arXiv subjects

Tomasz Trześniewski

Publications and source records attributed to Tomasz Trześniewski.

13 recordsLinked to original sources

arXiv2024

On Carrollian and Galilean contractions of BMS algebra in 3 and 4 dimensions

In this paper, we find a class of Carrollian and Galilean contractions of (extended) BMS algebra in 3+1 and 2+1 dimensions. To this end, we investigate possible embeddings of 3D/4D Poincaré into the BMS${}_3$ and BMS${}_4$ algebras, respectively. The contraction limits in the 2+1-dimensional case are then enforced by appropriate contractions of its Poincaré subalgebras. In 3+1 dimensions, we have to apply instead the analogy between the structures of Poincaré and BMS algebra. In the case of non-vanishing cosmological constant in 2+1 dimensions, we consider the contractions of $Λ$-BMS${}_3$ algebra in an analogous manner.

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arXiv2023

Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei

There is a surge of research devoted to the formalism and physical manifestations of non-Lorentzian kinematical symmetries, which focuses especially on the ones associated with the Galilei and Carroll relativistic limits (the speed of light taken to infinity or to zero, respectively). The investigations have also been extended to quantum deformations of the Carrollian and Galilean symmetries, in the sense of (quantum) Hopf algebras. The case of 2+1 dimensions is particularly worth to study due to both the mathematical nature of the corresponding (classical) theory of gravity, and the recently finalized classification of all quantum-deformed algebras of spacetime isometries. Consequently, the list of all quantum deformations of (anti-)de Sitter-Carroll algebra is immediately provided by its well-known isomorphism with either Poincaré or Euclidean algebra. Quantum contractions from the (anti-)de Sitter to (anti-)de Sitter-Carroll classification allow to almost completely recover the latter. One may therefore conjecture that the analogous contractions from the (anti-)de Sitter to (anti-)de Sitter-Galilei $r$-matrices provide (almost) all coboundary deformations of (anti-)de Sitter-Galilei algebra. This scheme is complemented by deriving (Carrollian and Galilean) quantum contractions of deformations of Poincaré algebra, leading to coboundary deformations of Carroll and Galilei algebras.

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arXiv2023

3D gravity, point particles and deformed symmetries

It is well known that gravity in 2+1 dimensions can be recast as Chern-Simons theory, with the gauge group given by the local isometry group, depending on the metric signature and the cosmological constant. Point particles are added into spacetime as (spinning) conical defects. Then, in principle, one may integrate out the gravitational degrees of freedom to obtain the effective particle action; the most interesting consequence is that the momentum space of a particle turns out to be curved. This is still not completely understood in the case of non-zero cosmological constant.

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arXiv2021

Hunting for gravitational quantum spikes

We present the result of our examination of quantum structures called quantum spikes. The classical spikes, that are known in gravitational systems, occur in the evolution of the inhomogeneous spacetimes. Different kind of spikes, which we name strange spikes, can be seen in the dynamics of the homogeneous sector of the Belinski-Khalatnikov-Lifshitz scenario. They can be made visible if the so-called inhomogeneous initial data are used. The question to be explored is whether the strange spikes may survive quantization. The answer is in the affirmative. However, this is rather a subtle effect that needs further examination using sophisticated analytical and numerical tools. The spikes seem to be of fundamental importance, both at classical and quantum levels, as they may serve as seeds of real structures in the universe.

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arXiv2020

General-relativistic spin system

The models of spin systems defined on Euclidean space provide powerful machinery for studying a broad range of condensed matter phenomena. While the non-relativistic effective description is sufficient for most of the applications, it is interesting to consider special and general relativistic extensions of such models. Here, we introduce a framework that allows us to construct theories of continuous spin variables on a curved spacetime. Our approach takes advantage of the results of the non-linear field space theory, which shows how to construct compact phase space models, in particular for the spherical phase space of spin. Following the methodology corresponding to a bosonization of spin systems into the spin wave representations, we postulate a representation having the form of the Klein-Gordon field. This representation is equivalent to the semi-classical version of the well-known Holstein-Primakoff transformation. The general-relativistic extension of the spin wave representation is then performed, leading to the general-relativistically motivated modifications of the Ising model coupled to a transversal magnetic field. The advantage of our approach is its off-shell construction, while the popular methods of coupling fermions to general relativity usually depend on the form of Einstein field equations with matter. Furthermore, we show equivalence between the considered spin system and the Dirac-Born-Infeld type scalar field theory with a specific potential, which is also an example of k-essence theory. Based on this, the cosmological consequences of the introduced spin field matter content are preliminarily investigated.

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arXiv2020

Gauge fields and quantum entanglement

The purpose of this letter is to explore the relation between gauge fields, which are at the base of our understanding of fundamental interactions, and the quantum entanglement. To this end, we investigate the case of ${\rm SU}(2)$ gauge fields. It is first argued that holonomies of the ${\rm SU}(2)$ gauge fields are naturally associated with maximally entangled two-particle states. Then, we provide some evidence that the notion of such gauge fields can be deduced from the transformation properties of maximally entangled two-particle states. This new insight unveils a possible relation between gauge fields and spin systems, as well as contributes to understanding of the relation between tensor networks (such as MERA) and spin network states considered in loop quantum gravity. In consequence, our results turn out to be relevant in the context of the emerging Entanglement/Gravity duality.

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arXiv2020

Spectral dimensions and dimension spectra of quantum spacetimes

Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterised by a discrete complex set -- the dimension spectrum. Here we show that these two notions complement each other and the dimension spectrum is very useful in unravelling the UV behaviour of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the $κ$-Minkowski spacetime, for a few different Laplacians. In particular, we find out that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the $κ$-Minkowski spacetime.

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arXiv2020

Quantum $D = 3$ Euclidean and Poincaré symmetries from contraction limits

Following the recently obtained complete classification of quantum-deformed $\mathfrak{o}(4)$, $\mathfrak{o}(3,1)$ and $\mathfrak{o}(2,2)$ algebras, characterized by classical $r$-matrices, we study their inhomogeneous $D = 3$ quantum IW contractions (i.e. the limit of vanishing cosmological constant), with Euclidean or Lorentzian signature. Subsequently, we compare our results with the complete list of $D = 3$ inhomogeneous Euclidean and $D = 3$ Poincaré quantum deformations obtained by P.~Stachura. It turns out that the IW contractions allow us to recover all Stachura deformations. We further discuss the applicability of our results in the models of 3D quantum gravity in the Chern-Simons formulation (both with and without the cosmological constant), where it is known that the relevant quantum deformations should satisfy the Fock-Rosly conditions. The latter deformations in part of the cases are associated with the Drinfeld double structures, which also have been recently investigated in detail.

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arXiv2018

Towards the map of quantum gravity

In this paper we point out some possible links between different approaches to quantum gravity and theories of the Planck scale physics. In particular, connections between Loop Quantum Gravity, Causal Dynamical Triangulations, Hořava-Lifshitz gravity, Asymptotic Safety scenario, Quantum Graphity, deformations of relativistic symmetries and nonlinear phase space models are discussed. The main focus is on quantum deformations of the Hypersurface Deformations Algebra and Poincaré algebra, nonlinear structure of phase space, the running dimension of spacetime and nontrivial phase diagram of quantum gravity. We present an attempt to arrange the observed relations in the form of a graph, highlighting different aspects of quantum gravity. The analysis is performed in the spirit of a mind map, which represents the architectural approach to the studied theory, being a natural way to describe the properties of a complex system. We hope that the constructed graphs (maps) will turn out to be helpful in uncovering the global picture of quantum gravity as a particular complex system and serve as a useful guide for the researchers.

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arXiv2018

Effective Chern-Simons actions of particles coupled to 3D gravity

Point particles in 3D gravity are known to behave as topological defects, while gravitational field can be expressed as the Chern-Simons theory of the appropriate local isometry group of spacetime. In the case of the Poincaré group, integrating out the gravitational degrees of freedom it is possible to obtain the effective action for particle dynamics. We review the known results, both for single and multiple particles, and attempt to extend this approach to the (anti-)de Sitter group, using the factorizations of isometry groups into the double product of the Lorentz group and AN(2) group. On the other hand, for the de Sitter group one can also perform a contraction to the semidirect product of AN(2) and the translation group. The corresponding effective action curiously describes a Carrollian particle with the AN(2) momentum space. We derive this contraction in a more rigorous manner and further explore its properties, including a generalization to the multiparticle case.

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arXiv2017

Nonlinear Field Space Cosmology

We consider the FRW cosmological model in which the matter content of universe (playing a role of inflaton or quintessence) is given by a novel generalization of the massive scalar field. The latter is a scalar version of the recently introduced Nonlinear Field Space Theory (NFST), where physical phase space of a given field is assumed to be compactified at large energies. For our analysis we choose the simple case of a field with the spherical phase space and endow it with the generalized Hamiltonian analogous to the XXZ Heisenberg model, normally describing a system of spins in condensed matter physics. Subsequently, we study both the homogenous cosmological sector and linear perturbations of such a test field. In the homogenous sector we find that nonlinearity of the field phase space is becoming relevant for large volumes of universe and then it can lead to a recollapse, and possibly also at very high energies, leading to the phase of a bounce. Quantization of the field is performed in the limit where nontrivial nature of its phase space can be neglected, while there is a non-vanishing contribution from the Lorentz symmetry breaking term of the Hamiltonian. As a result, in the leading order of the XXZ anisotropy parameter, we find that the inflationary spectral index remains unmodified with respect to the standard case but the total amplitude of perturbations is subject to a correction. The Bunch-Davies vacuum state also becomes appropriately corrected. The proposed new approach is bringing cosmology and condensed matter physics closer together, which may turn out to be beneficial for both disciplines.

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arXiv2017

Spectral dimension with deformed spacetime signature

Studies of the effective regime of loop quantum gravity (LQG) revealed that, in the limit of Planckian curvature scales, spacetime may undergo a transition from the Lorentzian to Euclidean signature. This effect is a consequence of quantum modifications of the hypersurface deformation algebra, which in the linearized case is equivalent to a deformed version of the Poincaré algebra. In this paper the latter relation is applied to the LQG-inspired hypersurface deformation algebra that is characterized by the above mentioned signature change. While the exact form of the deformed Poincaré algebra is not uniquely determined, the algebra under consideration is representative enough to capture a number of qualitative features. In particular, the analysis reveals that the signature change can be associated with two symmetric invariant energy scales, which separate three physically disconnected momentum subspaces. Furthermore, the invariant measure on momentum space is derived, which allows to properly define the average return probability, characterizing a fictitious diffusion process on spacetime. The diffusion is subsequently studied in the momentum representation for all possible variants of the model. Finally, the spectral dimension of spacetime is calculated in each case as a function of the scale parameter. In the most interesting case the deformation is of the asymptotically ultralocal type and the spectral dimension undergoes a reduction to $d_S = 1$ in the UV limit.

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arXiv2017

Nonlinear Field Space Theory and Quantum Gravity

Phase spaces with nontrivial geometry appear in different approaches to quantum gravity and can also play a role in e.g. condensed matter physics. However, so far such phase spaces have only been considered for particles or strings. We propose an extension of the usual field theories to the framework of fields with nonlinear phase space of field values, which generally means nontrivial topology or geometry. In order to examine this idea we construct a prototype scalar field with the spherical phase space and then study its quantized version with the help of perturbative methods. As the result we obtain a variety of predictions that are known from the quantum gravity research, including algebra deformations, generalization of the uncertainty relation and shifting of the vacuum energy.

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