Normal order

Abstract: In this paper, a method of generating separable forms of the wave-operator for incomplete model spaces is discussed. With a time-dependent access to the many-body perturbation and coupled-cluster theories, it is shown how one can extract the regular part of the wave-operator which consists of linked cluster-operators only in the adiabatic limit. The procedure naturally suggests a hierarchy of lower valence model spaces P (k) . once a particular m-valence incomplete model space P (m) is specified. The wave-operator Ω and the effective Hamiltonian H eff are linked in this development and are valence-universal in the sense of being valid for all P (k)' s. 0 k m. We have derived two distinct forms for Ω: (i) Ω = {exp(S)}, with { } as normal order with respect to suitable vacuum, where S are open operators inducing transitions from P (m) to outside it; (ii) Ω N = {exp(S + X)}, where X are additional closed operators which are introduced to maintain isometry of Ω N : P (k) Ω N + Ω N P (k) = P (k) . In neither of these choices do we have intermediate normalization. It is also possible to develop an alternative strategy with the complete model spaces, such that an effective valence-universal operator H may be found which generates roots, only a subset of which are equal to the eigenvalues of H. These subsets are the ones that H eff would have furnished. This may thus be viewed as a Fock-space realization of the intermediate Hamiltonian approach.

Abstract: A conventional context for supersymmetric problems arises when we consider systems containing both boson and fermion operators. In this note we consider the normal ordering problem for a string of such operators. In the general case, upon which we touch briefly, this problem leads to combinatorial numbers, the so-called Rook numbers. Since we assume that the two species, bosons and fermions, commute, we subsequently restrict ourselves to consideration of a single species, single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, specifically Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. In this note we concentrate on the combinatorial graph approach, showing how some important classical results of graph theory lead to transparent representations of the combinatorial numbers associated with the boson normal ordering problem.

Abstract: We present a simple procedure for constructing the complete cohomology of the BRST operator of the two-scalar and multi-scalar strings. The method consists of obtaining two level-15 physical operators in the two-scalar string that are invertible, and that can normal order with all other physical operators. They can be used to map all physical operators into non-trivial physical operators whose momenta lie in a fundamental unit cell. By carrying out an exhaustive analysis of physical operators in this cell, the entire cohomology problem is solved.