Poincaré, Scale and Conformal Symmetries: Gauge Perspective and Cosmological Ramifications
Georgios K. Karananas
备注：
PhD Thesis, 206 pages, 5 figures. Abstract shorter than in the file due to arXiv limit of 1920 characters. Based on arXiv:1212.4148, arXiv:1411.5613, arXiv:1510.07589, arXiv:1510.08042, arXiv:1601.03046, arXiv:1603.01274
In the first part of the thesis we focus on local symmetries. We review a self-consistent framework that we employed in order to discuss the dynamics of the theories of interest. Its merit lies in that we can make the symmetry group act internally and thus be effectively separated from coordinate transformations. We investigate under which conditions it is not needed to introduce extra compensating fields to make relativistic as well as nonrelativistic theories invariant under local symmetries and more precisely under scale transformations. We clarify the role that torsion plays in this context. We highlight the difference between Weyl and conformal invariance and we demonstrate that not all conformal theories can be coupled to gravity in a Weyl invariant way. Once this minimalistic treatment for gauging symmetries is left aside, new possibilities appear. Namely, if we consider the Poincaré group, the presence of the extra modes leads to nontrivial particle dynamics. We derive constraints such that the theory is free from pathologies. In the second part we make clear that even when not gauged, the presence of scale invariance is appealing. First, it makes possible for the dimensionful parameters that appear in a theory to be generated dynamically and be sourced by the vacuum expectation value of the dilaton. If the Standard Model is embedded into a scale-invariant framework, a number of interesting implications for cosmology arise. The inflationary stage of our Universe and its present-day acceleration become linked, a connection that might give us insight into the dark energy dynamics. We show that in the context of gravitational theories which are invariant under restricted coordinate transformations, the dilaton instead of being introduced ad hoc, can emerge from the gravitational part of a theory. We discuss the consequences of the nontrivial way this field emerges in the action.
Defining a new r-matrix compatible with the scalar product at the basis of the Chern-Simons action for a particle coupled to (2+1) Lorentzian gravity with cosmological constant, I show how deformed symmetries of $κ$-de Sitter and, in the vanishing cosmological limit, of $κ$-Poincaré kind, arise naturally as quantum-deformation of three dimensional gravity. I obtain moreover the non-commutative spacetime associated to these kinds of symmetries.
A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the $κ$-Poincaré model, analyzing the fuzzy properties of $κ$-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.
It is well-known that de Sitter Lie algebra $\mathfrak{o}(1,4)$ contrary to anti-de Sitter one $\mathfrak{o}(2,3)$ does not have a standard $\mathbb{Z}_2$-graded superextension. We show here that the Lie algebra $\mathfrak{o}(1,4)$ has a superextension based on the $\mathbb{Z}_2\times\mathbb{Z}_2$-grading. Using the standard contraction procedure for this superextension we obtain an {\it alternative} super-Poincaré algebra with the $\mathbb{Z}_2\times\mathbb{Z}_2$-grading.
A group theoretical mechanism is outlined, which can indecomposably extend the Poincaré group by the compact internal (gauge) symmetries at the price of allowing some nilpotent (or, more precisely: solvable) internal symmetries in addition. Due to the presence of this nilpotent part, the prohibitive argument of the well known Coleman-Mandula and McGlinn no-go theorems do not go through. In contrast to SUSY or extended SUSY, in our construction the symmetries extending the Poincaré group will be all internal, i.e. they do not act on the spacetime, merely on some internal degrees of freedom -- h…
We propose two different high-energy-theory correspondences with graphene (and related materials) scenarios, associated to grain boundaries, that are topological defects for which both Dirac points are necessary. The first correspondence points to a $(3+1)$-dimensional theory, with nonzero torsion, with spatiotemporal gauge group $SO(3,1)$, locally isomorphic to the Lorentz group in $(3+1)$ dimensions, or to the de Sitter group in $(2+1)$ dimensions. The other correspondence treats the two Dirac fields as an internal symmetry doublet, and it is linked here with unconventional supersymmetry wit…
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 furthe…
A spacetime interpretation of deformed relativity symmetry groups was recently proposed by resorting to Finslerian geometries, seen as the outcome of a continuous limit endowed with first order corrections from the quantum gravity regime. In this work we further investigate such connection between deformed algebras and Finslerian geometries by showing that the Finsler geometries associated to the generalisation of the Poincaré group (the so called $κ$-Poincaré Hopf algebra) are maximally symmetric spacetimes which are also of the Berwald type: Finslerian spacetimes for which the connections ar…
In this paper we revisit the model of $κ$-deformed complex scalar field. We find that this model possesses ten conserved Noether charges that form, under commutators, a representation of (undeformed) Poincaré algebra. It follows that the theory is relativistic and does not break Lorentz invariance. However the spacetime representation of boosts is not standard, and contains a non-local translation, different for positive and negative energy modes. It then follows that although the masses of particles and anti-particles are equal, the theory violates CPT symmetry in a subtle way. We explain why…
We study the symmetries of the Lorentz violating Randers-Finsler spacetime. The privileged frame defined by the background vector diagonalises the deformed mass-shell and provides an anisotropic observer transformations. The particle Randers transformations are achieved through the Finslerian Killing equation which also shows the equivalence between observer and particle Randers transformation. For a constant background vector, the Randers transformations can be regarded as a deformed directional-dependent Poincaré transformation. The Randers algebra is a deformed Poincaré algebra whose struct…
We show how the Weyl quantum walk derived from principles in Ref. [1], enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras--the usual Poincaré and the k-Poincaré algebras.
We study the explicit form of Poincaré and discrete transformations of flavor states in a two-flavor scalar model, which represents the simplest example of the field mixing. Because of the particular form of the flavor vacuum condensate, we find that the aforementioned symmetries are spontaneously broken. The ensuing vacuum stability group is identified with the Euclidean group E(3). With the help of Fabri-Picasso theorem, we show that flavor vacua with different time labels and in different Lorentz frames are unitarily inequivalent to each other and they constitute a manifold of zero-flavor-c…
We elaborate on the Green's functions that appeared in [1,2] when generalizing, from massless to massive particles, various equivalences between soft theorems and Ward identities of large gauge symmetries. We analyze these Green's functions in considerable detail and show that they form a hierarchy of functions which describe `boundary to bulk' propagators for large $U(1)$ gauge parameters, supertranslations and sphere vector fields respectively. As a consistency check we verify that the Green's functions associated to the large diffeomorphisms map the Poincare group at null infinity to the Po…
We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several exis…
We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumve…
We study systematically various extensions of the Poincaré superalgebra. The most general structure starting from a set of spinorial supercharges $Q_α$ is a free Lie superalgebra that we discuss in detail. We explain how this universal extension of the Poincaré superalgebra gives rise to many other algebras as quotients, some of which have appeared previously in various places in the literature. In particular, we show how some quotients can be very neatly related to Borcherds superalgebras. The ideas put forward also offer some new angles on exotic branes and extended symmetry structures in M-…
Recent work showed that $κ$-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter $Λ$, the Planck scale quantum deformation parameter $κ$ and the speed of light parameter $c$ to move to the well-studied $κ$-Poincaré algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these al…
We investigate how fields transform under the Poincaré group in nonrelativistic effective field theories of QCD. In constructing these transformations, we rely only on symmetries and field redefinitions to limit the number of allowed terms. By requiring invariance of the action under these transformations, nontrivial relations between Wilson coefficients for both nonrelativistic QCD and potential nonrelativistic QCD are derived. We show explicitly how the Poincaré algebra is satisfied, and how this gives complementary information on the Wilson coefficients. We also briefly discuss the implicat…
The canonical generator $G$ of local symmetries in Poincaré gauge theory is constructed as an integral over a spatial section $Σ$ of spacetime. Its regularity (differentiability) on the phase space is ensured by adding a suitable surface term, an integral over the boundary of $Σ$ at infinity, which represents the asymptotic canonical charge. For black hole solutions, $Σ$ has two boundaries, one at infinity and the other at horizon. It is shown that the canonical charge at horizon defines entropy, whereas the regularity of $G$ implies the first law of black hole thermodynamics.
Celestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional sphere describing Minkowski space at null infinity. We construct the generators of Poincare and conformal groups in the celestial representation and discuss how these symmetries are manifest in the amplitudes.
We construct 5D, N = 1 supergravity in a 4D, N = 1 superspace with an extra bosonic coordinate. This represents four of the supersymmetries and the associated Poincaré symmetries manifestly. The remaining four supersymmetries and the rest of the Poincaré symmetries are represented linearly but not manifestly. In the linearized approximation, the action reduces to the known superspace result. As an application of the formalism, we construct the $\int A\wedge R\wedge R$ invariant in this superspace.
We reveal an $\mathfrak{iso}(2,1)$ Poincaré algebra of conserved charges associated with the dynamics of the interior of black holes. The action of these Noether charges integrates to a symmetry of the gravitational system under the Poincaré group ISO$(2,1)$, which allows to describe the evolution of the geometry inside the black hole in terms of geodesics and horocycles of AdS${}_2$. At the Lagrangian level, this symmetry corresponds to Möbius transformations of the proper time together with translations. Remarkably, this is a physical symmetry changing the state of the system, which also nat…
Non Trivial Extension of the (1+2)-Poincaré Algebra and Conformal Invariance on the Boundary of $AdS_3$
利用$AdS_{3}\times N^d$上弦的最新结果，重新研究了Rausch de Traubenberg和Slupinsky在文献[1]和[29]中得到的（1+2）维Poincaré代数的非平凡扩张的推导.通过显式计算表明，这种新的扩张是一种特殊的分数阶超对称代数，它可能是由生活在$AdS_3$边界上的共形结构的变形而得到的.在我们的分析中，将用于建立推广的（1+2）Poincaré代数的自旋$\pm{1\over k}$的两个so（1，2）Lorentz模重新解释为$AdS_3$边界上左右Virasoro对称的最高权表示.利用仿射Kac-Moody谱流和超共形对称性，我们也完成了关于2d分数阶超对称性的已知结果.最后，我们对引入g-模的F-根将so（1，2）结果推广到高秩李代数g的技巧作了初步的评论。
By regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincaré. As a concrete example we focus on the new-minimal (12+12) off-shell version of simple four-dimensional supergravity obtained by tensoring the off-shell Yang-Mills multiplets (4+4,N_L =1)and(…
Asymptotic symmetries are a general and important feature of theories with long-ranging fields, such as gravity, electromagnetism, and Yang-Mills. They appear in the formalism once the analytic behaviour of fields near infinity is specified and have received a renewed interest in the last years after a possible connection with the information-loss paradox has been conjectured. One of the various methods used to study the asymptotic symmetries of field theories relies on the Hamiltonian formalism and was introduced in the seminal work of Henneaux and Troessaert, who successfully applied it to t…
