Are Einstein's equations of homogeneous multidimensional cosmology generic?

It is shown that Einstein's equations for multidimensional homogeneous block diagonal cosmology are generic only if the dimension of space-time is 4.     

On the continuity of causal automorphisms of space-time

We prove that any causal automorphism of the (curved or not) space-time in any dimension is continuous.     

The Feynman propagator from a single path

We show that it is possible to construct the Feynman propagator for a free particle in one dimension, without quantization, from a single continuous space-time path.     

Geometry of the Einstein and Yang-Mills equations

It is shown that the Einstein and Yang-Mills equations arise from the conditions for the space-time to be a submanifold of a pseudo-Euclidean space with dimension greater than 5. Some possible applications to cosmology, spin-2 fields, and geometrodynamics are discussed.     

A simple physical interpretation of the critical dimension of space-time in dual models

The energy due to zero point fluctuations of the dual string is calculated and shown to be divergent. We make it finite by introducing a cut-off. The result is non-covariant but invariant under reparametrization, so one has to modify the classical Lagrangian so as to get a covariant result. It is then shown that this leads to the well-known relation relating the mass squared of the lowest state to the number of dimensions space-time. This result is independent of the cut-off and shows clearly the physical significance of the critical dimension of space-time. Similar considerations for the Neveu-Schwarz model are also discussed.     

Generalization of Lagrangian dynamics

We speculate on a generalized dynamics described by an integral over action functionals that is a generalization of the standard functional integral. In a simple Gaussian case we obtain a certain differential equation for the measure of Feynman integral. We prove that the equation is satisfied for the spin zero field in one space-time dimension.     

Some special solutions of the elliptic-hyperbolic sine equation in 2 + 1 dimensions

Special solutions are given for the elliptic-hyperbolic sine equation in 2 + 1-dimensional space-time. One of these solutions is similar in form to the breather solution in one space and one time dimension.     

Holographic renormalization of foliation preserving gravity and trace anomaly

From the holographic renormalization group viewpoint, while the scale transformation plays a primary role in holographic dualities by providing the extra dimension, the special conformal transformation seems to only play a secondary role. We, however, claim that the space-time diffeomorphism is crucially related to the latter. For its demonstration, we study the holographic renormalization group flow of a foliation preserving diffeomorphic theory of gravity (a.k.a. space-time flipped Horava gravity). We find that the dual field theory, if any, is only scale invariant but not conformal invariant. In particular, we show that the holographic trace anomaly in four dimension predicts the Ricci scalar squared term that would be incompatible with the Wess–Zumino consistency condition if it were conformal. This illustrates how the foliation preserving diffeomorphic theory of gravity could be in conflict with a theorem of the dual unitary quantum field theory.     

A Possibility of Space-Time Discreteness

A possibility of existing minimum time interval is questioned owing to the uncertainty principle. Quantum mechanics on a discrete space (one dimension) and time is discussed. It is shown that the possibility of space-time discreteness provides a new understanding of confinement phenomena.     

Consistent ADD scenario with stabilized extra dimension

A model with one compact extra dimension and a scalar field of Brans–Dicke type in the bulk is discussed. It describes two branes with non-zero tension embedded into the space-time with flat background. This setup allows one to use a very simple method for stabilization of the size of extra dimension. It appears that the four-dimensional Planck mass is expressed only through parameters of the scalar field potentials on the branes.     

There are no scalar Lie fields in three or more dimensional space-time

We show that a Lie field structure is incompatible with a scalar relativistic quantum field theory if the dimension of space-time is greater than two. Our main argument is based on the Jacobi identity and the spectrum condition.     

Gauge Gravitational Field in a Fractal Space-Time

Considering the fractal structure of space-time, the scale relativity theory in the topological dimension D_T=2 is built. In such a conjecture, the geodesics of this space-time imply the hydrodynamic model of the quantum mechanics. Subsequently, the gauge gravitational field on a fractal space-time is given. Then, the gauge group, the gauge-covariant derivative, the strength tensor of the gauge field, the gauge-invariant Lagrangean, the field equations of the gauge potentials and the gauge energy-momentum tensor are determined. Finally, using this model, a Reissner-Nordström type metric is obtained.     

Kaluza-Klein space and strong gravity

A study of strong gravity theory is carried out in five-dimensional Kaluza-Klein space-time. A relation between the cosmological constant and the radius parameter of the fifth dimension is obtained. The effect of the extra dimension is seen through the generation of masses simulating a Regge-like mass spectrum. It is found that the confinement mechanism is built into the strong gravity formalism. We also discuss the trapping of quarks in the 5D background.     

Ghost triality and superstring partition functions

In three or more space-time dimensions a heterotic string can have just four kinds of space-time supersymmetry, corresponding to the exceptional algebras E₆, E₇, E₈ and F₄. We give a simple argument why this correspondence, first observed in the context of the covariant lattice construction, holds for any heterotic superstring, no matter how it is constructed. Using the fact that SO(8) triality is an inner automorphism of the exceptional algebras, we derive a set of identities for character-valued partition functions, valid for any superstring in any space-time dimension and at any genus. These identities have exactly the structure of the Riemann identities, and are identical to the Riemann identities in the case of ten-dimensional strings. Their relavance for proving vanishing of the cosmological constant and non-renormalization theorems is discussed.     

Infrared and ultraviolet dimensional meromorphy of Feynman amplitudes

By the concurrent use of dimensional and analytic regularizations with the complete Mellin (CM) representation, we find in a direct way the ultraviolet and infrared poles in space-time dimension, for any Feynman amplitude with an arbitrary subset of vanishing masses.     

The connection between an Euclidean Gauss Markov Vector Field and the Real Proca Wightman Field

We construct an Euclidean Gauss Markov Vector Field which leads to the Real Proca Wightman Field describing particles of mass >0 and spin 1, for space-time dimension equal to 4.     

Analysis of fractals with dependent branching by box counting, P-adic coverages, and systems of equations of P-adic coverages

Concept of the dimension of space-time in the general relativity theory and quantum theory is discussed. It is emphasized that the dimension of a discrete space can be defined based on the Hausdorff measure. The noninteger dimension is a typical characteristic of a fractal. The process of hadron formation in interactions between high-energy particles and nuclei is supposed to possess fractal properties. The following methods for analyzing fractals are considered: box counting (BC), method of P-adic coverages (PaC), and method of systems of equations of P-adic coverages (SePaC), for determining the fractal dimension. A comparative analysis of fractals with dependent branching is performed using these methods. We determine the optimum values of parameters permitting one to determine the fractal dimension D _F , number of levels N _lev, and the fractal structure with maximal efficiency. It is noted that the SePaC method has advantages in analyzing fractals with dependent branching.     

Five- and eight-vector extensions of relativistic quantum theory: The preferred reference frame

The complex quaternion approach to relativistic quantum theory is extended to include time, in the preferred cosmic reference frame, as a fifth dimension. Particle mass appears as an eigenvalue of this invariant operator. Closing the extended algebra also suggests three additional space-time components, possibly related to microscopic rotational degrees of freedom and the scale setting fundamental length. The Dirac, Klein-Gordon, Maxwell, and Einstein equations are generalized to accommodate 5- and 8-vector space-time, including gravitational curvature. The usual conserved probability density is shown to harbor fundamental difficulties.     

Anomalies in gravity-coupled SU(N) Yang-Mills theory

In a general even space-time dimension I consider spin-1/2 fermions coupled to SU(N) Yang-Mills fields and to gravity. It is shown (unexpectedly) that, once the chiral gauge anomaly is cancelled, gravitational anomalies do not further constrain the fermion representation.