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Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)?

The reason I am asking is because in quantum mechanics, the wavefunctions of spin 1/2 particles are invariant under 720° / 4$\pi$ rotations, but not under rotations of 360° / 2$\pi$, due to their spinorial nature. I've been wondering if these particles can be expressed in an easier form in this other space, which is then projected or folded down into something more "physical".

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Perhaps the Möbiusband? –  plusepsilon.de Apr 5 at 12:20

2 Answers 2

up vote 11 down vote accepted

This is quite classical.

The point is that the group $\operatorname{SO}(3)$ is not symply connected, in fact $$\pi_1(\operatorname{SO}(3)) \cong \mathbb{Z}/2 \mathbb{Z}.$$ Its universal cover is the group $$\operatorname{Spin}(3) \cong \operatorname{SU(2)}.$$ Hence we have an exact sequence $$1 \to \mathbb{Z}/2 \mathbb{Z} \to \operatorname{SU(2)} \to \operatorname{SO}(3) \to 1,$$ where the kernel $\mathbb{Z}/2 \mathbb{Z}$ is the subgroup $\{I, -I \}$.

Geometrically, this corresponds to the fact that $SU(2)$ is isomorphic to the group of unit quaternions, that in turn can be used to represent rotations in 3-dimensional space, but only up to sign.

See this page about "orientation entanglement" and the links contained there for a "physical explaination" of this phenomenon and more details about $SU(2)$, quaternions and rotations in $\mathbb{R}^3$.

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maybe one should also mentioned that continuous "transformations" in QM are projective unitary (or antiunitary) representations of some group and under some assumptions these lift to a representation of the universal covering group, which in the case of rotations $\mathrm{SU}(2)$ in which rotation by $4\pi$ equals the identity. –  Marcel Bischoff Jul 6 '11 at 8:40
    
You are right. My background is more in geometry rather than in physics, so I tend to "see" better the geometrical aspect of the problem. But of course the relationship with QM is the one you mentioned. Thank you for your comment. –  Francesco Polizzi Jul 6 '11 at 8:48
    
@Francesco: Thanks, although I'll need some time to digest / fully understand the answer, since I didn't pay enough attention in group theory lectures :-). You nicely answered my underlying question about spin, but I'm still wondering if there is a way to embed physical space in a higher dimensional space where spinors live more "naturally"... In the analogy of your link, a moebius strip projected onto 2D would be really odd, but in 3D is just an ordinary body. –  jdm Jul 6 '11 at 9:56
    
@Marcel: Funny to see you here. We were both in the same QM lectures in Göttingen. I guess you remember more of them, though :-) –  jdm Jul 6 '11 at 10:00
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Ok but i have no idea who you are... –  Marcel Bischoff Jul 6 '11 at 10:37

Not only spinors can be used to demonstrate that $360^\circ$ rotation is not an identity transformation. See http://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di991816.99p0601v.pdf (The Mercedes Knot Problem, by Aleksandar Jurisic) and http://www.pha.jhu.edu/~kgrizz/Documents/10-2-13/spinorSpanner.pdf (The Spinor Spanner, by Ethan D. Bolker).

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