Narrow Results

It is shown how a noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy $\frac{A}{4G}$, the logarithmic corrections, and further corrections, from the discrete mass transitions taking place among different mass states in $D=4$. The higher-dimensional generalization of the results in $D=4$ follows. The discretization of the entropy-mass relation $S=S(M)$ leads to an entropy quantization of the form $S=S(M_n)=n$, such that one may always assign $n$ “bits” to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [G. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D55 (1997) 6189.] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large $N$ Matrix model (fuzzy sphere) of C.-S. Chu [A matrix model proposal for QG and the QM of black holes, preprint, arXiv:2406.01466] leads to very similar results for the black hole entropy as the physical model described in this work which is based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.

In the first sections of this paper we give an elementary but rigorous approach to the construction of the quantum Bosonic and supersymmetric string system continuing the analysis of Dimock. This includes the construction of the DDF operators without using the vertex algebras. Next we give a rigorous proof of the equivalence between the light-cone and the covariant quantization methods. Finally, we provide a new and simple proof of the BRST quantization for these string models.

Boundary conditions and gluing conditions for open strings and D-branes in the SL(2,R) WZWN model, corresponding to AdS₃, are discussed. Some boundary conditions and gluing conditions previously considered in the literature are shown to be incompatible with the variation principle.

We then consider open string boundary conditions corresponding to a certain field-dependent gluing condition. This allows us to consider open strings with constant energy and angular momentum. Classically, these open strings naturally generalize the open strings in flat Minkowski space. For rigidly rotating open strings, we show that the torsion leads to a bending and an unfolding. We also derive the SL(2,R) Regge relation, which generalizes the linear Minkowski Regge relation. For "high" mass, it takes the form L ≈ ± M/H, where H is the scale of the SL(2,R) group manifold.

In this work we explain the construction of the thermal vacuum for the bosonic string, as well that of the thermal boundary state interpreted as a D_p-brane at finite temperature. In both case we calculate the respective entropy using the entropy operator of the Thermo Field Dynamics theory. We show that the contribution of the thermal string entropy is explicitly present in the D_p-brane entropy. Furthermore, we show that the Thermo Field approach is suitable to introduce temperature in boundary states.

We analyze the structure of heterotic M-theory on K3 orbifolds by presenting a comprehensive sequence of M-theoretic models constructed on the basis of local anomaly cancellation. This is facilitated by extending the technology developed in our previous papers to allow one to determine "twisted" sector states in nonprime orbifolds. These methods should naturally generalize to four-dimensional models, which are of potential phenomenological interest.

We present a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration relativity principle in the space–time tangent bundle and in phase spaces (cotangent bundle). The maximal proper-acceleration bound is a = c²/Λ in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. The group transformation laws of this maximal-acceleration phase space relativity theory under velocity and acceleration boosts are analyzed in full detail. For pure acceleration boosts it is shown why the minimal Planck-areas (maximal string tension) are universal invariant quantities in any frame of reference. Inspired by the maximal-acceleration corrections to the Lamb shifts of one-electron atoms by Lambiase, Papini and Scarpetta, we derive the exact integral equation that governs the renormalization-group-like scaling dependence of the fractional change of the fine structure constant as a function of the cosmological redshift factor and a cutoff scale L_c, where the maximal acceleration relativistic effects are dominant. A particular physical model exists dominated entirely by the vacuum energy, when the cutoff scale is the Planck scale, with Ω_Λ = 1. The cosmological implications of this extreme case scenario are studied.

Velocity-dependent interactions in a fundamental-string dominated universe lead quite naturally, with reasonable assumptions on initial conditions, to an accelerating expanding universe without assuming the existence of a cosmological constant. Using a mean-field approximation, exact solutions are obtained for various string/brane cosmological toy models. A multi-body model is also investigated. It is also shown that the repulsive velocity-dependent force arises in more general contexts and can lead to cosmic structure formation. Finally, we discuss a toy model including both ordinary and extremal matter, in which the latter accounts for dark matter while simultaneously acting as effective dark energy. Eternal acceleration is once more seen to arise in this case.

Using the ontological interpretation of quantum mechanics in a particular sense, we obtain the classical behavior of the scale factor and two scalar fields, derived from a string effective action for the Friedmann–Robertson–Walker (FRW) time dependent model. Besides, the Wheeler–DeWitt equation is solved exactly. We speculate that the same procedure could also be applied to S-branes.

A novel approach to evaluate the Wilson loops associated with a SU(∞) gauge theory in terms of pure string degrees of freedom is presented. It is based on the Guendelman–Nissimov–Pacheva formulation of composite antisymmetric tensor field theories of area (volume) preserving diffeomorphisms which admit p-brane solutions and which provide a new route to scale-symmetry breaking and confinement in Yang–Mills theory. The quantum effects are discussed and we evaluate the vacuum expectation values (VEV) of the Wilson loops in the large N limit of the quenched reduced SU(N) Yang–Mills theory in terms of a path integral involving pure string degrees of freedom. The quenched approximation is necessary to avoid a crumpling of the string worldsheet giving rise to very large Hausdorff dimensions as pointed out by Olesen. The approach is also consistent with the recent results based on the AdS/CFT correspondence and dual QCD models (dual Higgs model with dual Dirac strings). More general Loop wave equations in C-spaces (Clifford manifolds) are proposed in terms of generalized holographic variables that contain the dynamics of an aggregate of closed branes (p-loops) of various dimensionalities. This allows us to construct the higher-dimensional version of Wilson loops in terms of antisymmetric tensor fields of arbitrary rank which couple to p-branes of different dimensionality.

We provide a conceptual unified description of the quantum properties of black holes (BH), elementary particles, de Sitter (dS) and Anti-de Sitter (AdS) string states.The conducting line of argument is the classical–quantum (de Broglie, Compton) duality here extended to the quantum gravity (string) regime (wave–particle–string duality). The semiclassical (QFT) and quantum (string) gravity regimes are respectively characterized and related: sizes, masses, accelerations and temperatures. The Hawking temperature, elementary particle and string temperatures are shown to be the same concept in different energy regimes and turn out the precise classical–quantum duals of each other; similarly, this result holds for the BH decay rate, heavy particle and string decay rates; BH evaporation ends as quantum string decay into pure (nonmixed) radiation. Microscopic density of states and entropies in the two (semiclassical and quantum) gravity regimes are derived and related, an unifying formula for BH, dS and AdS states is provided in the two regimes. A string phase transition towards the dS string temperature (which is shown to be the precise quantum dual of the semiclassical (Hawking–Gibbons) dS temperature) is found and characterized; such phase transition does not occurs in AdS alone. High string masses (temperatures) show a further (square root temperature behavior) sector in AdS. From the string mass spectrum and string density of states in curved backgrounds, quantum properties of the backgrounds themselves are extracted and the quantum mass spectrum of BH, dS and AdS radii obtained.

This review is devoted to the Multiple Point Principle (MPP), according to which several vacuum states with the same energy density exist in Nature. The MPP is implemented to the Standard Model (SM), Family replicated gauge group model (FRGGM) and phase transitions in gauge theories with/without monopoles. Using renormalization group equations for the SM, the effective potential in the two-loop approximation is investigated, and the existence of its postulated second minimum at the fundamental scale is confirmed. Phase transitions in the lattice gauge theories are reviewed. The lattice results for critical coupling constants are compared with those of the Higgs monopole model, in which the lattice artifact monopoles are replaced by the point-like Higgs scalar particles with magnetic charge. Considering our (3+1)-dimensional space–time as, in some way, discrete or imagining it as a lattice with a parameter a = λ_P, where λ_P is the Planck length, we have investigated the additional contributions of monopoles to the β-functions of renormalization group equations for running fine structure constants α_i(μ) (i = 1, 2, 3 correspond to the U(1), SU(2) and SU(3) gauge groups of the SM) in the FRGGM extended beyond the SM at high energies. It is shown that monopoles have N_fam times smaller magnetic charge in the FRGGM than in the SM (N_fam is a number of families in the FRGGM). We have estimated also the enlargement of a number of fermions in the FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. We have reviewed that, in contrast to the case of the Anti-grand-unified-theory (AGUT), there exists a possibility of unification of all gauge interactions (including gravity) near the Planck scale due to monopoles. The possibility of the [SU(5)]³ or [SO(10)]³ unification at the GUT-scale ~10¹⁸GeV is briefly considered.

The quantum Yang–Mills theory, describing a system of fields with nondual (chromoelectric g) and dual (chromomagnetic ) charges and revealing the generalized dual symmetry, is developed by analogy with the Zwanziger formalism in QED. The renormalization group equations (RGE's) for pure non-Abelian theories are analyzed for both constants, α = g²/4π and . The pure gauge theory is investigated as an example. We consider not only monopoles, but also dyons. The behavior of the total SU(3) β-function is investigated in the whole region of α≡α_s: 0≤α < ∞. It is shown that this β-function is antisymmetric under the interchange α ↔ 1/α and is given by the well-known perturbative expansion not only for α≪1, but also for α≫1. Using an idea of the Maximal Abelian Projection by 't Hooft, we have considered the formation of strings — the ANO flux tubes — in the Higgs model of scalar monopole (or dyon) fields. In this model we have constructed the behavior of the β-function in the vicinity of the point α = 1, where it acquires a zero value. Considering the phase transition points at α≈0.4 and α≈2.5, we give the explanation of the freezing of α_s. The evolution of with energy scale μ and the behavior of V_eff(μ) are investigated for both, perturbative and nonperturbative regions of QCD. It was shown that the effective potential has a minimum, ensured by the dual sector of QCD. The gluon condensate , corresponding to this minimum, is predicted: , in agreement with the well-known results.

We study black hole-like solutions (space–times with singularities) of Einstein field equations in 3+1 and 2+2 dimensions. We find three different cases associated with hyperbolic homogeneous spaces. In particular, the hyperbolic version of Schwarzschild's solution contains a conical singularity at r = 0 resulting from pinching to zero size r = 0 the throat of the hyperboloid and which is quite different from the static spherically symmetric (3+1)-dimensional solution. Static circular symmetric solutions for metrics in 2+2 are found that are singular at ρ = 0 and whose asymptotic ρ→∞ limit leads to a flat (1+2)-dimensional boundary of topology S¹ × R². Finally we discuss the (1+1)-dimensional Bars–Witten stringy black hole solution and show how it can be embedded into our (3+1)-dimensional solutions. Black holes in a (2+2)-dimensional "space–time" from the perspective of complex gravity in 1+1 complex dimensions and their quaternionic and octonionic gravity extensions deserve furher investigation. An appendix is included with the most general Schwarzschild-like solutions in D ≥ 4.

We study the system formed by a gas of black holes and strings within a microcanonical formulation. The density of mass levels grows asymptotically as , (i = 1,…,N). We derive the microcanonical content of the system: entropy, equation of state, number of components N, temperature T and specific heat. The pressure and the specific heat are negative reflecting the gravitational unstability and a nonhomogeneous configuration. The asymptotic behavior of the temperature for large masses emerges as the Hawking temperature of the system (classical or semiclassical phase) in which the classical black hole behavior dominates, while for small masses (quantum black hole or string behavior) the temperature becomes the string temperature which emerges as the critical temperature of the system. At low masses, a phase transition takes place showing the passage from the classical (black hole) to quantum (string) behavior. Within a microcanonical field theory formulation, the propagator describing the string–particle–black hole system is derived and from it the interacting four-point scattering amplitude of the system is obtained. For high masses it behaves asymptotically as the degeneracy of states ρ(m) of the system (i.e. duality or crossing symmetry). The microcanonical propagator and partition function are derived from a (Nambu–Goto) formulation of the N-extended objects and the mass spectrum of the black hole–string system is obtained: for small masses (quantum behavior) these yield the usual pure string scattering amplitude and string–particle spectrum ; for growing mass the spectrum describes all the intermediate states up to the pure black hole behavior. The different black hole behaviors according to the different mass ranges: classical, semiclassical and quantum or string behaviors are present in the model.

Born's reciprocal relativity in flat space–times is based on the principle of a maximal speed limit (speed of light) and a maximal proper force (which is also compatible with a maximal and minimal length duality) and where coordinates and momenta are unified on a single footing. We extend Born's theory to the case of curved space–times and construct a reciprocal general relativity theory (in curved space–times) as a local gauge theory of the quaplectic group and given by the semidirect product , where the non-Abelian Weyl–Heisenberg group is H(1, 3). The gauge theory has the same structure as that of complex non-Abelian gravity. Actions are presented and it is argued why such actions based on Born's reciprocal relativity principle, involving a maximal speed limit and a maximal proper force, is a very promising avenue to quantize gravity that does not rely in breaking the Lorentz symmetry at the Planck scale, in contrast to other approaches based on deformations of the Poincaré algebra, quantum groups. It is discussed how one could embed the quaplectic gauge theory into one based on the U(1, 4), U(2, 3) groups where the observed cosmological constant emerges in a natural way. We conclude with a brief discussion of complex coordinates and Finsler spaces with symmetric and nonsymmetric metrics studied by Eisenhart as relevant closed-string target space backgrounds where Born's principle may be operating.

It is shown how w_∞, w_1+∞ gauge field theory actions in 2D emerge directly from 4D gravity. Strings and membranes actions in 2D and 3D originate as well from 4D Einstein gravity after recurring to the nonlinear connection formalism of Lagrange–Finsler and Hamilton–Cartan spaces. Quantum gravity in 3D can be described by a W_∞ matrix model in D = 1 that can be solved exactly via the collective field theory method. We describe why a quantization of 4D gravity could be attained via a 2D quantum W_∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that noncritical W_∞ (super)strings are devoid of BRST anomalies in dimensions D = 27(D = 11), respectively, follows and which coincide with the critical (super)membrane dimensions D = 27(D = 11). We establish the correspondence between the states associated with the quasifinite highest weights irreducible representations of W_∞, algebras and the quantum states of the continuous Toda molecule. Schrödinger-like quantum mechanics wave functional equations are derived and solutions are found in the zeroth-order approximation. Since higher-conformal spin W_∞ symmetries are very relevant in the study of 2DW_∞ gravity, the quantum Hall effect, large N QCD, strings, membranes, … it is warranted to explore further the interplay among all these theories.

We discuss tachyon-free examples of (Type IIB on) noncompact nonsupersymmetric orbifolds. Tachyons are projected out by discrete torsion between orbifold twists, while supersymmetry is broken by a Scherk–Schwarz phase (+1/-1 when acting on space–time bosons/fermions) accompanying some even order twists. The absence of tachyons is encouraging for constructing nonsupersymmetric D3-brane gauge theories with stable infrared fixed points. The D3-brane gauge theories in our orbifold backgrounds have chiral supersymmetric spectra, but nonsupersymmetric interactions.

Considering perturbative expansions around the topological sector described by the number of handles of the string worldsurface, we show that a spin-2 field is surrounding such a sector. This graviton-like field satisfies wave-like equations of motion on the worldsurface; the massive and massless modes are present and are discussed in detail. The action and the corresponding partition function are constructed and their properties are discussed. The results show how the perturbations of topological sectors may generate transitions from a state with no metric to another possessing such a structure, emerging spontaneously the unavoidable part of the spectrum of the string, a graviton-like field.

We review the recent results on a decay rate of metastable topological configurations such as strings and domain walls. The transition from a state with higher tension to a state with lower one proceeds through quantum tunneling or through thermally catalyzed quantum tunneling (at sufficiently small temperatures). It is shown that the effects of the motion in transverse direction lead to the renormalization of a mass (tension) parameter of a particle (string) associated with a boundary of a string (wall) in the semiclassical exponent. For a non-zero temperature we derive the catalysis factor for the decay rates. It is discovered that the catalysis factor is closely related to the probability (effective length) of the collision of the the Goldstone bosons, corresponding to the transverse waves on a string (wall). We find that the destruction of a string only takes place in collisions of even number of the bosons, while the destruction of the wall can occur in a collision of any number of particles.

We first discuss the problem of mass hierarchy and review briefly the main Beyond the Standard Model (BSM) proposals. We then describe the framework of strings, branes and large extra dimensions and give the main experimental predictions in both particle accelerators and microgravity experiments testing gravity at short distances. Finally, we present some models based on intersecting branes and discuss the issue of Standard Model embedding.