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

Natural quantum Lagrangians in Galilei quantum mechanics

Josef Janyška

Published Jan 1, 1995• 7 citations
OpenAlex

Abstract

The natural quantum Lagrangians which appear in Galilei general relativistic quantum mechanics are classified by using methods of gauge–natural bundles and natural operators. Namely, all natural quantum Lagrangians are multiples of the canonical scaled volume form, where multiplicative factors are invariant functions. A base of these invariant functions is described: the base is constituted by three functions and the unique non trivial function in the base is just the function corresponding to the natural quantum lagrangian considered by Jadczyk and Modugno.

Related papers

Symmetries in covariant quantum mechanics

Dirk Saller

The thesis is divided in two parts. In the first part of the thesis, I present an updated version of the theory of covariant quantum mechanics and of the associated classical background. This theory has originally been proposed by Jadczyk and Modugno and further developed in cooperation with other people. It provides a geometric model for quantum mechanics of a scalar and spin particle in a curved spacetime with absolute time equipped with a given classical gravitational and electromagnetic field. Starting from minimal axioms, this model yields in a covariant way the Schrödinger equation by means of a global Lagrangian formalism and the quantum operators associated with classical quantisable functions by means of classification of distinguished quantum vector fields. In the second part of the thesis, I investigate systematically the infinitesimal symmetries of the classical and quantum geometric structures. This part deals with the main original contributions of the thesis. Some results fit well-known facts of other mathematical models of quantum mechanics, but are derived by innovative procedures. Other results are completely new. The most important original contributions can be summarized by the following two statements: The quantum symmetries constitute Lie algebras, which are naturally isomorphic to subalgebras of the new classical Lie algebra of quantisable functions. Let me stress that this algebra is not a Poisson subalgebra. On the other hand, by means of the quantum Lagrangian formalism, we can associate a conserved quantum current with any element of a distinguished classical subalgebra. These results might be the prequantum source of the probabilistic interpretation of quantum mechanics.

•Jan 1, 2001•3 citations

Fibre bundle formulation of relativistic quantum mechanics. II. Covariant approach

Bozhidar Zakhariev Iliev

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

arXiv:quant-ph/0107002•Jun 30, 2001•0 citations

A Galilei-invariant Version of the Modal-Hamiltonian Interpretation

Olímpia Lombardi

Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.

•Feb 19, 2026•0 citations

Galilean relativity and the path integral formalism in quantum mechanics

Charles G Torre

Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.

arXiv:2306.13487•American Journal of Physics•Oct 23, 2023•0 citations

Fibre bundle formulation of relativistic quantum mechanics

Bozhidar Zakhariev Iliev

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

arXiv:quant-ph/0201085•Jan 19, 2002•2 citations

A Possible Generalization of Quantum Mechanics

Yu Peng Tian

A "minimal" generalization of Quantum Mechanics is proposed, where the Lagrangian or the action functional is a mapping from the (classical) states of a system to the Lie algebra of a general compact Lie group, and the wave function takes values in the corresponding group algebra. This formalism admits a probability interpretation and a suitable dynamics, but has no obvious classical correspondence. Allowing the Lagrangian or the action functional to take values in a general Lie algebra instead of only the real number field (actually the u(1) algebra) enlarges the extent of possible physical laws that can describe the real world. The generalized quantum dynamics of a point particle in a background gauge field is given as an example, which realizes the gauge invariance by a Wilson line structure and shows that some Schrodinger-like equation can be deduced within this formalism. Some possible developments of this formalism are also discussed.

arXiv:quant-ph/0512127•Dec 16, 2005•0 citations
The natural quantum Lagrangians which appear in Galilei general relativistic quantum mechanics are classified by using methods of gauge–natural bundles and natural operators. Namely, all natural quantum Lagrangians are multiples of the canonical scaled volume form, where multiplicative factors are invariant functions. A base of these invariant functions is described: the base is constituted by three functions and the unique non trivial function in the base is just the function corresponding to the natural quantum lagrangian considered by Jadczyk and Modugno.
The thesis is divided in two parts. In the first part of the thesis, I present an updated version of the theory of covariant quantum mechanics and of the associated classical background. This theory has originally been proposed by Jadczyk and Modugno and further developed in cooperation with other people. It provides a geometric model for quantum mechanics of a scalar and spin particle in a curved spacetime with absolute time equipped with a given classical gravitational and electromagnetic field. Starting from minimal axioms, this model yields in a covariant way the Schrödinger equation by means of a global Lagrangian formalism and the quantum operators associated with classical quantisable functions by means of classification of distinguished quantum vector fields. In the second part of the thesis, I investigate systematically the infinitesimal symmetries of the classical and quantum geometric structures. This part deals with the main original contributions of the thesis. Some results fit well-known facts of other mathematical models of quantum mechanics, but are derived by innovative procedures. Other results are completely new. The most important original contributions can be summarized by the following two statements: The quantum symmetries constitute Lie algebras, which are naturally isomorphic to subalgebras of the new classical Lie algebra of quantisable functions. Let me stress that this algebra is not a Poisson subalgebra. On the other hand, by means of the quantum Lagrangian formalism, we can associate a conserved quantum current with any element of a distinguished classical subalgebra. These results might be the prequantum source of the probabilistic interpretation of quantum mechanics.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.
Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
A "minimal" generalization of Quantum Mechanics is proposed, where the Lagrangian or the action functional is a mapping from the (classical) states of a system to the Lie algebra of a general compact Lie group, and the wave function takes values in the corresponding group algebra. This formalism admits a probability interpretation and a suitable dynamics, but has no obvious classical correspondence. Allowing the Lagrangian or the action functional to take values in a general Lie algebra instead of only the real number field (actually the u(1) algebra) enlarges the extent of possible physical laws that can describe the real world. The generalized quantum dynamics of a point particle in a background gauge field is given as an example, which realizes the gauge invariance by a Wilson line structure and shows that some Schrodinger-like equation can be deduced within this formalism. Some possible developments of this formalism are also discussed.