An eight-dimensional realization of the Clifford algebra in the five-dimensional Galilean covariant spacetime

We give an 8-dimensional realization of the Clifford algebra in the 5-dimensional Galilean space-time by using a dimensional reduction from the $(5+1)$ Minkowski space-time to the $(4+1)$ Minkowski space-time which encompasses the Galilean space-time. A set of solutions of the Dirac-type equation in the 5-dimensional Galilean space-time is obtained, based on the Pauli representation of $8\times 8$ gamma matrices. In order to find an explicit solution, we diagonalize the Klein-Gordon divisor by using the Galilean boost.