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arXiv:0705.1988 · Journal of Functional Analysis

The resolvent algebra: A new approach to canonical quantum systems

Detlev Buchholz A, Hendrik Grundling

Published Apr 11, 2008• 71 citations
arXivHTMLPDF
DOI
10.1016/j.jfa.2008.02.011

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The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on ad hoc prescriptions for the test of pertinent features, such as the appearance of Bose–Einstein condensates. The method is illustrated in the case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet, the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.
We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the “Grassmann algebra of canonical anticommutation relations” (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems.
The canonical anticommutation relations (CAR) for fermion systems can be represented by finite-dimensional matrix algebra, but it is impossible for canonical commutation relations (CCR) for bosons. After description of more simple case with representation of CAR and (bounded) quantum computational networks via Clifford algebras in the paper are discussed CCR. For representation of the algebra it is not enough to use quantum networks with fixed number of qubits and it is more convenient to consider Turing machine with essential operation of appending new cells for description of infinite tape in finite terms --- it has straightforward generalization for quantum case, but for CCR it is necessary to work with symmetrized version of the quantum Turing machine. The system is called here quantum abacus due to understanding analogy with the ancient counting devices (abacus).