Charge and/or spin limits for black holes at a non-commutative scale

We study the cosmological dynamics for R ^p exp(λ R) gravity theory in the metric formalism, using dynamical systems approach. Considering higher-dimensional FRW geometries in case of an imperfect fluid which has two different scale factors in the normal and extra dimensions, we find the exact solutions, and study its behaviour and stability for both vacuum and matter cases. It is found that stable solutions corresponding to accelerated expansion at late times exist, which can describe the inflationary era of the Universe. We also study the evolution of scale factors both in the normal and extra dimensions for different values of anisotropy parameter and the number of extra dimensions for such a scenario.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

Exact wave solutions of the five-dimensional einstein equations

Exact wave solutions are found for the five-dimensional Einstein equations in a vacuum. The solutions describe the consistent gravitational scalar wave processes [6] and generalize to five dimensions the known four-dimensional Einstein-Rosen and Bondi-Pirani-Robinson wave metrics and the Rosen metric. By means of the method of rotations involving the coordinates of x⁵, the solutions obtained are generalized to the case in which there is an electromagnetic field present. For each of the solutions, the three-dimensional electric and magnetic field strengths are found. Moscow University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 112–117, October, 1996.     

Domain with Noncompactified Extra Dimensions in the Multidimensional Universe with Compactified Extra Dimensions

It is argued that in our Universe with compactified extra dimensions (ED) also domains exist with noncompactified ED. The multidimensional gravity (MD) on the principal bundle with structural group SU(3) is considered and a spherically symmetric solution in this theory is obtained. This solution is a wormhole-like object located between two null surfaces ds² = 0 and can be considered as a domain with noncompactified ED which is sewn to a 4D spacetime with compactified ED. In some sense these solutions are dual to black holes: they represent static spherically symmetric solutions under null surfaces, whereas black holes are static spherically symmetric solutions outside the event horizon.     

Monte Carlo renormalization group study of φ4 field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

Nonexistence of Multiple-Instanton Solution of the Massless 4D ϕ4 Theory

The massless ϕ⁴ theory in four Euclidean dimensions is shown not to have a multiple-instanton solution — even though it does have a single-instanton solution. Our criterion for a multiple-instanton solution of this nontopological theory is that the energy-momentum tensor of the solution be identically zero. Arguments are given that a similar result holds in any number of dimensions for a one-component scale-invariant scalar theory.     

Monte Carlo renormalization group study of φ field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

A fixed point for quantum gravity

Using the 1/N expansion a fixed point of the renormalization group is found for quantized gravitational theories which is non-trivial in all dimensions, d, including four. Using the fixed point it is shown how Einstein's theory can be renormalized for 3<d<4. In four dimensions the pure Einstein theory does not exist, but the R + C_μναβ² theory does. It is shown how gravitational theories whose quantum lagrangians are scale invariant may be renormalized such that the scale invariance is broken only by the choice of the critical renormalization group trajectory. A comparison is made with the renormalization of four-fermion and Yukawa theories in 4?? dimensions which suggests that quantum gravity might exist in four dimensions even if those theories do not.     

Heterotic string from a higher dimensional perspective

The (abelian bosonic) heterotic string effective action, equations of motion and Bianchi identity at order α^′ in ten dimensions, are shown to be equivalent to a higher dimensional action, its derived equations of motion and Bianchi identity. The two actions are the same up to the gauge fields: the latter are absorbed in the higher dimensional fields and geometry. This construction is inspired by heterotic T-duality, which becomes natural in this higher dimensional theory.We also prove the equivalence of the heterotic string supersymmetry conditions with higher dimensional geometric conditions. Finally, some known Kähler and non-Kähler heterotic solutions are shown to be trivially related from this higher dimensional perspective, via a simple exchange of directions. This exchange can be encoded in a heterotic T-duality, and it may also lead to new solutions.     

Classical φ6-field theory in (1+1) dimensions. A model for structural phase transitions

The classical φ⁶-field theory in (1+1) dimensions, is considered as a model for the first order structural phase transitions. The equation of motion is solved exactly; and the presence of domain wall (kink) solutions at and below the transition point, in addition to the usual phonon-like oscillatory solutions, is demonstrated. The domain wall solutions are shown to be stable, and their mass and energies are calculated. Above the transition point there exists exotic unstable kink-like solutions which takes the particle from one hill top to the other of the potential. The partition function of the system is calculated exactly using the functional integral method together with the transfer matrix techniques which necessitates the determination of the eigenvalues of a Schrödinger-like equation. Thus the exact free energy is evaluated which in the low temperature limit has a phonon part and a contribution coming from the domain wall excitations. It was shown that this domain wall free energy differs from that calculated by the use of the domain wall phenomenology proposed by Krumhansl and Schrieffer. The exact solutions of the Schrödinger-like equation are also used to evaluate the displacement-displacement, intensity-intensity correlation functions and the probability distribution function. These results are compared with those obtained from the phenomenology as well as the φ⁴-field theory. A qualitative picture of the central peak observed in structural phase transitions is proposed.     

Higher dimensional cosmologies

Einstein equations are derived for D-dimensional space-time that spontaneously compactify to the product M⁴ × Π_{i = 1}^α M^d_i in which the metric is taken to be of the generalized Robertson-Walker form. Cosmological solutions for these equations are studied with power law, oscillatory and exponential behaviour for the D-dimensional Einstein-Maxwell, N = 2, D = 10 and N = 1, D = 11 supergravity models. In the Einstein-Maxwell case the presence of a cosmological constant forces the extra dimensions to be static. Nevertheless, it is possible to find solutions with vanishing effective 4 dimensional cosmological constant with an expanding 4-dimensional space-time. In the supergravity models the requirement of having compact extra dimensions restricts the solutions to have expansion only in the 4-dimensional space-time. Matter contribution is added to the energy-momentum tensor in an attempt to find new solutions.     

The <Emphasis Type="Italic">d</Emphasis>-dim Oscillator and Kepler Problems. The Potential 1/<Emphasis Type="Italic">x</Emphasis><Superscript>2</Superscript>

We compare the extra degeneracy of both the Oscillator and the Kepler problems in d dimensions as superintegrable systems. Addition of a centrifugal term like 1/r ² does not substantially change the solutions. We argue that both systems just describe a kind of rotation, and that the centrifugal potential is ineffective because its scale invariance.     

Large N 2D Yang-Mills Theory and Topological String Theory

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A= 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's “Ω^-1 points ”. We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach. Received: 3 March 1994 / Accepted: 2 February 1995     

Quantum Cosmology in Higher Derivative and Scalar-Tensor Gravity

The Bicknell theorem states that a non-linear Lagrangian can be recast in the form of a scalar-tensor theory, with a suitable potential, through a conformal transformation. In this paper, we first show that such classical equivalence remains valid at the level of the Wheeler—deWitt equation. Then, we consider a specific case, represented by a Lagrangian f(R) = R + l^–2(l²R)^4/3 whose vacuum cosmological solutions describe a non-singular Universe. The corresponding scalar-tensor theory and its cosmological solutions are written down. We find again non-singular solutions. The Wheeler—deWitt equation for this case is analyzed. The application of the Bicknell theorem leads to the interpretation of the behaviour of the scale factor in terms of the matter content, represented by the scalar field, and consequently to the energy conditions. The problem of classical and quantum regime is discussed and the classical behaviour is recovered, from the quantum solutions, near the maximum of the scale factor where the strong energy condition is satisfied.     

Riccion from higher-dimensional space-time with D-dimensional sphere as a compact manifold and one-loop renormalization

Lagrangian density of riccions is obtained with the quartic self-interacting potential using higher-derivative gravitational action in (4 +D)-dimensional space-time withS ^D as a compact manifold. It is found that the resulting four-dimensional theory for riccions is one-loop multiplicatively renormalizable. Renormalization group equations are solved and its solutions yield many interesting results such as (i) dependence of extra dimensions on the enegy mass scale showing that these dimensions increase with the increasing mass scale up toD = 6, (ii) phase transition at 3.05 × 10¹⁶ GeV and (iii) dependence of gravitational and other coupling constants on energy scale. Results also suggest that space-time above 3.05 × 10¹⁶ GeV should be fractal. Moreover, dimension of the compact manifold decreases with the decreasing energy mass scale such thatD = 1 at the scale of the phase transition. Results imply invisiblity of S¹ at this scale (which is 3.05 × 10¹⁶ GeV).     

Non-perturbative aspects in quantum field theory: Proceedings of Les Houches Winter Advanced Study Institute,March 1978

Until recent years field theories were only studied from standard perturbation series. These are equivalent to small fluctuation expansions around the classical ground state configurations in which the fields are space-time independent. It has been gradually realized that this procedure may miss some crucial physical features of the theory. In particular the absence of free quarks cannot be explained in quantum chromodynamics from this perturbative viewpoint. One is led to deal with the non-linearities in an essential way. This issue aims at covering the recent advances in this direction typically since 1975 and the issue 23C of Physics Reports.One way of going beyond perturbation theory is to make use of non-perturbative classical solutions which are in general very difficult to obtain. In Yang-Mills theories remarkable advances have been made in the search for self-dual solutions. They are described in the lectures (I, II) by E. Corrigan and R. Stora et al. The quantum meaning of these solutions and the technical problems raised by quantizing fluctuations around them are discussed by J.L. Gervais (III) and D. Gross (IV). The most recent topics of this subject deal with imaginary time solutions, the so-called instantons, which describe quantum tunneling in the semi-classical approximation. Other field configurations such as merons, are also discussed in relation with quark confinement. An attempt to build a general picture of strong interactions on these grounds is displayed in (IV).Recent advances concerning the solitons which are real time, classical Minkowski solutions, are discussed by D. Olive (V). These solutions describe generalized Dirac magnetic monopoles in non-Abelian gauge theories. The resulting model contains confined magnetic charges and unconfined electric charges. If the magnetic and elastic characters can be interchanged by a dual transformation, as discussed by F. Englert et al. in (VI), this would provide a model for quark confinement. This duality transformation has been spelled out in (IX) by C. Itzykson who proves the existence of a confinement phase transition in a lattice model with Z₂ gauge symmetry.Another non-linear feature of Yang-Mills theory is that no single gauge condition can be implemented over large field fluctuations. This is discussed in (VII) by Sciuto and also in (III).Another mechanism of confinement is proposed by McCoy and Wu (VIII) in which the propagator has a cut instead of a pole, as it does happen in the two-dimensional Ising model.The lectures of J. Zinn-Justin and G. Parisi, (X) and (XI) deal with the asymptotic estimation of the large order behaviour of perturbation theory, which one can obtain by semi-classical methods. The use of these methods is either to provide a way of improving practical perturbative calculations, or to characterize possible ambiguities due to vacuum tunneling. Additional problems raised in renormalizable theories are discussed in (XI).Another non-perturbative approach is to examine the large N limit of an SU(N) gauge theory. At leading order, only planar Feynman diagrams contribute. They are discussed in (XII) by E. Brézin. There are very subtle questions about the corresponding exact summation in two dimensions which are studied by T.T. Wu in (XV).In two-dimensional field theories exact solutions for the S-matrix have been recently discovered. They are discussed by Karowski in (XIII). In connection with this problem, Lüscher (XIV) has shown the existence of non-linear conserved charges which ensure the absence of particle production. These models are particularly interesting since the mass spectrum cannot be obtained from coupling constant expansion.Finally several related topics in statistical physics are discussed. D. Nelson (XVI) reviews the beautiful properties of the two-dimensional X-Y model which is an ideal example of mechanisms invoked in hadronic field theories. A. Luther (XVII) discussed an attempt to extend to a higher number of dimensions the bosonization of fermion theories which is so powerful in two dimensions. G. Toulouse (XVIII) describes modern ideas in spin glass phase transitions, a possible testing ground for gauge field theories besides its obvious intrinsic interest.     

Multi-black holes from nilpotent Lie algebra orbits

For N 3 2{\mathcal{N}\ge 2} supergravities, BPS black hole solutions preserving four supersymmetries can be superposed linearly, leading to well defined solutions containing an arbitrary number of such BPS black holes at arbitrary positions. Being stationary, these solutions can be understood via associated non-linear sigma models over pseudo-Riemannian spaces coupled to Euclidean gravity in three spatial dimensions. As the main result of this paper, we show that whenever this pseudo-Riemannian space is an irreducible symmetric space \mathfrakG/\mathfrakH^*{\mathfrak{G}/\mathfrak{H}^*}, the most general solutions of this type can be entirely characterised and derived from the nilpotent orbits of the associated Lie algebra \mathfrakg{\mathfrak{g}}. This technique also permits the explicit computation of non-supersymmetric extremal solutions which cannot be obtained by truncation to N=2{\mathcal{N}=2} supergravity theories. For maximal supergravity, we not only recover the known BPS solutions depending on 32 independent harmonic functions, but in addition find a set of non-BPS solutions depending on 29 harmonic functions. While the BPS solutions can be understood within the appropriate N=2{\mathcal{N}=2} truncation of N=8{\mathcal{N}=8} supergravity, the general non-BPS solutions require the whole field content of the theory.     

Quantum geometry of loops and the exact solubility of non-abelian gauge Chern-Simons theory-II

We quantize non-abelian Chern-Simons gauge theory in three dimensions in the presence of Wilson lines. We determine the theory dynamically in terms of the geometry of loops and show that it is exactly soluble. Remarkably the quantum loop equations are linear for S³ and they possess a class of solutions, among which is a non-critical Fermi string theory. Using these solutions we determine various important identities relevant to knot theory discovered recently by E. Witten, in particular, we show that the loop equation yields precisely the full exact skein relation of knot theory. As a byproduct we show that the partition function of an unknotted Wilson loop onS ³ is nothing but the character ofSU(2) in which the rotations areSU(N)-valued fractional angles. Furthermore, we generalize our solutions to the case where the manifoldM ₃ is oriented, closed, and non-simply connected withH ₁(M ₃)=0 (a homology 3-sphere).Address after 1^st October 1989: Physics Department, University of Florida, Gainsville, FL 32611, USA     

Dimensional regularization of massless theories in spherical space-time

The use of space-time curvature as an infra-red cut-off has been suggested for massless theories. In this paper we investigate the renormalization of massless theories in a spherical space-time (Euclidean version of de Sitter space) using dimensional regularization. Naive expectations are confirmed, namely that the coupling constant and wave-function renormalizations are independent of the curvature. Furthermore the curvature does not induce divergent mass terms or vacuum field values as would be possible on purely dimensional grounds. Although we have investigated only scalar field theories, φ⁴ theory in four dimensions and φ³ theory in six, these results are encouraging for an application of the method to gauge theories.Formally massless theories are conformally invariant so the formulation of the theory in a spherical space ought to be equivalent to its formulation in flat space. In fact the renormalization procedure breaks conformal invariance and removes this equivalence. We show that to achieve the flat space limit it is necessary to invoke the aid of the renormalization group. Thus the zero curvature limit can be achieved for infra-red stable theories (φ₄⁴) but not for infra-red unstable theories (φ₆³ as might be expected.     

Classical solutions and the energy-momentum tensor

A hyperelliptic two-meron solution of the massless scalar φ^N theory in $\text{n =}\text{2N}\text{(N ? 2)}$ Euclidean dimensions is given. This solution (which interpolates between the two-meron solution and the instanton solution of this theory) is used to illustrate several theory-independent statements which can be made about the energy-momentum tensor for instanton, meron and elliptic meron solutions of all scale invariant classical field theories.