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

Bogoliubov Hamiltonian as Derivative of Dirac Hamiltonian via Braid Relation