Coherent state quantization for conjugated variables

Abstract : In this work we focus on an alternative quantization method using generalized coherent states. The canonical method associates a pair of conjugated classical variables to their corresponding quantum observables identifying their Poisson bracket to a quantum commutator. These obervables, defined as self-adjoint operators acting on a particular Hilbert space, find their values in their spectral resolution an thus are linked to a Projection Valued (PV) measure. But there is an obstacle on the operator definition once we apply boundaries to the spectra. This restriction , described in a theorem by W. Pauli rises the question on the need of an alternative way of defining observables and opens the way to a new quantization protocol. Coherent state quantization proposes a quantum observable definition taking values through the mean value on a set of "coherent" non-orthogonal, overcomplete vectors in the Hilbert space H. These coherent states resolve the identity in H and are parametrized by a discrete parameter and a complex variable just as their homonyms for the harmonic oscillator. This last property makes them particularly useful to "translate" classical variables into well defined quantum operators. We have studied three particular cases where the self-adjoint operator definition is compromised. In first place we worked in a phase operator definition, corresponding to the angle-action classical pair. An example on how this idea could be extended to relative phases for the SU(N) group is given. The second example is the quantization of movement in an infinite well potential. Here the problematic momentum operator is defined properly. Finally we propose a time operator, conjugated to the Hamiltonian, for a free particle using SU(1,1) type coherent states on Poincaré half-planes
keyword : Quantification
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Pedro Lenin García de León. Coherent state quantization for conjugated variables. Computer Science [cs]. Université Paris-Est, 2008. English. ⟨NNT : 2008PEST0210⟩. ⟨tel-00432055⟩



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