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Publications of the Research Institute for Mathematical Sciences

Quantum Mechanics and Nilpotent Groups. I. The Curved Magnetic Field

Palle E. T. Jørgensen, W. H. Klink

Published Oct 31, 1985• 24 citations
Publisher

Abstract

The quantum mechanics of massive spinless particles in an external magnetic field polynomial in position variables is shown to be related to nilpotent Lie groups. By using the known representation structure of such groups, the Hilbert spaces of the quantum mechanical systems can be decomposed into irreducible representations of the nilpotent groups. Such a decomposition is given explicitly for constant and curved magnetic fields. Since the Hamiltonian for polynomial magnetic fields is quadratic in the Lie algebra elements, its spectrum can be found using the representation structure of nilpotent groups. The explicit time dependence of the system can also be found by solving the heat equation on nilpotent groups. These ideas are worked out for the constant magnetic field, where the solution is well known, and the curved magnetic field, where it is not. Generalizations to other systems whose interaction terms are polynomial are also given.

DOI
10.2977/prims/1195178792

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Quantum kinematics and the Lie group structure of non-Abelian quantum mechanics

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A formalism of quantum kinematics which rests on the regular representation of a Lie group is proposed. This representation leads to explicit canonical commutation relations for non-Abelian dynamical variables which, together with the Lie algebra, define the kinematic algebra. Generalised equations of motion for the group-parameter-dependent operators, as well as generalised wave equations, are introduced over the group manifold considered as a background arena. The formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system.

Journal of Physics A Mathematical and General•Jun 21, 1985•17 citations

Quasiclassical and Quantum Systems of Angular Momentum. Part II. Quantum Mechanics on Lie Groups and Methods of Group Algebras

Jan Jerzy Sławianowski, Vasyl Kovalchuk, Agnieszka Martens +2 more

In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Basing on differential geometry enables one to introduce explicitly representation of important physical quantities and formulate the general ideas of quasiclassical representation and classical analogy.

arXiv:1008.0512•Aug 3, 2010•12 citations

Canonically relativistic quantum mechanics: Casimir field equations of the quaplectic group

Stephen G. Low

The Hilbert space of the unitary irreducible representations of a Lie group that is a quantum dynamical group are identified with the quantum state space. Hermitian representation of the algebra are observables. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. A general class of dynamical groups are semidirect products K *s N for which the representations are given by Mackey's theory. The homogeneous group K must be a subgroup of the automorphisms of the normal group N. The archetype dynamical group is the Poincare group. The field equations defined by the representations of the Casimir operators define the basic equations of physics; Klein-Gordon, Dirac, Maxwell and so forth. This paper explores a more general dynamical group candidate that is also a semi-direct product but where the 'translation' normal subgroup N is now the Heisenberg group. The relevant automorphisms of the Heisenberg group are the symplectic group. This together with the requirement for an orthogonal metric leads to the pseudo-unitary group for the homogeneous group K. The physical meaning and motivation of this group, called the quaplectic group, is presented and the Hermitian irreducible representations of the algebra are determined. As with the Poincare group, choice of the group defines the Hilbert space of representations that are identified with quantum particle states. The field equations that are the eigenvalue equations for the representation of the Casimir operators, are obtained and investigated. The theory embodies the Born reciprocity principle and a new relativity principle.

arXiv:math-ph/0404077•Apr 30, 2004•0 citations

Group theoretical methods in physics: Proceedings of the IX international colloquium, held at Cocoyoc, Mexico, June 23-27, 1980

Kurt Bernardo Wolf

Group theory of the Morse potential.- Four Euclidean conformal group approach to the multiphoton processes in the H-atom.- Landau theory, symmetry breaking and the chain criterion.- The configuration dN in cubic symmetry: A symmetry adapted weak field approach.- Group theory of the effective potential.- Bloch theorem for crystals with structural distortions.- Global symmetries of spin systems defined on abelian manifolds.- Phases and conjugacy classes.- The probability density in non relativistic quantum mechanics and the (extended) Galilei group.- Quaternionic quantum mechanics and Adler's chromostatics.- Coherent angular momentum states for the isotropic oscillator.- How to measure the canonical momentum p and operators p + f(q).- A uniqueness result for the segal quantization of a classical system with symmetries.- On the singular points of the matrix elements of the resolvent operator.- Limitations on quantum measuring processes.- Classical and quantum physics on phase space: The state spaces.- Quantum dynamical semigroups and complete positivity. An application to isotropic spin relaxation.- On geometric quantization of the rigid body.- A unified theory of simple dynamical systems.- Geometry of the time-dependent variational principle in quantum mechanics.- Variational characterization of Hamiltonian vectorfields as lie algebra of canonical diffeomorphisms.- The lie theory of extended groups in classical mechanics - Is it of relevance to quantum mechanics?.- On the structure of phase space.- Quantization as mapping and as deformation.- Metric and group structures from quantum physics.- Spontaneous symmetry breaking and restricted superposition principle.- Many-body quantum mechanics as symplectic dynamics.- A nonlinear superposition principle for Riccati equations of the conformal type.- Backlund transformations, conservation laws and lie symmetries for the generalized nonlinear Schrodinger equation.- Killing tensors and variable separation for Hamilton-Jacobi equations.- Lie pseudogroups and gauge theory.- Conformal invariance and symmetry breaking for the one-dimensional wave equation.- The flavour sequence and superselection rules.- The Zitterbewegung of the electron and its compact phase space.- New experimental evidence for a new interpretation of flavor SU(3).- Baryon magnetic moments in broken SU(6).- The Racah-Wigner algebra and the masses of exotic hadrons.- Simple construction of a relativistic wave equation with arbitrary mass and spin and belinfante's 1/S-conjecture.- Poincare group, symplectic group, and relativistic extended hadrons consisting of spin 1/2 quarks.- An operator S matrix theory of the strongly interacting particles.- Generalized Galilei invariant partial wave expansions of the scattering amplitude for collisions between two particles with arbitrary spin.- Ternary algebras as the basis of a dynamical theory of subconstituents.- Solitons as sections in non trivial bundles.- Effects of heavy colored higgs scalars in grand unified theories.- Categories of nonlinear group realizations: A possible explanation for the multiple states of charge.- Classical solutions of non-linear ?-models and their quantum fluctuations.- Higgs representations for SO(n) gauges theories.- Multivortex configurations in the Abelian Higgs model.- Quark confinement in field theories with discrete gauge symmetry Z(3).- Proliferation of effective fields by spontaneous symmetry breakdown.- Explicit solutions for the scalar coupled Yang-Mills system in Minkowski space.- Dynamical supersymmetry SU(6/4) in nuclei.- Group theory of the IBA and its applications.- Bases for irreps of U(6) and three body problems.- Intrinsic sphape of nuclei.- Group theory and nuclear high spin phenomena.- SU3 symmetry in resonating group calculations. Toward a microscopic theory of 12C + 12C molecular resonances.- Collective motion, composite particle structure, and symplectic groups in nuclei.- Group theory and confrontation of nuclear collective models.- Matrix elements of generators of U(6) in a U(6)?U(3)?U(2) basis.- Hartree-Fock one-body dynamics and U(n) co-adjoint orbits.- Group-theoretical aspects of the many-body problem in nuclear theory.- The continuing quest for exact solutions of Einstein's field eouations - - new goals for the eighties.- All non-twisting N's with cosmological constant.- Can quantum effects prevent spacetime collapse?.- Complex plane representation of the Geroch group and a proof of a geroch conjecture.- The Geroch group and soliton solutions of the stationary axially symmetric Einstein's equations.- Geometry of projective asymptotic twistor space.- A truly relativistic approach of the concept of time.- Bases in Sp(2n) representation space.- Semiunitary projective representations of non-connected lie groups: A criterion for the choice of the unitary subgroup.- SU(mn) ? 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S(f1) x S(f2) isoscalar factors.- Structure of enveloping algebras of low-rank groups.- Computer programs for the reduction of symmetrised nth powers of space group irreducible representations using gard's subgroup method.- Some remarks on the uniqueness and reality of Clebsch-Gordan coefficients for corepresentations.- On some special relations involving 3-jm symbols.- Two problems related to the Stone-von Neumann-Mackey uniqueness theorem.- Non-complex representations and their relation to antiunitary symmetry.- Representation theory of compact groups.- Representation theory of compact groups / Shift operators for discrete unitary representations of O(p,q).- Representation theory of compact groups / Maximal Abelian subalgebras of sp(2n,?) and their applications in physics.- Representation theory of compact groups / Polynomial space group tensors.- A permutation for SUn consistent with duality.- Integral transform representation of SL (2,R).- The theory of G? supermanifolds.- Casimir operations and the relations of the simplest supersymmetric superalgebra.- On realizations of graded lie algebras and on properties of d-polynomials.- On gauge formulations of gravitation theories.- Lagrangian formalism for superfields with internal symmetry.- Causal propagation in second order Lorentz-invariant wave equations.- The exceptional group E8 for grand unification.- Completely integrable N-body problems in three-dimension and their relativistic generalization.- Ward-Takashi identities as a necessary and sufficient condition for gauce invariance.- Supersymmetric generalization of Riemannian symmetric pairs.- Composite gauge fields and Riemannian geometry.- Gauge-theory ghosts and unitarity - Progress in the geometrical treatment.- Color in internal supersymmetry unification.- Canonical (possibly Lagrangian) realizations of the Poincare' group with increasing mass-spin trajectories.- Similarity analysis of wave propagation in axially nonlinear nonhomogeneous systems.

•Jan 1, 1980•2 citations

Invariants and Lie algebraic solutions of Schrödinger equation

Franco Salmistraro, Riccardo Rosso

A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.

Journal of Mathematical Physics•Sep 1, 1993•8 citations

Path Integrals on Group Manifolds: The Representation Independent Propagator for General Lie Groups

Wolfgang A. Tomé

The quantization of physical systems moving on group and symmetric spaces has been an area of active research over the past three decades. This book shows that it is possible to introduce a representation-independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables.Furthermore, the book shows that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations, and although the configuration space is in general a multidimensional curved manifold, it is shown that the resulting lattice phase space path integral has the form of a lattice phase space path integral on a multidimensional flat manifold. Hence, a novel and extremely natural phase space path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phase space path integral quantization is (a) exact, (b) more general than, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds.To illustrate the general theory, a representation-independent propagator is explicitly constructed for SU(2) and the affine group

•Jun 15, 1998•0 citations
The quantum mechanics of massive spinless particles in an external magnetic field polynomial in position variables is shown to be related to nilpotent Lie groups. By using the known representation structure of such groups, the Hilbert spaces of the quantum mechanical systems can be decomposed into irreducible representations of the nilpotent groups. Such a decomposition is given explicitly for constant and curved magnetic fields. Since the Hamiltonian for polynomial magnetic fields is quadratic in the Lie algebra elements, its spectrum can be found using the representation structure of nilpotent groups. The explicit time dependence of the system can also be found by solving the heat equation on nilpotent groups. These ideas are worked out for the constant magnetic field, where the solution is well known, and the curved magnetic field, where it is not. Generalizations to other systems whose interaction terms are polynomial are also given.
A formalism of quantum kinematics which rests on the regular representation of a Lie group is proposed. This representation leads to explicit canonical commutation relations for non-Abelian dynamical variables which, together with the Lie algebra, define the kinematic algebra. Generalised equations of motion for the group-parameter-dependent operators, as well as generalised wave equations, are introduced over the group manifold considered as a background arena. The formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system.
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Basing on differential geometry enables one to introduce explicitly representation of important physical quantities and formulate the general ideas of quasiclassical representation and classical analogy.
The Hilbert space of the unitary irreducible representations of a Lie group that is a quantum dynamical group are identified with the quantum state space. Hermitian representation of the algebra are observables. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. A general class of dynamical groups are semidirect products K *s N for which the representations are given by Mackey's theory. The homogeneous group K must be a subgroup of the automorphisms of the normal group N. The archetype dynamical group is the Poincare group. The field equations defined by the representations of the Casimir operators define the basic equations of physics; Klein-Gordon, Dirac, Maxwell and so forth. This paper explores a more general dynamical group candidate that is also a semi-direct product but where the 'translation' normal subgroup N is now the Heisenberg group. The relevant automorphisms of the Heisenberg group are the symplectic group. This together with the requirement for an orthogonal metric leads to the pseudo-unitary group for the homogeneous group K. The physical meaning and motivation of this group, called the quaplectic group, is presented and the Hermitian irreducible representations of the algebra are determined. As with the Poincare group, choice of the group defines the Hilbert space of representations that are identified with quantum particle states. The field equations that are the eigenvalue equations for the representation of the Casimir operators, are obtained and investigated. The theory embodies the Born reciprocity principle and a new relativity principle.
Group theory of the Morse potential.- Four Euclidean conformal group approach to the multiphoton processes in the H-atom.- Landau theory, symmetry breaking and the chain criterion.- The configuration dN in cubic symmetry: A symmetry adapted weak field approach.- Group theory of the effective potential.- Bloch theorem for crystals with structural distortions.- Global symmetries of spin systems defined on abelian manifolds.- Phases and conjugacy classes.- The probability density in non relativistic quantum mechanics and the (extended) Galilei group.- Quaternionic quantum mechanics and Adler's chromostatics.- Coherent angular momentum states for the isotropic oscillator.- How to measure the canonical momentum p and operators p + f(q).- A uniqueness result for the segal quantization of a classical system with symmetries.- On the singular points of the matrix elements of the resolvent operator.- Limitations on quantum measuring processes.- Classical and quantum physics on phase space: The state spaces.- Quantum dynamical semigroups and complete positivity. An application to isotropic spin relaxation.- On geometric quantization of the rigid body.- A unified theory of simple dynamical systems.- Geometry of the time-dependent variational principle in quantum mechanics.- Variational characterization of Hamiltonian vectorfields as lie algebra of canonical diffeomorphisms.- The lie theory of extended groups in classical mechanics - Is it of relevance to quantum mechanics?.- On the structure of phase space.- Quantization as mapping and as deformation.- Metric and group structures from quantum physics.- Spontaneous symmetry breaking and restricted superposition principle.- Many-body quantum mechanics as symplectic dynamics.- A nonlinear superposition principle for Riccati equations of the conformal type.- Backlund transformations, conservation laws and lie symmetries for the generalized nonlinear Schrodinger equation.- Killing tensors and variable separation for Hamilton-Jacobi equations.- Lie pseudogroups and gauge theory.- Conformal invariance and symmetry breaking for the one-dimensional wave equation.- The flavour sequence and superselection rules.- The Zitterbewegung of the electron and its compact phase space.- New experimental evidence for a new interpretation of flavor SU(3).- Baryon magnetic moments in broken SU(6).- The Racah-Wigner algebra and the masses of exotic hadrons.- Simple construction of a relativistic wave equation with arbitrary mass and spin and belinfante's 1/S-conjecture.- Poincare group, symplectic group, and relativistic extended hadrons consisting of spin 1/2 quarks.- An operator S matrix theory of the strongly interacting particles.- Generalized Galilei invariant partial wave expansions of the scattering amplitude for collisions between two particles with arbitrary spin.- Ternary algebras as the basis of a dynamical theory of subconstituents.- Solitons as sections in non trivial bundles.- Effects of heavy colored higgs scalars in grand unified theories.- Categories of nonlinear group realizations: A possible explanation for the multiple states of charge.- Classical solutions of non-linear ?-models and their quantum fluctuations.- Higgs representations for SO(n) gauges theories.- Multivortex configurations in the Abelian Higgs model.- Quark confinement in field theories with discrete gauge symmetry Z(3).- Proliferation of effective fields by spontaneous symmetry breakdown.- Explicit solutions for the scalar coupled Yang-Mills system in Minkowski space.- Dynamical supersymmetry SU(6/4) in nuclei.- Group theory of the IBA and its applications.- Bases for irreps of U(6) and three body problems.- Intrinsic sphape of nuclei.- Group theory and nuclear high spin phenomena.- SU3 symmetry in resonating group calculations. Toward a microscopic theory of 12C + 12C molecular resonances.- Collective motion, composite particle structure, and symplectic groups in nuclei.- Group theory and confrontation of nuclear collective models.- Matrix elements of generators of U(6) in a U(6)?U(3)?U(2) basis.- Hartree-Fock one-body dynamics and U(n) co-adjoint orbits.- Group-theoretical aspects of the many-body problem in nuclear theory.- The continuing quest for exact solutions of Einstein's field eouations - - new goals for the eighties.- All non-twisting N's with cosmological constant.- Can quantum effects prevent spacetime collapse?.- Complex plane representation of the Geroch group and a proof of a geroch conjecture.- The Geroch group and soliton solutions of the stationary axially symmetric Einstein's equations.- Geometry of projective asymptotic twistor space.- A truly relativistic approach of the concept of time.- Bases in Sp(2n) representation space.- Semiunitary projective representations of non-connected lie groups: A criterion for the choice of the unitary subgroup.- SU(mn) ? SU(m) x SU(n) isoscalar factors and S(f1+f2) ? S(f1) x S(f2) isoscalar factors.- Structure of enveloping algebras of low-rank groups.- Computer programs for the reduction of symmetrised nth powers of space group irreducible representations using gard's subgroup method.- Some remarks on the uniqueness and reality of Clebsch-Gordan coefficients for corepresentations.- On some special relations involving 3-jm symbols.- Two problems related to the Stone-von Neumann-Mackey uniqueness theorem.- Non-complex representations and their relation to antiunitary symmetry.- Representation theory of compact groups.- Representation theory of compact groups / Shift operators for discrete unitary representations of O(p,q).- Representation theory of compact groups / Maximal Abelian subalgebras of sp(2n,?) and their applications in physics.- Representation theory of compact groups / Polynomial space group tensors.- A permutation for SUn consistent with duality.- Integral transform representation of SL (2,R).- The theory of G? supermanifolds.- Casimir operations and the relations of the simplest supersymmetric superalgebra.- On realizations of graded lie algebras and on properties of d-polynomials.- On gauge formulations of gravitation theories.- Lagrangian formalism for superfields with internal symmetry.- Causal propagation in second order Lorentz-invariant wave equations.- The exceptional group E8 for grand unification.- Completely integrable N-body problems in three-dimension and their relativistic generalization.- Ward-Takashi identities as a necessary and sufficient condition for gauce invariance.- Supersymmetric generalization of Riemannian symmetric pairs.- Composite gauge fields and Riemannian geometry.- Gauge-theory ghosts and unitarity - Progress in the geometrical treatment.- Color in internal supersymmetry unification.- Canonical (possibly Lagrangian) realizations of the Poincare' group with increasing mass-spin trajectories.- Similarity analysis of wave propagation in axially nonlinear nonhomogeneous systems.
A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.
The quantization of physical systems moving on group and symmetric spaces has been an area of active research over the past three decades. This book shows that it is possible to introduce a representation-independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables.Furthermore, the book shows that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations, and although the configuration space is in general a multidimensional curved manifold, it is shown that the resulting lattice phase space path integral has the form of a lattice phase space path integral on a multidimensional flat manifold. Hence, a novel and extremely natural phase space path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phase space path integral quantization is (a) exact, (b) more general than, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds.To illustrate the general theory, a representation-independent propagator is explicitly constructed for SU(2) and the affine group