In this article, we provide an alternative and general way to construct the result of the action of any particle-conserving bosonic or fermionic operator represented in second quantized form on a state vector, without resorting to the matrix representation of operators or even its elements. This approach is based on our proposal to compactly enumerate the configurations (i.e., determinants for fermions, permanents for bosons) that are the elements of the state vector. This extremely simplifies the calculation of the action of an operator on a state vector. The computations of statical properties and the evolution dynamics of a system become much more efficient, and applications to systems made of more particles become feasible. Explicit formulations are given for spin-polarized fermionic systems and spinless bosonic systems, as well as to general (two-component) fermionic systems, two-component bosonic systems, and mixtures thereof.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra
A^1, defined as the real subalgebra of the Weyl algebra
A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for
A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in
A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in
A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in
A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of
A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in
A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
Abstract The construction of subalgebras of the set of bilinear products of fermion operators which allow the formation of physically relevant solvable models is investigated. The subalgebras are assumed to be semisimple, hermitean closed and compatible with angular momentum. One obtains two classes of subalgebras, i) the wellknown set of spherical tensors of rank zero and ii) a new class of subalgebras which contain the angular momentum operators. The construction procedure for subalgebras of the second kind is presented in detail (hereby, one starts with the abstract type of the subalgebra and defines the physical model space by the carrier space of a representation) and a number of examples, which provide models not known from the literature, is given.
The paper considers quantal many-boson systems that are described by a rotationally invariant and boson-number conserving Hamiltonian. The properties of a generic model are studied which treats N bosons of p different kinds with non-zero angular momenta l_1,l_2,...,l_p, possibly augmented with a (number of) scalar s boson(s). The order k of the interaction between the bosons is arbitrary and closed formulas are given for matrix elements between N-boson states for any k if p=1 and p=2. A recursive procedure is defined for arbitrary k and p. With the expressions derived in the paper it is possible to express symbolically a Hamiltonian matrix element between N-boson states as a linear combination of k-body interaction matrix elements. More generally, the formulas allow the evaluation of matrix elements of tensor operators that are not necessarily scalar nor boson-number conserving. The numerical implementation of the formalism is discussed and illustrated with a few examples.
I derive in a novel fashion exact formulas for the calculation of general matrix elements, including the overlap (norm) matrix, between states constructed from fermion pairs. Mapping the fermion pairs to bosons, I show how to construct finite and exact (in the sense of preserving matrix elements) boson representations of the norm operator and one- and two-fermion operators. This may lead to a microscopic basis for the Interacting Boson Model, as well as new truncation schemes for the nuclear shell model.