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.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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsNov 15, 19915 citations

On the description of noise in quantum systems

Richard Bausch, Á. Stáhl

The European Physical Journal AFeb 1, 19679 citations

The Free Stochastic Limit of Interacting Quantum Fields

John Edward Gough

The problem of obtaining the quantum stochastic limit for a system interacting with a reservoir is investigated when both are considered as quantum fields. The pre-limit fields are as encountered in standard quantum field theory and in particular we do not resort to the simplifying responseless interaction for the reservoir field. In the separation of time scales between the unperturbed (slow) and interaction (fast) dynamics we find a new quantum stochastic noise emerging which describes multi-particle scatterings and has extended free statistics. We give explicit diagrammatic rules to compute the 2n-point correlators of the noise.The problem of obtaining the quantum stochastic limit for a system interacting with a reservoir is investigated when both are considered as quantum fields. The pre-limit fields are as encountered in standard quantum field theory and in particular we do not resort to the simplifying responseless interaction for the reservoir field. In the separation of time scales between the unperturbed (slow) and interaction (fast) dynamics we find a new quantum stochastic noise emerging which describes multi-particle scatterings and has extended free statistics. We give explicit diagrammatic rules to compute the 2n-point correlators of the noise.

Infinite Dimensional Analysis Quantum Probability and Related TopicsJul 1, 19983 citations

Hydrodynamic theory of scrambling in chaotic long-range interacting systems

Tianci Zhou, Andrew Y. Guo, Shenglong Xu +2 more

The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a mean-field theory of out-of-time-ordered commutators in locally interacting quantum chaotic systems at high energy density. In systems with power-law interactions, the corresponding fractional-derivative FKPP equation provides an analogous mean-field theory. However, the fractional FKPP description is potentially subject to strong quantum fluctuation effects, so it is not clear a priori if it provides a suitable effective description for generic chaotic systems with power-law interactions. Here we study this problem using a model of coupled quantum dots with interactions decaying as $1/{r}^{\ensuremath{\alpha}}$, where each dot hosts $N$ degrees of freedom. The large- $N$ limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by $1/N$ corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for $N=1$ and generic finite $N$ . In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a mean-field theory of out-of-time-ordered commutators in locally interacting quantum chaotic systems at high energy density. In systems with power-law interactions, the corresponding fractional-derivative FKPP equation provides an analogous mean-field theory. However, the fractional FKPP description is potentially subject to strong quantum fluctuation effects, so it is not clear a priori if it provides a suitable effective description for generic chaotic systems with power-law interactions. Here we study this problem using a model of coupled quantum dots with interactions decaying as $1/{r}^{\ensuremath{\alpha}}$, where each dot hosts $N$ degrees of freedom. The large-$N$ limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by $1/N$ corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for $N=1$ and generic finite $N$. In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.

arXiv:2208.01649Physical review. B./Physical review. BJan 3, 202310 citations

N

degrees of freedom. The large-

N

limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by

1/N

corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for

N=1

and generic finite

N

. In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.