Deformation Quantization for Systems with Fermions

Florian Becher, Nikolai Neumaier, Stefan Waldmann

Bing-Sheng Lin, Tai-Hua Heng

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.

I. Galaviz, H. García-Compeán, M. Przanowski +1 more

I. Galaviz

"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."

Anuar Kafuri

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.