Norman said:
I think the issue here is this notion of distinguishability/indistinguishability (sic). I have yet to hear Mike give a reasonable explanation how his "generalized exclusion principle" explains the spin triplet superconductors.
It has occurred to me that part of the difficulty you might be having in understanding this is not seeing the significance of the frame of reference. Unlike individual spins, which are
defined in a Lorentz-invariant way, spin orientations and composite spin are frame-dependent. So a spin triplet in one frame will not, in general be an exact spin triplet in another frame but a mixed state (due to the Wigner rotations on the individual spins). In the case of superconducting triplet pairs, it is the
total angular momentum in the CM frame (obtained by vector addition of the composite spin and orbital a.m.) not the composite spin (triplet obtained by vector addition of the individual spins), that defines the intrinsic angular momentum (spin) of the composite system.
My original rule, as I said several posts ago, is only really relevant in a frame in which the spatial co-ordinates or momenta can be the same, such as a canonical frame defined by (say) a nucleus at the origin. In the CM frame, the momenta are necessarily opposite, so they differ and the standard Pauli rule cannot be applied directly. That is why you need a CM frame rule, which turns out to be the even L+S condition. This has been known and understood for donkeys' years (although for reasons that were not completely correct).
But both versions have the same origin in permutation not being a physical transformation but an artifice of the state descriptions you employ. If you don't like these observable exclusion rules, but want to revert to the old but insufficiently precise Symmetrization Postulate instead, then here is the correctly qualified form of it:
When the rotation which takes the angular co-ordinates of one particle into the other are uniquely specified in an order-dependent way and the same canonical frame of reference is used for spin quantization of both particles, then the combined wavefunction for identical particles can be chosen anti-symmetric under permutation for half-integer spin particles and symmetric for integer spin particles.
(Note that the condition in bold is one that usually applies to the way people construct two-particle states, although it is seldom made explicit. Note also that I wrote "can be chosen" rather than "is" because, perverse though it may seem, one can always arbitrarily introduce an additional order-dependent phase into the direct product Hilbert space, giving a different direct product Hilbert space for each ordering. Of course, no one ever does this, so it is not critical in practice, only in terms of being precise and has no observable consequences.)
However, a simpler symmetrization rule, which gives the correct observable rules (and which follows from the physical non-significance of particle permutation) is:
When both particle states are
independently and fully described in an order-independent way,
sufficiently to define a unique wave function for the combined system, then that wavefunction can always be chosen to be permutation symmetric,
regardless of particle identity or spin. (See my paper cited earlier for the proof that this gives the observable rules and the physical equivalence to the (suitably qualified) conventional Symmetrization Postulate.)
All this is irrelevant, however, to my original point -- which was that the significant difference between bosons and fermions is that only the latter can give us chemistry, life and nuclear energy.
ne of weight 0 which is unidimensional and one irreducible space of weight 1 which is 3 dimensional.
