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Jan Govaerts

Publications and source records attributed to Jan Govaerts.

At least 37 records · Page 2Linked to original sources
arXiv2011

Quantized cosmological constant in 1+1 dimensional quantum gravity with coupled scalar matter

A two dimensional matter coupled model of quantum gravity is studied in the Dirac approach to constrained dynamics in the presence of a cosmological constant. It is shown that after partial fixing to the conformal gauge the requirement of a quantum realization of the conformal algebra for physical quantum states of the fields naturally constrains the cosmological constant to take values in a well determined and mostly discrete spectrum. Furthermore the contribution of the quantum fluctuations of the single dynamical degree of freedom in the gravitational sector, namely the conformal mode, to the cosmological constant is negative, in contrast to the positive contributions of the quantum fluctuations of the matter fields, possibly opening an avenue towards addressing the cosmological constant problem in a more general context.

hep-th↗
arXiv2011

Dressed Fermions, Modular Transformations and Bosonization in the Compactified Schwinger Model

The celebrated exactly solvable "Schwinger" model, namely massless two-dimensional QED, is revisited. The solution presented here emphasizes the non- perturbative relevance of the topological sector through large gauge transformations whose role is made manifest by compactifying space into a circle. Eventually the well-known non-perturbative features and solution of the model are recovered in the massless case. However the fermion mass term is shown to play a subtle role in order to achieve a physical quantization that accounts for gauge invariance under both small and large gauge symmetries. Quantization of the system follows Dirac's approach in an explicitly gauge invariant way that avoids any gauge fixing procedure.

hep-th↗
arXiv2010

Noncommutative Spherically Symmetric Spaces

We examine some noncommutative spherically symmetric spaces in three space dimensions. A generalization of Snyder's noncommutative (Euclidean) space allows the inclusion of the generator of dilations into the defining algebra of the coordinate and rotation operators. We then construct a spherically symmetric noncommutative Laplacian on this space having the correct limiting spectrum. This is presented via a creation and annihilation operator realization of the algebra, which may lend itself to a truncation of the Hilbert space.

hep-th↗
arXiv2010

Noncommuting Coordinates and Magnetic Monopoles

The appearance of noncommuting spatial coordinates is studied in quantum systems containing a magnetic monopole and under the influence of a radial potential. We derive expressions for the commutators of the coordinates that have been restricted to the lowest energy level. Quantum corrections are found to previous results by Frenkel and Pereira based on quantizing the Dirac brackets of the classical theory. For two different potentials, the modified harmonic oscillator potential and the modified Coulomb potential, we also calculate the commutators for a projection to a fixed energy level.

hep-th↗
arXiv2009

Variations on the Planar Landau Problem: Canonical Transformations, A Purely Linear Potential and the Half-Plane

The ordinary Landau problem of a charged particle in a plane subjected to a perpendicular homogeneous and static magnetic field is reconsidered from different points of view. The role of phase space canonical transformations and their relation to a choice of gauge in the solution of the problem is addressed. The Landau problem is then extended to different contexts, in particular the singular situation of a purely linear potential term being added as an interaction, for which a complete purely algebraic solution is presented. This solution is then exploited to solve this same singular Landau problem in the half-plane, with as motivation the potential relevance of such a geometry for quantum Hall measurements in the presence of an electric field or a gravitational quantum well.

hep-th↗
arXiv2009

The Klauder-Daubechies Construction of the Phase Space Path Integral and the Harmonic Oscillator

The canonical operator quantisation formulation corresponding to the Klauder-Daubechies construction of the phase space path integral is considered. This formulation is explicitly applied and solved in the case of the harmonic oscillator, thereby illustrating in a manner complementary to Klauder and Daubechies' original work some of the promising features offered by their construction of a quantum dynamics. The Klauder-Daubechies functional integral involves a regularisation parameter eventually taken to vanish, which defines a new physical time scale. When extrapolated to the field theory context, besides providing a new regularisation of short distance divergences, keeping a finite value for that time scale offers some tantalising prospects when it comes to strong gravitational quantum systems.

hep-th↗
arXiv2008

A Deformation of Quantum Dynamics through the Phase Space Path Integral

Using a regularised construction of the phase space path integral due to Ingrid Daubechies and John Klauder which involves a time scale ultimately taken to vanish, and motivated by the general programme towards a noncommutative space(time) geometry, physical consequences of assuming this time parameter to provide rather a new fundamental time scale are explored in the context of the one dimensional harmonic oscillator. Some tantalising results are achieved, which raise intriguing prospects when extrapolated to the quantum field theory and gravitational contexts.

hep-th↗
arXiv2008

A Pedestrian Introduction to the Mathematical Concepts of Quantum Physics

These notes offer a basic introduction to the primary mathematical concepts of quantum physics, and their physical significance, from the operator and Hilbert space point of view, highlighting more what are essentially the abstract algebraic aspects of quantisation in contrast to more standard treatments of such issues, while also bridging towards the path integral formulation of quantisation. A discussion of the (first) Noether theorem and Lie symmetries is also included to complement the presentation. Emphasis is put throughout, as illustrative examples threading the presentation, on the quantum harmonic oscillator and the dynamics of a charged particle coupled to the electromagnetic field, with the ambition to bring the reader onto the threshold of relativistic quantum field theories with their local gauge invariances as a natural framework for describing relativistic quantum particles in interaction and carrying specific conserved charges.

hep-th↗
arXiv2008

Classes of f-Deformed Landau Operators: Nonlinear Noncommutative Coordinates from Algebraic Representations

We consider, in a superspace, new operator dependent noncommutative (NC) geometries of the nonlinear quantum Hall limit related to classes of f-deformed Landau operators in the spherical harmonic well. Different NC coordinate algebras are determined using unitary representation spaces of Fock-Heisenberg tensored algebras and of the Schwinger-Fock realisation of the su(1,1) Lie algebra. A reduced model allowing an underlying N=2 superalgebra is also discussed.

hep-th↗
arXiv2007

World-line Quantisation of a Reciprocally Invariant System

We present the world-line quantisation of a system invariant under the symmetries of reciprocal relativity (pseudo-unitary transformations on ``phase space coordinates" $(x^μ(τ),p^μ(τ))$ which preserve the Minkowski metric and the symplectic form, and global shifts in these coordinates, together with coordinate dependent transformations of an additional compact phase coordinate, $θ(τ)$). The action is that of free motion over the corresponding Weyl-Heisenberg group. Imposition of the first class constraint, the generator of local time reparametrisations, on physical states enforces identification of the world-line cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group $Q(D-1,1)\cong U(D-1,1)\ltimes H(D)$, the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group (the central extension of the global translation group, with central extension associated to the phase variable $θ(τ)$). The spacetime spectrum of physical states is identified. Even though for an appropriate range of values the restriction enforced by the cosmological constant projects out negative norm states from the physical spectrum, leaving over spin zero states only, the mass-squared spectrum is continuous over the entire real line and thus includes a tachyonic branch as well.

hep-th↗
arXiv2007

The Weyl-Heisenberg Group on the Noncommutative Two-Torus: A Zoo of Representations

In order to assess possible observable effects of noncommutativity in deformations of quantum mechanics, all irreducible representations of the noncommutative Heisenberg algebra and Weyl-Heisenberg group on the two-torus are constructed. This analysis extends the well known situation for the noncommutative torus based on the algebra of the noncommuting position operators only. When considering the dynamics of a free particle for any of the identified representations, no observable effect of noncommutativity is implied.

hep-th↗
arXiv2007

Topologically Massive Gauge Theories and their Dual Factorised Gauge Invariant Formulation

There exists a well-known duality between the Maxwell-Chern-Simons theory and the self-dual massive model in 2+1 dimensions. This dual description has been extended to topologically massive gauge theories (TMGT) in any dimension. This Letter introduces an unconventional approach to the construction of this type of duality through a reparametrisation of the master theory action. The dual action thereby obtained preserves the same gauge symmetry structure as the original theory. Furthermore, the dual action is factorised into a propagating sector of massive gauge invariant variables and a sector with gauge variant variables defining a pure topological field theory. Combining results obtained within the Lagrangian and Hamiltonian formulations, a new completed structure for a gauge invariant dual factorisation of TMGT is thus achieved.

hep-th↗
arXiv2007

Gauge Invariant Factorisation and Canonical Quantisation of Topologically Massive Gauge Theories in Any Dimension

Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to generate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the 2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as particular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT) sector related to the topological mass term is not manifest in the original phase space. However through an appropriate canonical transformation, a gauge invariant factorisation of phase space into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate gauge invariant variables with free massive excitations. The second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated non dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also enables a definition of the usual projection from TMGT onto topological quantum field theories in a most natural and transparent way. None of these results rely on any gauge fixing procedure whatsoever.

hep-th↗
arXiv2006

A (p,q)-deformed Landau problem in a spherical harmonic well: spectrum and noncommuting coordinates

A (p,q)-deformation of the Landau problem in a spherically symmetric harmonic potential is considered. The quantum spectrum as well as space noncommutativity are established, whether for the full Landau problem or its quantum Hall projections. The well known noncommutative geometry in each Landau level is recovered in the appropriate limit p,q=1. However, a novel noncommutative algebra for space coordinates is obtained in the (p,q)-deformed case, which could also be of interest to collective phenomena in condensed matter systems.

hep-th↗
arXiv2006

(p,q)-Deformations and (p,q)-Vector Coherent States of the Jaynes-Cummings Model in the Rotating Wave Approximation

Classes of (p,q)-deformations of the Jaynes-Cummings model in the rotating wave approximation are considered. Diagonalization of the Hamiltonian is performed exactly, leading to useful spectral decompositions of a series of relevant operators. The latter include ladder operators acting between adjacent energy eigenstates within two separate infinite discrete towers, except for a singleton state. These ladder operators allow for the construction of (p,q)-deformed vector coherent states. Using (p,q)-arithmetics, explicit and exact solutions to the associated moment problem are displayed, providing new classes of coherent states for such models. Finally, in the limit of decoupled spin sectors, our analysis translates into (p,q)-deformations of the supersymmetric harmonic oscillator, such that the two supersymmetric sectors get intertwined through the action of the ladder operators as well as in the associated coherent states.

quant-ph↗