Speaker
Prof.
Iwo Białynicki-Birula
(Polish Academy of Sciences, Warsaw)
Description
Owing to the processes of particle creation and annihilation relativistic quantum mechanics does not exist as a consistent theory and it must be replaced by quantum field theory. Nevertheless the original construction of Wigner (and Szilard) can be extended in a natural manner to quantum field theory just by replacing wave functions by the field operators [1]. The expectation value in a one-particle state of such an object reduces in the nonrelativistc approximation to the standard Wigner function. Of course, in full quantum field theory the evolution equation for the Wigner function involves higher order functions that are built from products of more than two field operators and the hierarchy of such equations does not close. However, when the coupling to the external field plays the dominant role and the mutual interaction can be neglected, one obtains the evolution equation similar to the transport equation for the standard Wigner function that in some cases can be solved analytically. One important application of this construction is the description of electron-positron pair creation by an applied electromagnetic field. Every quantum field theory allows for the construction of the Wigner function from field operators but for the electromagnetic field (and also for other bosonic fields) there is also an entirely different construction that leads to a Wigner functional. The arguments of this functional are not points in the phase space but the electric and magnetic fields. Still, this functional shares many properties with the good old Wigner function and it enables one to learn something new about the properties of the quantized electromagnetic field. In particular, this construction gives a new insight into the statistical properties of the electromagnetic field at finite temperature.
References: [1] I. Bialynicki-Birula, P. Gornicki and J. Rafelski, Phys. Rev. D 44, 1825 (1991).
