Quantum Mechanics as a Generalization of Nambu Dynamics to the Weyl-Wigner Formalism

1997 ◽  
Vol 52 (1-2) ◽  
pp. 9-12
Author(s):  
Iwo Bialynicki-Birula ◽  
P.J. Morrison
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Lie Algebra ◽  
Hamiltonian Form ◽  
Prominent Feature ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Natural Choice ◽  
Wigner Representation

Abstract It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.

scholarly journals Dispersion Operators Algebra and Linear Canonical Transformations

2016 ◽  
Author(s):  
Raoelina Andriambololona ◽  
Ravo Tokiniaina Raymond Ranaivoson ◽  
Hasimbola Damo Emile Randriamisy ◽  
Hanitriarivo Rakotoson
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Lie Algebra ◽  
Canonical Transformation ◽  
Space Representation ◽  
Canonical Transformations ◽  
Linear Canonical Transformation ◽  
Multidimensional Generalization ◽  
Phase Space Representation ◽  
Operators Algebra

This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.

scholarly journals QUANTUM MECHANICS IN INFINITE SYMPLECTIC VOLUME

2004 ◽  
Vol 19 (05) ◽  
pp. 349-355
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Bergman Metric ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Classical Phase Space ◽  
Dimensional Projective Space

We quantise complex, infinite-dimensional projective space CP(ℋ). We apply the result to quantise a complex, finite-dimensional, classical phase space [Formula: see text] whose symplectic volume is infinite, by holomorphically embedding it into CP(ℋ). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on [Formula: see text] and the Fubini–Study metric on CP(ℋ). Then the Hilbert-space bundle over [Formula: see text] is the pullback, by the embedding, of the Hilbert-space bundle over CP(ℋ).

scholarly journals A free Lie algebra approach to curvature corrections to flat space-time

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Joaquim Gomis ◽  
Axel Kleinschmidt ◽  
Diederik Roest ◽  
Patricio Salgado-Rebolledo
Keyword(s):  
Lie Algebra ◽  
Flat Space ◽  
Geodesic Equation ◽  
Space Time ◽  
Free Lie Algebra ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Algebra Approach ◽  
Poincaré Algebra ◽  
Particle Action

Abstract We investigate a systematic approach to include curvature corrections to the isometry algebra of flat space-time order-by-order in the curvature scale. The Poincaré algebra is extended to a free Lie algebra, with generalised boosts and translations that no longer commute. The additional generators satisfy a level-ordering and encode the curvature corrections at that order. This eventually results in an infinite-dimensional algebra that we refer to as Poincaré∞, and we show that it contains among others an (A)dS quotient. We discuss a non-linear realisation of this infinite-dimensional algebra, and construct a particle action based on it. The latter yields a geodesic equation that includes (A)dS curvature corrections at every order.

scholarly journals QUANTUM MECHANICS AS A GAUGE THEORY OF METAPLECTIC SPINOR FIELDS

1998 ◽  
Vol 13 (22) ◽  
pp. 3835-3883 ◽  
Author(s):  
M. REUTER
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Phase Space ◽  
Hilbert Spaces ◽  
Canonical Quantization ◽  
Spinor Representation ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Spinor Fields ◽  
Matter Fields

A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.

scholarly journals Heisenberg Doubles for Snyder-Type Models

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1055
Author(s):  
Stjepan Meljanac ◽  
Anna Pachoł
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Dual Pair ◽  
One Dimension ◽  
Lorentz Algebra ◽  
Explicit Formulae ◽  
Heisenberg Double

A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

scholarly journals OFF-CRITICAL CURRENT ALGEBRAS

1995 ◽  
Vol 10 (12) ◽  
pp. 1717-1736 ◽  
Author(s):  
E. ABDALLA ◽  
M.C.B. ABDALLA ◽  
G. SOTKOV ◽  
M. STANISHKOV
Keyword(s):  
Abelian Group ◽  
Critical Current ◽  
General Structure ◽  
Dimensional Algebra ◽  
Fermionic Model ◽  
Infinite Dimensional ◽  
Conformally Invariant ◽  
Current Algebras

We discuss the infinite-dimensional algebras appearing in integrable perturbations of conformally invariant theories, with special emphasis on the structure of the consequent non-Abelian infinite-dimensional algebra generalizing W∞ to the case of a non-Abelian group. We prove that the pure left sector as well as the pure right sector of the thus-obtained algebra coincides with the conformally invariant case. The mixed sector is more involved, although the general structure seems to be near to being unraveled. We also find some subalgebras that correspond to Kac-Moody algebras. The constraints imposed by the algebras are very strong and, in the case of the massive deformation of a non-Abelian fermionic model, the symmetry alone is enough to fix the two- and three-point functions of the theory.

A REPRESENTATION OF THE BELAVKIN EQUATION VIA PHASE SPACE FEYNMAN PATH INTEGRALS

2004 ◽  
Vol 07 (04) ◽  
pp. 507-526 ◽  
Author(s):  
S. ALBEVERIO ◽  
G. GUATTERI ◽  
S. MAZZUCCHI
Keyword(s):  
Phase Space ◽  
Continuous Measurement ◽  
Quantum Particle ◽  
Oscillatory Integral ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Infinite Dimensional ◽  
Feynman Path Integrals ◽  
Characteristics Method ◽  
Stochastic Characteristics

The Belavkin equation, describing the continuous measurement of the momentum of a quantum particle, is studied. The existence and uniqueness of its solution is proved via analytic tools. A stochastic characteristics method is applied. A rigorous representation of the solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the phase space is also given.

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