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Journal of Mathematical Physics

A New Phase-Space Distribution Function in the Statistical Theory of the Electromagnetic Field

Yutaka Kano

Published Dec 1, 1965• 261 citations
Publisher

Abstract

In a previous paper a certain new probability distribution function q(z) relating to blackbody radiation was introduced. In the present paper the properties of this function for a general radiation field are studied. Unlike the phase-space distribution function of Sudarshan (1963), this function is nonnegative and is an ordinary function. A series expansion for q(z) is given, and it is shown that the series is absolutely convergent for all eigenvalues z of the destruction operator. It is also shown that the density matrix in the Fock representation can be uniquely determined from this probability distribution function, and vice versa. The relation between q(z) and the Sudarshan's phase-space distribution function is discussed.

DOI
10.1063/1.1704739

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•Nov 13, 2014•1 citation
In a previous paper a certain new probability distribution function q(z) relating to blackbody radiation was introduced. In the present paper the properties of this function for a general radiation field are studied. Unlike the phase-space distribution function of Sudarshan (1963), this function is nonnegative and is an ordinary function. A series expansion for q(z) is given, and it is shown that the series is absolutely convergent for all eigenvalues z of the destruction operator. It is also shown that the density matrix in the Fock representation can be uniquely determined from this probability distribution function, and vice versa. The relation between q(z) and the Sudarshan's phase-space distribution function is discussed.
Statistical properties of blackbody radiation are discussed by the techniques of second quantization. Various probability distribution functions are derived, in particular the Bose-Einstein distribution function and its generalization for an arbitrary number of momentum-spin states. The Sudarshan phase space distribution function for blackbody radiation is obtained. Although in general the Sudarshan distribution function appears to be a generalized function, it is found that for the radiation field considered here, it becomes an ordinary function and is in agreement with expressions derived recently somewhat heuristically by other authors. A certain new probability distribution function for a statistcal radiation field is introduced and is expressed in terms of the eigenvalues of the annihilation operator. Unlike the phase space distribution function of Sudarshan, this function is non-negative. Finally an expression is derived for this probability distribution function appropriate to blackbody radiation.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.
Expressions are derived for the electromagnetic correlation tensors of blackbody radiation defined on the basis of the theory of the quantized field. Correlation functions of all order are considered, but second-order ones are discussed in detail; it is found that these are identical with those obtained on the basis of semi-classical theory in part I of this investigation. This result illustrates a recent theorem of E.C.G. Sudarshan relating to the equivalence between semiclassical and quantum mechanical description of statistical light beams.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functionals of phase space fields, leading to expressions for quantum correlation functions in terms of phase space functional integrals, where the field functions—complex fields for bosons, Grassmann fields for fermions—replace field annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functionals, with distinct fields for annihilation and creation field operators; however, symmetrically ordered (Wigner type) distribution functionals are also considered, along with unnormalised B distribution functionals, which lead to phase space functional integrals for Fock position state populations and coherences. Characteristic functionals are first defined and then shown to be related to distribution functionals via phase space functional integrals. The existence of the distribution functionals is demonstrated from previous results for equivalent distribution functions by using mode expansions.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto ccc-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.