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Revista Mexicana de Física

Structure of global dynamics, star-conjugation symmetry, and stationarity in the theoretical description of closed quantum Liouvillian systems

William A. Wassam

Published Nov 1, 2002• 0 citations
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Abstract

We consider the symmetries required of quantum stochastic Liouvillian dynamical models intended to describe fluctuations about equilibrium in a closed system. Our considerations yield symmetry relations that represent necessary and sufficient conditions for the realization of the stationarity of symmetrized, anti-symmetrized, and one-sided equilibrium-averaged time correlation functions. Since stationarity is a physical requirement that must be satisfied by systems in equilibrium, these symmetry relations represent universal constraints on acceptable quantum stochastic Liouvillian dynamical models. In the classical limit, the symmetry relations reduce to the symmetry relation embodying stationarity for closed classical stochastic Liouvillian systems. [W.A. Wassam, Jr., J.A. Balderas-Lopez, and Go. Torres-Vega, J. Math. Phys. 31, 2466 (1990).] The symmetry relations are shown to lead to representation independent formulations of detailed balance and microscopic reversibility. In addition, it is shown that the symmetry relations, along with the basic structure of the global dynamics and star-conjugation symmetry, give rise to a number of generalized symmetry relations for symmetrized, anti-symmetrized, and one-sided equilibrium-averaged time correlation functions and their Fourier and Fourier-Laplace transforms. Apart from elucidating fundamental symmetries of acceptable dynamical models, the reported theory may be used to facilitate the solution of spectral and temporal problems. Also, the theory enables one to construct equilibrium solutions of dynamical models that satisfy the requirements of stationarity. Moreover, the theory may be used as a vehicle for correcting dynamical models that violate the stationarity requirement. These features are illustrated in an analysis of dynamical models for the damped harmonic oscillator and 2-state dissipative atom.

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We consider the symmetries required of quantum stochastic Liouvillian dynamical models intended to describe fluctuations about equilibrium in a closed system. Our considerations yield symmetry relations that represent necessary and sufficient conditions for the realization of the stationarity of symmetrized, anti-symmetrized, and one-sided equilibrium-averaged time correlation functions. Since stationarity is a physical requirement that must be satisfied by systems in equilibrium, these symmetry relations represent universal constraints on acceptable quantum stochastic Liouvillian dynamical models. In the classical limit, the symmetry relations reduce to the symmetry relation embodying stationarity for closed classical stochastic Liouvillian systems. [W.A. Wassam, Jr., J.A. Balderas-Lopez, and Go. Torres-Vega, J. Math. Phys. 31, 2466 (1990).] The symmetry relations are shown to lead to representation independent formulations of detailed balance and microscopic reversibility. In addition, it is shown that the symmetry relations, along with the basic structure of the global dynamics and star-conjugation symmetry, give rise to a number of generalized symmetry relations for symmetrized, anti-symmetrized, and one-sided equilibrium-averaged time correlation functions and their Fourier and Fourier-Laplace transforms. Apart from elucidating fundamental symmetries of acceptable dynamical models, the reported theory may be used to facilitate the solution of spectral and temporal problems. Also, the theory enables one to construct equilibrium solutions of dynamical models that satisfy the requirements of stationarity. Moreover, the theory may be used as a vehicle for correcting dynamical models that violate the stationarity requirement. These features are illustrated in an analysis of dynamical models for the damped harmonic oscillator and 2-state dissipative atom.
We analyze the symmetries of open dissipative models for quantum Liouvillian dynamical systems. Our analysis yields two representation independent symmetry relations. One of the symmetry relations, star-conjugation symmetry, is shown to represent a necessary and sufficient condition for ensuring the reality of observables and the Hermiticity of the system density operator. The other symmetry relation represents a necessary and sufficient condition for the realization of microscopic reversibility. These symmetry relations and the structure of the global dynamics are shown to give rise to a number of generalized symmetry relations for quantities expressible as matrix elements of the retarded, advanced, and global propagators. It is shown that the generalized symmetry relations may be used to facilitate the solution of spectral and temporal problems. This is demonstrated in a concrete way in an application entailing the use of dual Lanczos transformation theory to obtain an exact solution of a prototype model for excited state dynamics. Apart from elucidating the symmetries of open dissipative systems, the reported analysis reveals that there are two different and equally legitimate schemes for performing time-reversal transformations in the matrix formulation of quantum Liouvillian dynamics. The properties and ramifications of these time-reversal transformation schemes are discussed in detail.
We unveil the universal (model-independent) symmetry satisfied by Schwinger-Keldysh quantum field theories whenever they describe equilibrium dynamics. This is made possible by a generalization of the Schwinger-Keldysh path-integral formalism in which the physical time can be re-parametrized to arbitrary contours in the complex plane. Strong relations between correlation functions, such as the fluctuation-dissipation theorems, are derived as immediate consequences of this symmetry of equilibrium. In this view, quantum non-equilibrium dynamics – e.g. when driving with a time-dependent potential – are seen as symmetry-breaking processes. The symmetry-breaking terms of the action are identified as a measure of irreversibility, or entropy creation, defined at the level of a single quantum trajectory. Moreover, they are shown to obey quantum fluctuation theorems. These results extend stochastic thermodynamics to the quantum realm.
Extended quantum systems can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently describes both dynamical and static properties. We show here that in thermal equilibrium, defined by the validity of fluctuation-dissipation relations, the action of a quantum system is invariant under a certain symmetry transformation, and thus it is distinguished from generic systems. In turn, the fluctuation-dissipation relations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient test for the onset of thermodynamic equilibrium and it makes checking the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry reduces to the well-known one that characterizes equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.
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\hbar\mapsto0ℏ↦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.
We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.
Many physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and can be considered as in local thermal equilibrium. In this paper we show that any classical statistical system in local thermal equilibrium has an emergent supersymmetry at low energies. We use the framework of non-equilibrium effective field theory for quantum many-body systems defined on a closed time path contour and consider its classical limit. Unitarity of time evolution requires introducing anti-commuting degrees of freedom and BRST symmetry which survive in the classical limit. The local equilibrium is realized through a Z2 dynamical KMS symmetry. We show that supersymmetry is equivalent to the combination of BRST and a specific consequence of the dynamical KMS symmetry, to which we refer as the special dynamical KMS condition. In particular, we prove a theorem stating that a system satisfying the special dynamical KMS condition is always supersymmetrizable. We discuss a number of examples explicitly, including model A for dynamical critical phenomena, a hydrodynamic theory of nonlinear diffusion, and fluctuating hydrodynamics for relativistic charged fluids.