The topic of this post is torsors, which occur naturally throughout mathematics and physics whenever we have natural notions of relative — but not absolute — sizes, positions, temperatures and so forth. This post owes a lot to this 2009 post by John Baez, and so I’ll shamelessly
steal borrow some (but not all) examples from him:
- Even after choosing a unit of voltage (e.g. the SI unit, the volt), it makes no sense to say that the voltage at some point p in a circuit is, say, 7V. It does, however, make sense to say that the voltage at p, relative to that at another point q, is 7V. Relative to that chosen reference value at q, voltages are real numbers — but we are free to change the reference point, and without a reference point, voltages are not real numbers, but they do live in a real torsor.
- A good geographical example is longitude: we habitually describe longitude on Earth as longitude using units of degrees and relative to the Greenwich meridian. However, the choice of the Greenwich meridian is basically arbitrary, and if we were to change to the Cairo, Paris, or Washington meridian instead, it would not change the difference in longitude between any two points on Earth. Longitudes are not elements of the circle group S1 (angles); it is longitude differences that are angles in S1, whereas longitudes live in an S1-torsor.
- Both the previous two examples indicate that, whatever a “torsor” is, it’s like a well-behaved algebraic structure (like the real line ℝ or circle group S1) in which the usual reference point, the origin (0 in ℝ and 1 in S1) has been “forgotten”. The usual setting of plane geometry going all the way back to ancient Greece is like this, too: there is no preferred origin for plane Euclidean geometry: you are free to work relative to one corner of your graph paper, or relative to some point in the ground in your Athenian sand-pit.
So… what’s going on here?