Abstract
Inspired by the covering-space method in path integral on multiply connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. We show that the time-evolution kernel on Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on Λ/Γ, all of which are constructed in exactly the same way as for the time-evolution kernel.