We describe quantum-field-theoretical (QFT) techniques for mapping quantum problems onto c-number stochastic problems. This approach yields results which are identical to phase-space techniques [C.W. Gardiner, {\em Quantum Noise} (1991)] when the latter result in a Fokker-Planck equation for a corresponding pseudo-probability distribution. If phase-space techniques do not result in a Fokker-Planck equation and hence fail to produce a stochastic representation, the QFT techniques nevertheless yield stochastic difference equations in discretised time.
We describe quantum-field-theoretical (QFT) techniques for mapping quantum problems onto c-number stochastic problems. This approach yields results which are identical to phase-space techniques [C.W. Gardiner, {\em Quantum Noise} (1991)] when the latter result in a Fokker-Planck equation for a corresponding pseudo-probability distribution. If phase-space techniques do not result in a Fokker-Planck equation and hence fail to produce a stochastic representation, the QFT techniques nevertheless yield stochastic difference equations in discretised time.
We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations $(\mathrm{S}\ensuremath{\Delta}\mathrm{E}).$ Second, we show that introducing sources into the SDE's (or $\mathrm{S}\ensuremath{\Delta}\mathrm{E}\mathrm{'}\mathrm{s})$ generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.
Quasiprobability distributions emerging in the stochastic wave-function method of Carusotto et al. [Phys. Rev. A 63, 023606 (2001)] are investigated. We show that there are actually two types of quasiprobabilities. The first one, the ``diagonal Hartree-Fock state projection'' representation, is useful in representing the initial conditions for stochastic simulation in the most compact form. It defines antinormally ordered expansion of the density operator and normally ordered mapping of the observables to be averaged. We completely characterize the equivalence classes of this phase-space representation. The second quasiprobability distribution, the ``nondiagonal Hartree-Fock state projection'' representation, extends the first one in order to achieve stochastic representation of the quantum dynamics. We demonstrate how the differential identities of the stochastic ansatz generate the automorphisms of this phase-space representation. These automorphisms turn the stochastic representation into a gauge theory. The gauge transformations of the quasiprobability master equation are described. In particular, it is the analyticity of the stochastic ansatz that allows one to transform the master equation into the genuine Fokker-Planck equation. We demonstrate how the different variants of the stochastic wave-function method can be constructed, first by choosing a certain optimality criteria or constraints, and then by satisfying these criteria with a suitable choice of gauge. The problem of boundary terms is considered. It is demonstrated that the simple scheme with Fock states of Carusotto et al. is not subjected to this problem.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
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.
The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation.Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases.For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions.The solution corresponds to a c-number field equilibrating in an additional dimension.Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space.We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics.This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.
The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.