Talk:Parastatistics

This article should make clear that paraparticles are hypothesized and have not been experimentally observed; or so it appears based on a quick internet search. For example I came across the webpage of Charles Nelson of Binghampton University who says the following:

Paraparticles are particles which obey parastatistics which generalize the ordinary Fermi-Dirac and Bose-Einstein statistics (obeyed respectively by electrons and photons). Paraparticles are allowed in local relativistic quantum field theory but have not yet been experimentally discovered. As a major on-going program, we are analyzing the physical and mathematical properties of such theories. We are also constructing tests for the presence of paraparticles in reactions at high energy colliders and in astrophysics / cosmology.

Is this accurate? Judging by it, not only have the particles not been observed but the theory itself appears not to have been completely fleshed out. This is not bad in itself but making it clear is perhaps less "jarring" to those who otherwise might come away with the impression that, despite what they learned in graduate school, there are actually more classes of extant particles treated by quantum statistics than bosons and fermions.

Finally, I found a reference which could be made into a fairly lucid account of parastatistics. It's introductory paragraphs are readable by the non-specialist who has some familiarity with quantum statistics.

M. Cattani and J. M. F. Bassalo, Intermediate Statistics, Parastatistics, Fractionary Statistics and Gentilionic Statistics (date and journal unknown) Skinnerd (talk) 18:38, 19 October 2012 (UTC)

what statistics? what is the probability distribution?

I suspect it is a mistake to compare parastatistics with Fermi-Dirac (F-D) statistics (though I could be wrong). Parastatistics refers to an algebraic property rather than a statistical property, whereas F-D statistics are actual statistics: probability distributions for the special case of non-interacting quanta, at thermal equilibrium. F-D statistics is certainly a consequence of fermion algebra but by no means should the two be equated. For further discussion on this point see Talk:Anyon. Nanite (talk) 11:01, 15 February 2014 (UTC)

Some stuff is not explained

Consider the operator algebra of a system of N identical particles.

I don't think this operator algebra has been defined here.

Is it the algebra of all Hermitian operators on the Hilbert space of states of the system? If so, doesn't the definition of this Hilbert space depend on whether it's fermions, bosons, or maybe something else? Or do we mean the Hilbert space that is the raw tensor space of the Hilbert spaces of the individual particles?

In other words, the observable algebra would have to be a *-subalgebra invariant under the action of S^N (noting that this does not mean that every element of the operator algebra invariant under S^N is an observable).

Is the determination of particle statistics the same as specifying the observable algebra? If so, the article should come out and say this.

Therefore we can have different superselection sectors, each parameterized by a Young diagram of S^N.

I know I'll eventually be able to chase this through, but it really isn't explained here at all. The "therefore" is particularly annoying.

How do you connect the Young diagram with the sigma-algebra of observables? Is there a bijective correspondence between Young diagrams and superselection sectors (sums of isomorphic representations), which in turn have something to do with specifying a sigma-algebra?

Does the sigma algebra of observables correspond to a single superselection sector, or to a set of them? I notice that a few steps later we arrive at:

If we have N identical parabosons of order p, then the permissible Young diagrams are all those with p or fewer rows.

Does this mean that if we specify the set of all Young diagrams with p or fewer rows, then we have defined a superselection sector, or some collection of them, and this leads to an algebra of observables, which leads to N identical parabosons?

Do these parabosons exist in the original "raw" tensor product Hilbert space mentioned above, or in some subspace of tensors with a particular symmetry the way bosons or fermions do? So in the former case, it's only the algebra of observables that's different?

If all this is true, the introduction is a good place to say it.

There are a huge number of missing connections here.

178.38.85.195 (talk) 12:00, 13 April 2015 (UTC)

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Not physics.

According to

• Tolstoy, Valeriy N. "Once more on parastatistics." Physics of Particles and Nuclei Letters 11 (2014): 933-937,

Green's paper

• H.S. Green, A Generalized Method of Field Quantization. Phys. Rev. 90, 270–273 (1953).(c)

explores field quantization without the "symmetrization postulate". The entirety of Tolstoy's paper is math.

The review in

• Baker, David John, Hans Halvorson, and Noel Swanson. "The conventionality of parastatistics." The British Journal for the Philosophy of Science (2015).

makes it clear that this is entirely a math exercise: "Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. "

So this topic amounts to an exploration of the mathematical consequences of field theory absent the physical axiom that leads to observed effects. Johnjbarton (talk) 18:35, 18 February 2024 (UTC)