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arXiv:2509.17106

Canonical quantization of a product within the Cahill-Glauber correspondence

Lim, Hendry M.

Published Sep 21, 2025• 0 citations
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Abstract

The (canonical) quantization of a product of two phase-space functions can be expressed as a commutative mapping between the quantizations of its factors. In this note, we discuss such a mapping within the Cahill-Glauber $s$-parameterized correspondence framework, providing a heuristic (hence less mathematically rigorous, but arguably more easily understood) derivation for the bidifferential form. The term "hatted star product" is coined by virtue of the structural similarity to the star product in phase space, a special case of which is the celebrated Moyal star product.

DOI
10.48550/arxiv.2509.17106

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The (canonical) quantization of a product of two phase-space functions can be expressed as a commutative mapping between the quantizations of its factors. In this note, we discuss such a mapping within the Cahill-Glauber sss-parameterized correspondence framework, providing a heuristic (hence less mathematically rigorous, but arguably more easily understood) derivation for the bidifferential form. The term "hatted star product" is coined by virtue of the structural similarity to the star product in phase space, a special case of which is the celebrated Moyal star product.
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A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential form and that in some cases this approach provides more useful tools than the star-exponential form in finding the generating function for given canonical transformation or vice versa. It is also shown that transforms of c-number phase space functions under linear–nonlinear canonical transformations and intertwining method can be treated within this argument.
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The star product is the basic tool used in the phase-space formulation of quantum mechanics. We find a differential form of the star product for a class of \mbox{sss-parametrized} SU (2) Stratonovich-Weyl symbols which appear in the phase-space representation of spin-like systems. The limit of large spin is considered and the asymptotic form of the differential operator defining the star product is obtained.