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arXiv:quant-ph/9504016

The classical limit of quantum theory

Reinhard F. Werner

Published Apr 24, 1995• 20 citations
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Abstract

For a quantum observable $A_\hbar$ depending on a parameter $\hbar$ we define the notion ``$A_\hbar$ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that $A_\hbar\to0$ is equivalent with $\Vert A_\hbar\Vert\to0$. The $\hbar$-wise product of convergent observables converges to the product of the limiting phase space functions. $\hbar^{-1}$ times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the $\hbar$-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.

DOI
10.48550/arxiv.quant-ph/9504016

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For a quantum observable AℏA_\hbarAℏ​ depending on a parameter ℏ\hbarℏ we define the notion ``AℏA_\hbarAℏ​ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that Aℏ→0A_\hbar\to0Aℏ​→0 is equivalent with ∥Aℏ∥→0\Vert A_\hbar\Vert\to0∥Aℏ​∥→0. The ℏ\hbarℏ-wise product of convergent observables converges to the product of the limiting phase space functions. ℏ−1\hbar^{-1}ℏ−1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the ℏ\hbarℏ-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.
The correspondence principle between the quantum commutator $[\stackrel{^}{A},\stackrel{^}{B}]$ and the classical Poisson brackets $\ensuremath{\iota}\ensuremath{\hbar}{A,B}$ is examined in the context of response theory. The classical response function is obtained as the leading term of the $\ensuremath{\hbar}$ expansion of the phase space representation of the response function in terms of Weyl-Wigner transformations and is shown to increase without bound at long times as a result of ignoring divergent higher-order contributions. Systematical inclusion of higher-order contributions improves the accuracy of the $\ensuremath{\hbar}$ expansion at finite times. Resummation of all the higher-order terms establishes the classical-quantum correspondence $⟨v+n|\stackrel{^}{\ensuremath{\alpha}}(t)|v⟩\ensuremath{\leftrightarrow}{\ensuremath{\alpha}}_{n}{e}^{\ensuremath{\iota}n\ensuremath{\omega}t}{|}_{{J}_{v}+n\ensuremath{\hbar}/2}$. The time interval of the validity of the simple classical limit $[\stackrel{^}{A}(t),\stackrel{^}{B}(0)]\ensuremath{\rightarrow}\ensuremath{\iota}\ensuremath{\hbar}{A(t),B(0)}$ is estimated for quasiperiodic dynamics and is shown to be inversely proportional to anharmonicity.
We point out a correspondence between classical and quantum states, by showing that for every classical distribution over phase--space, one can construct a corresponding quantum state, such that in the classical limit of ℏ→0\hbar\to 0ℏ→0 the latter converges to the former with respect to all measurable quantities.
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative ⋆\star⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The considered class of deformations and the corresponding ⋆\star⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆\star⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆\star⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner-Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.
In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure. It will be shown that the star product in Berezin quantiza- tion is equivalent to the Poisson bracket on P(H) and the Berezin method to construction a correspondence between a given classical theory and a given quantum theory is used to define a classical limit for geometric quantum mechanics. the covariant symbols. These symbols are expectation values of the observables in terms of coherent states: the holomorphic functions on classical phase space M that is assumed to be aKmanifold. Berezin (2) showed that the covariant symbols form a ∗-algebra which in limit � → 0l eads to the Poisson algebra between the corresponding classical observables: The functions on the phase space M. In this paper we will see that the Berezin ∗-algebra is in fact a Poisson algebra which is induced by the Fubini-Study 2-form on space of coherent states. This space is defined as follows: coherent states span a dense subspace H of Hilbert space H. P( H), which is denoted by M, is aKmanifold with induced symplectic structure from P(H). Therefore the covariant symbols can be considered as functions on M. It is shown (13) that there exists an embedding mapping between the classical phase space M and M, by which to any point z ∈ M is associated ap ointZ ∈M. With this construction to all of the quantum observables are associated their covariant symbols, which form a Poisson algebra on M and since the corresponding classical observables form a Poisson algebra on M , the Berezin quantization is a systematic procedure to relate these two Poisson algebras. Also the relation of Berezin quantization and geometric for- mulation of quantum mechanics will be evident as follows. The geometric quantum mechanics is a formulation of quantum mechanics in projective Hilbert spaces. With our construction one sees that the Berezin quantization is an equivalent formulation and in addition gives a prescription as classical limit for geometric quantum mechanics (9).
Geometric Quantum Mechanics is a formulation demonstrating how quantum theory may be cast in the language of Hamiltonian phase-space dynamics. Within this framework, the classical properties of spin 1/2, spin 1 and spin 3/2 particles have been studied. The correspondence between the Poisson bracket and commutator algebras for these systems was shown by explicitly computing the value of the commutator of spin operators and comparing it with the Poisson bracket of the corresponding classical observables. This study was extended by comparing the Casimir operator and its classical counterpart. The results showed that there exists a correspondence between classical and quantum Casimir operators at least for the case of spin 1/2. This research clearly shows the limit of classical notions to describe the purely quantum concept.