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arXiv:1705.09960 · The European Physical Journal C

Covariant quantizations in plane and curved spaces

João Luis Meloni Assirati, Dmitri M Gitman

Published Jul 1, 2017• 3 citations
arXivHTMLPDF

Abstract

We present covariant quantization rules for nonsingular finite-dimensional classical theories with flat and curved configuration spaces. In the beginning, we construct a family of covariant quantizations in flat spaces and Cartesian coordinates. This family is parametrized by a function $$\omega (\theta )$$ , $$\theta \in (1,0)$$ , 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.

DOI
10.1140/epjc/s10052-017-5041-0

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We present covariant quantization rules for nonsingular finite-dimensional classical theories with flat and curved configuration spaces. In the beginning, we construct a family of covariant quantizations in flat spaces and Cartesian coordinates. This family is parametrized by a function ω(θ)\omega (\theta )ω(θ) , θ∈(1,0)\theta \in (1,0)θ∈(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 ω(θ).\omega (\theta ).ω(θ). Finally, we describe a more wide family of covariant quantizations in curved spaces. This family is already parametrized by two functions, the previous one ω(θ)\omega (\theta )ω(θ) and by an additional function Θ(x,ξ)\varTheta (x,\xi )Θ(x,ξ) . The above mentioned minimal family is a part at Θ=1\varTheta =1Θ=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.
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 H2{\mathit{H}}_{2}H2​, represented by a quadratic function of momenta with a nonsingular coefficient matrix depending on coordinates, and the operator of an arbitrary quadratic quantity F2{\mathit{F}}_{2}F2​ belonging to a system whose Hamiltonian is of the H2{\mathit{H}}_{2}H2​ type. It is shown that the operator of F2{\mathit{F}}_{2}F2​ depends also on H2{\mathit{H}}_{2}H2​ 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.
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.
This paper concerns the quantization of a rigid body in the framework of 'covariant quantum mechanics' on a curved spacetime with absolute time. We consider the configuration space of n classical particles as the n -fold product of the configuration space of one particle. Then, we impose a rigid constraint and the resulting space is dealt with as a configuration space of a single abstract 'particle'. This classical framework turns out to be suitable for the formulation of covariant quantum mechanics according to this scheme. Thus, we quantize such a 'particle' accordingly. This scheme can model, e.g., the quantum dynamics of extremely cold molecules. We provide a new mathematical interpretation of two-valued wavefunctions on SO (3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.
Canonical quantization entails using Cartesian coordinates, and Cartesian coordinates exist only in flat spaces. This situation can either be questioned or accepted. In this paper we offer a brief and introductory overview of how a flat phase space metric can be incorporated into a covariant, coordinate-free quantization procedure involving a continuous-time (Wiener measure) regularization of traditional phase space path integrals. Additionally we show how such procedures can be extended to incorporate systems with constraints and illustrate that extension for special systems.
Canonical quantization entails using Cartesian coordinates, and Cartesian coordinates exist only in flat spaces. This situation can either be questioned or accepted. In this paper we offer a brief and introductory overview of how a flat phase space metric can be incorporated into a covariant, coordinate-free quantization procedure involving a continuous-time (Wiener measure) regularization of traditional phase space path integrals. Additionally we show how such procedures can be extended to incorporate systems with constraints and illustrate that extension for special systems.
A geometrical framework for the de Broglie–Bohm quantum theory is presented, in which the trajectories of an N-particle system are interpretable as the integral curves of a particular vector field defined on a 3N-dimensional manifold [Formula: see text] constructed from physical space M. It is mathematically valid even when M is curved. If M is flat, the usual theory is recovered and automatically expressed in whatever curvilinear coordinates one may wish to choose. The general construction is illustrated by the case of a free particle moving on the surface of a sphere. (A modified Bohr quantization condition for angular momentum is obtained, with a first correction proportional to the curvature.) The Zeeman effect and some bound states on the sphere are also considered.