Hamiltonian of free field on infinite-dimensional hypercube

Abstract

The infinite-dimensional hypercube (IDH) is an infinite connected graph with infinite degree at each its vertex, and can be viewed as an infinite-dimensional analog of finite-dimensional hypercubes. In this paper, we investigate a self-adjoint operator determined by the topology of the IDH and a function on the nonnegative integers, which can be interpreted as the Hamiltonian of a free fermion field. We prove that, under some mild conditions, the operator has only pure point spectrum and its spectrum is even a compact interval of the real line. We also obtain some commutation relations concerning the operator, which are of physical interest.