Probability Representation in Quantum Field Theory

The recently proposed probability representation of quantum mechanics is generalized to quantum field theory. We introduce a probability distribution functional for field configurations and find an evolution equation for such a distribution. The connection to the time-dependent generating functional of Green's functions is elucidated and the classical limit is discussed. The use of statistical methods for describing quantum physics gives the opportunity to describe classical and quantum phenomena in a unified approach. Since the beginning of quantum mechanics there have been attempts to understand its nature in a classical-like context , namely to describe quantum states in terms of a classical distribution of probability. It is this philosophy which inspired the so-called quasi-probability distribution functions of Wigner, Husimi, Glauber and Sudarshan [1]. The original goal was not completely achieved (the above distribution functions are not always positive defined or they do not describe measurable variables) until Cahill and Glauber in [2] introduced a class of distribution functions, known as marginal distribution functions (MDF), which enjoyed all the properties of a density of probability. Nevertheless, it was realized only recently [3] that quantum mechanics could be described entirely in terms of a distribution of such a family, suitably defined for a random variable , which we will specify below. In [3] a consistent scheme has been proposed, the so-called probability representation, which has been shown to be completely equivalent to the ordinary formulation. Quantum states are described by a distribution of probability, the MDF, and the time evolution by an integro-differential equation for the MDF. Invertible relations have been established between the MDF and the density matrix [3, 4] and between the Green's functions of the related evolution equations [5].