Spin without quantum mechanics?

Question

In Emergence of spin from special relativity some answers discuss how spin can arise in non-relativistic quantum mechanics (let's not enter into those details here). However it is also argued that you do not even need quantum mechanics as there are some relativistic constructions.

Roger Penrose & Wolfgang Rindler's "Spinors and Space-Time" construct some spinorial space-time. What does this imply for spin? Can you have spin without quantum mechanics? Spin more precisely in the context of describing the spin of particles (if that makes any sense without quantumness).

Disclaimer: I guess spinors can be an interesting mathematical tool (as described by the comments) but I hope there will be some kind of correspondence between the not-quantum and quantum. Does the quantum mechanical spin lead to some classical spinor quantity at the not-quantum limit? I guess not.

Answer 1

In the context of 2-spinor formalism and Twistor theory, we have two separate pictures of "helicity" appearing in classical relativistic mechanics: (1) One can associate a helicity $|s|=n/2$ (in units of $\hbar$) to the free massless spin-n/2 field $\phi_{ABC...L}$ satisfying the equation $$\nabla^{AA'}\phi_{ABC...L}=0$$ (2) There is also a helicity picture associated to dynamics of massless particles as briefly mentioned below:

Consider a finite system of relativistic particles in flat space-time. If $(P_a,M^{ab})$ represents momentum and angular momentum of center of mass, we can find the trajectory of center of mass from the equation $P_aM^{ab}=0$. We can also define the spin vector $S^a=-\frac{1}{2}\eta_{abcd}P^bM^{cd}$. For massless particle, we have the following relation $$S^a=sP^a$$ where s is the helicity and $|s|$ is the spin (in units of $\hbar$) of massless particles. This can be better represented using twistor $Z^{\alpha}=(\omega^A,\pi_{A'})$ where $P_a=\pi_A\bar{\pi}_{A'}$ and $M^{ab}=i\bar{\pi}^{(A}\omega^{B)}\epsilon^{A'B'}+c.c.$. Helicity in terms of twistors is given by $$s=\frac{1}{2}Z^{\alpha}\bar{Z}_{\alpha}$$The above construction is purely classical.

In order to identify the two helicity pictures appearing in (1) and (2) one needs to invoke "Quantisation" on this Twistor space (see section 2.4 of :https://doi.org/10.1016/0370-1573(73)90008-2). We define operators $Z^{\alpha}\to \hat{Z}^{\alpha}$ and $\bar{Z}_{\alpha}\to -\frac{\partial}{\partial Z^{\alpha}}$. Then the "Spin operator" $S:=\frac{1}{2}(\hat{Z}^{\alpha}\bar{Z}_{\alpha}-2)$ acts on the Twistor function $g(Z)$ corresponding to spinor field $\phi_{ABC...L}$ to give the helicity:$$S g(Z)=\frac{1}{2}((n+2)-2)g(Z)=sg(Z)$$This is the same helicity which appears in QM. Thus, the two pictures of helicity appears within the formulation of classical relativistic mechanics and we need quantization to equate these two seemingly different pictures.

*I should mention that the distinction b/w Quantum and classical picture in Twistor space is "hazy", because certain aspects of this Quantized twistor space is also required to generate classical vacuum Einstein's solution in space-time manifold(like Non-linear graviton construction, Palatial Twistor theory).

Answer 2

In general relativity textbooks, it will be mentioned that general covariance can be achieved easily if the equations are tensorial. But tensor equations are not the only equations that have general covariance. Spinor equations also satisfy general covariance. In curved spacetime, spinors are defined using fiber bundles.

Quoting Robert M. Wald from chapter 13 of General Relativity called Spinors

Spinors arise most naturally in the context of quantum theory..... However, we should emphasize that the notion of spinors has proven to be an extremely powerful tool for analyzing purely classical problems. Perhaps the most dramatic example of this is Witten's (1981) spinorial proof of the positive mass conjecture. In section 13.2 we shall give further examples of this by deriving a useful spinorial decomposition of the curvature tensor and obtaining the existence and properties of the principal null directions of the Weyl tensor in a manner far simpler than can be achieved by tensor methods.

You can find more information in that chapter. Also check this and this which seem more intuitive than Wald.

Answer 3

Quick answer to final part of your question: yes quantum mechanical spin can lead to a classical spin at the non-quantum limit. It is similar to the way you can get a wavepacket to behave more and more like a classical particle if you use a Glauber coherent state (i.e. superpose states close in momentum with a Poissonian distribution of amplitudes). In the angular momentum case the result is a state for which the angular momentum uncertainty is small compared to the mean, so that all three components of angular momentum can be simultaneously well-defined up to some $\Delta S_i \ll S$. I forget the details (or where I saw them) but perhaps this answer will encourage you to keep looking. The resulting vector behaves just like a classical vector (well a pseudo-vector because it is angular momentum) in the limit, but it can be totally made of spin! There doesn't need to be any orbital angular momentum contributing. Such states are unlikely to occur naturally but I vaguely recall that they are artificially generated in some experiments involving clouds of cold atoms.