785 research outputs found
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.Comment: 5 pages, 3 figure
A generic Hopf algebra for quantum statistical mechanics
In this paper, we present a Hopf algebra description of a bosonic quantum
model, using the elementary combinatorial elements of Bell and Stirling
numbers. Our objective in doing this is as follows. Recent studies have
revealed that perturbative quantum field theory (pQFT) displays an astonishing
interplay between analysis (Riemann zeta functions), topology (Knot theory),
combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure).
Since pQFT is an inherently complicated study, so far not exactly solvable and
replete with divergences, the essential simplicity of the relationships between
these areas can be somewhat obscured. The intention here is to display some of
the above-mentioned structures in the context of a simple bosonic quantum
theory, i.e. a quantum theory of non-commuting operators that do not depend on
space-time. The combinatorial properties of these boson creation and
annihilation operators, which is our chosen example, may be described by
graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf
algebra structure. Our approach is based on the quantum canonical partition
function for a boson gas.Comment: 8 pages/(4 pages published version), 1 Figure. arXiv admin note: text
overlap with arXiv:1011.052
Wigner quantization of some one-dimensional Hamiltonians
Recently, several papers have been dedicated to the Wigner quantization of
different Hamiltonians. In these examples, many interesting mathematical and
physical properties have been shown. Among those we have the ubiquitous
relation with Lie superalgebras and their representations. In this paper, we
study two one-dimensional Hamiltonians for which the Wigner quantization is
related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the
Hamiltonian H = xp, is popular due to its connection with the Riemann zeros,
discovered by Berry and Keating on the one hand and Connes on the other. The
Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we
will examine. Wigner quantization introduces an extra representation parameter
for both of these Hamiltonians. Canonical quantization is recovered by
restricting to a specific representation of the Lie superalgebra osp(1|2)
Hierarchical Dobinski-type relations via substitution and the moment problem
We consider the transformation properties of integer sequences arising from
the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form
exp(x (a*)^r a), r=1,2,..., under the composition of their exponential
generating functions (egf). They turn out to be of Sheffer-type. We demonstrate
that two key properties of these sequences remain preserved under
substitutional composition: (a)the property of being the solution of the
Stieltjes moment problem; and (b) the representation of these sequences through
infinite series (Dobinski-type relations). We present a number of examples of
such composition satisfying properties (a) and (b). We obtain new Dobinski-type
formulas and solve the associated moment problem for several hierarchically
defined combinatorial families of sequences.Comment: 14 pages, 31 reference
Dobinski-type relations: Some properties and physical applications
We introduce a generalization of the Dobinski relation through which we
define a family of Bell-type numbers and polynomials. For all these sequences
we find the weight function of the moment problem and give their generating
functions. We provide a physical motivation of this extension in the context of
the boson normal ordering problem and its relation to an extension of the Kerr
Hamiltonian.Comment: 7 pages, 1 figur
Combinatorics and Boson normal ordering: A gentle introduction
We discuss a general combinatorial framework for operator ordering problems
by applying it to the normal ordering of the powers and exponential of the
boson number operator. The solution of the problem is given in terms of Bell
and Stirling numbers enumerating partitions of a set. This framework reveals
several inherent relations between ordering problems and combinatorial objects,
and displays the analytical background to Wick's theorem. The methodology can
be straightforwardly generalized from the simple example given herein to a wide
class of operators.Comment: 8 pages, 1 figur
Spin operator and spin states in Galilean covariant Fermi field theories
Spin degrees of freedom of the Galilean covariant Dirac field in (4+1)
dimensions and its nonrelativistic counterpart in (3+1) dimensions are
examined. Two standard choices of spin operator, the Galilean covariant and
Dirac spin operators, are considered. It is shown that the Dirac spin of the
Galilean covariant Dirac field in (4+1) dimensions is not conserved, and the
role of non-Galilean boosts in its nonconservation is stressed out. After
reduction to (3+1) dimensions the Dirac field turns into a nonrelativistic
Fermi field with a conserved Dirac spin. A generalized form of the Levy-Leblond
equations for the Fermi field is given. One-particle spin states are
constructed. A particle-antiparticle system is discussed.Comment: Minor corrections in the text; journal versio
Topological Expansion and Exponential Asymptotics in 1D Quantum Mechanics
Borel summable semiclassical expansions in 1D quantum mechanics are
considered. These are the Borel summable expansions of fundamental solutions
and of quantities constructed with their help. An expansion, called
topological,is constructed for the corresponding Borel functions. Its main
property is to order the singularity structure of the Borel plane in a
hierarchical way by an increasing complexity of this structure starting from
the analytic one. This allows us to study the Borel plane singularity structure
in a systematic way. Examples of such structures are considered for linear,
harmonic and anharmonic potentials. Together with the best approximation
provided by the semiclassical series the exponentially small contribution
completing the approximation are considered. A natural method of constructing
such an exponential asymptotics relied on the Borel plane singularity
structures provided by the topological expansion is developed. The method is
used to form the semiclassical series including exponential contributions for
the energy levels of the anharmonic oscillator.Comment: 46 pages, 22 EPS figure
Path-integral quantization of Galilean Fermi fields
The Galilei-covariant fermionic field theories are quantized by using the
path-integral method and five-dimensional Lorentz-like covariant expressions of
non-relativistic field equations. Firstly, we review the five-dimensional
approach to the Galilean Dirac equation, which leads to the Levy-Leblond
equations, and define the Galilean generating functional and Green's functions
for positive- and negative-energy/mass solutions. Then, as an example of
interactions, we consider the quartic self-interacting potential , and we derive expressions for the 2- and 4-point
Green's functions. Our results are compatible with those found in the
literature on non-relativistic many-body systems. The extended manifold allows
for compact expressions of the contributions in space-time. This is
particularly apparent when we represent the results with diagrams in the
extended manifold, since they usually encompass more diagrams in
Galilean space-time.Comment: LATEX file, 27 pages, 8 figures; typos in the journal version are
removed, equation (1) in Introduction is correcte
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