1,721,001 research outputs found
On the quasi-position representation in theories with a minimal length
No full textQuantum mechanical models with a minimal length are often described by modifying the commutation relation between position and momentum. Although this represents a small complication when described in momentum space, at least formally, the (quasi-)position representation acquires numerous issues, source of misunderstandings. In this work, we review these issues, clarifying some of the aspects of minimal length models, with particular reference to the representation of the position operator
Space and time transformations with a minimal length
No full textPhenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics to introduce a minimal uncertainty in position in quantum mechanics. In the present work, we show the influence of space and time transformations in shaping quantities such as momentum, energy, and their relations with the generators of transformations. Thus, such an influence determines, among other aspects, the time evolution of a quantum system. In the exemplary case of Galilean transformations, the Schrödinger equation is identical to the ordinary case
Position in Minimal Length Quantum Mechanics
No full textSeveral approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems
Minimal length effects on quantum cosmology and quantum black hole models
No full textA Kantowski-Sachs model with a modified quantization prescription is considered. Such quantization rules, inspired by the so-called generalized uncertainty principle, correspond to a modified commutation relation between minisuperspace variables and their conjugate momenta. For a wide range of the modification parameter, this approach differentiates from the standard results by the presence of a potential well in the corresponding Wheeler-DeWitt equation. This then produces the appearance of a set of wave functions, with corresponding discrete energy spectrum
Generalized ladder operators for the perturbed harmonic oscillator
No full textIn this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that corrections to ladder operators for a harmonic oscillator with a generic perturbation and to an arbitrary order of perturbation theory have been constructed
Generalized uncertainty principle: from the harmonic oscillator to a QFT toy model
No full textSeveral models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg uncertainty principle into the generalized uncertainty principle. In this work, we study the implications of a polynomial generalized uncertainty principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the generalized uncertainty principle
Quantum field theory with the generalized uncertainty principle II: Quantum Electrodynamics
No full textContinuing our earlier work on the application of the Relativistic Generalized Uncertainty Principle (RGUP) to quantum field theories, in this paper we study Quantum Electrodynamics (QED) with minimum length. We obtain expressions for the Lagrangian, Feynman rules and scattering amplitudes of the theory, and discuss their consequences for current and future high energy physics experiments. We hope this will provide an improved window for testing Quantum Gravity effects in the laboratory
Relativistic generalized uncertainty principle
No full textThe Generalized Uncertainty Principle and the related minimum length are normally considered in non-relativistic Quantum Mechanics. Extending it to relativistic theories is important for having a Lorentz invariant minimum length and for testing the modified Heisenberg principle at high energies. In this paper, we formulate a relativistic Generalized Uncertainty Principle. We then use this to write the modified Klein–Gordon, Schrödinger and Dirac equations, and compute quantum gravity corrections to the relativistic hydrogen atom, particle in a box, and the linear harmonic oscillator
Response to Comments on the paper “Relativistic generalized uncertainty principle”
No full textWe address the comments on our paper (Todorinov et al., 2019) presented in Chargui (2020). We show that the points raised in Chargui (2020) do not contain any new results or valid criticisms
Potential tests of the generalized uncertainty principle in the advanced LIGO experiment
No full textThe generalized uncertainty principle and a minimum measurable length arise in various theories of gravity and predict Planck-scale modifications of the canonical position-momentum commutation relation. Postulating a similar modified commutator between the canonical variables of the electromagnetic field in quantum optics, we compute Planck-scale corrections to the radiation pressure noise and shot noise of Michelson–Morley interferometers, with particular attention to gravity wave detectors such as LIGO. We show that advanced LIGO is potentially sensitive enough to observe Planck-scale effects and thereby indirectly a minimal length. We also propose estimates for the bounds on quantum gravity parameters from current and future advanced LIGO experiments
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