Unitary quantization and para-Fermi statistics of order two

TL;DR

This paper explores the connection between unitary quantization and para-Fermi statistics of order two, introducing an extension of Green's ansatz, revealing new relations, and constructing explicit transformations and coherent states.

Contribution

It proposes an extended Green's ansatz for para-Fermi fields, linking trilinear relations to unitary equivalence, and constructs explicit Klein transformations and coherent states for para-Fermi oscillators.

Findings

Extended Green's ansatz for para-Fermi fields.

Explicit Klein transformation reducing to Fermi relations.

Existence of two alternative coherent state definitions.

Abstract

A connection between a unitary quantization scheme and para-Fermi statistics of order 2 is considered. An appropriate extension of Green's ansatz is suggested. This extension allows one to transform bilinear and trilinear commutation relations for the annihilation and creation operators of two different para-Fermi fields $\phi_{a}$ and $\phi_{b}$ into identity. The way of incorporating para-Grassmann numbers $\xi_{k}$ into a general scheme of uniquantization is also offered. For parastatistics of order 2 a new fact is revealed, namely, the trilinear relations containing both the para-Grassmann variables $\xi_{k}$ and the field operators $a_{k}$, $b_{m}$ under a certain invertible mapping go over into the unitary equivalent relations, where commutators are replaced by anticommutators and vice versa. It is shown that the consequence of this circumstance is the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green's components of the operators $a_{k}$, $b_{m}$ is constructed in an explicit form that enables us to reduce the initial commutation rules for the components to the normal commutation relations of ordinary Fermi fields. A nontrivial connection between trilinear commutation relations of the unitary quantization scheme and so-called Lie-supertriple system is analysed. A brief discussion of the possibility of embedding the Duffin-Kemmer-Petiau theory into the unitary quantization scheme is provided.