Quantum creation of multidimensional universes

A non zero probability amplitude for the appearance of a multidimensional universe of (1+d) dimensions is found. This can happen either in a “symmetric phase”, in which all dimensions are in an exponential expansion, or else in a “broken phase”, withd ₁ dimensions inflating exponentially andd ₂ forming a sphere of constant radius. The value of these amplitude for different total number of dimensions is discussed, and so physical consequences for Kaluza-Klein cosmologies are drawn.     

Dimensional reduction of Weyl,Majorana and Majorana-Weyl spinors

We discuss the dimensional reduction for Weyl, Majorana, or Majorana-Weyl spinors coupled to pure d-dimensional (d ? 4) gravity. The only case where a realistic four-dimensional low-energy spectrum for the fermions may be obtained, is for a Majorana-Weyl spinor in d = 2 mod 8 dimensions. Chiral massless fermions are not excluded in this case. The minimal number of dimensions for the construction of a realistic theory out of pure gravity is d = 18.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

Heat-kernel expansion and counterterms of the Faddeev–Popov determinant in Coulomb and Landau gauge

The Faddeev–Popov determinant of Landau gauge in d dimensions and Coulomb gauge in d+1 dimensions is calculated in the heat-kernel expansion up to next-to-leading order. The UV-divergent parts in d=3,4 are isolated and the counterterms required for a non-perturbative treatment of the Faddeev–Popov determinant are determined.     

Exact solution of the Percus-Yevick equation for a hard-core fluid in odd dimensions

It is shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard-core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions. The nonlinear integral equation is reduced to an algebraic equation of order d?3 for odd dimensions d greater than three. As an example the direct correlation function in five dimensions is derived explicitly.     

On the renormalization of the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation of nonlinear stochastic growth in d dimensions is studied using the mapping onto a system of directed polymers in a quenched random medium. The polymer problem is renormalized exactly in a minimally subtracted perturbation expansion about d = 2. For the KPZ roughening transition in dimensions d > 2, this renormalization group yields the dynamic exponent z = 2 and the roughness exponent χ = 0, which are exact to all orders in ε ≡ (2 − d)/2. The expansion becomes singular in d = 4. If this singularity persists in the strong-coupling phase, it indicates that d = 4 is the upper critical dimension of the KPZ equation. Further implications of this perturbation theory for the strong-coupling phase are discussed. In particular, it is shown that the correlation functions and the coupling constant defined in minimal subtraction develop an essential singularity at the strong-coupling fixed point.     

Instability of Heisenberg-spin-glass-phase transitions

It is pointed out that there should be no stable phase transitions for XY- and Heisenberg spin glasses with d ? 4 dimensions, and for Ising spin glasses with d ? 2, in the presence of arbitrarily small random magnetic fields. In the absence of such fields the critical dimensions are 2 and 1, respectively.     

Dynamical realization of l-conformal Galilei algebra and oscillators

The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.     

Is there a c-theorem in four dimensions?

The difficulties of extending Zamolodchikov's c-theorem to dimensions d ≠ 2 are discussed. It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, S^d, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly. It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.     

Discrete symmetries in Kaluza-Klein theories

We investigate discrete symmetries in theories of higher-dimensional (d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d ? 4 dimensional internal space. If the action of pure d-dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d-dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor.A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong-CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.     

Non critical super strings on world sheets of constant curvature

We consider correlation functions in Neveu-Schwarz string theory coupled to two dimensional gravity. The actionfor the 2D gravity consists of the string induced Liouville action and the Jackiw-Teitelboim action describing pure 2D gravity. Then gravitational dressed dimensions of vertex operators are equal to their bare conformal dimensions. There are two possible interpretations of the model. Considering the 2D dilaton and the Liouville field as additional target space coordinates one gets ad+2-dimensional critical string. In thed-dimensional non critical string picture gravitational fields retain their original meaning and ford=4 one can get a mass spectrum via consistency requirements. In both cases a GSO projection is possible.     

Ramond-Neveu-Schwarz string with new boundary conditions

For the Ramond-Neveu-Schwarz string in D space-time dimensions we seek boundary conditions which preserve Poincaré invariance in d dimensions, d<D. We obtain twisted closed and twisted open strings preserving Gervais-Sakita supersymmetry. Covariant BRST quantization yields D=10. For some boundary conditions, partition functions exhibit space-time supersymmetry.     

Critical dimensions for strings,bags,...

Consider extendedd-dimensional objects such as strings (d=1), membranes (d=2), ... It is argued that their quantum theories cannot be consistent in space-times of dimension exceeding the critical valueD=2+24/d (orD=2+8/d in the supersymmetric case). This rules out consistent theories for objects extended in more than 5 (3 in the supersymmetric case) space-like dimensions.     

Gauge theory of gravitation in higher-dimensional space-time and its dimensional reduction

It is shown that the Euler invariant type action of an O(d) gauge theory in d dimensions is a surface term and becomes the (d − 1)-dimensional gauge theory of gravitation. Its Kaluza-Klein dimensional reduction gives a theory which contains the ordinary Einstein gravity.     

Variational approach for the N-state spin and gauge Potts model

A hamiltonian variational treatment is applied both to the spin Potts model and to its gauge version for any number of states N and spatial dimensions d?2. Regarding the former we reproduce the correct critical coupling and latent heat for not too low N and d. For the latter, our approach turns the gauge theory into an equivalent d-dimensional classical spin model, which evaluated for d + 1 = 4 gives results in agreement with 1/N expansions.     

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.     

The XY model and Kadanoff's variational real space renormalization group

The XY model in d dimensions is studied by means of a variational real space renormalization group transformation. Contrary to an earlier computation in the same framework for the d = 2 case, we find that a low order operator basis truncation is highly unstable. For certain values of the variational parameter p the renormalization group flow can display period doubling sequences towards a chaotic regime. The behavior in the d = 3 case is very similar.     

Replica symmetry breaking in and around six dimensions

Two, replica symmetry breaking specific, quantities of the Ising spin glass — the breakpoint x₁ of the order parameter function and the Almeida-Thouless line — are calculated in six dimensions (the upper critical dimension of the replicated field theory used), and also below and above it. The results confirm that replica symmetry breaking does exist below d=6, and also the tendency of its escalation for decreasing dimension continues. As a new feature, x₁ has a nonzero and universal value for d<6 at criticality. Near six dimensions we have x1c=3(6−d)+O[²(6−d)]. A method to expand a generic theory with replica equivalence around the replica symmetric one is also demonstrated.     

Energy relations in the density functional theory of d-dimensional heavy positive ions

By means of the virial theorem, the total energy E, eigenvalue sum E_s and chemical potential μ are related in a heavy positive ion in d dimensions. The scaling properties of these quantities in two dimensions are thereby established.