On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Abstract

Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by construction. However, only a Zn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cℓ(V ) ↔ V and an ordering, guarantees a multivector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Zn -grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Zn -grading which we now call Clifford algebras of multivectors or quantum Clifford algebras. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cℓp,q which can be decomposed in the symmetric case into a tensor product Cℓp−1,q−1 ⊗ Cℓ1,1. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q -deformed structures through nontrivial vacuum states in quantum theories is outlined.

2-gradings of Clifford algebras and multivector structures

“…Applications range from models in QFT [15] to q-quantization of Clifford algebras [16] (see also [11] and references therein).…”

“…It is important to note that such Z-grading for Λ(V ) is by no means unique [10,11]. Nevertheless, suppose one wants to identify V with the tangent space (at a certain point) of a spacetime M .…”

Quantum Clifford Hopf gebra for quantum field theory

“…The definition of QCAs can be found in [25,14,17]. We give the basic definitions to show how this algebras fit for QFT.…”

Products, Coproducts, and Singular Value Decomposition

“…The new eigenbasis introduces a f -dependent filtration of the algebra. This filtration can be turned into a gradation which was described by dotted wedge products in previous works [11,12,15,14]. Exactly this new filtration establishes the Wick reordering of quantum field theory [13].…”