Reduction Procedures in Classical and Quantum Mechanics

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

doi:10.1142/s0219887807002594

Cited by 17 publications

(18 citation statements)

References 51 publications

(62 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…gives a regular Lagrangian function on T Q which is evidently invariant under the left action of SU(2) on itself and under the right action of the gauge group 16 . In order to describe the corresponding equations of the motions we notice that S 3 ≃ SU(2) is parallelisable (as it is a Lie group manifold).…”

Section: 3mentioning

confidence: 99%

Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture

marmo,

schiavone,

zampini

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: 3mentioning

confidence: 99%

Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture

marmo,

schiavone,

zampini

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is just a very quick summary of the framework which has been developed in the last 30 years and which can be found in Refs. [29,7,30,31,32,33,34,35,36] and references therein. For the sake of simplicity, we shall focus only on the finite dimensional case.…”

Section: Summary Of Geometric Quantum Mechanicsmentioning

confidence: 99%

Statistics and Nosé formalism for Ehrenfest dynamics

Alonso¹,

Castro²,

Clemente-Gallardo³

et al. 2011

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract. Quantum dynamics (i.e., the Schrödinger equation) and classical dynamics (i.e., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of a Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature Molecular Dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.

show abstract

“…From a mathematical point of view it is natural to try to see how much of this theory may be extended to closed forms of higher degree. A number of authors have already made attempts at generalising the Hamiltonian picture to higher-degree, or multi-, phase spaces, often motivated by the interest in various field theories [11,12,19,1,2]. Indeed string-and M-theories with fluxes give a number of geometries equipped with closed differential forms of varying degrees, see [20] for one such example.…”

Section: Introductionmentioning

confidence: 99%

Closed forms and multi-moment maps

Madsen

Swann

2012

Geom Dedicata

View full text Add to dashboard Cite

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.Comment: 36 page

show abstract

Reduction Procedures in Classical and Quantum Mechanics

Cited by 17 publications

References 51 publications

Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture

Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture

Statistics and Nosé formalism for Ehrenfest dynamics

Closed forms and multi-moment maps

Contact Info

Product

Resources

About