Fock Space - Definition

Definition

Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space

Here represents the states of no particles, the state of one particle, the states of two identical particles etc.

A typical state in is given by

where

is a vector of length 1, called the vacuum state and is a complex coefficient,

is a state in the single particle Hilbert space,

, and is a complex coefficient

etc.

The convergence of this infinite sum is important if is to be a Hilbert space. Technically we require to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples $|\Psi\rangle_\nu = (|\Psi_0\rangle_\nu, |\Psi_1\rangle_\nu , |\Psi_2\rangle_\nu, \ldots)$ such that the norm, defined by the inner product is finite

where the particle norm is defined by

i.e. the restriction of the norm on the tensor product

For two states

, and

the inner product on is then defined as

where we use the inner products on each of the -particle Hilbert spaces. Note that, in particular the particle subspaces are orthogonal for different .

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