The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn’t change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives.The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn’t change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives.

International Journal of Geometric Methods in Modern PhysicsJul 20, 20231 citation

On the covariant formulation of quantum mechanics

U. Kasper, Eckhard Kreisel, H.‐J. Treder

Foundations of PhysicsJun 1, 19773 citations

Quantum potential in covariant quantum mechanics

Josef Janyška, Marco Modugno

Differential Geometry and its ApplicationsApr 5, 20171 citation

Generalized covariance condition and quantization in curved configuration space

Ze-sen Yang, Li Xianhui, Qi Hui +1 more

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.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.

Physical Review AApr 1, 19930 citations

Covariant Formulation of Non-Relativistic Quantum Mechanics

N. Ionescu-Pallas, I. Gottlieb

General Relativity and Gravitation, Volume 1Jul 1, 19830 citations

Lorentz-covariant quantum mechanics and preferred frame

Paweł Caban, Jakub Rembieliński

In this paper the relativistic quantum mechanics (QM) is considered in the framework of the nonstandard synchronization scheme, which preserves the Poincar\'e covariance but (at least formally) distinguishes an inertial frame. This enables one to avoid the problem with a strong formulation of local causality related to the breaking of Bell's inequalities in QM. Our analysis has been focused mainly on the problem of the existence of a proper position operator for massive particles. We have proved that in our framework such an operator exists for particles with arbitrary spin. This operator is Hermitian and covariant, it has commuting components, and its eigenvectors (localized states) are covariant too. We have found an explicit form of the position operator and demonstrated that it coincides with the Newton-Wigner one in the preferred frame. We have also defined a covariant spin operator and constructed an invariant spin squared operator. Moreover, full algebra of observables consisting of position, four-momentum, and spin operators is manifestly Poincar\'e covariant in this framework. Our results support expectations of other authors [J. S. Bell, in Quantum Gravity, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford University Press, Oxford, 1981), p. 611; P. H. Eberhard, Nuovo Cimento B 46, 392 (1978)] that a consistent formulation of quantum mechanics demands the existence of a preferred frame.In this paper the relativistic quantum mechanics (QM) is considered in the framework of the nonstandard synchronization scheme, which preserves the Poincar\'e covariance but (at least formally) distinguishes an inertial frame. This enables one to avoid the problem with a strong formulation of local causality related to the breaking of Bell's inequalities in QM. Our analysis has been focused mainly on the problem of the existence of a proper position operator for massive particles. We have proved that in our framework such an operator exists for particles with arbitrary spin. This operator is Hermitian and covariant, it has commuting components, and its eigenvectors (localized states) are covariant too. We have found an explicit form of the position operator and demonstrated that it coincides with the Newton-Wigner one in the preferred frame. We have also defined a covariant spin operator and constructed an invariant spin squared operator. Moreover, full algebra of observables consisting of position, four-momentum, and spin operators is manifestly Poincar\'e covariant in this framework. Our results support expectations of other authors [J. S. Bell, in Quantum Gravity, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford University Press, Oxford, 1981), p. 611; P. H. Eberhard, Nuovo Cimento B 46, 392 (1978)] that a consistent formulation of quantum mechanics demands the existence of a preferred frame.

arXiv:quant-ph/9808013Physical Review AJun 1, 199946 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${\mathrm{^}}_{2}$ 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.