Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1302.0194 · Physica Scripta

Tensorial description of quantum mechanics

Jesús Clemente-Gallardo, Giuseppe Marmo

Published Mar 1, 2013• 2 citations
arXivHTMLPDF

Abstract

Relevant algebraic structures for the description of Quantum Mechanics in the Heisenberg picture are replaced by tensorfields on the space of states. This replacement introduces a differential geometric point of view which allows for a covariant formulation of quantum mechanics under the full diffeomorphism group.
DOI
10.1088/0031-8949/2013/t153/014012

Related papers

Tensorial dynamics on the space of quantum states

José F. Cariñena, Jesús Clemente-Gallardo, Jorge Alberto Jover-Galtier +1 more

Abstract A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski–Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skew-symmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products.

arXiv:1705.05186•Journal of Physics A Mathematical and Theoretical•Jul 24, 2017•12 citations

Geometric formulation of quantum mechanics

Hoshang Heydari

Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.

arXiv:1503.00238•Mar 1, 2015•16 citations

Introduction to Quantum Mechanics and the Quantum-Classical transition

José F. Cariñena, Jesús Clemente-Gallardo, Giuseppe Marmo

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level.

arXiv:0707.3539•Jul 24, 2007•4 citations

Quantum Mechanics in the Heisenberg (Matrix) Picture

Ana Marı́a Cetto, Luis de la Peña

Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.

Cambridge University Press eBooks•Dec 18, 2025•0 citations

The Mathematics of Quantum Mechanics

Jeffrey Alan Barrett

Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.

•Dec 26, 2019•1 citation

Classical Geometric Interaction- picture-like Description

Shao, M. X, Zhenduo Zhu

In order to get the classical analogue of quantum interaction picture in classical symplectic geometric description, the space of solutions of free equations of motion is suggested to replace the phase space in $T^{*}Q$ description or the space of motions in usual classical symplectic geometric description. The way to determine measured values of observables in this scheme is worked out.

arXiv:math-ph/9807008•Jul 10, 1998•0 citations
Abstract A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski–Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skew-symmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products.
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.
Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
In order to get the classical analogue of quantum interaction picture in classical symplectic geometric description, the space of solutions of free equations of motion is suggested to replace the phase space in T∗QT^{*}QT∗Q description or the space of motions in usual classical symplectic geometric description. The way to determine measured values of observables in this scheme is worked out.