Reconstruction of Quantum Fields: CCR, CAR and Transfields

TL;DR

This paper develops a new framework for quantum field reconstruction by generalizing creation-annihilation algebras through quotient spaces of distinguishable particle states, leading to novel algebraic structures.

Contribution

It introduces a method to derive new creation-annihilation algebras from quotient spaces, extending the traditional boson and fermion formalisms.

Findings

Reproduces partition functions of transtatistics

Generalizes bosons and fermions within operational principles

Provides a new algebraic approach to quantum field construction

Abstract

One of the traditional ways of introducing bosons and fermions is through creation-annihilation algebras. Historically, these have been associated with emission and absorption processes at the quantum level and are characteristic of the language of second quantization. In this work, we formulate the transition from first to second quantization by taking quotients of the state spaces of distinguishable particles, so that the resulting equivalence classes identify states that contain no information capable of distinguishing between particles, thereby generalising the usual symmetrisation procedure. Assuming that the resulting indistinguishable-particle space (i) admits an ordered basis compatible with how an observer may label the accessible modes, (ii) is invariant under unitary transformations of those modes, and (iii) supports particle counting as a mode-wise local operation, we derive a new class of creation-annihilation algebras. These algebras reproduce the partition functions of transtatistics, the maximal generalisations of bosons and fermions consistent with these operational principles.