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Is it possible to measure the effects of $SU(2)$ rotations acting on spinor wave functions $\psi$ in the fundamental representation? That means, is it possible to extract information that distinguishes $SU(2)$ from $SO(3)$ rotations?

These are basically two questions:

  1. How would an observable, a matrix element … look like mathematically?
  2. What would be an appropriate experimental setup?

In principle the effects of the $SU(2)$ rotations can be calculated quite easily, but it is not clear to me how they can be measured. The following is just a collection of ideas.

Regarding (1) a first idea – but there may be others – goes into the direction of the Aharonov-Bohm effect, but including a non-vanishing, constant magnetic field. The reason is that an $SU(2)$ rotation

$$ U_\alpha = e^{-i \sigma_k\alpha_k /2} $$

has the same structure as one of the interaction terms

$$ H_\text{int} \sim \sigma_k B_k $$

for a spinor coupled to a constant magnetic field $B$ in the Pauli-equation, when promoted to the time evolution operator

$$ U(t) = e^{-iHt} = e^{-iH_At} \otimes e^{-i \sigma_k\alpha_k /2} $$

where the first exponential depending on the vector potential $A$ acts on position space only, the second one depending on the magnetic field $B$ acts on spin space only, and where I used $\alpha_k \sim B_k \, t$.

If we are able to let an appropriate interaction rotate the spin in just one one of two components of a spinor wave function

$$ \psi_0 = \psi_{0,1} + \psi_{0,2} $$

$$ \psi(t) = U_{B=0}(t) \, \psi_{0,1} + U_{B \neq 0}(t) \, \psi_{0,2} $$

and to measure the interference pattern of $|\psi(t)|^2$ that may demonstrate the effect created by the $SU(2)$ rotation.

Splitting the wave functions into position and spin part

$$ \psi_\chi(r) = u(r) \otimes \chi $$

and using $B$ in y-direction we get

$$ |\psi|^2 = |u_1|^2 + |u_2|^2 + (u_1^\ast u_2 + u_2^\ast u_1 ) \, \cos(\alpha /2) $$

where the cosine is caused by the $SU(2)$ rotation, $\alpha \sim B \, t$.

As said, this is just an idea, there may be others to address question (1).

Regarding an experiment (2) to measure the effects derived from the idea (1) I see several problems: The idea looks feasible only in the case of a constant magnetic field $B = \text{const}$. The effect of the magnetic field $B$ rotating the spin part $\chi$ is mixed with the interaction caused by the vector potential $A$ acting on the position space wave function $u$; therefore, it is not clear how to extract the sole effect of the $SU(2)$ rotation. The ansatz with the interference of two components requires us to confine $B$ to a spatial region, such that only one component is affected by the rotation, whereas the other component does not feel any interaction with $B$ (but still with $A$).

TomS
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2 Answers2

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Let us analyze a more general setup. Suppose you have two spins, one has spin-$S_1$ and the other has spin-$S_2$. No restriction on $S_1$ or $S_2$. We can then consider any state in the two-spin Hilbert space. To be explicit, we can expand them in the $S^z$ eigenbasis:

$|\psi\rangle=\sum_{m_1,m_2}c_{m_1m_2}|m_1m_2\rangle$

Now we apply $2\pi$ rotation to just one of the spin, say the first spin. Let us assume that this can be done, without worrying about practical issues It transforms $|\psi\rangle$ in the following way:

$|\psi\rangle\rightarrow \sum_{m_1,m_2}c_{m_1m_2}(-1)^{2S_1}|m_1m_2\rangle=(-1)^{2S_1}|\psi\rangle$

Still just an overall phase, which is not observable.

Apparently, the same is true even when we have more spins.

The only way out is that somehow, you can have a "spin" whose Hilbert space is a direct sum of both half-integer spin and integer spin. However, if we are only allowed to build things out of tensor products of spin-$1/2$'s (or for that matter, any spin-$S$), we will never get such direct sum. This is the superselection rule.

Meng Cheng
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I haven't checked all details and referenced papers, but that goes into the right direction:

4π-Periodicity of the spinor wave function under space rotation

We report the results of an experiment which observed the 4π-symmetry of the neutron wave function under space rotation by the use of a slowly rotating magnetic field which trapped the precessing neutron spin and turned it in space.

TomS
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