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Matrix geometry for ellipsoids

Asato Tsuchiya, Kaho Matsuura

Published Mar 26, 2020• 0 citations
OpenAlex

Abstract

We study description of ellipsoids by matrices to gain insights into emergence of space in matrix models.We apply the coherent state method to the fuzzy ellipsoids and perform the Berezin-Toeplitz quantization for ellipsoids.We see that the manifold described by the coherent state method is an ellipsoid.We find that the Toeplitz operators obtained in the Berezin-Toeplitz quantization agree with the matrices for the fuzzy ellipsoids used in the coherent state method and that the wave functions in the above two methods coincide...........

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We study description of ellipsoids by matrices to gain insights into emergence of space in matrix models.We apply the coherent state method to the fuzzy ellipsoids and perform the Berezin-Toeplitz quantization for ellipsoids.We see that the manifold described by the coherent state method is an ellipsoid.We find that the Toeplitz operators obtained in the Berezin-Toeplitz quantization agree with the matrices for the fuzzy ellipsoids used in the coherent state method and that the wave functions in the above two methods coincide...........
Abstract We study description of ellipsoids by matrices to gain insights into emergence of space in matrix models. We apply the coherent state method to the fuzzy ellipsoids and perform the Berezin-Toeplitz quantization for ellipsoids. We see that the manifold described by the coherent state method is an ellipsoid. We find that the Toeplitz operators obtained in the Berezin-Toeplitz quantization agree with the matrices for the fuzzy ellipsoids used in the coherent state method and that the wave functions in the above two methods coincide.
In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in configuration space. To each of these states, we associate a classs of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.
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).
This paper defines coherent manifolds and discusses their properties and their application in quantum mechanics. Every coherent manifold with a large group of symmetries gives rise to a Hilbert space, the completed quantum space of ZZZ, which contains a distinguished family of coherent states labeled by the points of the manifold. The second quantization map in quantum field theory is generalized to quantization operators on arbitrary coherent manifolds. It is shown how the Schrödinger equation on any such completed quantum space can be solved in terms of computations only involving the coherent product. In particular, this applies to a description of Bosonic Fock spaces as completed quantum spaces of a class of coherent manifolds called Klauder spaces.