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.
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.
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.
Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Abstract Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. As its title suggests, this book presents the phase space theory approach to treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single-particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phase space variables which replace mode annihilation and creation operators, the dynamical equation for the density operator determines a Fokker–Planck equation for the distribution function, and measurable quantities such as quantum correlation functions are given as phase space integrals. Finally, the phase space variables are replaced by time-dependent stochastic variables satisfying Langevin stochastic equations obtained from the Fokker–Planck equation, with stochastic averages giving the measurable quantities. Second, a quantum field approach is then treated, the density operator being represented by a distribution functional of field functions which replace field annihilation and creation operators, the distribution functional satisfying a functional Fokker–Planck equation, etc. A novel feature of this book is that the phase space variables for fermions are Grassmann variables, not c-numbers. However, the book shows that Grassmann distribution functions and functionals still provide equations for obtaining both analytic and numerical solutions.
Abstract This chapter deals with deriving functional Fokker–Planck equations (FFPEs) that govern the behaviour of phase space distribution functionals (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by field annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functionals, these being functional differentiations and/or multiplications by field functions (left or right for fermions). Derivations are carried out by applying mode expansions to previously established results for distribution functions. Specific forms are obtained for FFPEs relating drift vector and diffusion matrix elements in the FFPE to corresponding quantities in equivalent Fokker–Planck equations and to sums over mode functions. Symmetry (bosons) and antisymmetry (fermions) properties of the diffusion matrix are established. Results are generalised to treat cases with several sets of field operators.
In this chapter, we will develop methods for mapping operator equations to equivalent c-number equations. This results in a continuous phase-space representation of a many-body quantum system, using phase-space distributions instead of density matrices. In order to do this, we will first find c-number representations of operators. We have already seen one such representation, the Glauber–Sudarshan P -representation for the density matrix. We now introduce several more such representations. The main focus will be on the truncated Wigner representation, valid at large photon number, and the positive P -representation, which uses a double-dimensional phase space and exists as a positive probability for all quantum density matrices. Phase-space techniques have a great advantage over conventional matrix-type solutions to the Schrödinger equation, in that they do not have an exponential growth in complexity with mode and particle number. Instead, the equations that describe the dynamics of these c-number representations of the density matrix are Fokker–Planck equations. These have equivalent stochastic differential equations, which behave as c-number analogs of Heisenberg-picture equations of motion. This means that problems that would be essentially impossible to solve using conventional number-state representations can be transformed into readily soluble differential equations. In many cases, no additional approximations, such as perturbation theory or factorization assumptions, are needed.