# What is the single particle Hilbert space?

I know what an Hilbert Space is, but I'm not sure what exactly is the single particle hilbert space; I understand it as the space of all possible states of the particle; does it matter if you're talking about an electron, a neutron or a quark, or is it a particle in the 'abstract'? How does one construct it?

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The single-particle space depends on mass, spin, and other quantum numbers of the particle.

In general, a single-particle space is a positive energy irreducible unitary representation of the symmetry group considered, thus the Poincare group for relativistic particles, but in the nonrelativistic case the Galilei group, and in some cases an extra group (typically a $SU(n)$, accounting for flavors, etc.)

The simplest single-particle space is that of a scalar (= spin 0) particle, which is given by the L^2 functions of momenta $p$ on a mass shell ($p^2=m^2$, $p_0>0$), integrated with the Lorentz invariant measure. The others are more complicated versions of it.

The irreducible unitary representations of the Poincare group with positive energy were classified by Wigner, and are characterized by mass and spin; for each such combination there is one single-particle space. For constructions for arbitrary spin $>0$, see Weinberg's book on QFT, which has perhaps the clearest discussion in textbook form.

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The single-particle Hilbert space is the Hilbert space of all states that may be classified as one-particle states; it is a subset of the Hilbert space of a full theory containing states with $N=1$.

If we consider the flat space, then on this Hilbert space, one may find the independent momentum operators $p_x, p_y, p_z$ and discrete operators of spin that depend on the particle. The Hamiltonian is typically $$E = \frac{p^2}{2m}$$ in non-relativistic approximations or $$E = \sqrt{p^2c^2+m^2 c^4}$$ in special relativity except that we usually write the Hamiltonian in prettier ways, e.g. as the Dirac Hamiltonian for the spin-1/2 particles.

The particles may have some extra features or interact with the background fields (in the environment etc.). Or the background spacetime may be compact, and so on. All these things affect on the context. It makes no sense to ask about the context; the context should be a part of the question if it were fully well-defined.

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