Understanding the second quantization of fermions in Clifford and in Grassmann space -- New way of second quantization of fermions, Part I

TL;DR

This paper introduces a novel approach to second quantization of fermions using Clifford and Grassmann algebras, explaining their properties and relations to Dirac's postulates, with a focus on integer spin fermions in Part I.

Contribution

It proposes a new method for second quantization of fermions based on algebraic basis vectors, avoiding Dirac's conditions, and explores properties of integer spin fermions in Grassmann space.

Findings

Grassmann fermions manifest integer spins.

Operators satisfy anticommutation relations on vacuum and Hilbert space.

Solutions to equations of motion for Grassmann fermions are presented.

Abstract

Both algebras, Clifford and Grassmann, offer "basis vectors" for describing the internal degrees of freedom of fermions. The oddness of the "basis vectors", transferred to the creation operators, which are tensor products of the finite number of "basis vectors" and the infinite number of momentum basis, and to their Hermitian conjugated partners annihilation operators, offers the second quantization of fermions without postulating the conditions proposed by Dirac, enabling the explanation of the Dirac's postulates. But while the Clifford fermions manifest the half integer spins -- in agreement with the observed properties of quarks and leptons and antiquarks and antileptons -- the "Grassmann fermions" manifest the integer spins. In Part I properties of the creation and annihilation operators of integer spins "Grassmann fermions" are presented and the proposed equations of motion solved. The anticommutation relations of second quantized integer spin fermions are shown when applying on the vacuum state as well as when applying on the Hilbert space of the infinite number of "Slater determinants" with all the possibilities of empty and occupied "fermion states". In Part II the conditions are discussed under which the Clifford algebras offer the appearance of the second quantized fermions, enabling as well the appearance of families. In both parts, Part I and Part II, the relation between the Dirac way and our way of the second quantization of fermions is presented.