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arXiv:hep-th/0112049 · Journal of Physics A Mathematical and General

Deformation quantization of geometric quantum mechanics

H García-Compeán, J F Plebanski, M Przanowski, F J Turrubiates

Published May 3, 2002• 6 citations
arXivHTMLPDF

Abstract

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrödinger spinless field is considered. Under the assumption that the phase space of the Schrödinger field is ∞ , both the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is P ∞ endowed with the Fubini-Study Kählerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new `second quantization' leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always 1/ℏ.

DOI
10.1088/0305-4470/35/19/311

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Deformation Quantization of Geometric Quantum Mechanics

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Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \\over \\hbar}$.

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We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.

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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).

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Semiclassical Methods of Deformation Quantisation in Transport Theory

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Among the equivalent formulations of quantum mechanics, Heisenberg's matrix mechanics and quantum theory, based on the Schrödinger wave equation, are the most technically advanced.Since the advent of quantum mechanics, these two schemes have provided the mathematical tools and the primary basis for the description of quantum phenomena.The path integral method developed by Dirac and Feynman has important advantages in the quantisation of gauge theories.We discuss here the formulation of quantum mechanics in phase space, known as deformation quantisation.Deformation quantisation uses the Wigner-Weyl association rule (Weyl, 1927(Weyl, , 1929(Weyl, , 1931;;Wigner, 1932) to establish a one-to-one correspondence between the functions in the phase space and the operators in the Hilbert space.Wigner's function appears as the Weyl symbol of the density matrix.A consistent dynamical description of the systems with the help of the Wigner function leads to deformation quantization.A useful formulation of the Wigner-Weyl association rule was proposed by Groenewold (1946) andStratonovich (1957).Groenewold introduced a non-commutative associative -product (star-product) of the functions in the phase space (Groenewold, 1946).The evolution of the quantum systems is determined by the antisymmetric part of the -product (Groenewold, 1946;Moyal, 1949), known as the Moyal bracket.The Moyal bracket represents the quantum deformation of the Poisson bracket.Deformation quantisation preserves many features of classical Hamiltonian dynamics.The formulation of deformation quantisation is based on the Wigner function and the Moyal bracket (i.e., the -product).The Wigner-Weyl association rule is necessary to prove the equivalence of deformation quantisation and the standard formalisms of quantum mechanics.Extensive literature has reported on the formulation of quantum mechanics in the phase space and the -product.We refer the reader to excellent reviews by Bayen et al. 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Dorje C. Brody, Lane P. Hughston

The phase space Γ of quantum mechanics can be viewed as the complex projective space CPn endowed with a Kählerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrödinger equation as generating a Hamiltonian dynamics on Γ. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical Γ-ensemble differs from the density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field.

arXiv:quant-ph/9709048•Journal of Mathematical Physics•Dec 1, 1998•43 citations
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrödinger spinless field is considered. Under the assumption that the phase space of the Schrödinger field is ∞ , both the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is P ∞ endowed with the Fubini-Study Kählerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new `second quantization' leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always 1/ℏ.
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is CinftyC^{\\infty}Cinfty, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is CPinftyCP^{\\infty}CPinfty endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always 1overhbar{1 \\over \\hbar}1overhbar.
We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.
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).
Among the equivalent formulations of quantum mechanics, Heisenberg's matrix mechanics and quantum theory, based on the Schrödinger wave equation, are the most technically advanced.Since the advent of quantum mechanics, these two schemes have provided the mathematical tools and the primary basis for the description of quantum phenomena.The path integral method developed by Dirac and Feynman has important advantages in the quantisation of gauge theories.We discuss here the formulation of quantum mechanics in phase space, known as deformation quantisation.Deformation quantisation uses the Wigner-Weyl association rule (Weyl, 1927(Weyl, , 1929(Weyl, , 1931;;Wigner, 1932) to establish a one-to-one correspondence between the functions in the phase space and the operators in the Hilbert space.Wigner's function appears as the Weyl symbol of the density matrix.A consistent dynamical description of the systems with the help of the Wigner function leads to deformation quantization.A useful formulation of the Wigner-Weyl association rule was proposed by Groenewold (1946) andStratonovich (1957).Groenewold introduced a non-commutative associative -product (star-product) of the functions in the phase space (Groenewold, 1946).The evolution of the quantum systems is determined by the antisymmetric part of the -product (Groenewold, 1946;Moyal, 1949), known as the Moyal bracket.The Moyal bracket represents the quantum deformation of the Poisson bracket.Deformation quantisation preserves many features of classical Hamiltonian dynamics.The formulation of deformation quantisation is based on the Wigner function and the Moyal bracket (i.e., the -product).The Wigner-Weyl association rule is necessary to prove the equivalence of deformation quantisation and the standard formalisms of quantum mechanics.Extensive literature has reported on the formulation of quantum mechanics in the phase space and the -product.We refer the reader to excellent reviews by Bayen et al. (1978a, 1978b), Carruthers & Zachariasen (1983), Balazs & Jennings (1984), Hillery et al. (1983), Karasev & Maslov (1991), and Osborn & Molzahn (1995), where one may find additional references.Wigner's function, as a fundamental object of deformation quantisation, has numerous applications in many-body physics, kinetic theory, collision theory, and quantum chemistry.Transport models, originally created to simulate chemical reactions, have been modified and are widely used to describe heavy-ion collisions.www.intechopen.comSome Applications of Quantum Mechanics 68 Deformation quantisation does not have many recognised successful applications in quantum theory.Recently, attempts have been made to use specific properties of the formalism to investigate semiclassical expansion (Osborn & Molzahn, 1995;McQuarrie et al., 1998) and to calculate the determinants of one-loop operators in field theory (Pletnev & Banin, 1999;Banin et al., 2001) and the high-order corrections to the Bohr-Sommerfeld quantisation rule (Gracia-Saz, 2004;Cargo et al., 2005).Potential applications of deformation quantisation in transport models are presented in this review.Transport models for heavy-ion physics are designed for phenomenological descriptions of the complex dynamics of nuclear collisions.Several types of advanced transport models are based on the Boltzmann-Uhlenbeck-Ueling equations (BUU) (Blaettel et al., 1993), (relativistic) quantum molecular dynamics (QMD/RQMD) (Sorge et al., 1989;Aichelin, 1991;Faessler, 1992), or antisymmetrised molecular dynamics (AQMD) (Feldmeier & Schnack, 1997).These approaches have the correct classical limit and they contain special plug-ins and quantum-mechanical attributes, such as Pauli blocking for binary collisions of fermions.Numerical solutions are implemented through the distribution of test particles (BUU) or centroids of wave packets (QMD, AQMD) for classical trajectories in the phase space.For AQMD, the wave packets are antisymmetrised in their parameters.The transport models provide a solid basis for a phenomenological description of a variety of complex nuclear collisions phenomena.However, quantum coherence effects and non-localities are beyond the scope of these models.The internal consistencies of the approximations in the models remain a subject of debate, promoting further developments (see, for example, papers by Kohler (1995) andFeldmeier &Schnack (1997) and references therein).The most striking feature of the transport models is the depiction of the trajectories in the phase space, which are test particles or centroids of wave packets.The evolution of the system of classical particles can be calculated using standard programs to solve first-order ordinary differential equations (ODEs).At the same time, the evolution of wave functions of many-body systems is a field-theoretic problem with an infinite number of degrees of freedom that cannot be solved either analytically or numerically.Any simulation of many-body quantum dynamics must be based on a concept of trajectories, which is the only attribute allowing access to an approximate description of complex quantum systems.The concept of phase-space trajectories arises naturally in the formalism of deformation quantisation through the Wigner transformation of the operators of the canonical coordinates and momenta in the Heisenberg representation.These trajectories satisfy the quantum version of Hamilton's equations (Osborn & Molzahn, 1995;Krivoruchenko & Faessler, 2006b) and are the characteristics by which the time-dependent Weyl's symbols for the other operators can be determined (Krivoruchenko & Faessler, 2006b;Krivoruchenko et al., 2006c Krivoruchenko et al., , 2007)).In the classical limit, quantum characteristics reduce to classical trajectories.Knowledge of the quantum phase flow, i.e., the quantum trajectories, is equivalent to a complete knowledge of the quantum dynamics.In this chapter, we provide an introduction to deformation quantisation and demonstrate the usefulness of the formalism in solving the evolution problem for many-body systems in terms of semiclassical expansion.We show that, in any fixed order of expansion over the Planck's constant, the evolution problem can be reduced to a statistical-mechanics problem of calculating an ensemble of quantum characteristics in the phase space and their Jacobi fields.In comparison with the corresponding rules of classical statistical mechanics, the rules for computing the probabilities and time-dependent averages of observables are www.intechopen.com
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.
The phase space Γ of quantum mechanics can be viewed as the complex projective space CPn endowed with a Kählerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrödinger equation as generating a Hamiltonian dynamics on Γ. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical Γ-ensemble differs from the density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field.