Helicity -- from Clifford to graphene (Q2890790)

The paper explores the link between two seemingly disjoint ways of thinking of helicity. First, the definition of helicity is discussed, that is, the helicity operator and its corresponding eigenvalues and eigenvectors, which are prevalent in quantum mechanics and related theories. In the discussion Pauli spin matrices and Dirac gamma matrices are used. Then, an alternative definition of helicity is presented which expresses the concept using the study of Clifford algebras. The authors define a relation between vector- and pseudo-vector-type quantities which describes properties of Fermions in the context of the Dirac equation. These two quantities form a part of a set of bilinear covariants which are expressed as inner products of Dirac spinors and their adjoints. In order to establish the definition of helicity, Dirac gamma matrices are used to generate the Clifford algebra. Then, bilinear covariants are constructed, two of which are used to define helicity in an alternative way. By writing the bilinear covariant equation in the matrix form, using projection and charge conjugation operators, the 2-spinor representation is stated. Further, considering a purely right-handed state (i.e., positive helicity) and purely left-handed state (i.e., negative helicity), the authors obtain the main results of the paper. They recover the correct definition for the quantum mechanical helicity operator acting on the right-handed (or the left-handed) states. They obtain additionally two pairs of equations which explain how the conjugate states behave. The first pair of the equations represents exactly the relations which define helicity as an eigenvalue equation, and the second pair of equations represents the equivalent definition of helicity for anti-particles. Then, by considering the Hamiltonian for the particles in the vicinity of a vertex \(V\) in one of the hexagonal lattices of graphene, it is shown that the Hamiltonian for the Fermion in the vicinity of \(V\) is proportional to the 2D helicity operator. By rewriting the graphene Hamiltonian, it is possible to write the Hamiltonian in terms of the possible eigenstates of a Fermion travelling through graphene near the vertex \(V\). Finally, the Hamiltonian is formulated in terms of objects used in the classification of Clifford algebras.