Eisenhart lifts and symmetries of time-dependent systems |
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Authors: | M Cariglia C Duval GW Gibbons PA Horváthy |
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Institution: | 1. DEFIS, Universidade Federal de Ouro Preto, MG, Brazil;2. Dipartimento di Fisica, Universitá degli Studi di Camerino, Italy;3. Centre de Physique Théorique, Aix Marseille Université & Université de Toulon & CNRS UMR 7332, Case 907, 13288 Marseille, France;4. D.A.M.T.P., Cambridge University, UK;5. Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, France;6. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China |
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Abstract: | Certain dissipative systems, such as Caldirola and Kannai’s damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with n degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in n+2 dimensions, equipped with its covariantly constant null Killing vector field. Reparametrisation of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola–Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schrödinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space. |
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Keywords: | Caldirola&ndash Kanai model Damped oscillator Bargmann space Eisenhart lift Hubble model Dmitriev&ndash Zel&rsquo dovich equations |
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