Time-Dependent Variational Principle for ϕ4\phi^4 Field Theory: RPA Approximation and Renormalization (II)

The Gaussian-time-dependent variational equations are used to explored the physics of $(\phi^4)_{3+1}$ field theory. We have investigated the static solutions and discussed the conditions of renormalization. Using these results and stability analysis we show that there are two viable non-trivial versions of $(\phi^4)_{3+1}$. In the continuum limit the bare coupling constant can assume $b\to 0^{+}$ and $b\to 0^{-}$, which yield well defined asymmetric and symmetric solutions respectively. We have also considered small oscillations in the broken phase and shown that they give one and two meson modes of the theory. The resulting equation has a closed solution leading to a ``zero mode'' and vanished scattering amplitude in the limit of infinite cutoff.Comment: 29 pages, LaTex file, to appear in Annals of Physic

Random phase approximation and its extension for the quantum O(2) anharmonic oscillator

We apply the random phase approximation (RPA) and its extension called renormalized RPA to the quantum anharmonic oscillator with an O(2) symmetry. We first obtain the equation for the RPA frequencies in the standard and in the renormalized RPA approximations using the equation of motion method. In the case where the ground state has a broken symmetry, we check the existence of a zero frequency in the standard and in the renormalized RPA approximations. Then we use a time-dependent approach where the standard RPA frequencies are obtained as small oscillations around the static solution in the time-dependent Hartree-Bogoliubov equation. We draw a parallel between the two approaches.Comment: 26 pages, Latex file, no figur

Gaussian Time-Dependent Variational Principle for Bosons I - Uniform Case

We investigate the Dirac time-dependent variational method for a system of non-ideal Bosons interacting through an arbitrary two body potential. The method produces a set of non-linear time dependent equations for the variational parameters. In particular we have considered small oscillations about equilibrium. We obtain generalized RPA equations that can be understood as interacting quasi-bosons, usually mentioned in the literature as having a gap. The result of this interaction provides us with scattering properties of these quasi-bosons including possible bound-states, which can include zero modes. In fact the zero mode bound state can be interpreted as a new quasi-boson with a gapless dispersion relation. Utilizing these results we discuss a straightforward scheme for introducing temperature.Comment: 28 pages, 1 figure to appear in Annals of Physic

Length minimizing Hamiltonian paths for symplectically aspherical manifolds

In this paper we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the role of the fixed global extrema in the Floer complex of the generating Hamiltonian. Our main result determines a natural condition on a fixed global maximum of a Hamiltonian which implies that the corresponding path minimizes the positive Hofer length. We use this to prove that a quasi-autonomous Hamiltonian generates a length minimizing path if it has under-twisted fixed global extrema and no periodic orbits with period one and action greater than the fixed extrema. This, in turn, allows us to produce new examples of autonomous Hamiltonian flows which are length minimizing for all times. These constructions are based on the geometry of coisotropic submanifolds. Finally, we give a new proof of the recent theorem of D. McDuff which states that quasi-autonomous Hamiltonians generate length minimizing paths over short time intervals.Comment: 23 pages, references added and final revisions made for publicatio