strangerep said:
The analogy certainly is not complete since the angular momentum case does not involve inserting half-integers somewhere by hand.
In the case of the Heisenberg group (or equivalently but better, the slightly bigger oscillator group which also contains ##N## as a generator) of
a single oscillator, nothing is inserted by hand, except for the choice of the Hermitian operator. This is because there is more freedom than in the ##SO(3)## case, due to the non-compactness of the Heisenberg group. If you pick ##q## or ##p## to determine the spectrum you get all reals; if you pick ##N##, you get only the nonnegative integers.
From the interpretation in terms of a harmonic osciilator, the meaning of the eigenvalues of ##N## is the number of excitations of the eigenfunctions. Not the number of particles - after all, we have only a single oscillator, not enough to make up a particle. In a classical analogy, they count overtones - the number of zeros of a standing harmonic wave clamped at both ends.
When you increase the number of oscillators,
the eigenvalues of ##N## (now summed over the oscillators) still
count the number of excitations. Why should this interpretation suddenly change in the limit of infinitely many oscillators? It doesn't. Therefore
the eigenvalues of the number operator in a free quantum field theory count the number of excitations, and nothing else.
In particular,
they never count the number of particles, since so far, particles don't even make sense in our construction. To make sense of it we must impose a - somewhat weird and only historically justified - particle interpretation. In this particle interpretation, one says that
an elementary excitation of the quantum field (i.e., a state in the ##N=1## eigenspace)
constitutes an elementary particle, and defines the meaning of a single particle in this way! It is an arbitrary (only historically sanctioned) name for these states. It just amounts to using the word ''particle'' for ''elementary excitation'', thereby suggesting a sometimes appropriate, sometimes very misleading imagery.
Note that the particle interpretation is possible only when ##N## exists as an operator - i.e., in the free case, or, in the interacting case, asymptotically in the limit of infinite times for bound clusters in a scattering experiment! Therefore, in the real world, where one can never scatter in infinite time,
the resulting particle picture is strictly speaking never appropriate - except in an approximate way!
Poincare invariance, Locality, and the uniqueness of the vacuum state now imply that the newly christened single particle space furnishes a causal unitary irreducible representation of the Poincare group, which were classified by Wigner in 1939. This is why
particle theorists say that elementary particles are causal unitary irreducible representations of the Poincare group, Thus elementary particles are
something exceedingly abstract, not tiny, fuzzy quantum balls!
For spin ##\le 1##, these representations happen to roughly match the solution space of certain wave equations for a single relativistic particle in the conventional sense of quantum mechanics, but only if one discards the contributions of all negative energy states of the latter. This already shows that
there is something very unnatural about the relativistic particle picture. Problems abound if one tries to push the analogies further, and quantum field theorists in their right mind will never do so.
Thus from a quantum field perspective,
particles are ghosts from the past still haunting us as long as we continue to believe in them. It is historical baggage that carries no real weight - except in the terminology, which grew historically and is difficult to change.
To be fair, the particle picture has a very practical use. But only as an approximate, semiclassical concept valid when the fields are concentrated along a single (possibly bent) ray and the resolution is coarse enough. But whenever these conditions apply, one is no longer in the quantum domain and can already describe everything classically, perhaps with small quantum corrections. Thus
the particle concept is useful when and only when the semiclassical description is already adequate. Note that this domain of validity excludes experiments with beam-splitters, half-silvered mirrors, double slits, diffraction, long-distance entanglement, and the like. Thus it is no surprise that in the interpretation of experimens involving these, particle imagery leads to mind-boggling features otherwise only knowns from dreams and ghost stories. The latter they are!