So if spin isn't really spin

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  • #51
strangerep
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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).
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.

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.

In 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.
As Jabs' notes in his paper, we understand that bosonic (resp. fermionic) statistics go with
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.
 
  • #52


someone please help me out if I'm wrong; but can't we record spins from wavelengths in an angular momentum barrier through 3d scanning to determine an objects rate to absorption?
 
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  • #53


Is this not the correct procedure for converting quantum into classical?
 
  • #54


how about converting half spins into hertz pendulum effect to where if f denotes the frequency the period is T=1/f
 
  • #55
Fra
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I think it's not correct to say they're "mathematically equivalent". A spin-0 rep of
the rotation group is not equivalent to a spin-1/2 rep (nor to a spin-1 rep, etc).
Yes your right in what you say, but I meant it in a different way. With mathematically equivalent I did not refer to state space information but to the entire theory as an interaction tool. Both are "formally" possible, it's just they they are not equally efficient in a deeper view.

The information encoded in the dirac state is different than the KG state, but there is more implicit information that in the traditional view is not acknowledged. I do acknowledge it though.

The traditional picture is that only information encoded in the initial state is acknowledged. Other constraints such as dynamical evolution rules are not thought of as "information", it's thought of as just timeless elements of reality - not sujbect to inferencial query in the physical sense of measurement.

In my own view, this is inconsistent because on symmetry grounds there simply is no good reason why some information is subject to inferencial constraints and some information is not. This connects very much to the foundation of QM and notion of "theory" in general as well.

To use the KG picture, there are further constraints in the initial state (corresponding to specifying the spinor components). If you specify these, we have en equivalent description of the system, that makes the same predictions and without problems of probability non-conservation. It's only if you ignore the additional constraints in the initial KG state that this is an issue.

It's just that in the dirac state space picture, the notion of spin ½ appears on it's own in the explicit sense. In the KG picture the information about the spin½ requires information both from the state space and history (derivatives), like an extended state space. And I interpret it simply as a property of the way information is encoded by the observer. No need to even bring in classical analogies. The question I am instead facing is; why is it the case that apparently all observers in nature "choose" to encode it this way? I think it's when you look at the observer as a informaiotn processing and encoding structure it may be easier to see that it's simply a more "economic" way to represent information in the diracy picture, and this I think ultimaltey can be understood not in terms of "mathematical simplicity" which was to me always a very volatile argument, but "simplicity" in hte sense that a bounded observer with limited resourses simply will always CHOOSE the code-wise "simplest" representation in a context of evolving law and interacting observers.

I also suspect this is also connects to why noone yet observed fundamental scalar bosons, we have yet to see also higgs boson. The interesting angle I propose is that instead of asking in a realist sense why are there no scalar boson, one can ask this: How would information about a scalar boson be processed and REpresented by an observer? would this observer be stable?

This is how my take on this has alwas been. But what I miss, is a formalised presentation of things as per this view. Since some years ago I deicded to try to work out this on my own... but I still have plenty of work left. Nothing published yet.

/Fredrik
 
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  • #56


Hypothetically speaking, if we achieve quantum teleportation to transport people; wouldn't they rule the world. A good way to get rid of some 'ol school bullies I guess lol
 

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