Nonpointlike Particles in Harmonic Oscillators

Quantum mechanics ordinarily describes particles as being pointlike, in the sense that the uncertainty $\Delta x$ can, in principle, be made arbitrarily small. It has been shown that suitable correction terms to the canonical commutation relations induce a finite lower bound to spatial localisation. Here, we perturbatively calculate the corrections to the energy levels of an in this sense nonpointlike particle in isotropic harmonic oscillators. Apart from a special case the degeneracy of the energy levels is removed.Comment: LaTeX, 9 pages, 1 figure included via epsf optio

Comment on "Quantum mechanics of smeared particles"

In a recent article, Sastry has proposed a quantum mechanics of smeared particles. We show that the effects induced by the modification of the Heisenberg algebra, proposed to take into account the delocalization of a particle defined via its Compton wavelength, are important enough to be excluded experimentally.Comment: 2 page

Perturbation spectrum in inflation with cutoff

It has been pointed out that the perturbation spectrum predicted by inflation may be sensitive to a natural ultraviolet cutoff, thus potentially providing an experimentally accessible window to aspects of Planck scale physics. A priori, a natural ultraviolet cutoff could take any form, but a fairly general classification of possible Planck scale cutoffs has been given. One of those categorized cutoffs, also appearing in various studies of quantum gravity and string theory, has recently been implemented into the standard inflationary scenario. Here, we continue this approach by investigating its effects on the predicted perturbation spectrum. We find that the size of the effect depends sensitively on the scale separation between cutoff and horizon during inflation.Comment: 6 pages; matches version accepted by PR

Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework

In the context of a two-parameter $(\alpha, \beta)$ deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using techniques of supersymmetric quantum mechanics combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters $\alpha$, $\beta$ vanishes, in which case shape invariance under parameter translation occurs. In the special case where $\alpha = \beta \ne 0$, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with $q > 1$ and its eigenvectors are therefore $n$-$q$-boson states. In the general case where $0 \ne \alpha \ne \beta \ne 0$, the eigenvectors are constructed as linear combinations of $n$-$q$-boson states by resorting to a Bargmann representation of the latter and to $q$-differential calculus. They are finally expressed in terms of a $q$-exponential and little $q$-Jacobi polynomials.Comment: LaTeX, 24 pages, no figure, minor changes, additional references, final version to be published in JP

Maximally localized states and causality in non commutative quantum theories

We give simple representations for quantum theories in which the position commutators are non vanishing constants. A particular representation reproduces results found using the Moyal star product. The notion of exact localization being meaningless in these theories, we adapt the notion of ``maximally localized states'' developed in another context . We find that gaussian functions play this role in a 2+1 dimensional model in which the non commutation relations concern positions only. An interpretation of the wave function in this non commutative geometry is suggested. We also analyze higher dimensional cases. A possible incidence on the causality issue for a Q.F.T with a non commuting time is sketched.Comment: 11 pages, Revtex. The presentation has been improved, the subsection on high dimensions has been modified. This version will appear in PR

Harmonic oscillator with minimal length uncertainty relations and ladder operators

We construct creation and annihilation operators for harmonic oscillators with minimal length uncertainty relations. We discuss a possible generalization to a large class of deformations of cannonical commutation relations. We also discuss dynamical symmetry of noncommutative harmonic oscillator.Comment: 8 pages, revtex4, final version, to appear in PR

Mode Generating Mechanism in Inflation with Cutoff

In most inflationary models, space-time inflated to the extent that modes of cosmological size originated as modes of wavelengths at least several orders of magnitude smaller than the Planck length. Recent studies confirmed that, therefore, inflationary predictions for the cosmic microwave background perturbations are generally sensitive to what is assumed about the Planck scale. Here, we propose a framework for field theories on curved backgrounds with a plausible type of ultraviolet cutoff. We find an explicit mechanism by which during cosmic expansion new (comoving) modes are generated continuously. Our results allow the numerical calculation of a prediction for the CMB perturbation spectrum.Comment: 9 pages, LaTe

A note on inflation and transplanckian physics

In this paper we consider the influence of transplanckian physics on the CMBR anisotropies produced by inflation. We consider a simple toy model that allows for analytic calculations and argue on general grounds, based on ambiguities in the choice of vacuum, that effects are expected with a magnitude of the order of $H/\Lambda$, where $H$ is the Hubble constant during inflation and $\Lambda$ the scale for new physics, e.g. the Planck scale.Comment: 12 pages. v2: typos corrected and references added. v3: final version accepted for publication by PRD. Improved discussion of adiabatic vacuu

Signatures in the Planck Regime

String theory suggests the existence of a minimum length scale. An exciting quantum mechanical implication of this feature is a modification of the uncertainty principle. In contrast to the conventional approach, this generalised uncertainty principle does not allow to resolve space time distances below the Planck length. In models with extra dimensions, which are also motivated by string theory, the Planck scale can be lowered to values accessible by ultra high energetic cosmic rays (UHECRs) and by future colliders, i.e. $M_f\approx$ 1 TeV. It is demonstrated that in this novel scenario, short distance physics below $1/M_f$ is completely cloaked by the uncertainty principle. Therefore, Planckian effects could be the final physics discovery at future colliders and in UHECRs. As an application, we predict the modifications to the $e^+e^- \to f^+f^-$ cross-sections.Comment: 14 pages, 4 figures, typos corrected, references adde