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

On projective bundle representations and Gauge equivalence

Henk Hoogland

Published Jan 1, 1980• 0 citations
OpenAlex

Abstract

The projective equivalence relation for representations of symmetry groups in quantum mechanics is refined to a so-called gauge equivalence relation. An example is given where this leads to a physically more relevant classification of elementary quantum mechanical systems. It is indicated how this idea might be applied to projective bundle .representations.

Related papers

Gauge equivalence of representations of symmetry groups in quantum mechanics

Henk Hoogland

The equivalence of representations of symmetry groups operating upon wavefunctions in configuration space is studied with regard to the (intuitive) notion of physical equivalence. A refinement of the usual projective equivalence relation is introduced, called gauge equivalence, for which the allowed unitary equivalence transformations are gauge transformations. For a Euclidean as well as for a Newton-Hooke symmetry group the gauge equivalence classes of unitary multiplier representations are determined. These examples support the assertion that equivalence from a physical viewpoint corresponds better to this new gauge equivalence concept than to the usual notion of projective equivalence.

Journal of Physics A Mathematical and General•Aug 1, 1978•10 citations

On local and global equivalence of wave functions in quantum mechanics

Henk Hoogland

de totstandkordng van dit proefschriftThis thesis consists of four parts.Notes and references are given at the end of each part.In part I we will give a review of the various notions of "representation" and "equivalence" that are used in theoretical physics and we will give a precise definition of the concept of "gauge equivalence".There we will meet the notion of gauge matrix.A method will be developed for obtaining the gauge matrices up to equivalence, as this is necessary in order to find the gauge equivalence classes of representations.In this method we will use the concept of superequivalence of group exponents, generalised here to group multipliers.Part I has a rather abstract mathematical character.Its results will be applied to physically interesting symmetry groups in the subsequent parts, where the notation and terminology of part I will be used freely.In part II we will consider the Poincaré as well as the Galilei symmetry group of uniform parallel electromagnetic fields.We will see that there are representations of these groups that are projectively equivalent, but that describe particles which arc, indeed, not "equivalent in every physical aspect".Specifically, particles in different Landau levels are described by projectively equivalent representations.However, those particles are not at all equivalent from the physical point of view; a transition between different Landau levels is not a trivial transformation of the mathematical description but it is a physical process that involves emission or absorption of energy.This is one of the most striking examples showing that, in general, projective equivalence of representations does not correspond to equivalence of physical systems.On the other hand, the different -13 -Landau levels are inequivalent with respect to the gauge equivalence concept.Another example which shows that gauge equivalence corresponds better than projective equivalence to "equivalence in every physical aspect" is provided by particles with different masses in uniform parallel electromagnetic fields (with Ε φ 0) in the Poincaré theory.Surprisingly, these particles are described by projectively equivalent representations; these representations are, however, gauge inequivalent.As a by-product we will find in part II what might be called a group theoretical derivation of the minimal coupling of a spinless particle in uniform parallel electromagnetic fields.Traditionally, the minimal coupling in quantum mechanics is "derived" by correspondence with classical mechanics.A derivation has also been given by a pseudo-invariance argument using instantaneous transformations.Although the latter argument may be the best that is available for arbitrary electromagnetic fields (that break space-time symmetry) it is in fact a "waste" of invariance in those cases where a special field is present that preserves a part of the symmetry.Indeed, in part II we obtain the minimal coupling in the Schrödinger and Klein-Gordon equations from the (exact rather than pseudo) symmetry group of the uniform parallel fields, without the detour via the free particle equations and without the "trick" of the substitution ІЭ ->• ІЭ -eA.This may be seen as a contribution to the process of remodelling traditional quantum mechanics by modern (group) theoretical tools.In part III other uniform electromagnetic fields are considered which are particular, in the sense that their connected symmetry groups are not isomorphic to the connected symmetry group of the uniform parallel fields.These are the uniform crossed fields under the Poincaré group and the uniform electric fields under the Galilei group.The equations -14of motion, including the minimal coupling, will be derived from the symmetry, analogous to part II.We will find a "Galilean subdynamics" for a Poincaré particle in uniform crossed fields.The exponents of the symmetry groups will give rise to an unexpected parameter in the equations of motion, which can be interpreted in terms of an anisotropic mass, corresponding to the anisotropy of the physical system due to the external field.In part IV we apply the notion of gauge equivalence of locally operating representations to some cosmologica!symmetry groups, operating on curved universes.For these groups the notion of superequivalence of group exponents plays a non-trivial role in the determination of a complete set of inequivalent gauge matrices.We will find that in cosmologica!models with one space dimension a free particle and a particle in an external field are described by projective representations of the symmetry group that are projectively equivalent but gauge inequivalent.This shows once more that the concept of gauge equivalence of locally operating representations is a more natural concept for the description of physical systems than that of the simple projective equivalence.As a by-product we will find that the minimal coupling is not necessarily the correct description of the interaction of a charged particle in an external field in a (relativistic) cosmologica!space-time.

•Jan 1, 1977•2 citations

Relativistic implications of the quantum phase

Stephen G Low

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations. We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.

arXiv:0903.4641•Journal of Physics Conference Series•Feb 8, 2012•1 citation

A gauge theory for a quantum system with isospin

En-Bing Lin

A quantum system with external symmetry is made into a principal fibre bundle with a nonAbelian Lie group as the structure group, and is equipped with a connection. A gauge theory for a nonrigid molecule on the basis of the observation that the vector bundle associated with the principal fibre bundle provides a setting for quantum mechanics of the internal molecular motion.

Journal of Physics A Mathematical and General•Sep 7, 1991•0 citations

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

Paolo Aniello

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.

arXiv:1811.00858•Journal of Physics Conference Series•Apr 1, 2019•5 citations

Projective Statistics and Spinors in Hilbert Space

Frank A. Wilczek

In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension $2^{(N-1)/2}$, and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian quantum Hall state realize this representation. Projective statistics is a consistent theoretical possibility in any dimension.

arXiv:hep-th/9806228•Jun 29, 1998•12 citations
The projective equivalence relation for representations of symmetry groups in quantum mechanics is refined to a so-called gauge equivalence relation. An example is given where this leads to a physically more relevant classification of elementary quantum mechanical systems. It is indicated how this idea might be applied to projective bundle .representations.
The equivalence of representations of symmetry groups operating upon wavefunctions in configuration space is studied with regard to the (intuitive) notion of physical equivalence. A refinement of the usual projective equivalence relation is introduced, called gauge equivalence, for which the allowed unitary equivalence transformations are gauge transformations. For a Euclidean as well as for a Newton-Hooke symmetry group the gauge equivalence classes of unitary multiplier representations are determined. These examples support the assertion that equivalence from a physical viewpoint corresponds better to this new gauge equivalence concept than to the usual notion of projective equivalence.
de totstandkordng van dit proefschriftThis thesis consists of four parts.Notes and references are given at the end of each part.In part I we will give a review of the various notions of "representation" and "equivalence" that are used in theoretical physics and we will give a precise definition of the concept of "gauge equivalence".There we will meet the notion of gauge matrix.A method will be developed for obtaining the gauge matrices up to equivalence, as this is necessary in order to find the gauge equivalence classes of representations.In this method we will use the concept of superequivalence of group exponents, generalised here to group multipliers.Part I has a rather abstract mathematical character.Its results will be applied to physically interesting symmetry groups in the subsequent parts, where the notation and terminology of part I will be used freely.In part II we will consider the Poincaré as well as the Galilei symmetry group of uniform parallel electromagnetic fields.We will see that there are representations of these groups that are projectively equivalent, but that describe particles which arc, indeed, not "equivalent in every physical aspect".Specifically, particles in different Landau levels are described by projectively equivalent representations.However, those particles are not at all equivalent from the physical point of view; a transition between different Landau levels is not a trivial transformation of the mathematical description but it is a physical process that involves emission or absorption of energy.This is one of the most striking examples showing that, in general, projective equivalence of representations does not correspond to equivalence of physical systems.On the other hand, the different -13 -Landau levels are inequivalent with respect to the gauge equivalence concept.Another example which shows that gauge equivalence corresponds better than projective equivalence to "equivalence in every physical aspect" is provided by particles with different masses in uniform parallel electromagnetic fields (with Ε φ 0) in the Poincaré theory.Surprisingly, these particles are described by projectively equivalent representations; these representations are, however, gauge inequivalent.As a by-product we will find in part II what might be called a group theoretical derivation of the minimal coupling of a spinless particle in uniform parallel electromagnetic fields.Traditionally, the minimal coupling in quantum mechanics is "derived" by correspondence with classical mechanics.A derivation has also been given by a pseudo-invariance argument using instantaneous transformations.Although the latter argument may be the best that is available for arbitrary electromagnetic fields (that break space-time symmetry) it is in fact a "waste" of invariance in those cases where a special field is present that preserves a part of the symmetry.Indeed, in part II we obtain the minimal coupling in the Schrödinger and Klein-Gordon equations from the (exact rather than pseudo) symmetry group of the uniform parallel fields, without the detour via the free particle equations and without the "trick" of the substitution ІЭ ->• ІЭ -eA.This may be seen as a contribution to the process of remodelling traditional quantum mechanics by modern (group) theoretical tools.In part III other uniform electromagnetic fields are considered which are particular, in the sense that their connected symmetry groups are not isomorphic to the connected symmetry group of the uniform parallel fields.These are the uniform crossed fields under the Poincaré group and the uniform electric fields under the Galilei group.The equations -14of motion, including the minimal coupling, will be derived from the symmetry, analogous to part II.We will find a "Galilean subdynamics" for a Poincaré particle in uniform crossed fields.The exponents of the symmetry groups will give rise to an unexpected parameter in the equations of motion, which can be interpreted in terms of an anisotropic mass, corresponding to the anisotropy of the physical system due to the external field.In part IV we apply the notion of gauge equivalence of locally operating representations to some cosmologica!symmetry groups, operating on curved universes.For these groups the notion of superequivalence of group exponents plays a non-trivial role in the determination of a complete set of inequivalent gauge matrices.We will find that in cosmologica!models with one space dimension a free particle and a particle in an external field are described by projective representations of the symmetry group that are projectively equivalent but gauge inequivalent.This shows once more that the concept of gauge equivalence of locally operating representations is a more natural concept for the description of physical systems than that of the simple projective equivalence.As a by-product we will find that the minimal coupling is not necessarily the correct description of the interaction of a charged particle in an external field in a (relativistic) cosmologica!space-time.
The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations. We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.
In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension 2(N−1)/22^{(N-1)/2}2(N−1)/2, and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian quantum Hall state realize this representation. Projective statistics is a consistent theoretical possibility in any dimension.