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Russian Journal of Mathematical Physics

Quantum geometry and quantum mechanics of integrable systems. II

Mihail Vladimirovich Karasev

Published Jun 1, 2010• 1 citation
Publisher

Abstract

For a generic quantum integrable system, we describe the asymptotics of the eigenstate density and of the trace of the evolution operator in all orders of the quantization parameter. This is done by using quantum symplectic geometry, which makes the given quantum system to be equivalent to a deformed classical system with arbitrary accuracy with respect to the quantization parameter. The asymptotics is explicitly given via the deformed symplectic form, deformed Liouville-Arnold tori, and deformed Maslov class.

DOI
10.1134/s1061920810020056

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For a generic quantum integrable system, we describe the asymptotics of the eigenstate density and of the trace of the evolution operator in all orders of the quantization parameter. This is done by using quantum symplectic geometry, which makes the given quantum system to be equivalent to a deformed classical system with arbitrary accuracy with respect to the quantization parameter. The asymptotics is explicitly given via the deformed symplectic form, deformed Liouville-Arnold tori, and deformed Maslov class.
Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ∞) and are invariant under quantum dynamics in the long-time range O(ħ −∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.
We study the relation between the mathematical structures of statistical mechanics on an infinite-dimensional phase space (denoted by ) and quantum mechanics. It is shown that quantum averages (given by the von Neumann trace formula) can be obtained as the main term of the asymptotic expansion of Gaussian functional integrals with respect to a small parameter . Here is the dispersion of the Gaussian measure. The symplectic structure on the infinite-dimensional phase space plays a crucial role in our considerations. In particular, the Gaussian measures that induce quantum averages must be consistent with the symplectic structure. The equations of Schrodinger, Heisenberg and von Neumann are images of the Hamiltonian dynamics on .
A bstract The construction of exactly-solvable models has recently been advanced by considering integrable TT‾ T\overline{T} TT T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse TT‾ T\overline{T} TT T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space. We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes. Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure. These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.
This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a two-fold way. The first part is devoted in establishing an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold. This correspondence enables a geometric quantization scheme that identifies a family of classical Hamiltonian systems admitting exponentially compressed quantum representations-appropriate for quantum simulation. In the second part we demonstrate that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding. This allows efficient quantum representation and parallel evolution of large phase-space ensembles, where entangled encodings provide exponential compression in ensemble size and enable quantum speed-ups in observable estimation via amplitude estimation techniques. For non-integrable systems, Lie canonical perturbation theory is incorporated to construct near-symplectic transformations that map dynamics to approximately integrable forms, preserving unitary evolution up to a controlled error. We derive the resulting quantum computational complexity of the proposed quantum-symplectic scheme, revealing both an exponential compression in memory requirements and a potential polynomial speed-up with respect to the system size. Finally, the transport evolution equation governing the quantum phase-space observables is obtained.