Space with 720° / not 2$\pi$ rotational symmetry? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:55:58Z http://mathoverflow.net/feeds/question/69611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69611/space-with-720-not-2-pi-rotational-symmetry Space with 720° / not 2$\pi$ rotational symmetry? jdm 2011-07-06T08:09:37Z 2011-07-06T08:33:36Z <p>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)?</p> <p>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".</p> http://mathoverflow.net/questions/69611/space-with-720-not-2-pi-rotational-symmetry/69613#69613 Answer by Francesco Polizzi for Space with 720° / not 2$\pi$ rotational symmetry? Francesco Polizzi 2011-07-06T08:33:36Z 2011-07-06T08:33:36Z <p>This is quite classical.</p> <p>The point is that the group $\operatorname{SO}(3)$ is <em>not</em> 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 $\{I, -I \}$.</p> <p>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 <em>only up to sign</em>.</p> <p>See <a href="http://en.wikipedia.org/wiki/Orientation_entanglement" rel="nofollow">this page about "orientation entanglement"</a> and the links contained there for a "physical explaination" of this phenomenon and more details about $SU(2)$, quaternions and rotations in $\mathbb{R}^3$.</p>