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arXiv:quant-ph/0301044 · Journal of Physics A Mathematical and General

Mixing quantum and classical mechanics and uniqueness of Planck's constant

Debendranath Sahoo

Published Jan 6, 2004• 41 citations
arXivHTMLPDF

Abstract

Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras (HAs), respectively (Grgin and Petersen 1974 J. Math. Phys. 15 764, Sahoo 1977 Pramana 8 545). We show that the tensor product of two quantum HAs, each characterized by a different Planck's constant (PC), is an algebra of the same type characterized by yet another PC. The algebraic structure of mixed quantum and classical systems is then analysed by taking the limit of vanishing PC in one of the component algebras. This approach provides new insight into failures of various formalisms dealing with mixed quantum–classical systems. It shows that in the interacting mixed quantum–classical description, there can be no back-reaction of the quantum system on the classical system. A natural algebraic requirement involving restriction of the tensor product of two quantum HAs to their components proves that PC is unique.

DOI
10.1088/0305-4470/37/3/031

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Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras (HAs), respectively (Grgin and Petersen 1974 J. Math. Phys. 15 764, Sahoo 1977 Pramana 8 545). We show that the tensor product of two quantum HAs, each characterized by a different Planck's constant (PC), is an algebra of the same type characterized by yet another PC. The algebraic structure of mixed quantum and classical systems is then analysed by taking the limit of vanishing PC in one of the component algebras. This approach provides new insight into failures of various formalisms dealing with mixed quantum–classical systems. It shows that in the interacting mixed quantum–classical description, there can be no back-reaction of the quantum system on the classical system. A natural algebraic requirement involving restriction of the tensor product of two quantum HAs to their components proves that PC is unique.
We provide an answer to the long-standing problem of mixing quantum and classical dynamics within a single formalism. The construction is based on p -mechanical derivation of quantum and classical dynamics from the representation theory of the Heisenberg group. To achieve a quantum-classical mixing we take the product of two copies of the Heisenberg group which represent two different Planck's constants. In comparison with earlier guesses our answer contains an extra term of analytical nature, which was not obtained before in purely algebraic setup.
The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h0h_0h0​. For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operatorial framework.
The main objective of this dissertation is the study of the theory of Lie-Jordan Banach algebras, their role in the framework of classical and quantum mechanics, and their applications to different aspects of quantum systems. One of the main results of this thesis is the novel proof of a theorem which characterizes the Jordan (-Banach) algebras that are in a unique correspondence with C*-algebras. The novelty lies in the introduction of a Lie structure on the algebra, which is required to be compatible with the Jordan one. The physical interpretation of this connection with the Lie structure reflects the dual role played by the observables: they are measurable quantities but also the generators of motions of the state space. Whenever constraints are imposed on a quantum system or symmetries are present, both dynamical or gauge, some reduction on the state space must be considered either because not all states are physical and/or because families of states are equivalent. We address this problem in algebraic terms and obtain a reduction procedure which is free from the problems of gauge anomalies which is often encountered in a heuristic approach to the quantization of constraints. Then we study the composition of physical systems and we prove, from algebraic considerations only, the uniqueness of the Planck's constant. This is a strong result, since he existence of multiple Planck's constants would lead to violations of basic space-time conservation laws, which are not observed experimentally. Finally we address the problem of characterizing classical and quantum states, which has become particularly relevant for the field of quantum information and quantum computation, since a number of tasks in computation and communication can be performed only if quantum resources are available. By using the Jordan product we are able to provide a general definition of classical states and the associated quantumness witness which is suitable to infinite-dimensional systems. Then we propose a measure of nonclassicality based on the incompatibility of states relative to each other, rather than on correlations. Surprisingly, this new measure turns out to be experimentally suitable to a direct estimation by using a quantun circuit based on an interference experiment. We finally describe this experiment which is realizable with current technology.
The generator aspect of observables in classical mechanics leads naturally to a generalized classical mechanics, of which quantum mechanics is shown to be a particular case. Basic features of quantum mechanics follow, such as the identification of observables with self-adjoint operators, and canonical quantization rules. This point of view also gives a new insight on the geometry of quantum theory: Planck's constant is related for instance to the curvature of the quantum-mechanical space of states, and the uniqueness of quantum mechanics can be proved. Finally, the origin of the probabilistic interpretation is discussed.