# Why the Universal Covering Space of a (Spacetime) Symmetry Group?

Potential philosophical issues notwithstanding, it is commonly said that the definition of an elementary particle is an irreducible, unitary representation of the Poincaré group (times a gauge group like $U(1)\times SU(2)\times SU(3)$, of course, but for simplicity I'll just leave that out for the moment).

A key feature of this lies in the fact that this is the universal cover of the group of isometries of Minkowski spacetime, and is in particular a double cover of said group -- this is what allows us to describe the particle's spin in terms of irreps. All this being said...

My Question: Imagine we find ourselves living in a universe where we have a good understanding of how particles/fields/whatever can be described in terms of representations of the symmetry group of the underlying spacetime we inhabit, but we have yet to learn about something like spin. Suppose we have some preferred choice of spacetime, with an associated isometry group that is not simply connected, and as far as we can tell thus far at that this point in history the irreps of this group seems to be fine and dandy for describing the universe around us. Is there some sort of physical reason/insight that might compel us to say, "Hey! Maybe we should look into irreps of this universal cover thingy... perhaps that will reveal something new about nature!"?

In other words, why irreps of the universal cover? Do we just use them because they happen to describe something we know about, or is there reason to suspect they should hold crucial information about the universe?

Granted, this question may be silly and quite likely doesn't really have an answer... but I'm curious nonetheless to have a nice, convincing way of working with irreps of the universal cover other than "it works."

EDIT: I suppose I should say at the get-go that this question is definitely verging on being a duplicate of this one. I will just preemptively say that I think that hinges more on one already having the ideas of quantum mechanics in play (in particular the need for projective representations to deal with the fact that states are only unique up to phase) -- I am asking a bit more generally, even if one knows nothing about quantum mechanics, why they might be lead to consider the universal cover rather than the group itself.

• Possible duplicates: physics.stackexchange.com/q/96045/50583, physics.stackexchange.com/q/203944/50583 – ACuriousMind Oct 16 '16 at 14:26
• Thanks for the possible duplicates (adding to the one I found myself). I think that those are more like the possible duplicate I mentioned in my edit, in that they are (or at least the given answers are) focused on QM, whereas I don't intend to be. Perhaps this is not a substantive distinction -- I'll leave that up to the stackexchange overlords -- but I, at least, feel there is something different I'm grasping at here. :) – OperaticDreamland Oct 16 '16 at 14:32
• I'm not sure what you're asking for, then - outside of QM, i.e. in classical mechanics, we don't look at universal covers. – ACuriousMind Oct 16 '16 at 16:06
• "in classical mechanics, we don't look at universal covers." Is that totally true? What about the use of quarternions by the computer gaming industry? en.wikipedia.org/wiki/Charts_on_SO(3) – Bruce Greetham Oct 16 '16 at 17:22
• @ACuriousMind Well, then I would say that your answer to my question is: "We use irreps of the universal cover to get projective representations, giving us states as equivalent if they differ by a phase -- if we don't want that, we don't care about the universal cover." – OperaticDreamland Oct 18 '16 at 10:42