An alternative approach to canonical quantizationof the radiation damping

Abstract. Some aspects of the "exotic" particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.

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“…It is worth mentioning that the acceleration-dependent model presented in this Section can be related to radiation damping [35].…”

Section: Commutative -Versus Noncommutative Planementioning

confidence: 99%

Exotic Galilean Symmetry and Non-Commutative Mechanics

Horváthy

Stichel

Martina

2010

SIGMA

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“…This Lagrangian is invariant under time translations and depending on the form of the potential V (q), it could have other symmetries. In particular, the kinetic term and the nonconservative potential have an SO(1, 1) symmetry [44]…”

Section: A Linear Dissipative Forcesmentioning

confidence: 99%

“…We give a formulation for Noether theorem, considering the transformations which let invariant only the conservative Lagrangian, as well as the transformations which let invariant the whole nonconservative Lagrangian. Among the last, there may be transformations that mix both types of variables [44,47]. The application of Noether theorem for the symmetries of the nonconservative Lagrangian leads to conserved quantities that are generators of the corresponding transformations.…”

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confidence: 99%

On the Lagrangian description of dissipative systems

Martínez-Pérez

Ramı́rez

2018

Journal of Mathematical Physics

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We consider the Lagrangian formulation with duplicated variables of dissipative mechanical systems. The application of Noether theorem leads to physical observable quantities which are not conserved, like energy and angular momentum, and conserved quantities like the Hamiltonian, that generate symmetry transformations and do not correspond to observables. We show that there are simple relations among the equations satisfied by these two types of quantities. In the case of the damped harmonic oscillator, from the quantities obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky. Further, if we consider the whole dynamics, the degrees of freedom separate into a physical and an unphysical sector. We analyze several cases, with linear and nonlinear dissipative forces; the physical consistency of the solutions is ensured observing that the unphysical sector has always the trivial solution.PACS numbers: 04.20.Fy,12.60.Jv * Electronic address: nephtalieliceo@hotmail.com † Electronic address: cramirez@fcfm.buap.mx 2 Hamilton variational principle. In this approach the variation of the action is done with boundary conditions only at the initial time, independently for each of both variables, and at the final time these variables must coincide. A similar development for classical and quantum mechanics was given by means of an extension of the Closed Time Path formalism to classical mechanics by Polonyi [40,41], who in [42] considers the issue of breaking of time reversal symmetry.The main interest in the study of phenomenological dissipative systems is on their quantum description. Classically, the doubled variable formalism allows to write the equations of motion and after that the additional variables are somehow discarded. In fact, these variables are considered as an artifice which takes account of the dissipative external influence, the whole system being isolated. However, from its construction, the nonconservative Lagrangian has not the standard form due to the time reversed characteristics of the additional sector, i.e. the kinetic term is not positive definite and the potential appears with an unstable term. Thus, an interpretation of its outcome as a whole is not obvious. On the other side, in a quantum theory every interacting degree of freedom in general contributes to the probabilities, spectra and mean values, as they form part of the operator algebra. Thus, it would be desirable to consider the classical theory taking into account the whole dynamics. Moreover, a general knowledge of the relevant quantities in the theory, as delivered e.g. by Noether theorem, is necessary for the definition of the Hilbert space. Actually, in the doubled variable approach, Noether theorem has been applied considering the conservation laws of the conservative part, and these laws are violated due to the dissipative terms [41,43]. Furthermore, Noether theorem has been applied in similar approaches to the symmetries of the whole doubled variables action in [44,45], and for time dependent lagrangians in [46]. ...

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“…On the hyperbolic plane, the equations (12) shows that the dissipative term actully acts as a coupling between the systems r (1) and r (2) . Recently, one of us, in [28] have studied the canonical quantization of the radiation damping. A Hamiltonian analysis is done in commutative and noncommutative scenarios, what leads to the quantization of the system, where the dynamical group structure associated with our system is that of SU (1, 1).…”

Section: Indirect Lagrangian Representation Of the Radiation Dampingmentioning

confidence: 99%

“…Its symmetries and the corresponding conserved Noether charges were discused. It is shown that this supersymmetric version provides a supersymmetric generalization of the Galilei algebra of the model [28], where the supersymmetric action can be split into dynamically independent external and internal sectors.…”

Section: Indirect Lagrangian Representation Of the Radiation Dampingmentioning

confidence: 99%