A New Perspective on Quantum Field Theory Revealing Possible Existence of Another Kind of Fermions Forming Dark Matter

Abstract

Quantum fields are considered as generators of infinite-dimensional Clifford algebra Cl(∞), which can be either orthogonal (in case of fermions) or symplectic (in case of bosons). A generic quantum state can be expressed as a superposition of the basis elements of Cl(∞), the superposition coefficients being multiparticle complex-valued wave functions. The basis elements, that are products of the generators of Cl(∞) in the Witt basis, act as creation and annihilation operators. They create positive and negative energy states that include the bare and the Dirac vacuum as special cases. It is shown that the nonvanishing electric charge arises from an extra dimension or from doubling the number of creation and annihilation operators, which brings an extra imaginary unit ī into the description. A further extension is to consider the ī as one of the quaternionic imaginary units and consider a generic state as having values in the quaternionic algebra or, equivalently, in the complexified 2-dimensional Clifford algebra, Cl(2) ⊗ C. It contains two distinct fundamental representations of SU (2), one associated with the weak isospin doublet (ν e , e − ), and the other one with the doublet of new leptons, denoted ( + , ν ), that together with the new quarks (u , d ) can be identified with dark matter.