Quantum-field-theoretical approach to phase-space techniques: Generalizing the positive-Prepresentation

Abstract

We present an introduction to phase-space techniques ͑PST͒ based on a quantum-field-theoretical ͑QFT͒ 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 doubling͒ and hence fail to produce a stochastic differential equation ͑SDE͒, we show how the system in question may be approximated via stochastic difference equations (S⌬E). Second, we show that introducing sources into the SDE's ͑or S⌬E's͒ generalizes them to a full quantum nonlinear stochastic response problem ͑thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory͒. 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.

Causal signal transmission by quantum fields. I: Response of the harmonic oscillator

“…endnote [19]. In (18), D R is, up to overall sign, the retarded Green's function of the c-number equation for the displacement of a classical oscillator under the influence of an external force f (t) = −j(t)…”

Numerical representation of quantum states in the positive-P and Wigner representations

“…and comparison with (39) shows that the Gaussian approximation is exact for the mean and variance, and accurate to O(1/N 2 ) for m = 3. It is easily shown by induction on m that the exact moments satisfy…”

“…The independence of the variables can cause serious stability problems with the numerical integration, but for problems where the integration converges, the positive-P representation is an extremely powerful theoretical tool [38]. As a final remark, we note that a method has been developed for mapping Hamiltonians which would give higher than second order derivatives in a generalised Fokker-Planck equation onto stochastic difference equations [39], which is useful for analysing processes such as third harmonic generation [40] and others which go beyond the common three and four-wave mixing processes of quantum optics and trapped ultra-cold gases.…”

Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea

“…Papers of the present authors [40][41][42] aside, the only work we are aware of, that can be seen as a predecessor of this investigation, is the article of Aurenche and Becherrawy [28]. These authors replace the Keldysh rotation by diagonalisation of the 2 × 2 matrix propagator.…”

“…This transforms the Perel-Keldysh series into a causal (Wyld) diagram series [42,73], with propagator D R and three vertices:…”