Salient features of the application of corepresentation theory to the lattice dynamics problem

The group-theoretical scheme generally employed for analysis of lattice vibrations involves irreducible corepresentations and is regarded as being incomplete and somewhat cumbersome. The four papers presented here suggest a new scheme based on the theories of irreducible and induced corepresentations (CR). In this scheme, the reality of the displacements and the microstructure of a crystal are easily and clearly taken into account, all possible types of secular equations are revealed, and a method is provided for solving the secular equations in cases not considered earlier. This paper presents a formal version of the CR theory, the only version possible in approximate calculations. To be specific, unlike the usual practice, the basis vectors of CR theory are not assumed to be the eigenvectors of the energy operator; instead, we first develop a theory of CR matrix forms, use vectors of a sufficiently arbitrary corresponding space (displacements, local functions) to construct CR basis vectors from the CR matrices and then use the basis vectors to derive the eigenvectors by diagonalizing the Hamiltonian. In this version of the theory we prove several theorems and describe a universal procedure for carrying out the transformation to the real form of corepresentations. We propose that the energy be represented as a combination of invariant forms constructed from displacements which transform according to the irreducible corepresentations. A theorem on the number of second-order invariants is presented, and all of the invariants are written out. In particular, the fact that four mixed invariants correspond to two identical corepresentations of type b necessitates a restructuring of the group-theoretical schemes for the analysis of quasiparticles in a crystal.

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