# Spin-1/2 rotational symmetry

1. Feb 5, 2012

### dudy

Hey,
We saw in class that rotating a spin state with an angle of 2pi returns minus the state, and so it has to be rotated 4pi rad in order to return to the original state.
However, we also saw that the expected value of the spin DOES return to its original value after a rotation of 2pi rad.
My question is- Being that we can only measure expected values, and not the states themselves, what physical meaning is there to the "double rotation" symmetry of a spin state?
I mean, how is this theoretical result manifested in experiments ?
thanks

2. Feb 5, 2012

### questionpost

Spin I think has to do with vector states. It has to do with the sort of pattern of how a particle oscillates. If your familiar with polar graphs or parametric, you'll notice that depending on how you make a trigonometric equation, it will take more or less "time" to repeat. It isn't much more than a mathematical pattern, since no one knows exactly what spin is in the real world.

3. Feb 5, 2012

### Bill_K

Not exactly. Spinor wavefunctions are double-valued functions. That is, ψ(x,t) and -ψ(x,t) denote the same state. For a normal wavefunction we would say that ψ and -ψ differ by a phase factor, but for spinors it is even stronger: they are the same. When you rotate a half-integer spin state by 2π, the wavefunction changes sign but the state is the same.

4. Feb 5, 2012

### Matterwave

The reason behind this, as far as I know, is purely mathematical. The mapping between SO(3), the group of real rotations in 3 space (rotating real 3-vectors), and SU(2) the group of rotations in this 2 state spin space (rotating complex 2-spinors) is not one to one. SU(2) double covers SO(3), and so literally there are 2 elements in SU(2) that correspond to any one element in SO(3). This property carries over to quantum mechanics in the definition of a spinor.

Since a spinor is a 2 component complex vector, then the group of isometries preserving the inner product is SU(2), and we have to use SU(2) matrices to denote rotations of the spinor.

Real observables MUST return to the original configuration after a rotation of 360 degrees. After all, I don't change the universe simply by spinning in a 360 degree circle.

5. Feb 7, 2012

### PhilDSP

I suppose if you think entirely relativistically (purely local considerations) rather than considering global connections you would miss a physical manifestation of the 4pi rotation invariance.

In "Rotations, Quaternions, and Double Groups" Simon Alltmann takes up the question of what physical situations lead to a 4pi rather than 2pi invariance. He summarizes the analysis of 2 or 3 papers from other authors and comes to the conclusion that it is a consequence of a connection to an external object that shows the 4pi dependence. That fact is demonstrated in the various "scissors" illustrations invented by Dirac such as this one:

Last edited: Feb 7, 2012
6. Feb 8, 2012

### houseii

Spinor wavefunctions are double-valued functions.http://www.bosin.info/g.gif [Broken]

Last edited by a moderator: May 5, 2017
7. Feb 8, 2012

### strangerep

I vaguely recall that this can be demonstrated with a neutron interferometer (similar idea to an ordinary interferometer for light, but using neutrons instead). It's possible (using certain configurations of magnetic fields to rotate a neutron. So performing such a $2\pi$
on one arm of the interferometer, gives cancellations when the two sides are recombined.