In a free field theory, at least, we have mathematically a state of N particles with momenta [tex]k_j,j=1,..,N[/tex] being constructed from the vacuum state according to [tex]|k_1,...,k_N\rangle\propto a^\dag(k_1) \cdot\cdot\cdot a^\dag(k_N)|0\rangle[/tex] where [tex]a^\dag(k)[/tex] is the suitable creation operator. (I'm not sure yet about what interactions do to this. I'm not that far along yet.) So from a theoretical perspective, what prevents us from applying the creation operator a non-integral number of times? To put it another way, the particle density operator (in the momentum representation) acting on a state of N particles yields something like [tex]n(k)|k_1,...,k_N\rangle\propto \Sigma_j\delta(k-k_j)|k_1,...,k_N\rangle[/tex] Why are the eigenvalues here restricted to delta functions? Why not something like [tex]n(k)|\rho\rangle\propto \rho(k)|\rho\rangle[/tex] where [tex]\rho(k)[/tex] is a continuous function? I realize a continuous number of particles is an even bigger jump than just a fractional number. But still why not? Why are "partices"--excitations of the field--discrete? Constructing such a state out of the creation operators is a little tricky. It would involve something in the way of an infinitely continuous product of the creation operators, the product analog to the passing from Reimann sums to integration. But just starting from the vacuum state and the creation & annihilation operators, I don't see why it would not be allowed. Of course, observation requires integral particles. But where does it come into the theory? As an analogy, there is integral spin, which is demanded by the algebra of the generators of rotation. Is there something like that for particle number? Or is it one more postulate tacked on?