Quantum field theory with minimum length

In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N > 1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N = 2. We further show that this theorem does not hold for N > 2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical position expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity.

Quantum field theory with the generalized uncertainty principle II: Quantum Electrodynamics

One can use the generalized uncertainty principle (GUP) to incorporate the minimum measurable length in quantum gravity. It may be interesting to have a minimal time interval as well as the minimal length in the relativistic version of quantum mechanics in the presence of the gravitational effects. In this paper, we considered a covariant version of the GUP to investigate the effects of both the minimal time and the minimal length. Using the covariant GUP, the energy–momentum dispersion relation was modified. Starting with the modified dispersion relation, the corrections to the wave function were obtained and the problems of the particle in a box and the hydrogen atom were revisited. Considering the minimal time may be an interesting new topic, which is not widely studied before. It may lead to new results that can open a new window into quantum gravity.

We study the correction of the energy spectrum of a gravitational quantum well due to the combined effect of the braneworld model with infinite extra dimensions and generalized uncertainty principle. The correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed by the Schrödinger–Newton equation based on a minimal length framework. The two fold correction in the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory.

Dirac particle in gravitational quantum mechanics

Inspired by string theory, Heisenberg's uncertainty principle can be generalized to include the photon-electron gravitational interaction, which leads to the Generalized Uncertainty Principle (GUP). Although GUP considers gravitational uncertainty at the minimum fundamental length scale in physics, it does not consider the effects of spacetime curvature on quantum mechanical uncertainty relations. The Extended Uncertainty Principle (EUP) is a generalization of Heisenberg's Uncertainty Principle that, unlike the GUP, applies to large length scales. GEUP is also a linear combination of EUP and GUP that creates minimal uncertainty on large length scales. The Einstein-Gauss-Bonnet theory (EGB) can be considered as one of the most promising candidates for modified gravity. In this paper, by using GUP, EUP, and GEUP, we intend to obtain the Hawking temperature of a four-dimensional EGB black hole in the asymptotically flat and (Anti)-de Sitter spacetime. We show that coupling constant, cosmological constant, mass, and radius significantly affect Hawking temperature and decrease or increase Hawking temperature depending on the chosen horizons.

This paper is with the permission of Stepan Moskaliuk similar to what he will put in the confer-ence proceedings of the summer teaching school and workshop for Ukrainian PhD physics stu-dents as given in Bratislava, as of summer 2015. With his permission, this paper will be in part reproduced here for this journal. First of all, we restate a proof of a highly localized special case of a metric tensor uncertainty principle first written up by Unruh. Unruh did not use the Roberson-Walker geometry which we do, and it so happens that the dominant metric tensor we will be examining, is variation in δgtt. The metric tensor variations given by δgrr, and are negligible, as compared to the variation δgtt. Afterwards, what is referred to by Barbour as emergent duration of time is from the Heisenberg Uncertainty principle (HUP) applied to δgtt in such a way as to give, in the Planckian space-time regime a nonzero minimum non zero lower ground to a massive graviton, mgraviton. The lower bound to the massive graviton is influenced by δgtt and kinetic energy which is in the Planckian emergent duration of time δt as (E-V) . We find from δgtt version of the Heisenberg Uncertainty Principle (HUP), that the quantum value of the Δt·ΔE Heisenberg Uncertainty Principle (HUP) is likely not recoverable due to δgtt ≠ Ο(1)~gtt ≡ 1. i.e. δgtt≠ Ο(1) . i.e. is consistent with non-curved space, so Δt · ΔE ≥ no longer holds. This even if we take the stress energy tensor approximation Tii= diag (ρ ,-p,-p,-p) where the fluid approximation is used. Our treatment of the inflaton is via Handley et al., where we consider the lower mass limits of the graviton as due to when the inflaton is many times larger than a Potential energy, with a kinetic energy (KE) proportional to ρw ∝ a-3(1-w) ~ g*T4 , with g* initial degrees of freedom, and T initial temperature. Leading to non-zero initial entropy as stated in Appendix A. In addition we also examine a Ricci scalar value at the boundary between Pre Planckian to Planckian regime of space-time, setting the magnitude of k as approaching flat space conditions right after the Planck regime. Furthermore, we have an approximation as to initial entropy production N~Sinitial(graviton)~1037. Finally, this entropy is N, and we get an initial version of the cosmological “constant” as Appendix D which is linked to initial value of a graviton mass. Appendix E is for the Riemannian-Penrose inequality, which is either a nonzero NLED scale factor or quantum bounce as of LQG. Note that, Appendix F gives conditions so that a pre Planckian kinetic energy (inflaton) value greater than Potential energy occurs, which is foundational to the lower bound to Graviton mass. We will in the future add more structure to this calculation so as to confirm via a precise calculation that the lower bound to the graviton mass, is about 10-70 grams. Our lower bound is a dimensional approximation so far. We will make it exact. We conclude in this document with Appendix G, which is comparing our Pre Planckian space-time metric Heisenberg Uncertainty Principle with the generalized uncertainty principle in quantum gravity. Our result is different from the one given by Ali, Khali and Vagenas, in which our energy fluctuation is not proportional to that of processes of energy connected to Black hole physics, and we also allow for the possibility of Pre Planckian time. Whereas their result (and the generalized string theory Heisenberg Uncertainty principle) have a more limited regime of interpolation of final results. We do come up with equivalent bounds to recover δgtt ~ small-value ≠ O(1) and the deviation of fluctuations of energy, but with very specific bounds upon the parameters of Ali, Khali, and Vegenas, but this has to be more fully explored. Finally, we close with a comparison of what this new Metric tensor uncertainty principle presages as far as avoiding the Bicep 2 mistake, and the different theories of gravity, as reviewed in Appendix H.

Several phenomenological approaches to quantum gravity predict the existence of a minimal measurable length and/or a maximum measurable momentum near the Planck scale. When embedded into the framework of quantum mechanics, such constraints induce a modification of the canonical commutation relations and thus a generalization of the Heisenberg uncertainty relations, commonly referred to as generalized uncertainty principle (GUP). Different models of quantum gravity imply different forms of the GUP. For instance, in the framework of string theory the GUP is quadratic in the momentum operator, while in the context of doubly special relativity it includes an additional linear dependence. Among the possible physical consequences, it was recently shown that the quadratic GUP induces a universal decoherence mechanism, provided one assumes a foamy structure of quantum spacetime close to the Planck length. Along this line, in the present work we investigate the gravitational decoherence associated to the linear-quadratic GUP and we compare it with the one associated to the quadratic GUP. We find that, despite their similarities, the two generalizations of the Heisenberg uncertainty principle yield decoherence times that are completely uncorrelated and significantly distinct. Motivated by this result, we introduce a theoretical and experimental scheme based on cavity optomechanics to measure the different time evolution of nonlocal quantum correlations corresponding to the two aforementioned decoherence mechanisms. We find that the deviation between the two predictions occurs on time scales that are macroscopic and thus potentially amenable to experimental verification. This scenario provides a possible setting to discriminate between different forms of the GUP and therefore different models of quantum gravity.

The effects of gravity which manifest themselves when performing the simultaneous measurement of two non-commuting observables in the quantum theory are discussed. Matter and gravity are considered as quantum fields. The Schr¨odinger-type time equation is given for the case of a finite number of degrees of freedom: one for the matter field and one for geometry. For a spatially closed system filled with dust and radiation being in definite quantum states, the solutions to the quantum equations are found, and the existence of the minimum measurable length and the minimum momentum is shown. It appears that the simultaneous measurement of fluctuations of the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime cannot be performed with an accuracy exceeding the Planck constant. Unruh’s and Bronstein’s uncertainty relations are discussed.

Inflationary universe in the presence of a minimal measurable length

Doubly special relativity (DSR) is usually regarded as a low-energy limit of a quantum gravity theory with testable predictions. On the other hand, non-local quantum field theories have been presented as a solution to the inconsistencies arising when quantizing gravity. Here, we present a new formulation of quantum field theories in DSR with non-local behavior. Our construction restricts the models to those showing linear Lorentz invariance. We derive the deformed Klein–Gordon, Dirac, and electromagnetic Lagrangians, as well as the deformed Maxwell equations. We also discuss the electric potential of a point charge. Finally, we analyze the connection between the nonlocality of field theories and DSR.

We study the consequences of the generalized Heisenberg uncertainty relation which admits a minimal uncertainty in length such as the case in a theory of quantum gravity. In particular, the theory of quantum harmonic oscillators arising from such a generalized uncertainty relation is examined. We demonstrate that all the standard properties of the quantum harmonic oscillators prevail when we employ a generalized momentum. We also show that quantum electrodynamics and coherent photon states can be described in the familiar standard manner despite the generalized uncertainty principle.

We show that the Equivalence Principle is violated by Quantum Gravity (QG) effects. The predicted violations are compared to experimental observations for Gravitational Redshift, Law of Reciprocal Action and Universality of Free Fall. This allows us to derive explicit bounds for β—the QG scale. In our approach, there appears a deviation in the geodesic motion of a particle. This deviation is induced by a non-commutative spacetime, consistent with a Generalized Uncertainty Principle (GUP). GUP admits the presence of a minimum length scale, that is advocated by QG theories. Remarkably, the GUP induced corrections are quite robust since the bound on β obtained by us, in General Relativity scenario in an essentially classical setting of modified geodesic motion, is closely comparable to similar bounds in recent literature (Das and Vagenas 2008 Phys. Rev. Lett. 101 221301). The latter are computed in purely quantum physics domain in flat spacetime.

Black holes represent fundamental laboratories for testing quantum gravity theories, where extreme curvature and thermal physics provide unique opportunities to probe Planck-scale modifications to space time. We investigate quantum gravity corrections to black hole thermodynamics through the Generalized Uncertainty Principle (GUP), focusing on modifications to Hawking radiation and accretion disk physics. Our comprehensive analysis reveals that GUP effects systematically alter the temperature-mass relationship for black holes, leading to observable signatures in both the thermal emission spectrum and accretion disk properties. Through detailed calculations incorporating relativistic effects and advanced statistical mechanics, we demonstrate that quantum gravity modifications introduce spectral distortions in the X-ray regime that are potentially detectable with current and future high-energy missions. The modified thermodynamics also affects accretion efficiency, disk temperature profiles, and iron line emission, producing characteristic signatures in the observed continuum spectra. Our analysis of recent data from NuSTAR, XRISM, and other X-ray observatories places new constraints on the GUP parameter β≤1500 for stellar-mass black holes, representing the most stringent limits from black hole observations to date. These results demonstrate the potential for black hole observations to probe fundamental quantum gravity effects and provide complementary constraints to those derived from neutron star studies. The framework developed here establishes black hole systems as viable laboratories for testing quantum gravity theories through multi-messenger astronomy.

Various theories of quantum gravity argue that near the Planck scale, the Heisenberg uncertainty principle should be replaced by the so called generalized uncertainty principle (GUP). We show that the GUP gives rise to two additional terms in any quantum mechanical Hamiltonian, proportional to βp4 and β2p6, respectively, where β ∼ 1/(MPlc)2 is the GUP parameter. These terms become important at or above the Planck energy. Considering only the first of these and treating it as a perturbation, we show that the GUP affects the Lamb shift, Landau levels, reflection and transmission coefficients of a potential step and potential barrier, and the current in a scanning tunnel microscope (STM). Although these are too small to be measurable at present, we speculate on the possibility of extracting measurable predictions in the future.