here is the mass of a particle, and are the boson field operators that annihilate and create a particle at the position . If the gas is dilute and cold, then the two-body potential can be replaced by the pseudopotential which is fixed by a single parameter, the -wave scattering length , through the coupling constant

The Hamiltonian (1.1) takes the form

The field operator can decomposed as , where are single-particle wavefunctions with quantum number . The bosonic creation and annihilation operators and are defined in Fock space through the relations

(1.4) | |||

(1.5) |

where are the eigenvalues of the operator giving the number of atoms in the single-particle state . The operators and obey the usual bosonic commutation rules

Bose-Einstein condensation occurs when the number of particles in one particular single-particle state becomes very large . In this limit the states with and correspond to the same physical configuration and, consequently, the operators and can be treated as complex numbers

For a uniform gas in a volume the good single-particle states correspond to momentum states and BEC occurs in the single-particle state having zero momentum. Thus, the field operator can be decomposed in the form . The generalization for the case of nonuniform and time-dependent configurations is given by

(1.8) |

where the Heisenberg representation for the field operators is used. Here is a complex function defined as the expectation value of the field operator . The function is a classical field having the meaning of an order parameter and is often called the