Quantum electrodynamics in the squeezed vacuum state: Electron mass shift
Volkmar Putz, Karl Svozil · 2004
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 δm(Ω, s)/m = α(2/π)(Ω/m) 2 sinh 2 (s), where Ω and s stand for the mode frequency and the squeeze parameter and α for the fine structure constant, respectively. 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 |i 〉 = a (r)† e (⃗q)|sv 〉 be the initial state, r the incoming electron’s spin, ⃗q its momentum and |sv 〉 the squeezed vacuum state. |sv 〉 is a pure photonic state and behaves like an ordinary Fock-vacuum regarding the electron creation and annihilation operators. The final state is 〈f | = 〈sv|a (r ′) e (⃗q ′). It is important to remark that the initial squeezed vacuum state will be assumed 1 to be the same as the final one. Hence, in this approximation, |sv 〉 is time independent. The scattering matrix is given by 〈f|S|i〉 = 〈f|Te i