Fock space (${\mathcal{F}}_{\pm }(H)$)

For a one-particle Hilbert space $H$, the bosonic or fermionic Fock space is ${\mathcal{F}}_{\pm }(H)={⨁}_{n=0}^{\infty }{H}^{{\otimes }_{\pm }n}$, using symmetric or antisymmetric tensor powers. Its zeroth summand is the one-dimensional vacuum sector. Creation operators and annihilation operators change particle number. The bosonic Fock space and fermionic Fock space implement the corresponding exchange statistics.

The eigenvalue of a mode number operator counts excitations in that mode. Bosonic occupation numbers range over all nonnegative integers; fermionic occupation numbers are zero or one.

For $\{f,f^{†}{\}}_{+}=1$ and $f^2=0$, the occupation number operator $f^{†}f$ has eigenvalues zero and one. This gives the single-mode exclusion principle.

For $[a,a^{†}]=1$, the occupation number operator $a^{†}a$ has eigenvalues $0,1,2,\dots$ on the bosonic oscillator Fock space.