# $SU(2)$ vs. $SO(3)$ transformation, spinor rotation and measurement

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 $$A$$).

• A few things in this question confuse me. The matrices from $SU(2)$ act on vectors from $\mathbb C^2$ thereby changing more than just the phase. The Lie algebra of $SU(2)$ is generated by the (hermitian) Pauli matrices that measure spin, as we know. The observable that measures if a state is $\psi$ is the projection $|\psi\rangle\langle\psi|$. See this post. Commented Aug 3, 2023 at 15:40
• Sorry for the confusion. In the end I am interested in the difference of SU(2) vs. SU(3) rotations. So I would like to measure the difference of the effects of two rotations $U_1, U_2 \in \text{SU(2)}$ acting on a spinor in the fundamental representation of SU(2), which are mapped to the same $U \in \text{SO(3)}$.
– TomS
Commented Aug 3, 2023 at 16:28
• The observable you are proposing is not sufficient b/c it is bilinear and therefore not able to detect any phase.
– TomS
Commented Aug 3, 2023 at 16:32
• Anyway, I am more interested in the experimental setup. Calculating effects of the phase mathematically is not the problem.
– TomS
Commented Aug 3, 2023 at 16:39
• Before we get to $SU(2)$ vs. $SO(3)$ let me ask again about phase. This answer in the post I linked earlier says imho correctly that phase has no direct physical meaning and that states which just differ by a phase are identical. Why do you want to measure that? Commented Aug 3, 2023 at 17:44

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.

• Thanks you. That’s clear.
– TomS
Commented Aug 4, 2023 at 19:07

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.