Physical Review A

Generalized covariance condition and quantization in curved configuration space

Ze-sen Yang, Li Xianhui, Qi Hui, Wei-Zhen Deng

Published Apr 1, 1993• 0 citations

Abstract

We expound the generalized covariance condition in quantization. This condition is combined with the other standard conditions to determine the quantum-mechanical operator of a quadratic Hamiltonian ${\mathit{H}}_{2}$, represented by a quadratic function of momenta with a nonsingular coefficient matrix depending on coordinates, and the operator of an arbitrary quadratic quantity ${\mathit{F}}_{2}$ belonging to a system whose Hamiltonian is of the ${\mathit{H}}_{2}$ type. It is shown that the operator of ${\mathit{F}}_{2}$ depends also on ${\mathit{H}}_{2}$ and that H${\mathrm{^}}_{2}$ or F${\mathrm{^}}_{2}$ cannot contain a curvature term. We also present a modified ordering method with certain conditions under which the operator of a quadratic quantity can be found by replacing some linear functions of momenta with their operators. Finally, a path-integration formula is formed and shown to yield the expression of H${\mathrm{^}}_{2}$ that we derive by the canonical quantization method.

DOI: 10.1103/physreva.47.2574

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Physical Review A

Generalized covariance condition and quantization in curved configuration space

Ze-sen Yang, Li Xianhui, Qi Hui, Wei-Zhen Deng

Published Apr 1, 1993• 0 citations

Publisher

Abstract

We expound the generalized covariance condition in quantization. This condition is combined with the other standard conditions to determine the quantum-mechanical operator of a quadratic Hamiltonian

{\mathit{H}}_{2}

, represented by a quadratic function of momenta with a nonsingular coefficient matrix depending on coordinates, and the operator of an arbitrary quadratic quantity

{\mathit{F}}_{2}

belonging to a system whose Hamiltonian is of the

{\mathit{H}}_{2}

type. It is shown that the operator of

{\mathit{F}}_{2}

depends also on

{\mathit{H}}_{2}

and that H

{\mathrm{^}}_{2}

or F

{\mathrm{^}}_{2}

cannot contain a curvature term. We also present a modified ordering method with certain conditions under which the operator of a quadratic quantity can be found by replacing some linear functions of momenta with their operators. Finally, a path-integration formula is formed and shown to yield the expression of H

{\mathrm{^}}_{2}

that we derive by the canonical quantization method.

DOI: 10.1103/physreva.47.2574

Related papers

Covariant quantizations in plane and curved spaces

João Luis Meloni Assirati, Dmitri M Gitman

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Apr 15, 20210 citations

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Jean Zinn‐Justin

Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.

Oxford University Press eBooksJun 6, 20020 citations

Quantization of a general dynamical system

Shelomo Izhaq Ben-Abraham, A. Lonke

Journal of Mathematical PhysicsDec 1, 197312 citations

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Эрнест Ахмеджанович Тагиров

Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the Rimannian configurational spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for non-invariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between coordinates and momenta of a prticle. A possible cosmological consequence of the deformation is presented in short. Key words: quantum mechanics, Riemannian space, geodesic motion, deformation.Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the Rimannian configurational spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for non-invariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between coordinates and momenta of a prticle. A possible cosmological consequence of the deformation is presented in short. Key words: quantum mechanics, Riemannian space, geodesic motion, deformation.

arXiv:gr-qc/0212076Dec 18, 20021 citation

Path Integral Quantization and Riemannian-Symplectic Manifolds

Sergei V. Shabanov, John R. Klauder

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce aWe develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a

Jan 1, 19986 citations

We expound the generalized covariance condition in quantization. This condition is combined with the other standard conditions to determine the quantum-mechanical operator of a quadratic Hamiltonian

{\mathit{H}}_{2}

, represented by a quadratic function of momenta with a nonsingular coefficient matrix depending on coordinates, and the operator of an arbitrary quadratic quantity

{\mathit{F}}_{2}

belonging to a system whose Hamiltonian is of the

{\mathit{H}}_{2}

type. It is shown that the operator of

{\mathit{F}}_{2}

depends also on

{\mathit{H}}_{2}

and that H

{\mathrm{^}}_{2}

or F

{\mathrm{^}}_{2}

{\mathrm{^}}_{2}

that we derive by the canonical quantization method.

\omega (\theta )

\theta \in (1,0)

, which describes an ambiguity of the quantization. We generalize this construction presenting covariant quantizations of theories with flat configuration spaces but already with arbitrary curvilinear coordinates. Then we construct a so-called minimal family of covariant quantizations for theories with curved configuration spaces. This family of quantizations is parametrized by the same function

\omega (\theta ).

Finally, we describe a more wide family of covariant quantizations in curved spaces. This family is already parametrized by two functions, the previous one

\omega (\theta )

and by an additional function

\varTheta (x,\xi )

. The above mentioned minimal family is a part at

\varTheta =1

of the wide family of quantizations. We study constructed quantizations in detail, proving their consistency and covariance. As a physical application, we consider a quantization of a non-relativistic particle moving in a curved space, discussing the problem of a quantum potential. Applying the covariant quantizations in flat spaces to an old problem of constructing quantum Hamiltonian in polar coordinates, we directly obtain a correct result.