Let the complete set of 1-particle eigenstates be arranged in a sequence according to some rules so that each of them can be referred to by a single integer . For example, one can arrange them in the order of increasing energies so that is the ground state. The case of continuous quantum numbers can be treated either by replacing a with real numbers or replacing the quantum numbers with their discrete approximations. Thus, for the case of the momentum and spin eigenstates, a given value of a will represent a state with quantum numbers .
The basis states in the number representation are written as where is the number of particles in the 1-particle state a. The canonical (generalized coordinate and momentum) operators in the number representation are the creation and annihilation operators with , where is the adjoint (hermitian conjugate) operator. Note that we have suppressed the ^ on both and since they are always used as operators. By definition, all dynamical operators of the system can be written as functions of and . For example, .