Galilean covariant harmonic oscillator

A Galilean covariant approach to classical mechanics of a single particle is described. Within the proposed formalism, all non-covariant force laws defining acting forces which become to be defined covariantly by some differential equations are rejected. Such an approach leads out of the standard classical mechanics and gives an example of non-Newtonian mechanics. It is shown that the exactly solvable linear system of differential equations defining forces contains the Galilean covariant description of harmonic oscillator as its particular case. Additionally, it is demonstrated that in Galilean covariant classical mechanics the validity of the second Newton law of dynamics implies the Hooke law and vice versa. It is shown that the kinetic and total energies transform differently with respect to the Galilean transformations

Classical confined particles

An alternative picture of classical many body mechanics is proposed. In this picture particles possess individual kinematics but are deprived from individual dynamics. Dynamics exists only for the many particle system as a whole. The theory is complete and allows to determine the trajectories of each particle. It is proposed to use our picture as a classical prototype for a realistic theory of confined particles

Comments on the Properties of Mittag-Leffler Function

The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schr\"odinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Sch\"odinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.Comment: 16 pages, 9 figure

Normal Order: Combinatorial Graphs

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.Comment: 7 pages, 15 references, 2 figures. Presented at "Progress in Supersymmetric Quantum Mechanics" (PSQM'03), Valladolid, Spain, July 200

Combinatorial algebra for second-quantized Quantum Theory

We describe an algebra G of diagrams that faithfully gives a diagrammatic representation of the structures of both the Heisenberg–Weyl algebra H – the associative algebra of the creation and annihilation operators of quantum mechanics – and U(LH), the enveloping algebra of the Heisenberg Lie algebra LH. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(LH). While both H and U(LH) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation–annihilation system, of which it provides a concrete model

A multipurpose Hopf deformation of the Algebra of Feynman-like Diagrams

We construct a three parameter deformation of the Hopf algebra $\mathbf{LDIAG}$. This new algebra is a true Hopf deformation which reduces to $\mathbf{LDIAG}$ on one hand and to $\mathbf{MQSym}$ on the other, relating $\mathbf{LDIAG}$ to other Hopf algebras of interest in contemporary physics. Further, its product law reproduces that of the algebra of polyzeta functions.Comment: 5 page

On certain non-unique solutions of the Stieltjes moment problem

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form (2rn)! and [(rn)!]2. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for r > 1 both forms give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems

A Three Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams

We construct a three-parameter deformation of the Hopf algebra \LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the {\em product formula} in a simplified version of Quantum Field Theory. This new algebra is a true Hopf deformation which reduces to \LDIAG for some parameter values and to the algebra of Matrix Quasi-Symmetric Functions (\MQS) for others, and thus relates \LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler-Zagier sums

Heisenberg–Weyl algebra revisited: Combinatorics of words and paths

The Heisenberg–Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg–Weyl algebra, which offers novel perspectives, methods and applications