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Revista Mexicana de Física

Deformation quantization for fermionic fields

I. Galaviz

Published Aug 1, 2007• 4 citations
OpenAlex

Abstract

"The deformation quantization of any Grassmann free field, or fermionic free field, in particular, the Dirac free field is discussed. Stratonovich-Weyl quantizer, Moyal product andWigner functionals are obtained for this field by deforming suitable Fermi oscillator variables. In additionthe propagator of the Dirac field is computed in this context."

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Rarita-Schwinger (RS) quantum free field is reexamined in the context of deformation quantization.It is found out that the subsidiary condition does not introduce any change either in the Wigner function or in other aspects of the deformation quantization formalism, in relation to the Dirac field case.This happens because the vector structure of the RS field imposes constraints on the space of wave function solutions and not on the operator structure.The RS propagator was also calculated within this formalism.

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"The deformation quantization of any Grassmann free field, or fermionic free field, in particular, the Dirac free field is discussed. Stratonovich-Weyl quantizer, Moyal product andWigner functionals are obtained for this field by deforming suitable Fermi oscillator variables. In additionthe propagator of the Dirac field is computed in this context."
Rarita-Schwinger (RS) quantum free field is reexamined in the context of deformation quantization.It is found out that the subsidiary condition does not introduce any change either in the Wigner function or in other aspects of the deformation quantization formalism, in relation to the Dirac field case.This happens because the vector structure of the RS field imposes constraints on the space of wave function solutions and not on the operator structure.The RS propagator was also calculated within this formalism.
In order to quantize systems involving second-class constraints, one should use Dirac bracket instead of Poisson bracket. Furthermore, one can specify a star product in which the term linear in [Formula: see text] is proportional to the Dirac bracket. In this way an oscillator system in a deformed fermionic phase space is analyzed and the corresponding energy level and Wigner functions are evaluated according to scheme of deformation quantization. We also study the entanglement entropy induced by the deformation of the fermionic phase space.
This thesis addresses a fundamental problem in deformation quantization: the difficulty of calculating the star-exponential, the symbol of the evolution operator, due to convergence issues. Inspired by the formalism that connects the star-exponential with the quantum propagator for bosonic systems, this work develops the analogous extension for the fermionic case. A rigorous method, based on Grassmann variables and coherent states, is constructed to obtain a closed-form expression for the fermionic star-exponential from its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, allowing for the calculation of the ground state energy directly in phase space. Finally, the method is validated by successfully applying it to the simple and driven harmonic oscillators, where it is demonstrated that a simplified ("naive") approach (with an ad-hoc "remediation") is a valid weak-coupling limit of the rigorous ("meticulous") formalism, thereby providing a new and powerful computational tool for the study of fermionic systems.
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is CinftyC^{\\infty}Cinfty, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is CPinftyCP^{\\infty}CPinfty endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always 1overhbar{1 \\over \\hbar}1overhbar.
This chapter is an introduction to some computation techniques for fermionic states. After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin [23, 26], we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formula for fermionic observables. In particular there exists a Moyal product formula. As an application we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators.
Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.