We study the effect of quantum fluctuations in a ${\phi }^4$ field theory using a Hartree-type approximation. We reduce the operator field equations into a set of coupled $c$-number (infinite in number) equations. We show that these equations can also be derived from a variational principle. We carry out the renormalization procedure in the one-dimensional theory in detail and demonstrate how the renormalization can be done in a fashion consistent with the Hartree approximation. We then apply the technique to study the effective potential and the stability of the vacuum. We find that the abnormal vacuum is unstable as the coupling becomes strong, and a transition between the abnormal ($〈\phi 〉\ne 0$) and the normal ($〈\phi 〉\ne 0$) vacua occurs in both the one- and the three-dimensional theory. A generalization of the theory to include internal symmetries is briefly outlined.

• Received 23 April 1975

DOI:https://doi.org/10.1103/PhysRevD.12.1071