# What are the natural generalizations (if any) to Bose and Fermi statistics?

1. Jan 25, 2012

### alemsalem

What are the "natural" generalizations (if any) to Bose and Fermi statistics?

fermions: 1 particle per state
Bosons: unlimited number of particles per state
do people consider things in between like states with a capacity n? are there other generalizations of these statistics?

Thanks!!

2. Jan 25, 2012

### Bill_K

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

At one time there was some attention given to this idea, under the name of quasistatistics. However they turned out to be equivalent to fermions with an unobserved internal degree of freedom. That is, if you have fermions with an unobserved color (red, blue, green) it looks like you can have three particles in the same state.

3. Jan 25, 2012

### clem

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

I saw it called 'parastatistics', with parastatistics of order n allowing n particles in each state. As applied to quarks ('paraquarks'), it is not exactly equivalent to fermions with an unobserved internal degree of freedom. For instance paraquarks of order three would allow states like uuds that color forbids.

Last edited: Jan 25, 2012
4. Jan 25, 2012

### Cthugha

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

There are some proposals that one might find non-Abelian anyons in topological insulators, but that is pretty far-out stuff.

5. Jan 25, 2012

### Bill_K

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

Sorry: parastatistics it is. The Wikipedia article states that QCD is equivalent to a theory with parafermions of order 3 and parabosons of order 8. Is this correct?

6. Jan 25, 2012

### clem

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

Not as parafermions were originally proposed.
That is misinformation that has become the standard.
Paraquarks of order three would allow states like uuds that color forbids.
Parabosons of order 8 makes no sense to me even after seeing the Wicki.
I know there are 8 gluons, but I don't see the connection with parabosons.

Last edited: Jan 25, 2012
7. Jan 25, 2012

### questionpost

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

Don't bosons have half integer spins and fermions have whole integer spins?

8. Jan 26, 2012

### DrDu

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

I'd rather say that parastatistics is equivalent to fermions or bosons and an unobservable internal degree of freedom. However, in QCD there is an additional restriction that observable particles have to have no colour. So parastatistics is not equivalent to QCD.

9. Jan 26, 2012

### DrDu

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

All possibilities for statistics in 4 dimensional spacetime are classifyable as irreducible representations of the symmetric group. Besides parastatistics, there are projective representations of the symmetric group. However, they are not possible for elementary particles as these won't obey the cluster principle.
However they appear e.g. in molecular physics where they give rise to so called "double groups" for electronic wavefunctions or half integer angular momentum representations of the molecular rotation groups.

10. Jan 26, 2012

### DrDu

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

No, it's the other way round.

11. Jan 26, 2012

### Physics Monkey

Re: What are the "natural" generalizations (if any) to Bose and Fermi statistics?

In two dimensions bosons and fermions can be very naturally generalized into anyons. These particles can have spin intermediate between integer and half integer e.g. a rational number in between 1/2 and 1. They are strongly believed to occur in two dimensional electron gases in high magnetic field known as fractional quantum hall liquids. They can carry fractional charge in addition to fractional angular momentum. Anyons also generalize the notion of Pauli exclusion as shown by Haldane http://prl.aps.org/abstract/PRL/v67/i8/p937_1

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