arXiv subjects

Michele Arzano

Publications and source records attributed to Michele Arzano.

At least 55 records · Page 3Linked to original sources

arXiv2015

Quantization of fluctuations in DSR: the two-point function and beyond

We show that the two-point function of a quantum field theory with de Sitter momentum space (herein called DSR) can be expressed as the product of a standard delta function and an energy-dependent factor. This is a highly non-trivial technical result in any theory without a preferred frame. Applied to models exhibiting running of the dimensionality of space, this result is essential in proving that vacuum fluctuations are generally scale-invariant at high energies whenever there is running to two dimensions. This is equally true for theories with and without a preferred frame, with differences arising only as we consider higher order correlators. Specifically, the three-point function of DSR has a unique structure of "open triangles", as shown here.

gr-qc↗

arXiv2015

Vacuum fluctuations in theories with deformed dispersion relations

We examine vacuum fluctuations in theories with modified dispersion relations which represent dimensional reduction at high energies. By changing units of energy and momentum we can obtain a description rendering the dispersion relations undeformed and transferring all the non-trivial effects to the integration measure in momentum space. Using this description we propose a general quantization procedure, which should be applicable whether or not the theory explicitly introduces a preferred frame. Based on this scheme we evaluate the power spectrum of quantum vacuum fluctuations. We find that in {\it all} theories which run to 2 dimensions in the ultraviolet the vacuum fluctuations, in the ultraviolet regime, are scale-invariant. This is true in flat space but also for "inside the horizon" modes in an expanding universe. We spell out the conditions upon the gravity theory for this scale-invariance to be preserved as the modes are frozen-in outside the horizon. We also digress on the meaning of dimensionality (in momentum and position space) and suggest that the spectral index could itself provide an operational definition of dimensionality.

gr-qc↗

arXiv2015

Gravity as the breakdown of conformal invariance

We propose that at the beginning of the universe gravity existed in a limbo either because it was switched off or because it was only conformally coupled to all particles. This picture can be reverse-engineered from the requirement that the cosmological perturbations be (nearly) scale-invariant without the need for inflation. It also finds support in recent results in quantum gravity suggesting that spacetime becomes two-dimensional at super-Planckian energies. We advocate a novel top-down approach to cosmology based on the idea that gravity and the Big Bang Universe are relics from the mechanism responsible for breaking the fundamental conformal invariance. Such a mechanism should leave clear signatures in departures from scale-invariance in the primordial power spectrum and the level of gravity waves generated.

gr-qc↗

arXiv2014

Planck-scale phenomenology with anti-de Sitter momentum space

We investigate the anti-de Sitter (AdS) counterpart to the well studied de Sitter (dS) model for energy-momentum space, viz "$κ$-momentum space" space (with a structure based on the properties of the $κ$-Poincaré Hopf algebra). On the basis of previous preliminary results one might expect the two models to be "dual": dS exhibiting an invariant maximal spatial momentum but unbounded energy, AdS a maximal energy but unbounded momentum. If that were the case AdS momentum space could be used to implement a principle of maximal Planck-scale energy, just as several studies use dS momentum space to postulate of maximal Planck-scale spatial momentum. However several unexpected features are uncovered in this paper, which limit the scope of the expected duality, and interestingly they take different forms in different coordinatizations of AdS momentum space. "Cosmological" AdS coordinates mimic the dS construction used for $κ$-momentum space, and produce a Carrol limit in the ultraviolet. However, unlike the $κ$-momentum space, the boundary of the covered patch breaks Lorentz invariance, thereby introducing a preferred frame. In "horospherical" coordinates we achieve full consistency with frame independence as far as boost transformations are concerned, but find that rotational symmetry is broken, leading to an anisotropic model for the speed of light. Finally, in "static" coordinates we find a way of deforming relativistic transformations that successfully enforces frame invariance and isotropy, and produces a Carrol limit in the ultraviolet. However, the phenomenological implications appear to be too weak for any realistic chance of detection. Our results are also relevant for a long-standing debate on whether or not coordinate redefinitions in momentum space lead to physically equivalent theories: our three proposals are evidently physically inequivalent (abridged)

gr-qc↗

arXiv2014

Localization and diffusion in polymer quantum field theory

Polymer quantization is a non-standard approach to quantizing a classical system inspired by background independent approaches to quantum gravity such as loop quantum gravity. When applied to field theory it introduces a characteristic polymer scale at the level of the fields classical configuration space. Compared with models with space-time discreteness or non-commutativity this is an alternative way in which a characteristic scale can be introduced in a field theoretic context. Motivated by this comparison we study here localization and diffusion properties associated with polymer field observables and dispersion relation in order to shed some light on the novel physical features introduced by polymer quantization. While localization processes seems to be only mildly affected by polymer effects, we find that polymer diffusion differs significantly from the "dimensional reduction" picture emerging in other Planck-scale models beyond local quantum field theory.

gr-qc↗

arXiv2014

Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum space describing point particles coupled to Einstein gravity in three space-time dimensions. We introduce different sets of coordinates on the group manifold and discuss their properties under Lorentz transformations. In particular we show how a certain set of coordinates exhibits an upper bound on the energy under deformed Lorentz boosts which saturate at the Planck energy. We discuss how this deformed symmetry framework is generally described by a quantum deformation of the Poincaré group: the quantum double of ${\rm SL}(2,\mathbb{R})$. We then illustrate how the space of functions on the group manifold momentum space has a dual representation on a non-commutative space of coordinates via a (quantum) group Fourier transform. In this context we explore the connection between Weyl maps and different notions of (quantum) group Fourier transform appeared in the literature in the past years and establish relations between them.

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arXiv2014

Purity is not eternal at the Planck scale

Theories with Planck-scale deformed symmetries exhibit quantum time evolution in which purity of the density matrix is not preserved. In particular we show that the non-trivial structure of momentum space of these models is reflected in a deformed action of translation generators on operators. Such action in the case of time translation generators leads to a Lindblad-like evolution equation for density matrices when expanded at leading order in the Planckian deformation parameter. This evolution equation is covariant under the deformed realization of Lorentz symmetries characterizing these models.

hep-th↗

arXiv2014

Diffusion on $κ$-Minkowski space

We study the spectral dimension associated with diffusion processes on Euclidean $κ$-Minkowski space. We start by describing a geometric construction of the "Euclidean" momentum group manifold related to $κ$-Minkowski space. On such space we identify various candidate Laplacian functions, i.e. deformed Casimir invariants, and calculate the corresponding spectral dimension for each case. The results obtained show a variety of running behaviours for the spectral dimension according to the choice of deformed Laplacian, from dimensional reduction to super-diffusion.

hep-th↗

arXiv2013

Planck-scale dimensional reduction without a preferred frame

Several approaches to quantum gravity suggest that the standard description of spacetime as probed at low-energy, with four dimensions, is replaced in the Planckian regime by a spacetime with a spectral dimension of two. The implications for relativistic symmetries can be momentous, and indeed the most tangible picture for "running" of the spectral dimension, found within Horava-Lifschitz gravity, requires the breakdown of relativity of inertial frames. In this Letter we incorporate running spectral dimensions in a scenario that does not require the emergence of a preferred frame. We consider the best studied mechanism for deforming relativistic symmetries whilst preserving the relativity of inertial frames, based on a momentum space with curvature at the Planck scale. We show explicitly how running of the spectral dimension can be derived from these models.

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arXiv2013

Black-hole entropy and minimal diffusion

The density of states reproducing the Bekenstein-Hawking entropy-area scaling can be modeled via a nonlocal field theory. We define a diffusion process based on the kinematics of this theory and find a spectral dimension whose flow exhibits surprising properties. While it asymptotes four from above in the infrared, in the ultraviolet the spectral dimension diverges at a finite (Planckian) value of the diffusion length, signaling a breakdown of the notion of diffusion on a continuum spacetime below that scale. We comment on the implications of this minimal diffusion scale for the entropy bound in a holographic and field-theoretic context.

hep-th↗

arXiv2013

The measure matters

We adopt a framework where quantum-gravity's dynamical dimensional reduction of spacetime at short distances is described in terms of modified dispersion relations. We observe that by subjecting such models to a momentum-space diffeomorphism one obtains a "dual picture" with unmodified dispersion relations, but a modified measure of integration over momenta. We then find that the UV {\it Hausdorff} dimension of momentum space which can be inferred from this modified integration measure coincides with the short-distance {\it spectral} dimension of spacetime. This result sheds light into why scale-invariant fluctuations are obtained if the original model for two UV spectral dimensions is combined with Einstein gravity. By studying the properties of the inner product we derive the result that it is only in 2 energy-momentum dimensions that microphysical vacuum fluctuations are scale-invariant. This is true ignoring gravity, but then we find that if Einstein gravity is postulated in the original frame, in the dual picture gravity switches off, since all matter becomes conformally coupled. We argue that our findings imply that the following concepts are closely connected: scale-invariance of vacuum quantum fluctuations, conformal invariance of the gravitational coupling, UV reduction to spectral dimension 2 in position space and UV reduction to Hausdorff dimension 2 in energy-momentum space.

gr-qc↗

arXiv2013

Rainbow gravity and scale-invariant fluctuations

We re-examine a recently proposed scenario where the deformed dispersion relations associated with a flow of the spectral dimension to a UV value of 2 leads to a scale-invariant spectrum of cosmological fluctuations, without the need for inflation. In that scenario Einstein gravity was assumed. The theory displays a wavelength-dependent speed of light but by transforming to a suitable "rainbow frame" this feature can be removed, at the expense of modifying gravity. We find that the ensuing rainbow gravity theory is such that gravity switches off at high energy (or at least leads to a universal conformal coupling). This explains why the fluctuations are scale-invariant on all scales: there is no horizon scale as such. For dispersion relations that do not lead to exact scale invariance we find instead esoteric inflation in the rainbow frame. We argue that these results shed light on the behaviour of gravity under the phenomenon of dimensional reduction.

gr-qc↗

arXiv2013

Beyond Fock space in three dimensional semiclassical gravity

Quantization of relativistic point particles coupled to three-dimensional Einstein gravity naturally leads to field theories living on the Lorentz group in their momentum representation. The Lie group structure of momentum space can be traced back to the classical phase space of the particles coupled to topological gravity. In this work we show how the non-trivial structure of momentum space leads to an unusual description of Fock space. The latter is reflected in a deformed algebra of creation and annihilation operators which reduces to the ordinary algebra when momentum space "flattens" to Minkowski space in the limit in which the three-dimensional Newton's constant vanishes. The construction is covariant under the action of relativistic symmetries acting on the Lorentz group-momentum space. This shows how it is possible to build a Fock space on a group manifold momentum space in a way consistent with the underlying (deformed) relativistic symmetries.

hep-th↗

arXiv2013

From bricks to quasinormal modes: A new perspective on black hole entropy

Calculations of black hole entropy based on the counting of modes of a quantum field propagating in a Schwarzschild background need to be regularized in the vicinity of the horizon. To obtain the Bekenstein-Hawking result the short distance cut-off needs to be fixed by hand. In this note we give an argument for obtaining this cut-off in a natural fashion. We do this by modelling the black hole by its set of quasinormal modes. The horizon then becomes a extended region: the quantum ergosphere. The interaction of the quantum ergosphere and the quantum field provides a natural regularization mechanism. The width of the quantum ergosphere provides the right cut-off for the entropy calculation. We arrive at a dual picture of black hole entropy. The entropy of the black hole is given both by the entropy of the quantum field in the bulk and the dynamical degrees of freedom on the horizon.

gr-qc↗

arXiv2013

Dimensional reduction in the sky

We explore the cosmological implications of a mechanism found in several approaches to quantum-gravity, whereby the spectral dimension of spacetime runs from the standard value of 4 in the infrared (IR) to a smaller value in the ultraviolet (UV). Specifically, we invoke the picture where the phenomenon is associated with modified dispersion relations. With minimal assumptions, we find that UV behaviour leading to 2 spectral dimensions results in an exactly scale-invariant spectrum of vacuum scalar and tensor fluctuations, regardless of the equation of state. The fluctuation production mechanism is analogous to the one known for varying speed of sound/light models and, unlike in inflation, the spectrum is already scale-invariant before leaving the horizon, remaining so after freeze-in. In the light of Planck's recent results we also discuss scenarios that break exact scale-invariance, such as the possibility that the spectral dimension runs down to a value slightly higher than 2, or runs down to 2 but with an extremely slow transient. We further show that the tensor to scalar ratio is fixed by the UV ratio between the speed of gravity and the speed of light. Not only does our model not require inflation, but at its most minimal it seems incompatible with it. In contrast, we find that running spectral dimensions can improve the outlook of the cyclic/ekpyrotic scenario, solving the main problems present in its simplest and most appealing realisations.

gr-qc↗

arXiv2012

Weaving commutators: beyond Fock space

The symmetrization postulate and the associated Bose/Fermi (anti)-commutators for field mode operators are among the pillars on which local quantum field theory lays its foundations. They ultimately determine the structure of Fock space and are closely connected with the local properties of the fields and with the action of symmetry generators on observables and states. We here show that the quantum field theory describing relativistic particles coupled to three dimensional Einstein gravity as a topological defect must be constructed using a deformed algebra of creation and annihilation operators. This reflects a non-trivial group manifold structure of the classical momentum space and a modification of the Leibniz rule for the action of symmetry generators governed by Newton's constant. We outline various arguments suggesting that, at least at the qualitative level, these three-dimensional results could also apply to real four-dimensional world thus forcing us to re-think the ordinary multiparticle structure of quantum field theory and many of the fundamental aspects connected to it.

hep-th↗

arXiv2012

The DSR-deformed relativistic symmetries and the relative locality of 3D quantum gravity

Over the last decade there were significant advances in the understanding of quantum gravity coupled to point particles in 3D (2+1-dimensional) spacetime. Most notably it is emerging that the theory can be effectively described as a theory of free particles on a momentum space with anti-deSitter geometry and with noncommutative spacetime coordinates of the type $[x^μ,x^ν]=i \hbar \ell \varepsilon^{μν}_{\phantom{μν}ρ} x^ρ$. We here show that the recently proposed relative-locality curved-momentum-space framework is ideally suited for accommodating these structures characteristic of 3D quantum gravity. Through this we obtain an intuitive characterization of the DSR-deformed Poincaré symmetries of 3D quantum gravity, and find that the associated relative spacetime locality is of the type producing dual-gravity lensing.

hep-th↗

arXiv2011

Fractional and noncommutative spacetimes

We establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of κ-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncommutative spacetime defined by a certain nonlinear algebra of the coordinates, which interpolates between κ-Minkowski and canonical spacetime. These results are based upon a braiding formula valid for any nonlinear algebra which can be mapped onto the Heisenberg algebra.

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