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 .

Read more about this topic:  Fock Space

Famous quotes containing the word definition:

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)