Relative Entropy of Fermion Excitation States on the CAR Algebra

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED
Stefano Galanda, Albert Much, Rainer Verch
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引用次数: 4

Abstract

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

CAR代数上费米子激发态的相对熵
本文研究了正则反对易关系(CAR)代数上某些态的相对熵。CAR代数用于描述量子力学和量子场论中的费米子自由度。研究相对熵的状态是相对于由CAR代数的单粒子希尔伯特空间上的幺正动力群引起的时间演化所定义的KMS状态的多激发态(类似于多粒子态)。如果KMS态是准自由的,则多激发态的相对熵可以用2点函数显式计算,2点函数完全由定义CAR代数的单粒子希尔伯特空间和单粒子希尔伯特空间上动力群的Hamilton算子定义。这也适用于单粒子希尔伯特空间汉密尔顿算符具有连续谱的情况,因此多激发态的相对熵不能用冯·诺伊曼熵来定义。本文得到的CAR代数上的多激发态相对熵的结果可以看作是最近出现的正则交换关系代数上相干态相对熵的结果的对应。利用Araki引入的自对偶CAR代数的设置是有用的。
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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
审稿时长
>12 weeks
期刊介绍: MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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