- #51

strangerep

Science Advisor

- 3,166

- 1,009

I wouldn't call Jabs' argument "hand-wavy", although certainly it is written for a particular group of readers and more elaboration might be helpful to widen that group.I cannot say I understand the details [of Arthur Jabs' paper] but I will say that it does not surprise me that such an argument can be constructed. If spin can be derived from a non-relativistic wave equation (cf Greiner link above), notwithstanding that the derivation is 'ad-hoc' or 'hand-waving', then it seems reasonable that a non-relativistic argument exists for the exclusion principle, even though it might appear 'unnatural' (ie hand-wavy).

Maybe I'll try to write an elaborated version in a separate thread if I find the time. Until then I'll just offer a few more observations on what's needed to grasp Jabs' argument.

1) We must understand that, in QM, one models two indistinguishable particles via a tensor product space of (identical) one-particle Hilbert spaces [itex]H[/itex]. I'll denote the tensor product (2-particle) Hilbert space as [itex]H \otimes H[/itex]. Actually, I'll go further and give labels to the component spaces: [itex]H_a \otimes H_b[/itex]. (But note that they're

*not*(skew-)symmetrized, at least not yet.)

2) Then we must clarify exactly what "exchange" means in the context of a tensor product space. Lets pick two state vectors [itex]\psi_1(x_1,...) \in H_a[/itex] and [itex]\psi_2(x_2,...) \in H_b[/itex], where the x's denote a position coordinate and the "..." denote other quantum numbers, including spin, spin-orientations, and (possibly) a pose angle [itex]\chi[/itex].

3) What then does it mean to "exchange" the particles in a way that relates obviously to physical transformations. I think it means that we must apply a transformation in [itex]H_a[/itex] such that [itex]\psi_1(x_1,...) \to \psi_2(x_2,...)[/itex] and another transformation in [itex]H_b[/itex] such that [itex]\psi_2(x_2,...) \to \psi_1(x_1,...)[/itex]

4) To perform the translation [itex]x_1 \to x_2[/itex], in [itex]H_a[/itex] we use an operator like [itex]e^{iP\cdot (x_2 - x_1)}[/itex] (and vice-versa in [itex]H_b[/itex]). But what about the rotation transformations? (For simplicity, restrict here to the case where both particles are spin-1/2 at rest). There's now a difficulty because of double-valuedness of the rotation group. For spin-1/2, we confront a 2-sheeted complex function, so it's possible that the transformation might change sheets in [itex]H_a[/itex], but not in [itex]H_b[/itex], depending on where we take the branch cut. Often, one takes a branch cut along the +ve real axis, but this is arbitrary. So one thing at least is certain: the physically measurable consequences of the theory must not depend on where we choose the arbitrary branch cut. IOW, they must not depend on which part of the Hilbert space we call the "1st sheet", and which we call the "2nd sheet".

5) Arthur Jabs' solution to this is to demand that the both rotation transformations be performed in the same sense (i.e., both clockwise or both anticlockwise). The familiar spin-statistics result then follows straightforwardly from this demand by cranking the mathematical handle.

6) The thing that still leaves me a little perplexed is this: although demanding a consistent sense for the rotation transformations sounds asthetically pleasing, I have trouble seeing why it's essential (a priori) from a physical perspective. But hey, the double-valuedness of rotations is tricky at the best of times -- needing the "Dirac belt trick" or similar devices to illiustrate it.

As Jabs' notes in his paper, we understand that bosonic (resp. fermionic) statistics go withIn fact the relativistic derivation is not so 'natural' either, and this would suggest that spin (and associated statistics) is not a so well understood a physical phenomenon.

integral (resp. half-integral) spin, and that other choices are inconsistent. But the older proofs don't really give a deeply satisfying insight into why this is so. I found Jabs' approach interesting for exactly this reason.