X iv : h ep - p h / 97 06 37 2 v 1 1 6 Ju n 19 97 Reduction of the wave packet of a relativistic charged particle by emission of a photon

The problem of reduction of the wave packet of a relativistic charged particle by emission of a photon is studied with help of the path integral approach. A general expression for arbitrary order correlation function of the electromagnetic field is obtained. As a specific example an ultrarelativistic electron circulating in a storage ring is considered. It is shown that the longitudinal width of the electron wave packet defined by characteristic difference of time between registrations of two photons emitted by the electron is of the order of the wave length of the photons. 1 Introduction When detecting a photon emitted by the charged particle we measure the coordinate of the particle. Following the general principles of quantum measurement theory it results to the reduction of the wave packet of the particle. However in our case the reduction of the wave packet occurs in the process of emission of the photon and not in the process of measurement, because the photodetector may be placed sufficiently far from the particle path. The goal of the paper is to study which way the reduction of the wave packet of the particle occurs and to calculate the scale of the localization of the wave packet. The technique used in the paper is the functional integral formulation of relativistic quantum mechanics. Within this approach the interaction of the particle with quantum electromagnetic field can be described with the help of the " influence functional " technique of Feynman and Vernon [1, 2]. 2 Correlation function of electromagnetic field We consider a relativistic charged particle moving in external classical potentials and interacting with quantum electromagnetic field. The state of the system is described by a wave function which is the function of the coordinate of the particle q and the functional of variables of quantum electromagnetic field Ψ(q, t, {A}) ≡ q, A|Ψ. (1) The wave function is the projection of the state-vector of the system |Ψ onto the state of particle with given coordinate q| and the state of electromagnetic field – eigenstate of the field operatorˆA(x) with the eigenvalue A(x): A|ˆA(x) = A|A(x) .