The Principle of Relativity: Aut(Geom)= Sym(L) Jeffrey Ketland June 5, 2021 1 Some Puzzles When I was an undergraduate physics student, a number of things baffled me. I list them: (P1) The Mott Problem (P2) How can the |x〉’s and |p〉’s form a “Hilbert space”? (P3) Why can we “add” quantum states? (P4) Coordinate transformation calculations. (P5) We get the Poincare group from Maxwell’s equations and we get the same group by studying the invariants of Minkowski spacetime. Why? (P6) Why (or how) are isomorphic spacetimes “the same world”? The first of these, (P1), really did leave me puzzled, as I sat in lecture theatres in The Mott Building in the mid 1980s. For first, I learnt that, according to quantum theory, a quantum mechanical particle doesn’t have a trajectory. In classical physics, a particle moves along a trajectory, x(t) (1) so that, at time t, the particle is at x(t) (i.e., has coordinates x(t)) and its velocity is ˙ x(t). In quantum theory, however, a particle doesn’t have a trajectory. For if it it did, it would have a definite position and momentum, thereby violating Heisenberg’s principle ΔxΔp ≥  2 (2) On the other hand, in my lectures, I was also shown cloud chamber particle tracks, like this: 1