By means of Dirac procedure, we re-examine Yang's quantized space-time model, its relation to Snyder's model, the de Sitter special relativity and their UV-IR duality. Starting from a dimensionless dS_5-space in a 5+1-d Mink-space a complete Yang model at both classical and quantum level can be presented and there really exist Snyder's model, the dS special relativity and the duality.
We study the deformed conformal-Poincare symmetries consistent with the Snyder--de Sitter space. A relativistic particle model invariant under these deformed symmetries is given. This model is used to provide a gauge independent derivation of the Snyder--de Sitter algebra. Our results are valid in the leading order in the parameters appearing in the model.
We show that if Yang's model as the earliest TSR is completed, it should contain both Snyder's quantized space-time model as the earliest DSR, the de Sitter SR and their duality.
There is a one-to-one correspondence between Snyder's model in de Sitter space of momenta and the \dS-invariant special relativity. This indicates that physics at the Planck length $\ell_P$ and the scale $R=3/Λ$ should be dual to each other and there is in-between gravity of local \dS-invariance characterized by a dimensionless coupling constant $g=\ell_P/R\sim 10^{-61}$.
The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic…
In terms of the Beltrami model of de Sitter space we show that there is an interchangeable relation between Snyder's quantized space-time model in dS-space of momenta at the Planck length $\ell_P=(G\hbar c^{-3})^{1/2}$ and the dS-invariant special relativity in dS-spacetime of radius $R\simeq(3Λ^{-1})^{1/2}$, which is another fundamental length related to the cosmological constant. Here, the cosmological constant $Λ$ is regarded as a fundamental constant together with the speed of light $c$, Newton constant $G$ and Planck constant $\hbar$. Furthermore, the physics at two fundamental scales of …
Three dimensional Dirac oscillator was considered in deformed space obeyed to deformed commutation relations known as Snyder-de Sitter algebra. Snyder-de Sitter commutation relations gives rise to appearance minimal uncertainty in position as well as in momentum. To derive energy spectrum and wavefunctions of the Dirac oscillator supersymmetric quantum mechanics and shape invariance technique was applied.
The Snyder-de Sitter (SdS) model is a generalization of the Snyder model to a spacetime background of constant curvature. It is an example of noncommutative spacetime admitting two fundamental scales besides the speed of light, and is invariant under the action of the de Sitter group. Here we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to a three-dimensional sphere, and the related model obtained by considering the anti-Snyder model on a pseudosphere, that we call anti-Snyder-de Sitter (aSdS). By means of a nonlinear transformation relating the SdS phase space va…
One of the foremost goals of theoretical modern physics is to obtain a reliable theory of quantum gravity. In so doing, new fundamental scales are being proposed. For example, in the Snyder - de Sitter model, two scales manifest which relate measurement limits of the position and momentum. In this work, we study the thermodynamic functions of the Scharwzschild black hole in the Snyder - de Sitter model within the presence of position-independent and dependent gravity's rainbow. Heuristically, we prove the presence of a non-zero lower bound value of the horizon, mass, and temperature of the bla…
We study the two-dimensional version of a quartic self-interacting quantum scalar field on a curved and noncommutative space (Snyder-de Sitter). We show that the model is renormalizable at the one-loop level and compute the beta functions of the related couplings. The renormalization group flow is then studied numerically, arriving at the conclusion that noncommutative-curved deformations can yield both relevant and irrelevant contributions to the one-loop effective action.
Valtancoli in his paper entitled [P. Valtancoli, Canonical transformations, and minimal length J. Math. Phys. 56, 122107 (2015)] has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the …
We consider generalizations of the Snyder algebra to a curved spacetime background with de Sitter symmetry. As special cases, we obtain the algebras of the Yang model and of triply special relativity. We discuss the realizations of these algebras in terms of canonical phase space coordinates, up to fourth order in the deformation parameters. In the case of triply special relativity we also find exact realization, exploiting its algebraic relation with the Snyder model.
We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra, and exploiting the geometrical properties of the phase space, that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.
The Snyder-de Sitter model is an extension of the Snyder model to a de Sitter background. It is called triply special relativity (TSR) because it is based on three fundamental parameters: speed of light, Planck mass, and the cosmological constant. In this paper, we study the three dimensional DKP oscillator for spin zero and one in the framework of Snyder-de Sitter algebra in momentum space. By using the technique of vector spherical harmonics the energy spectrum and the corresponding eigenfunctions are obtained for both cases.
The relativistic $D=4$ Snyder model is formulated in terms of $D=4$ $dS$ algebra $o(4,1)$ generators, with noncommutative Lorentz-invariant Snyder quantum space-time provided by $\frac{O(4,1)}{O(3,1)}$ coset generators. Analogously, in relativistic $D=4$ Yang models the quantum-deformed relativistic phase space is described by the algebras of coset generators $\frac{O(5,1)}{O(3,1)}$ or $\frac{O(4,2)}{O(3,1)}$. We extend these algebraic considerations by using respective $dS$ superalgebras, which provide Lorentz-covariant quantum superspaces (SUSY Snyder model) as well as relativistic quantum p…
We study an interacting $λ\,ϕ^4_{\star}$ scalar field defined on Snyder-de Sitter space. Due to the noncommutativity as well as the curvature of this space, the renormalization of the two-point function differs from the commutative case. In particular, we show that the theory in the limit of small curvature and noncommutativity is described by a model similar to the Grosse-Wulkenhaar one. Moreover, very much akin to what happens in the Grosse-Wulkenhaar model, our computation demonstrates that there exists a fixed point in the renormalization group flow of the harmonic and mass terms.
We analyze the model of a self-interacting $ϕ^4_{\star}$ scalar field theory in Snyder-de Sitter space. After analytically computing the one-loop beta functions {in the small noncommutativity and curvature limit}, we solve numerically the corresponding system of differential equations, showing that in this limit the model possesses at least one regime in which the theory is asymptotically free. Moreover, in a given region of the parameter space we also observe a peculiar running of the parameter associated to the curvature, which changes its sign and therefore can be interpreted as a transitio…
We study the classical dynamics of a particle in nonrelativistic Snyder-de Sitter space. We show that for spherically symmetric systems, parametrizing the solutions in terms of an auxiliary time variable, which is a function only of the physical time and of the energy and angular momentum of the particles, one can reduce the problem to the equivalent one in classical mechanics. We also discuss a relativistic extension of these results, and a generalization to the case in which the algebra is realized in flat space.
The Snyder-de Sitter (SdS) model which is invariant under the action of the de Sitter group, is an example of a noncommutative spacetime with three fundamental scales. In this paper, we considered the massless Dirac fermions in graphene layer in a curved Snyder spacetime which are subjected to an external magnetic field. We employed representation in the momentum space to derive the energy eigenvalues and the eigenfunctions of the system. Then, we used the deduced energy function obtaining the internal energy, heat capacity, and entropy functions. We investigated the role of the fundamental sc…
Long time ago, C.N. Yang proposed a model of noncommutative spacetime that generalized the Snyder model to a curved background. In this paper we review his proposal and the generalizations that have been suggested during the years. In particular, we discuss the most general algebras that contain as subalgebras both de Sitter and Snyder algebras, preserving Lorentz invariance, and are generated by a two-parameter deformation of the canonical Heisenberg algebra. We also define their realizations on quantum phase space, giving explicit examples, both exact and in terms of a perturbative expansion…
After a brief summary of Double Special Relativity (DSR), we concentrate on a five dimensional procedure, which consistently introduce coordinates and momenta in the corresponding four-dimensional phase space, via a Hamiltonian approach. For the one particle case, the starting point is a de Sitter momentum space in five dimensions, with an additional constraint selected to recover the mass shell condition in four dimensions. Different basis of DSR can be recovered by selecting specific gauges to define the reduced four dimensional degrees of freedom. This is shown for the Snyder basis in the o…
We introduce three space-times that are discrete in time and compatible with the Lorentz symmetry. We show that these spaces are no commutative, with commutation relations similar to the relations of the Snyder and Yang spaces. Furthermore, using a reparametrized relativistic particle we obtain a realization of the Snyder type spaces and we construct an action for them.
We describe, in an algebraic way, the $κ$-deformed extended Snyder models, that depend on three parameters $β, κ$ and $λ$, which in a suitable algebra basis are described by the de Sitter algebras ${o}(1,N)$. The commutation relations of the algebra contain a parameter $λ$, which is used for the calculations of perturbative expansions. For such $κ$-deformed extended Snyder models we consider the Heisenberg double with dual generalized momenta sector, and provide the respective generalized quantum phase space depending on three parameters mentioned above. Further, we study for these models an a…
In this paper we discuss the dynamics of the cosmological Bartnick-McKinnon analogue with $n=1$ and $Λ=Λ_{reg}(n)$ . We derive boundary conditions from energy considerations. Numerical simulations are carried out to show the existence of a non-stationary solution.
A new topological conformal field theory in four Euclidean dimensions is constructed from N=4 super Yang-Mills theory by twisting the whole of the conformal group with the whole of the R-symmetry group, resulting in a theory that is conformally invariant and has two conformally invariant BRST operators. A curved space generalisation is found on any Riemannian 4-fold. This formulation has local Weyl invariance and two Weyl-invariant BRST symmetries, with an action and energy-momentum tensor that are BRST-exact. This theory is expected to have a holographic dual in 5-dimensional de Sitter space.
We obtain D=4 de Sitter gravity coupled to SU(2) Yang-Mills gauge fields from an explicit and consistent truncation of D=11 supergravity via Kaluza-Klein dimensional reduction on a non-compact space. The ``internal'' space is a smooth hyperbolic 7-space (H^7) written as a foliation of two 3-spheres, on which the SU(2) Yang-Mills fields reside. The positive cosmological constant is completely fixed by the SU(2) gauge coupling constant. The explicit reduction ansatz enables us to lift any of the D=4 solutions to D=11. In particular, we obtain dS_2 in M-theory, where the nine-dimensional transver…
Instantons and their quantisation in pure Yang-Mills theory formulated in the background of de Sitter spacetime represented by spatially-closed ($k = 1$) Friedmann-Robertson-Walker metric are discussed. As for the classical treatment of the instanton physics, first, explicit instanton solutions are found and next, quantities like Pontryagin index and the semiclassical approximation to the inter-vacua tunnelling amplitude are evaluated. The Atiyah-Patodi-Singer index theorem is checked as well by constructing explicitly the normalizable fermion zero modes in this de Sitter spacetime instanton b…
We test the conjectured relationship between N=4 super Yang-Mills theory in four dimensions and IIB supergravity compactified on $AdS_5\times S_5$ by computing the two- and three-point functions of R-symmetry currents. We observe that the integral expressions describing the general three-point correlator on the supergravity side have a structure similar to one-loop triangle diagrams in N=4 super Yang-Mills theory. This allows us to compare the expressions on both sides of the AdS/CFT correspondence without the technical complications of the loop integrations. We confirm that the two- and three…
Recently, a relation between N=4 Super Yang Mills in 3+1 dimensions and supergravity in an $AdS_5$ background has been proposed. In this paper we explore the idea that the correspondence between operators in the Yang Mills theory and modes of the supergravity theory can be obtained by using the D3 brane action. Specifically, we consider two form gauge fields for this purpose. The supergravity analysis predicts that the operator which corresponds to this mode has dimension six. We show that this is indeed the leading operator in the three brane Dirac-Born-Infeld and Wess-Zumino action which cou…
We compute the imaginary part of scalar four-point functions in the AdS/CFT correspondence relevant to N=4 super Yang-Mills theory. Unitarity of the AdS supergravity demands that the imaginary parts of the correlation functions factorize into products of lower-point functions. We include the exchange diagrams for scalars as well as gravitons and find explicit expressions for the imaginary parts of these correlators. In momentum space these expressions contain only rational functions and logarithms of the kinematic invariants, in such a manner that the correlator is not a free-field result. The…
In this paper, we analyze the thermodynamic properties of the Anti de Sitter black hole in the Einstein-Maxwell-Yang-Mills-AdS gravity (EMYM) via many approaches and in different thermodynamical ensembles (canonical/ grand canonical). First, we give a concise overview of this phase structure in the entropy-thermal diagram for fixed charges then we investigate this thermodynamical structure in fixed potentials ensemble. The Next relevant step is recalling the nonlocal observables such as holographic entanglement entropy and two-point correlation function to show that both observables exhibit a …
We construct new axially symmetric solutions of SU(2) Yang-Mills theory in a four dimensional anti-de Sitter spacetime. Possessing nonvanishing nonabelian charges, these regular configurations have also a nonzero angular momentum. Numerical arguments are also presented for the existence of rotating solutions of Einstein-Yang-Mills equations in an asymptotically anti-de Sitter spacetime.
We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant …
From Cosmological Cuts to Yang--Mills Wavefunctions in de Sitter Space
New Cornerstone Laboratory, Institute of Theoretical Physics, Chinese Academy of Sciences（新Cornerstone实验室，中国科学院理论物理研究所）；Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）
We demonstrate that world-volume actions describing probe branes in anti de Sitter space have an infinite-dimensional symmetry algebra that is reminiscent of Yangian symmetry. Two essential ingredients are responsible for this symmetry. The first is an inversion symmetry of anti de Sitter space. The second is the compensating gauge transformations that are required to properly describe global symmetries after gauge fixing the general coordinate invariance.
This paper investigates an integrable system which is related to hyperbolic monopoles; ie the Bogomolny Yang-Mills-Higgs equations in (2+1) anti-de Sitter space which are integrable and whose solutions can be obtained using analytical methods. In particular, families of soliton solutions have been constructed explicitly and their dynamics has been investigated in some detail.
A continuum of new monopole and dyon solutions in the Einstein-Yang-Mills theory in asymptotically anti-de Sitter space are found. They are regular everywhere and specified with their mass, and non-Abelian electric and magnetic charges. A class of monopole solutions which have no node in non-Abelian magnetic fields are shown to be stable against spherically symmetric linear perturbations.
We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter dS$_4$ and anti-de Sitter AdS$_4$ spaces and construct various solutions to the Yang-Mills equations. On de Sitter space we reduce the Yang-Mills equations via an SU(2)-equivariant ansatz to Newtonian mechanics of a particle moving in ${\mathbb R}^3$ under the influence of a quartic potential. Then we describe magnetic and electric-magnetic solutions, both Abelian and non-Abelian, all having finite energy and finite action. A similar reduction on anti-de Sitter space also yields Yang-Mills solutions with finite energy and a…
We consider (3 + 1)-dimensional N=4 super Yang-Mills theory with a nonvanishing scalar Higgs vacuum expectation value, and compare this theory to AdS supergravity with branes in the bulk. We show that the one-loop effective potential for excitations of the Yang-Mills field agrees with the classical linearized potential for brane waves in the AdS picture in the limit of long wavelengths. This supports the idea that the AdS/CFT correspondance fits into string theory as expected from previous work.
We consider particle-like and black holes solutions of the Einstein-Yang-Mills system with positive cosmological constant in d>4 spacetime dimensions. These configurations are spherically symmetric and present a cosmological horizon for a finite value of the radial coordinate, approaching asymptotically the de Sitter background. In the usual Yang--Mills case we find that the mass of these solutions, evaluated outside the cosmological horizon at future/past infinity generically diverges for d>4. Solutions with finite mass are found by adding to the action higher order gauge field terms belongin…
We consider an Einstein-Yang-Mills Lagrangian in a five dimensional space-time including a cosmological constant. Assuming all fields to be independent of the extra coordinate, a dimensional reduction leads to an effective (3+1)-dimensional Einstein-Yang-Mills-Higgs-dilaton model where the cosmological constant induces a Liouville potential in the dilaton field. We construct spherically symmetric solutions analytically in specific limits and study the generic solutions for vanishing dilaton coupling numerically. We find that in this latter case the solutions bifurcate with the branch of (Anti-…
The properties of hairy black holes in Einstein-Yang-Mills (EYM) theory are reviewed, focusing on spherically symmetric solutions. In particular, in asymptotically anti-de Sitter space (adS) stable black hole hair is known to exist for su(2) EYM. We review recent work in which it is shown that stable hair also exists in su(N) EYM for arbitrary N, so that there is no upper limit on how much stable hair a black hole in adS can possess.
We investigate the thermodynamics of spherically symmetric black hole solutions in a four-dimensional Einstein--Yang-Mills-SU(2) theory with a negative cosmological constant. Special attention is paid to configurations with a unit magnetic charge. We find that a set of Reissner-Nordström--Anti-de Sitter black holes can become unstable to forming non-Abelian hair. However, the hairy black holes are never thermodynamically favoured over the full set of abelian monopole solutions. The thermodynamics of the generic configurations possessing a noninteger magnetic charge is also discussed.
We study the braneworld solutions based on a solvable model of 5d gauged supergravity with two scalars of conformal dimension three, which correspond to bilinear operators of fermions in the dual $\mathcal{N}=4$ super Yang-Mills theory on the boundary. An accelerating braneworld solution is obtained when both scalars are taken as the form of deformations of the super Yang-Mills theory and the bulk supersymmetry is broken. This solution is smoothly connected to the Poincare invariant brane in the limit of vanishing cosmological constant. The stability of this brane-solution and the corresponden…
The analyticity property of de Sitter's quantum Yang-Mills theory in the framework of Kerin space quantization, including quantum metric fluctuation, is demonstrated. This property completes our previous work regarding quantum Yang-Mills theory in de Sitter's ambient space formalism, and we can construct an axiomatic quantum field theory similar to Wightman's axioms. The color confinement is proven for the general case, which was previously approved in the early universe. It is shown by using the interaction between gluon fields and the conformal sector of the gravitational field, which is a m…
The propagators of the Faddeev-Popov (FP) ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. We point out, however, that the modes responsible for these divergences will not contribute to loop diagrams in computations of time-ordered products in either Yang-Mills theories or perturbative quantum gravity. Therefore we propose that the IR divergent FP-ghost propagator should be regularized by a small mass term that is sent to zero in the end of any perturbative calculations. This proposal is equivalent to usi…
T-Duality on a timelike circle does not interchange IIA and IIB string theories, but takes the IIA theory to a type $IIB^*$ theory and the IIB theory to a type $IIA^*$ theory. The type $II^*$ theories admit E-branes, which are the images of the type II D-branes under timelike T-duality and correspond to imposing Dirichlet boundary conditions in time as well as some of the spatial directions. The effective action describing an E$n$-brane is the $n$-dimensional Euclidean super-Yang-Mills theory obtained by dimensionally reducing 9+1 dimensional super-Yang-Mills on $9-n$ spatial dimensions and on…
We present new soliton and hairy black hole solutions of su(N) Einstein-Yang-Mills theory in asymptotically anti-de Sitter space. These solutions are described by N+1 independent parameters, and have N-1 gauge field degrees of freedom. We examine the space of solutions in detail for su(3) and su(4) solitons and black holes. If the magnitude of the cosmological constant is sufficiently large, we find solutions where all the gauge field functions have no zeros. These solutions are of particular interest because we anticipate that at least some of them will be linearly stable.
We demonstrate five-dimensional anti-de Sitter black hole emerges as dual geometry holographic to weakly interacting N=4 superconformal Yang-Mills theory. We first note that an ideal probe of the dual geometry is the Yang-Mills instanton, probing point by point in spacetime. We then study instanton moduli space at finite temperature by adopting Hitchin's proposal that geometry of the moduli space is definable by Fisher-Rao "information geometry". In Yang-Mills theory, the information metric is measured by a novel class of gauge-invariant, nonlocal operators in the instanton sector. We show tha…
Here we study the global existence of `hairy' dyonic black hole and dyon solutions to four dimensional, anti-de Sitter Einstein-Yang-Mills theories for a general simply-connected and semisimple gauge group $G$, for so-called topologically symmetric systems, concentrating here on the regular case. We generalise here cases in the literature which considered purely magnetic spherically symmetric solutions for a general gauge group and topological dyonic solutions for $\mathfrak{su}(N)$. We are able to establish the global existence of non-trivial solutions to all such systems, both near existing …
