Chirality of Dirac Spinors Revisited

Abstract

We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators encountered in the context of Dirac algebra. Then we show how a recent general mathematical definition of chirality unifies the chirality concepts and resolve conflicting conclusions about symmetry operators, and particularly about the so-called chirality operator. The proofs are based on group theory rather than on Clifford algebras. The results are independent on the representations of Dirac gamma matrices, and stand for higher dimensional ones.

Molecular Chirality in Classical Spacetime: Solving the Controversy about the Spinning Cone Model of Rotating Molecules

“…This controversy about such a simple system is clearly about a case of chirality in spacetime, for which Lord Kelvin's definition of chirality does not suffice. We show further that this controversy can be solved through the new definition of chirality, encompassing Lord Kelvin's one, that we published recently, and that in fact both authors were right. In the rest of this paper, we discuss chirality only in the controversial case of the non‐translating spinning cone rotating about its symmetry axis, and not chirality of more complex models of molecules involving charges, electric or magnetic fields, spins, or other physical aspects of molecules: mathematical models involving these advanced concepts are out of the scope of this paper.…”

“…For example, consider the definition of a distance function (or metric ): there is only one mathematical definition, but the space underlying this definition can be any the user wishes, so that it is a powerful modeling tool, able to target a huge range of physical situations. A related example is that of symmetry: a single definition suffices for Euclidean spaces (with color constraints or not), functions, probability distributions, matrices, strings, graphs, etc …”

“…This terminology recovers the usual concepts, and in particular we retrieve the chirality concept of Lord Kelvin. The reader is directed to the original papers for a detailed presentation.…”

Comment on “Bad Language”: Resolving some Ambiguities about Chirality

“…The above‐mentioned description should not be seen as simply laboring the point, because its superficial analysis can lead many people to erroneously consider that chirality cannot be defined in a space which is not oriented, at least locally. A result from 2010 shows that chirality can be defined in any metric space, and therefore can be defined in non‐oriented spaces . The proof was also recently presented in a mathematics journal, but it was never mentioned in a chemistry journal.…”

“…The lack of a definition in the sense of modern mathematics is not reserved for physicists. The definition of chirality in references and could not have been produced without a general definition of symmetry. And it happens that intuitive definitions involving “ geometrical figures ” still appear frequently in the symmetry literature.…”