Narrow Results

Order-p parasupersymmetric and orthosupersymmetric quantum mechanics are shown to be fully reducible when they are realized in terms of the generators of a generalized deformed oscillator algebra and a ℤ_p+1-grading structure is imposed on the Fock space. The irreducible components provide p + 1 sets of bosonized operators corresponding to both unbroken and broken cases. Such a bosonization is minimal.

Fractional supersymmetric quantum mechanics of order λ is realized in terms of the generators of a generalized deformed oscillator algebra and a ℤ_λ-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing λ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of ℤ_λ-graded uniform topological symmetry of type (1, 1, …, 1) with topological invariants generalizing the Witten index.

A short review of Schrödinger Hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose–Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach.

Hamiltonians with inverse square interaction potential occur in the study of a variety of physical systems and exhibit a rich mathematical structure. In this talk we briefly mention some of the applications of such Hamiltonians and then analyze the case of the N-body rational Calogero model as an example. This model has recently been shown to admit novel solutions, whose properties are discussed.

We calculate the quantum statistical force acting on a partition wall that divides a one-dimensional box into two halves. The two half boxes containing the same (fixed) number of noninteracting bosons are kept at the same temperature, and admit the same boundary conditions at the outer walls; the only difference is the distinct boundary conditions imposed at the two sides of the partition wall. The net force acting on the partition wall is nonzero at zero temperature and remains almost constant for low temperatures. As the temperature increases, the force starts to decrease considerably, but after reaching a minimum it starts to increase, and tends to infinity with a square-root-of-temperature asympotics. This example demonstrates clearly that distinct boundary conditions cause remarkable physical effects for quantum systems.

We quantise complex, infinite-dimensional projective space CP(ℋ). We apply the result to quantise a complex, finite-dimensional, classical phase space whose symplectic volume is infinite, by holomorphically embedding it into CP(ℋ). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on and the Fubini–Study metric on CP(ℋ). Then the Hilbert-space bundle over is the pullback, by the embedding, of the Hilbert-space bundle over CP(ℋ).

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e. independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Y_{n m}(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u(1,1). Then by applying the lowering operator of the Lie algebra u(1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Y_{m m}(θ,φ) and Y_{m+1 m}(θ,φ).

The stationary phase method is applied to diffusion by a potential barrier for an incoming wave packet with energies greater than the height of the barrier. It is observed that a direct application leads to paradoxical results. The correct solution, confirmed by numerical calculations is the creation of multiple peaks as a consequence of multiple reflections. Lessons concerning the use of the stationary phase method are drawn.

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H₄ with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h₄ not only by the lowest Landau levels, but also by their corresponding standard coherent states.

Using the partition of the number p-1 into p-1 real parts which are not equal with each other necessarily, we develop the unitary parasupersymmetry algebra of arbitrary order p so that the well-known Rubakov–Spiridonov–Khare parasupersymmetry becomes a special case of the developed one. It is shown that the developed algebra is realized by simple harmonic oscillator and Landau problem on a flat surface with the symmetries of h₃ and h₄ Heisenberg–Lie algebras. For this new parasupersymmetry, the well-known unitary condition is violated, however, unitarity of the corresponding algebra is structurally conserved. Moreover, the components of the bosonic Hamiltonian operator are derived as functions from the mean value of the partition numbers with their label weight function.

An Abelian gerbe is constructed over classical phase space. The two-cocycles defining the gerbe are given by Feynman path integrals whose integrands contain the exponential of the Poincaré–Cartan form. The U(1) gauge group on the gerbe has a natural interpretation as the invariance group of the Schrödinger equation on phase space.

We show that an unification of quantum mechanics and general relativity implies that there is a fundamental length in Nature in the sense that no operational procedure would be able to measure distances shorter than the Planck length. Furthermore we give an explicit realization of an old proposal by Anderson and Finkelstein who argued that a fundamental length in nature implies unimodular gravity. Finally, using hand waving arguments we show that a minimal length might be related to the cosmological constant which, if this scenario is realized, is time dependent.

A one-dimensional quantum mechanical model possessing mass-gap, a gapless excitation, and an approximate parity doubling of energy levels is constructed based on heuristic QCD-inspired arguments. The model may serve for illustrative purposes when considering the related dynamical phenomena in particle and nuclear physics.

Information measures for relativistic quantum spinors are constructed to satisfy various postulated properties such as normalization invariance and positivity. Those measures are then used to motivate generalized Lagrangians meant to probe shorter distance physics within the maximum uncertainty framework. The modified evolution equations that follow are necessarily nonlinear and simultaneously violate Lorentz invariance, supporting previous heuristic arguments linking quantum nonlinearity with Lorentz violation. The nonlinear equations also break discrete symmetries. We discuss the implications of our results for physics in the neutrino sector and cosmology.

Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial g_μ arising in the denominator of the potentials.

In this paper, we propose to solve the relativistic Klein–Gordon and Dirac equations subjected to the action of a uniform electromagnetic field with a generalized uncertainty principle in the momentum space. In both cases, the energy eigenvalues and their corresponding eigenfunctions are obtained. The limit case is then deduced for a small parameter of deformation.

We show that this problem gives rise to the same differential equation of a well known potential of ordinary quantum mechanics. However there is a subtle difference in the choice of the parameters of the hypergeometric function solving the differential equation which changes the physical discussion of the spectrum.

We present the quantum dynamics of a charged particle interacting simultaneously with background electromagnetic field and linearized gravitational waves in the long wave-length and low-velocity limit. We start from the geodesic deviation equation, considering its close association with the proper detector frame, rather than the usual geodesic equation which is solved classically in the existing literature. The Hamiltonian is obtained, quantized and solved by using algebraic iterative methods. The solution shows conformity with the classical analysis by exhibiting the same resonance condition.

Interaction of linearized gravitational waves with a otherwise free particle has been studied quantum mechanically in a noncommutative (NC) phase-space to examine whether the particle's response to the gravitational wave gets modified due to spatial and/or momentum noncommutativity. The result shows that momentum noncommutativity introduces a oscillatory noise with a specific frequency determined by the fundamental momentum scale and particle mass. Because of the global nature of the phase-space noncommutativity such noise will have similar characteristics for all detector sites and thus will stand out in a data cross-correlation procedure. If detected, this noise will provide evidence of momentum noncommutativity and also an estimation of the relevant noncommutative parameter.