Chirality in Geometric Algebra

Abstract

We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used.

Chirality in Affine Spaces and in Spacetime

“…Our approach was based on an extension of the usual chirality concept in Euclidean spaces to spacetime [12,13], but it should not be confused with the chirality concept specific to quantum field theory [32].…”

“…Direct and indirect isometries of the inhomogeneous Lorentz group were exhibited [12]. They were also exhibited for the orthogonal group in finite-dimensional real quadratic spaces [13]. The symmetry invariance may be based neither on a metric nor on a pseudometric.…”

“…No reflection is supported by a vector of null square [13]. Then, a single reflection is an indirect isometry (Theorem 5a), and O(p, q) is generated by the reflections (see Cartan-Dieudonné theorem [17,18]).…”

Global Symmetries, Local Symmetries and Groupoids

“…Definitions 1 and 2 are based on the assumption of the existence of a metric, but none of the axioms defining a true metric were necessary to define isometries [33]. Therefore, these definitions were retained in the case of the Minkowski spacetime [35], and, more generally, in geometric algebra [36], for which intervals are preserved rather than distances.…”

“…As mentioned in Section 2.1, Definition 1 extends to the case where the metric is not a true one. Thus, Definition 1 is relevant in the case of a field: this latter receives a value at each point of the spacetime [35,36]. This value may be a scalar, a vector, a tensor, or else.…”