Interesting to describe the interaction between 2 pieces. And of course the classical buzzwords must appear here: particle-wave dualism, uncertainty principle, Heisenberg Lie algebra. Measuring the momentum results in a translation ('vector' groups), and looking at the faces changes the momentum, or not?!
Quantum field theory is perhaps what we need. Well, the Chess pieces are Fermions (obeying the Pauli principle), and now what about those Bosons? And the Chess pieces of foreign colour, are they sort of antimatter?