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Journal of Physics A Mathematical and General

The RPA equation embedded into infinite-dimensional Fock spaceF∞

Seiya Nishiyama, João da Providência, Takao Komatsu

Published Jul 13, 2005• 2 citations
Publisher

Abstract

To clear up both algebraic and geometric structures for integrable systems derived from self-consistent field theory, in particular, a geometrical aspect of the random phase approximation (RPA) equation is presented from the viewpoint of symmetry of the evolution equation. The RPA equation for an infinite-dimensional Grassmannian is constructed. It gives us a simple geometrical interpretation that the collective submanifold is a rotator on a curved surface.

DOI
10.1088/0305-4470/38/30/010

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To clear up both algebraic and geometric structures for integrable systems derived from self-consistent field theory, in particular, a geometrical aspect of the random phase approximation (RPA) equation is presented from the viewpoint of symmetry of the evolution equation. The RPA equation for an infinite-dimensional Grassmannian is constructed. It gives us a simple geometrical interpretation that the collective submanifold is a rotator on a curved surface.
The integrability-condition method is regarded as a mathematical tool to describe the symmetry of collective sub-manifold. We here adopt the particle-hole representation. In the conventional time-dependent (TD) self-consistent field (SCF) theory, we take the one-form linearly composed of the TD SCF Hamiltonian and the infinitesimal generator induced by the collective-variable differential of canonical transformation on a group. Standing on the differential geometrical viewpoint, we introduce a Lagrange-like manner familiar to fluid dynamics to describe collective coordinate systems. We construct a geometric equation, noticing the structure of coset space SU(m+n)/S(U(m) x U(n)). To develop a perturbative method with the use of the collective variables, we aim at constructing a new fermion SCF theory, i.e., renewal of TD Hartree-Fock (TDHF) theory by using the canonicity condition under the existence of invariant subspace in the whole HF space. This is due to a natural consequence of the maximally decoupled theory because there exists an invariant subspace, if the invariance principle of Schredinger equation is realized. The integrability condition of the TDHF equation determining a collective sub-manifold is studied, standing again on the differential geometric viewpoint. A geometric equation works well over a wide range of physics beyond the random phase approximation.
In a slight different way from the previous one, we propose a modified non-Euclidean transformation on the [Formula: see text] Grassmannian which gives the projected [Formula: see text] Tamm–Dancoff equation. We derive a classical time-dependent (TD) [Formula: see text] Lagrangian which, through the Euler–Lagrange equation of motion for [Formula: see text] coset variables, brings another form of the previous extended-TD Hartree–Bogoliubov (HB) equation. The [Formula: see text] random phase approximation (RPA) is derived using Dyson representation for paired and unpaired operators. In the [Formula: see text] HB case, one boson and two boson excited states are realized. We, however, stress non-existence of a higher RPA vacuum. An integrable system is given by a geometrical concept of zero-curvature, i.e. integrability condition of connection on the corresponding Lie group. From the group theoretical viewpoint, we show the existence of a symplectic two-form [Formula: see text].
An application of a self-consistent version of RPA to quantum field theory with broken symmetry is presented. Although our approach can be applied to any bosonic field theory, we specifically study the ϕ4ϕ^4ϕ4 theory in 1+1 dimensions. We show that standard RPA approach leads to an instability which can be removed when going to a superior version,i.e. the renormalized RPA. We present a method based on the so-called charging formula of the many electron problem to calculate the correlation energy and the RPA effective potential.