# Is there a proof for Fermi and Bose statistics?

Is there a proof for Fermi and Bose statistics? What is the background of this proof? To what extent it is mathematically strict? Can one prove that no other statistics is posible?

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Gokul43201
Staff Emeritus
Gold Member
Ruslan_Sharipov said:
Is there a proof for Fermi and Bose statistics? What is the background of this proof?
You will find the derivation in any Stat Mech textbook (such as Reif, Pathria, or Hwang)
To what extent it is mathematically strict?
What does this question mean ? Are you asking if there are approximations used or do you mean something else?
Can one prove that no other statistics is posible?
No other quantum statistics of indistinguishable particles can exist. Since the probability density should be invariant under interchange of indistinguishable particles, the wavefunction can only be one of two things :
(i) symmetric, or (ii) anti-symmetric. Those are the only possibilities. The first leads to BE statistics and the second, to FD statistics.

Gokul43201 said:
No other quantum statistics of indistinguishable particles can exist. Since the probability density should be invariant under interchange of indistinguishable particles, the wavefunction can only be one of two things :
(i) symmetric, or (ii) anti-symmetric. Those are the only possibilities. The first leads to BE statistics and the second, to FD statistics.
I am not sure it's that simple. I am not in a position to discuss parastatistics or parasupersymmetry (I just don't know any details). But there are composite particles. Is deuton a boson? Not quite (see, e.g., H.J.Lipkin, Quantum Mechanics (North-Holland/American Elsevier, 1973 (to R.S.: there is a Russian translation)).

Physics Monkey
Homework Helper
In three or more dimensions, Bose and Fermi statistics are the only possibilities. The parastatistics you speak of, where higher dimensional reps of the permutation group are used, are equivalent to Bose or Fermi statistics plus extra internal degress of freedom. A good example of this is the color quantum number in QCD where quarks can be cast as parafermions of order 3 and gluons as parabosons of order 8. In lower spacetime dimensions, there are other kinds of exotic statistics that are associated with non-trivial topologies. Such statistics occur in the fractional quantum hall effect, for example. As for composite particles, I think its fair to say that they behave as good bosons or fermions so long as their composite nature is not important.

This is what I know about it, I would be happy to hear from someone more informed if I have it wrong.

Gokul43201
Staff Emeritus
Gold Member
I guess I should have thrown in a disclaimer saying I was talking about the weakly interacting limit. Yes, in strongly correlated systems, you can have other exotic statistics like the fractional (anyonic) statistics for a strongly interacting 2D electron gas (as PM pointed out) in a magnetic field. There are (tricks and) limits, though, (such as low number density in a composite boson gas) where the approximation of one or other of FD/BE statistics works pretty well.

So, it really comes down to how you want to do the math - even a seemingly strongly interacting system can be cleverly magicked into a weakly interacting system by renormalizing the correlations into say, an effective mass/charge. Having done that, you can use the simpler statistics of the quasiparticles to deal with the behavior of the system.

Physics Monkey said:
In three or more dimensions, Bose and Fermi statistics are the only possibilities. .
Hi, just to provide a reference. Like Gokul43201 indirectly suggests, statistics is a symmetry (under the permutation group) of the wave function under *idealized* interaction assumptions. In the same way is the kinematical decoupling of the internal degrees of freedom from external ones just that : a simplifying assumption (which is quite dangerous in some circumstances in my mind). So, using statistics is a matter of good judgement; however it is not a *fundamental* principle in my mind (while flat space particle physicists promote it as such), just a very good effective way of understanding more complicated things. A good reference for starters (albeit a bit mathematical) in that respect is : particle statistics in three dimensions´´ Phys Rev. D. Vol 27, number 3, 1787...1797. Sorkin has developped the statistics of topological three geons in the context of canonical quantum gravity.

dextercioby
Homework Helper
Ruslan_Sharipov said:
Is there a proof for Fermi and Bose statistics? What is the background of this proof? To what extent it is mathematically strict? Can one prove that no other statistics is posible?
Well, Fermi-Dirac and Bose-Einstein statistics combined with the special theory of relativity lead in a natural way to the Pauli-Lüders theorem of spin & statistics. Violations of this theorem, beside the ones already stated, appear in the quantum field theory of gauge systems where one introduces (anti)ghost fields (in the Hamiltonian BRST quantization) or antifields and ghosts in the Batalin-Vilkovisky (or the antifield-antibracket) formalism...

Daniel.

dextercioby said:
Well, Fermi-Dirac and Bose-Einstein statistics combined with the special theory of relativity lead in a natural way to the Pauli-Lüders theorem of spin & statistics. Violations of this theorem, beside the ones already stated, appear in the quantum field theory of gauge systems where one introduces (anti)ghost fields (in the Hamiltonian BRST quantization) or antifields and ghosts in the Batalin-Vilkovisky (or the antifield-antibracket) formalism...
Daniel.
I guess the intuitive understanding of this is that prensence of gauge fieds make the particles (slightly) distinguishable ?

dextercioby
Homework Helper
Nope, gauge symmetry is bad for physics, it yields an infinite path integral, because the number of degrees of freedom is less than the number implied in the functional integration. Technically, ghost fields are used in "gauge fixing".Distinguishibility has nothing to do here, basically some virtual particles, Lorentz even spin fermions are involved in Feynman diagrams, nothing more...

Daniel.

dextercioby said:
Nope, gauge symmetry is bad for physics, it yields an infinite path integral, because the number of degrees of freedom is less than the number implied in the functional integration. Technically, ghost fields are used in "gauge fixing".Distinguishibility has nothing to do here, basically some virtual particles, Lorentz even spin fermions are involved in Feynman diagrams, nothing more...
Daniel.
So could you illuminate then what precisely goes wrong (I knew of what you said regarding field theory and gauge fixing procedures, but there should be a better way to understand this) ? You seem to say for example that the commutation relations for two integer spin field operators at spacelike separated events do not hold anymore. So what identity between them does hold such that causality is not violated ??

I was more hinting at the fact that you can derive the possible *statistics* for N particles without using quantum field theory at all, but that you need to assume that particles are perfectly indistinguishable or distinguishable (that is what Sorkin does in this paper). Now, for a relativist these are very fishy concepts ... and much has been written on this issue (unfortunatly QFT theorists have a rather one sided view on this matter).

Gokul43201 said:
I guess I should have thrown in a disclaimer saying I was talking about the weakly interacting limit. Yes, in strongly correlated systems, you can have other exotic statistics like the fractional (anyonic) statistics for a strongly interacting 2D electron gas (as PM pointed out) in a magnetic field. There are (tricks and) limits, though, (such as low number density in a composite boson gas) where the approximation of one or other of FD/BE statistics works pretty well.
So, it really comes down to how you want to do the math - even a seemingly strongly interacting system can be cleverly magicked into a weakly interacting system by renormalizing the correlations into say, an effective mass/charge. Having done that, you can use the simpler statistics of the quasiparticles to deal with the behavior of the system.
I agree. I'd like just to emphasize that there are extremely important exceptions (not just some exotic systems), such as BCS theory of superconductifity, where Cooper pairs may be considered as composite particles, and their density is high, so Fermi or Bose statistics is not appropriate (see the above-quoted Lipkin's book).

dextercioby said:
Nope, gauge symmetry is bad for physics,
:rofl: hmm, many people will disagree

it yields an infinite path integral, because the number of degrees of freedom is less than the number implied in the functional integration.
Are you saying this happens all the time ? I mean in every interaction and at every energy scale ? I hope not, because it is wrong.

Technically, ghost fields are used in "gauge fixing".
Nope, you can fix any gauge without using ghost fields and such fields are also used outside gauge fixing.

Ofcourse, the Fadeev Poppov ghost fields are used to get rid of the unphysical degrees of freedom that arise due to gauge fixing.

marlon

marlon said:
:rofl: hmm, many people will disagree
Are you saying this happens all the time ? I mean in every interaction and at every energy scale ? I hope not, because it is wrong.
Nope, you can fix any gauge without using ghost fields and such fields are also used outside gauge fixing.
Ofcourse, the Fadeev Poppov ghost fields are used to get rid of the unphysical degrees of freedom that arise due to gauge fixing.
marlon
Yes, our hungarian collegue does not seem to have learned that gauge symmetries improve renormalizabilty of the corresponding QFT. As a small remark: the other gauge fixing procedure you are referring to can be technically rather problematic, no?

Careful said:
As a small remark: the other gauge fixing procedure you are referring to can be technically rather problematic, no?
Certainly. That fact that Fadeev-Popov ghosts "exist" should say enough

marlon

edit : isn't Dexter Romanian ?

marlon said:
edit : isn't Dexter Romanian ?
Well, I must have made that mistake by observing this creature from outer space