White noise and heating of quantum field theory in an open system

Abstract

The time evolution of the density matrix of a relativistic quantum field theory coupled to random noise is discussed. Using path-integral techniques, we solve analytically for the density matrix of a system of free fields coupled to noise. We show that in the limit of large time the density matrix evolves to a thermal state and we compute the temperature of the state in terms of the characteristics of the noise. While the noise is coupled the temperature increases steadily in time. Also, it is necessary to choose a specific spectrum for the noise in order that all of the modes of the free field theory come to the same temperature after some given time. We then present an alternative derivation of these results using time-path techniques. In this case we consider a system where noise couples for a finite time interval and find that the propagator interpolates between finite-temperature field-theory propagators with the temperatures of the system at initial and final times. We argue that this can be used as a model of heating in an interacting field theory.

Nonclassical time correlation functions in continuous quantum measurement

“…As always, the signal-to-noise ratio can be improved by repeated measurements. The fact that the backaction contribution (50) to the noise grows with time reflects the continuous pumping of energy to the system affected by the measurement process [35]. This does not happen in the two-level system because of its bounded spectrum; still also in that system the temperature grows to infinity (ρ(t) → (1/2)1) as implied by equations (35)-(37).…”

Classical fields near thermal equilibrium

“…In particular, the noise terms will continuously 'heat' the system (for field theories see ref. [41]) whereas the dissipative part of the response function counteracts. Equilibrium is achieved when the system has thermalized to the temperature dictated by the bath.…”

“…For a detailed discussion and interpretation of such discontinuities in finite temperature field theory we refer the reader to the article by Weldon [34]. From the expressions (41)(42)(43) we see that each kernel is composed of two contributions: the direct part Γ d (k, ω), or the "loss" term, and the inverse part Γ i (k, ω), or "gain" term. Following [34], we write…”

“…One notices that the contributions with an odd power in φ contain the on-shell reaction rates (41)(42)(43). Thus we expect them to describe the dissipative aspects of the equation of motion for the soft modes as energy is transferred by scattering (or production) processes between the soft and hard modes.…”

Environmental gravitational decoherence and a tensor noise model

“…Although the relation (21) has not been derived from fundamental physics and although it is not clear yet, whether an intrinsic decoherence exists in nature at all [18], we want to address the question, whether this decoherence rate can be derived in some limit from a noise model. Scalar noise models have been studied in the literature, see, for example [19]. However, we want to couple the energy-momentum tensor to the noise which requires a tensor random current.…”