Abstract:

The present work concerns nonperturbative variational studies of the effective potential beyond the Gaussian effective potential (GEP) approximation. In the Hamiltonian formalism, we study the method of nonlinear canonical transformations (NLCT) which allows one to perform variational calculations with nonGaussian trial states, constructed by nonlinear unitary transformations acting on Gaussian states. We consider in detail a particular transformation that leads to qualitative as well as quantitative improvement over the Gaussian approximation. In particular we obtain a nontrivial correction to the Gaussian mass renormalization. For a general NLCT state, we present formulas for the expectation value of the $O(N)$symmetric $\gamma(\phi\sp2)\sp2$ Hamiltonian, and also for the oneparticle NLCT state energy.
We also report on the development of a manifestly covariant formulation, based on the Euclidian path integral, to construct lowerbound approximations to $\Gamma\sb{1PI}$, the generating functional of oneparticleirreducible Green's functions. In the Gaussian approximation the formalism leads to the Gaussian effective action (GEA), as a natural variational bound to $\Gamma\sb{1PI}$. We obtain, nontrivially, the proper vertex functions at nonzero momenta, and nonzero values of the classical field. In general, the formalism allows improvement beyond the Gaussian approach, by applying nonlinear measurepreserving field transformations to the path integral. We apply this method to the $O(N)$symmetric $\lambda(\phi\sp2)\sp2$ theory. In 4 dimensions, we consider two applications of the GEA. First, we consider the N = 1 $\lambda\phi\sp4$ theory, whose renormalized GEA seems to suggest that the theory undergoes SSB, but has noninteracting particles in its SSB phase. Second, we study the Higgs mechanism in scalar quantum electrodynamics (i.e., $O(2)$ $\lambda\phi\sp4$ coupled to a U(1) gauge field) in a general covariant gauge. In our variational scheme we can optimize the gauge parameter, leading to the Landau gauge as the optimal gauge. We derive optimization equations for the GEA and obtain the renormalized effective potential explicitly. 