Magnetic translation group as group extensions

The idea of magnetic translation groups, appearing in considerations of movement of electron in an external magnetic field, was proposed independently by Brown [1] and Zak [2]. From those works follows that the magnetic translation group is an image of Weyl–Heisenberg group [3] obtained by imposing the Born–von Kármán periodic boundary conditions. The general description of similar problems has been presented by Schwinger [4], who considered unitary operator bases. One of the considered cases can be interpreted as a description of finite phase space. This two-dimensional space is spanned by one space (positional) dimension and the other corresponding to kinetic momentum. Two unitary translation operators acting in these two dimensions are given by exponential functions of Hermitian operators of momentum and position, respectively, and, of course, they do not commute. Algebraic structure generated by such operators resembles an extension of a direct product of two Abelian (translation) groups by a group G containing commutators (or simply factors since, in general, it is a subgroup of the field of complex numbers). Therefore, such extensions are throughout studied in this work and the physical relevance is indicated. In the next section some basic ideas of the Weyl–Heisenberg group, finite phase spaces and magnetic translations groups are briefly presented. All possible (central) extensions of a direct product T of two finite cyclic groups ZZN1 and ZZN2 by an Abelian group G are determined in Sec 3. The Mac Lane method [8, 9] (see also Lulek [10, 11, 12]), has been applied and the solution can be given in a general (analytic) form (see also the appendix). In Sec 4 the cases of G = U(1) and G = ZZN are discussed as the simplest, and the most important, examples of possible groups of factors (gauge groups). This work is ended by a short discussion and remarks.