feynman-rules by

iDmn(x-y) = ásv|T[Am(x)An(y)]|svñ.

This expression can be evaluated as follows:

ásv|T[Am (x)An (y)]|svñ = = 1 (2p)3 ó õ ó õ d3kd3k¢ 2(Ek Ek¢)1/2 ásv|q(x0 - y0)[e(r)m( ® k   ) a-r( ® k   )e(l)n( ® k   ¢)afl ( ® k   ¢)e-i(kx-k¢y) + e(r)m( ® k   )afr( ® k   ) e(l)n( ® k   ¢)a-l ( ® k   ¢)ei(kx-k¢y)] + (x « y)|svñ,

where

ásv|af r( ® k   )a-l( ® k   ¢)|sv ñ = -grld3( ® k   - ® k   ¢)n(k), [ a-r( ® k   ) , a f l( ® k   ¢)] = -grld3( ® k   - ® k   ¢), grle(r)m( ® k   )e(l)n( ® k   ) = gmn.

Then,

ásv|T[Am (x)An (y)]|svñ = = -gmn ó õ d3k (2p)3 2Ek [q(x0-y0)e-ik(x-y)+ q(y0-x0)eik(x-y)] - gmn ó õ d3k (2p)3 2Ek × n(k) [q(x0-y0)e-ik(x-y) + q(x0-y0)eik(x-y) + q(y0-x0)e-ik(y-x) + q(y0-x0)eik(y-x)] [eik(x-y) + e-ik(x-y)]  .

Hence we obtain

iDmn(x-y) = ásv|T[Am (x)An (y)]| svñ = -gmn 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)

Notice, as remarked above, 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 within the appropriate spatial and temporal ranges.) The propagator can be rewritten using contour-integral techniques

iDmn(x-y) = i ó õ d4k (2p)4 e-ik(x-y)Dmn (k) iDmn(k) = -igmn 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(E_k -W), where E_k 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 g_m 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.

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

-iS(p;W,s) = ó õ d4k (2p)4 [iDmn(k;W,s)](-ie)gm i \rlap/p-\rlap/k-m (-ie)gn     .

(3)

The physical mass is interpreted 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 nonsqueezed 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 | \rlap/p = m = ó õ d4kd(k2)n(k)gm \rlap/p-\rlap/k+m (p-k)2 -m2+ie gm|\rlap/p = m = -2 ó õ d4kd(k2)n(k) \rlap/k+m 2pk-ie |\rlap/p = m = - ó õ d3 ® k   dk0[ d(k0-| ® k   |) |2k0| + d(k0+| ® k   |) |2k0| ]n(k) k0g0- ® k   ® g   + m k0 p0- ® k   ® p   -ie |\rlap/p = m .

As the epsilon is not needed, it will be dropped.

= - ó õ d3k n(| ® k   |)[ | ® k   |g0- ® k   ® g   +2m 2 | ® k   |(| ® k   |- ® k   ® p   ) + -| ® k   |g0- ® k   ® g   + 2m 2 | ® k   |(-| ® k   |p0- ® k   ® p   ) [k\vec] ® -k  ]|\rlap/p = m = - ó õ d3k n(| ® k   |)[ | ® k   |g0- ® k   ® g   | ® k   |(| ® k   |p0- ® k   ® p   ) |\rlap/p = m = - ó õ d3k n(| ® k   |) kmgm | ® k   |(pk) |\rlap/p = m, e.o.m.: k2 = 0 = a 2p2 Im (p)pm m |p2 = m2    , (5)

where d(k²) = d(k⁰-|[k\vec]|)/2k⁰ + d(k⁰+|[k\vec]|)/2k⁰ and Gordon's identity, which reduces to g_m = p_m /m (remind p_m g^m = m, p² = m²), have 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) |e.o.m.: k2 = 0     .

(6)

In the rest frame of the squeezed vacuum 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).

One has to bear in mind that the above calculation did not take into explicit account the spatial 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.

However, even the above rather simple model calculations suggest that physical parameters such as electron mass, charge and magnetic moment dependent on external conditions. The squeezed vacuum is arguably the simplest theoretically treatable yet experimentally realizable state. Here, we have just evaluated the electron-mass-shift. Measuring the renormalization effects on the electron mass due to the squeezed vacuum is certainly a challenging yet difficult task beyond the scope of this presentation. Calculations of charge-shift and of corrections to the magnetic moment will be presented in a forthcoming paper.

References

[1]

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

[2]

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

[3]

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).

[4]

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).

[5]

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

[6]

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

[7]

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

[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 generalization 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.