We investigate the properties of a functional representation of states for a self-coupled scalar field theory. The assumption is made that all states can be generated by applying functionals of the field at a fixed time ($t=0\mathrm{}$) to the vacuum state. It is shown that for the class of models considered the Hamiltonian is uniquely determined by the vacuum functional. The calculation of scalar products between states leads to functional integrals. The measure in this integration over function space is also determined by the vacuum state. Two methods for the evaluation of the functional integrals are discussed. The first one reduces the problem in some simple cases to the solution of an eigenvalue problem for a Hilbert-Schmidt kernel plus a finite number of ordinary integrations. The other one gives a perturbation series.

• Received 26 August 1959

DOI:https://doi.org/10.1103/PhysRev.117.1137