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.
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl [Formula: see text]-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
We want to point out thc following strcngthcning of the classical theorem of Grocnewold and van Hove:There exists no ,napping Op from polynomial observables f (p, q) on the phase space R2n into linear operators on L2(Rn)which would map Poisson brackets into commutators, the position and momentum obscrvablcs p and q into the usual(Schrodinger) position and momentum operators, and would obey the yon Neumann rifle 0p(cfk) = c 0p(f)k for k = 1,2, 3and c R. The point is that neither linearity, nor continuity etc. of Op are assumed.
Abstract In the present paper, we extend the notion of quantum time shift, and the related results obtained in [Accardi, Barhoumi and Ouerdiane, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9: 215–248, 2006], from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular, this class of processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally, we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way comes somehow as a surprise.
We propose an algebraic approach to stochastic graph-rewriting which extends the classical construction of the Heisenberg-Weyl algebra and its canonical representation on the Fock space. Rules are seen as particular elements of an algebra of "diagrams": the diagram algebra D. Diagrams can be thought of as formal computational traces represented in partial time. They can be evaluated to normal diagrams (each corresponding to a rule) and generate an associative unital non-commutative algebra of rules: the rule algebra R. Evaluation becomes a morphism of unital associative algebras which maps general diagrams in D to normal ones in R. In this algebraic reformulation, usual distinctions between graph observables (real-valued maps on the set of graphs defined by counting subgraphs) and rules disappear. Instead, natural algebraic substructures of R arise: formal observables are seen as rules with equal left and right hand sides and form a commutative subalgebra, the ones counting subgraphs forming a sub-subalgebra of identity rules. Actual graph-rewriting is recovered as a canonical representation of the rule algebra as linear operators over the vector space generated by (isomorphism classes of) finite graphs. The construction of the representation is in close analogy with and subsumes the classical (multi-type bosonic) Fock space representation of the Heisenberg-Weyl algebra.