Quantum F-un: the q=1 limit of Galois field quantum mechanics, projective geometry and the field with one element

Abstract

We argue that the q = 1 limit of Galois Field Quantum Mechanics, which was constructed on a vector space over the Galois Field Fq = GF(q), corresponds to its ‘classical limit,’ where superposition of states is disallowed. The limit preserves the projective geometry nature of the state space, and can be understood as being constructed on an appropriately defined analogue of a ‘vector’ space over the ‘field with one element’ F1.