This came up during a Particle Physics discussion :

Here is a proof that the wavefunction of n identical particles has to be either completely symmetric or anti-symmetric.

Wavefunction ψ=ψ(a_{1},a_{2},……..,a_{n})

Let **P**_{ij} be the operator which flips particles i and j

Now indistinguishability requires

**P**_{ij}ψ=c ψ

Since flipping twice results in the same wavefunction

**P**_{ij} **P**_{ij} ψ = ψ

Implies c=1 or -1

Now consider this relation

**P**_{ij}=**P**_{ik} **P**_{kj} **P**_{ki}

One can easily veryify that this true by looking at the action of **P**_{ij} on the ordrerd set (i,j,k).

Since **P**_{ik} **P**_{ik}=1

**P**_{ij}=**P**_{kj}

Extending this simlarly one gets

**P**_{ij} = **P**_{ik} =** P**_{kl} for any k,l

Thus **P**_{fg} = c for all f,g

Thus the wavefunction has to be **completely **symmetric or antisymmetric.

Venkateshan

PS: I took the liberty of tidying up the notation, hope I didn't make any mistake/ change your argument. –Shanth

How do you make the above claim? The Pab's don't commute. So you can't pull out the Pkj out from the middle. Or am I missing something in your argument?Another comment, please try to typeset the equations you write a bit more neatly for the blog, I tidied up the thing this time anyway. Also, more importantly, mark the categories for your post when you make a new post. This is there in the right side of the screen when you are composing a post. Anway, nice to see this forum being used. Thanks Tom.

Shanth

PS: Tom/Ravi do any of you have an idea of how to enable HTML in the comments?

P_ik u= su (u is the wavefunction and s is either 1 or -1)

P_kj u = s’ u (again s’ is either +1 or -1)

Pik PkjPik u =s’ u (independent of s)

Implies

Pij u =s’ u

And regarding typing , yes, I have no idea how to include mathemtaical expressions or even use the super/sub script.

Maybe one of you can explain .

Venky : You should be a bit more careful than this. For one, your “proof” doesnot allow for anyons which have been observed in 2-D systems ! Before setting out to prove symmetric/antisymmetric result , you should remember that it doesn’t hold in 2-D, and that fact should enter your proof in some fundamental way. In quantum mechanics, every single equality is equality modulo a phase, for example Pij=(some phase)Pik Pkj Pki. All these phases should be clearly exhibited and then exorcised before your proof falls through. ðŸ™‚

Just to give you an idea of what is involved, I quote from Weinberg-section 4.1,

<snip>

Shanth: As you can see for yourself you CAN type html in comments. But, I don’t think you would like to hear how I do it. ðŸ˜‰ This is the path reserved for perfect masochists….

The statement that only symmetric/antisymmetric wavefunctions are allowed goes by the name of symmetrisation postulate.

See for example

Symmetrization Postulate and Its Experimental Foundation Phys. Rev. 136, B248â€“B267 (1964)

(Should be accessible inside IITK)

and especially the problems with the kind of proof that you have presented above are detailed in the paper

Permutation Symmetry of Many-Particle Wave Functions-

Phys. Rev. 139, B500â€“B508 (1965)(Should be accessible inside IITK)

I meant to say the other day that for this to happen the corresponding group must admit a projective representation ( by that I strictly mean U(T1)U(T2) U(T1*T2) )

and this would require non -zero central charge in the Lie algebra and the topological space associated with the group must not be simply connected (straight from Weinberg) .