We introduce an area operator for the Moyal noncommutative plane. We find that the spectrum is discrete, but, contrary to the expectation formulated by other authors, not characterized by a "minimum-area principle". We show that an intuitive analysis of the uncertainty relations obtained from Moyal-plane noncommutativity is fully consistent with our results for the spectrum, and we argue that our area operator should be generalizable to several other noncommutative spaces. We also observe that the properties of distances and areas in the Moyal plane expose some weaknesses in the line of reasoning adopted in some of the heuristic analyses of the measurability of geometric spacetime observables in the quantum-gravity realm.
Violation of unitarity for noncommutative field theory on compact space-times is considered. Although such theories are free of ultraviolet divergences, they still violate unitarity while in a usual field theory such a violation occurs when the theory is nonrenormalizable. The compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in the absence of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. Violation of causal…
In this article, we study the quantum theory of gravitational boundary modes on a null surface. These boundary modes are given by a spinor and a spinor-valued two-form, which enter the gravitational boundary term for self-dual gravity. Using a Fock representation, we quantise the boundary fields, and show that the area of a two-dimensional cross section turns into the difference of two number operators. The spectrum is discrete, and it agrees with the one known from loop quantum gravity with the correct dependence on the Barbero--Immirzi parameter. No discrete structures (such as spin network …
Diverse theories of Quantum Gravity expect modification of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle.It was shown by some authors that the Generalized uncertainty principle gives rise to corrections to Schrodinger,Klien-Gordon and Dirac equations.By solving the Generalized uncertainty principle corrected equations,the authors arrived at quantization not only of energy but also of box length,area and volume. We extend the above result to the case of curved Space-Time(Schwarzschild metric). We showed that we arrived at the Quantiz…
One key element to calculate thermodynamical properties for a black hole is the partition function. In this paper we have incorporated the idea of a two dimensional area in a noncommutative space and were able to calculate the partition function with such a spectra. Employing the canonical quantum statistics formalism we compute the temperature, entropy and time of evaporation for a Schwarzschild black hole.
We present some evidence that noncommutative Yang-Mills theory in two dimensions is not invariant under area preserving diffeomorphisms, at variance with the commutative case. Still, invariance under linear unimodular maps survives, as is proven by means of a fairly general argument.
We present an investigation on the invariance properties of noncommutative Yang-Mills theory in two dimensions under area preserving diffeomorphisms. Stimulated by recent remarks by Ambjorn, Dubin and Makeenko who found a breaking of such an invariance, we confirm both on a fairly general ground and by means of perturbative analytical and numerical calculations that indeed invariance under area preserving diffeomorphisms is lost. However a remnant survives, namely invariance under linear unimodular tranformations.
We use twist deformation techniques to analyse the behaviour under area-preserving diffeomorphisms of quantum averages of Wilson loops in Yang-Mills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analysing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, su…
We study the effects of noncommutative spaces on the horizon, the area spectrum and Hawking temperature of a Schwarzschild black hole. The results show deviations from the usual horizon, area spectrum and the Hawking temperature. The deviations depend on the parameter of space-space noncommutativity.
Quantum gravity effects modify the Heisenberg's uncertainty principle to a generalized uncertainty principle (GUP). Earlier work showed that the GUP-induced corrections to the Schrödinger equation, when applied to a non-relativistic particle in a one-dimensional box, led to the quantization of length. Similarly, corrections to the Klein-Gordon and the Dirac equations, gave rise to length, area and volume quantizations. These results suggest a fundamental granular structure of space. In this work, it is investigated how spacetime curvature and gravity might influence this discreteness of space.…
We emphasize the importance of the Voros product in defining a noncommutative inspired Reissner-Nordström black hole. The entropy of this black hole is then computed in the tunneling approach and is shown to obey the area law at the next to leading order in the noncommutative parameter $θ$. Modifications to entropy/area law is then obtained by going beyond the semi-classical approximation. The leading correction to the semiclassical entropy/area law is found to be logarithmic and its coefficient involves the noncommutative parameter $θ$.
We show the importance of the Voros product in defining a noncommutative Schwarzschild black hole. The corrected entropy/area-law is then computed in the tunneling formalism. Two types of corrections are considered; one, due to the effects of noncommutativity and the other, due to the effects of going beyond the semiclassical approximation. The leading correction to the semiclassical entropy/area-law is logarithmic and its coefficient involves the noncommutative parameter.
Taking resort to Haldane's spherical geometry we can visualize fractional quantum Hall effect on the noncommutative manifold $M_4 \times Z_N$ with $N>2$ and odd. The discrete space leads to the deformation of symplectic structure of the continuous manifold such that the symplectic area is given by $\triangle p.\triangle q=2πm \hbar$ with $m$ an odd integer which is related to the Berry phase and the filling factor is given by $\frac{1}{m}$. We here argue that this is equivalent to the noncommutative field theory as prescribed by Susskind and Polychronakos which is characterized by area preserv…
Using a graphical analysis, we show that for the horizon radius $r_h\gtrsim 4.8\sqrtθ$, the standard semiclassical Bekenstein-Hawking area law for noncommutative Schwarzschild black hole exactly holds for all orders of $θ$. We also give the corrections to the area law to get the exact nature of the Bekenstein-Hawking entropy when $r_h<4.8\sqrtθ$ till the extremal point $r_h=3.0\sqrtθ$.
``Extremely'' localized wavefunctions in noncommutative geometry have disturbances that are localized to distances smaller than $\sqrtθ$, where $θ$ is the ``area'' parameter that measures noncommutativity. In particular, distributions such as the sign function or the Dirac delta function are limiting cases of extremely localized wavefunctions. It is shown that Moyal star products of extremely localized wavefunctions cannot be correctly computed perturbatively in powers of $θ$. Nonperturbative effects as a function of $θ$ are explicitly displayed through exact computations in several examples. …
It is known that, in the noncommutative Schwarzschild black hole spacetime, the point-like object is replaced by the smeared object, whose mass density is described by a Gaussian distribution of minimal width $\sqrtθ$ with $θ$ the noncommutative parameter. The elimination of the point-like structures makes it quite different from the conventional Schwarzschild black hole. In this paper, we mainly investigate the area spectrum and entropy spectrum for the noncommutative Schwarzschild black hole with $0\leq θ\leq (\frac{M}{1.90412})^{2}$. By the use of the new physical interpretation of the quas…
Field theory and gauge theory on noncommutative spaces have been established as their own areas of research in recent years. The hope prevails that a noncommutative gauge theory will deliver testable experimental predictions and will thus be a serious candidate for an extension of the Standard Model. This note contains the results for expanded gauge theory actions on a noncommutative space with constant theta, up to second order, together with a discussion of the ambiguities of the expanded theory and how they affect the action.
We discuss Wilson loop averages in 4-dimensional non-commutative superYang-Mills theory using the dual supergravity description. We postulate that the Wilson loops are located at the mimimum length scale $R$ in the fifth radial coordinate. We find that they exhibit a crossover from Coulomb type of behaviour for large loops, for which non-commutativity is unimportant, to area law for small loops, for which non-commutativity effects are large. The string tension, which can be read off from the area law, is controlled by the non-commutativity scale. The crossover itself, however, appears to invol…
Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a loga…
We argue that Yang-Mills theory on noncommutative torus, expressed in the Fourrier modes, is described by a gauge theory in a usual commutative space, the gauge group being a generalization of the area-preserving diffeomorphisms to the noncommutative case. In this way, performing the loop calculation in this gauge theory in the continuum limit we show that this theory is {\it one loop renormalizable}, and discuss the UV and IR limits. The moduli space of the vacua of the noncommutative super Yang-Mills theories in (2+1) dimensions is discussed.
In confrontation with serious and fundamental problems towards ultimate theory of quantum gravity and physics of Planck scale, we emphasize the importance of underlying noncommutative space-time such as Snyder's or Yang's Lorentz-covariant quantized space-time. The background of Bekenstein-Hawking's Area-entropy law and Holographic principle is now substantially understood in terms of {\it Kinematical} Holographic Relation [KHR], which holds in Yang's quantized space-time as the result of the kinematical reduction of spatial degrees of freedom caused by its own nature of noncommutative geometr…
We have constructed a noncommutative deformation of the holographic QCD (Sakai-Sugimoto) model and evaluated the mass spectrum of low spin vector mesons at finite temperature. The masses of light vector- and pseudovector-meson in the noncommutative holographic QCD model reduces over the whole area in the intermediate-temperature regime compared to the commutative case. However, the space noncommutativity does not change the properties of temperature dependence for the mass spectrum of low spin mesons. The masses of meson also decrease with increasing temperature in noncommutative case.
Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle (GUP). In some recent papers, we showed that the GUP gives rise to corrections to the Schrodinger equation, which in turn affect all quantum mechanical Hamiltonians. In particular, by applying it to a particle in a one dimensional box, we showed that the box length must be quantized in terms of a fundamental length (which could be the Planck length), which we interpreted as a signal of fundamental discreteness of space itself. …
In this paper we consider the generalized uncertainty principle (GUP) in the tunneling formalism via Hamilton-Jacobi method to determine the quantum-corrected Hawking temperature and entropy for noncommutative BTZ black hole. In our results we obtain several types of corrections including the expected logarithmic correction to the area entropy associated with the noncommutative BTZ black holes. We also show that the area entropy product of the noncommutative BTZ black holes is dependent on mass and by analyzing the nature of the specific heat capacity we have observed that the noncommutative B…
We consider canonical/Weyl-Moyal type noncommutative (NC) spaces with rectilinear coordinates. Motivated by the analogy of the formalism of the quantum mechanical harmonic oscillator problem in quantum phase-space with that of the canonical-type NC 2-D space, and noting that the square of length in the latter case is analogous to the Hamiltonian in the former case, we arrive at the conclusion that the length and area are quantized in such an NC space, if the area is expressed entirely in terms of length. We extend our analysis to 3-D case and formulate a ladder operator approach to the quantiz…
The three original publications in this thesis encompass various aspects in the still developing area of noncommutative quantum field theory, ranging from fundamental concepts to model building. One of the key features of noncommutative space-time is the apparent loss of Lorentz invariance that has been addressed in different ways in the literature. One recently developed approach is to eliminate the Lorentz violating effects by integrating over the parameter of noncommutativity. Fundamental properties of such theories are investigated in this thesis. Another issue addressed is model building,…
In the Seiberg-Witten limit, the low-energy dynamics of N weakly coupled identical open strings on a D3-brane can behave as two-dimensional incompressible hydrodynamics. Classical vortices are frozen in the fluid and described by an action expressed in terms of two canonical conjugate fields, which can be taken as the new space coordinate. The noncommutative space naturally arises when this pair of conjugate fields are quantized. To the lowest order of $\hbar$, the vorticity can replace the background $B$-field on the D3 brane, thereby yielding a spatially and temporally varying noncommutative…
Four years ago the Extended Scale Relativity (ESR) theory in C-spaces (Clifford manifolds) was proposed as the plausible physical foundations of string theory. In such theory the speed of light and the minimum Planck scale are the two universal invariants. All the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using polyvector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,.....We review the derivation of the mini…
We emphasized the importance of underlying noncommutative geometry or Lorenz-covariant quantized space-time towards ultimate theory of quantum gravity and Planck scale physics. We focused there our attention on the statistical and substantial understanding of Bekenstein-Hawking's Area-Entropy Law of black holes in terms of Kinematical Holographic Relation (KHR). KHR manifestly holds in Yang's quantized space-time as the result of kinematical reduction of spatial degrees of freedom caused by its own nature of noncommutative geometry and plays an important role in our approach without any recour…
We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutativ…
We find the moduli space of multi-solitons in noncommutative scalar field theories at large theta, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/theta is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the theta=infinity solitons. In two spatial dimensions, the parameter space for k solitons is a Kahler de-singularization of the symmetric product (R^2)^k/S_k. We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: R^2/Z_k, cylinder, and T^2. However, we show that tori of area less than …
We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the ``heat kernel'' semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler charact…
In the preceding paper, we derived a kind of kinematical holographic relation (KHR) in the Lorentz-covariant Yang's quantized space-time algebra (YSTA). It essentially reflects the fundamental nature of the noncommutative geometry of YSTA and its representation, that is, a definite kinematical reduction of spatial degrees of freedom in comparison with the ordinary lattice space. On the basis of the relation and its extension to various spatial dimensions, we derive a new area-entropy relation in a simple $D_0$ brane gas system subject to YSTA, following the idea of M-theory. Furthermore, we ma…
We study strong coupling phenomena in U(1) gauge theory on the non-commutative plane. To do so, we make use of a T-dual description in terms of an $N\to\infty$ limit of U(N) gauge theory on a commutative torus. The magnetic flux on this torus is taken to be $m=N-1$, while the area scales like 1/N, keeping $Λ_{QCD}$ fixed. With a few assumptions, we argue that the speed of high frequency light in pure non-commutative QED is modified in the non-commutative directions by the factor $1 + Λ_{QCD}^4 θ^2$, where $θ$ is the non-commutative parameter. If charged flavours are included, there is an upper…
In this letter, we provide an alternative method to study the area spectrum of certain classes of extremal black holes which have near-horizon geometry as warped AdS. We argue that previous methods which are based on the existence of quasinormal modes may not be applicable in the extremal limit. The topology difference of the near-horizon geometry between non-extremal and extremal black holes implies a separate treatment is needed to study the area discreteness in the extremal limit. To be specific, we will study area spectrum of supersymmetric BMPV black holes/black rings and Reissner-Norstro…
The study of the heat-trace expansion in noncommutative field theory has shown the existence of Moyal nonlocal Seeley-DeWitt coefficients which are related to the UV/IR mixing and manifest, in some cases, the non-renormalizability of the theory. We show that these models can be studied in a worldline approach implemented in phase space and arrive to a master formula for the $n$-point contribution to the heat-trace expansion. This formulation could be useful in understanding some open problems in this area, as the heat-trace expansion for the noncommutative torus or the introduction of renormal…
For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $θ= \frac{Z}{n}$ and the area of ${\cal T}$ is an integer, we construct the basis of Hilbert space ${\cal H}_n$ in terms of $θ$ functions of the positions $z_i$ of $n$ solitons. The loop wrapping around the torus generates the algebra ${\cal A}_n$. We show that ${\cal A}_n$ is isomorphic to the $Z_n \times Z_n$ Heisenberg group on $θ$ functions. We find the explicit form for the local operators, which is the generators $g$ of an elliptic $su(n)$, and transforms covariantly by the global gauge transformation of the Wilson lo…
A regularized model of the double compactified D=11 supermembrane with nontrivial winding in terms of SU(N) valued maps is obtained. The condition of nontrivial winding is described in terms of a nontrivial line bundle introduced in the formulation of the compactified supermembrane. The multivalued geometrical objects of the model related to the nontrivial wrapping are described in terms of a SU(N) geometrical object which in the $ N\to \infty$ limit, converges to the symplectic connection related to the area preserving diffeomorphisms of the recently obtained non-commutative description of th…
We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty …
It is a very well known fact that the energy conditions concerning traversable wormhole (WH) solutions of Einstein equations are violated. Consequently, attempts to avoid the violation of the energy conditions constitutes one of the main areas of research in WH physics. On the other hand, the current literature shows us that noncommutativity is one of the most promising candidates to help us to understand the physics of the early Universe. However, since noncommutativity does not change the commutative results, we also can expect that energy conditions violation near the throat must occur. We …
In this thesis some systems with exotic symmetries, which are a peculiarity in 2+1 space-time dimensions, are studied. Coded in the exotic structure, there appear noncommutative coordinates and a phases structure. This kind of systems has attracted attention from different areas of physics independently. Among them, we can mention: theory of ray representations of Lie groups, anyon physics, some condensed matter systems, for instance the quantum Hall effect, planar gauge and gravitation theories, noncommutative field theory, noncommutative geometry, and noncommutative quantum mechanics. We foc…
In this paper canonical quantum gravity on noncommutative space-time is considered. The corresponding generalized classical theory is formulated by using the moyal star product, which enables the representation of the field quantities depending on noncommuting coordinates by generalized quantities depending on usual coordinates. But not only the classical theory has to be generalized in analogy to other field theories. Besides, the necessity arises to replace the commutator between the gravitational field operator and its canonical conjugated quantity by a corresponding generalized expression …
We study the time-dependent Aharonov-Bohm effect on the noncommutative space. Because there is no net Aharonov-Bohm phase shift in the time-dependent case on the commutative space, therefore, a tiny deviation from zero indicates new physics. Based on the Seiberg-Witten map we obtain the gauge invariant and Lorentz covariant Aharonov-Bohm phase shift in general case on noncommutative space. We find there are two kinds of contribution: momentum-dependent and momentum-independent corrections. For the momentum-dependent correction, there is a cancellation between the magnetic and electric phase sh…
Our main focus is to explore different models in noncommutative spaces in higher dimensions. We provide a procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of noncommutative spaces, first by util…
A semi-classical reasoning leads to the non-commutativity of the space and time coordinates near the horizon of Schwarzschild black hole. This non-commutativity in turn provides a mechanism to interpret the brick wall thickness hypothesis in 't Hooft's brick wall model as well as the boundary condition imposed for the field considered. For concreteness, we consider a noncommutative scalar field model near the horizon and derive the effective metric via the equation of motion of noncommutative scalar field. This metric displays a new horizon in addition to the original one associated with the S…
Metrics structures stemming from the Connes distance promote Moyal planes to the status of quantum metric spaces. We discuss this aspect in the light of recent developments, emphasizing the role of Moyal planes as representative examples of a recently introduced notion of quantum (noncommutative) locally compact space. We move then to the framework of Lorentzian noncommutative geometry and we examine the possibility of defining a notion of causality on Moyal plane, which is somewhat controversial in the area of mathematical physics. We show the actual existence of causal relations between the …
This paper establishes a link between Noncommutative Geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac type operator which resembles a global functional derivation operator. The interaction between the Dirac operator and the algebra reproduces the Poisson structure of General Relativity. Moreover, the associated Hilbert space corresponds, up to a discrete symmetry group, to the Hilbert space of diffeomorphism invariant states known from Loop Quantum Gravity. Correspondingl…
