Torsors Made Easy

John Baez:

In Newtonian mechanics, we can only measure energy differences, not energies themselves. The reason is that we can add any real number to our definition of energy without changing any of the physics. This means it doesn't make much sense to ask what the energy of a system is - we can answer this question only after picking an arbitrary convention about what counts as "zero energy". What makes more sense is to talk about the difference between the energy of a system in one state and the energy of that system in some other state.

We can express this in terms of torsors as follows: energy differences lie in the group of real numbers R, but energies themselves do not: they lie in an "R-torsor".

In quantum mechanics, we can only measure relative phases, not phases themselves. The reason is that we can multiply the phase of a quantum state by any unit complex number without changing any of the physics. So, it doesn't make much sense to ask what the phase of a quantum state is - we can answer this question only after picking an arbitrary convention. What makes sense more sense is to talk about the relative phase between two states that differ only by a phase.

We can express this in terms of torsors as follows: relative phases lie in the group of unit complex numbers, which is called U(1), but phases themselves do not: they lie in a "U(1)-torsor".