This paper is the first of two papers devoted to formulation of quantum mechanics of a particle in a normal geodesic frame of reference in the general Riemannian space-time. Here canonical quantization of geodesic motion in the 1+3-formalism is considered, the result of which will be compared in the subsequent paper II with that of the field-theoretical approach, see also gr-qc/9807030. The Schrödinger representation of quantum-mechanical kinematics and dynamics is presented in the general-covariant form and a physical interpretation of the state vectors and the position operators is discussed.This paper is the first of two papers devoted to formulation of quantum mechanics of a particle in a normal geodesic frame of reference in the general Riemannian space-time. Here canonical quantization of geodesic motion in the 1+3-formalism is considered, the result of which will be compared in the subsequent paper II with that of the field-theoretical approach, see also gr-qc/9807030. The Schrödinger representation of quantum-mechanical kinematics and dynamics is presented in the general-covariant form and a physical interpretation of the state vectors and the position operators is discussed.

arXiv:gr-qc/9812084Dec 24, 19982 citations

Generalized coordinate representation in quantum mechanics

Cheng-Shing Wang

American Journal of PhysicsJan 1, 19897 citations

Quantum mechanics on Riemannian Manifold in Schwinger's Quantization Approach, 2

N.M. Chepilko, A V Romanenko

Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold $M$ is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of $M$ with the quotient space $G/H$ , where $H$ is the isotropy group of an arbitrary fixed point of $M$ , we show that quantum mechanics on $G/H$ possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle $G(G/H, H)$ . The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold $M$ is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of $M$ with the quotient space $G/H$, where $H$ is the isotropy group of an arbitrary fixed point of $M$, we show that quantum mechanics on $G/H$ possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle $G(G/H, H)$. The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.

Jan 1, 20010 citations

Polymer quantum mechanics on compact configuration spaces

Maxwell R. Siebersma, Basie Seibert, Samuel Shuman +1 more

"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts."Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.

quant-pharXiv:2606.06019UniverseJun 5, 20260 citations

Representations of Quantum Mechanics

John C. Inkson

Springer eBooksJan 1, 19840 citations

Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold

M

is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of

M

with the quotient space

G/H

, where

H

is the isotropy group of an arbitrary fixed point of

M

, we show that quantum mechanics on

G/H

possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle

G(G/H, H)

. The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.