Bogoliubov Hamiltonian as Derivative of Dirac Hamiltonian via Braid Relation

In this paper we discuss a new type of 4-dimensional representation of the braid group. The matrices of braid operations are constructed by q-deformation of Hamiltonians. One is the Dirac Hamiltonian for free electron with mass m, the other, which we find, is related to the Bogoliubov Hamiltonian for quasiparticles in $^3$He-B with the same free energy and mass being m/2. In the process, we choose the free q-deformation parameter as a special value in order to be consistent with the anyon description for fractional quantum Hall effect with $\nu = 1/2$.