The \Phi^4 quantum field in a scale invariant random metric

We discuss a D-dimensional Euclidean scalar field interacting with a scale invariant quantized metric. We assume that the metric depends on d-dimensional coordinates where d<D. We show that the interacting quantum fields have more regular short distance behaviour than the free fields. A model of a Gaussian metric is discussed in detail. In particular, in the \Phi^4 theory in four dimensions we obtain explicit lower and upper bounds for each term of the perturbation series. It turns out that there is no coupling constant renormalization in the \Phi^4 model in four dimensions. We show that in a particular range of the scale dimension there are models in D=4 without any divergencies

Quantization of the scalar field in a static quantum metric

We investigate the Hamiltonian formulation of quantum scalar fields in a static quantum metric. We derive a functional integral formula for the propagator. We show that the quantum metric substantially changes the behaviour of the scalar propagator and the effective Yukawa potential.Comment: Latex, 12 page

Minimal gauge-Higgs unification with a flavour symmetry

We show that a flavour symmetry a la Froggatt-Nielsen can be naturally incorporated in models with gauge-Higgs unification, by exploiting the heavy fermions that are anyhow needed to realize realistic Yukawa couplings. The case of the minimal five-dimensional model, in which the SU(2)_L x U(1)_Y electroweak group is enlarged to an SU(3)_W group, and then broken to U(1)_em by the combination of an orbifold projection and a Scherk-Schwarz twist, is studied in detail. We show that the minimal way of incorporating a U(1)_F flavour symmetry is to enlarge it to an SU(2)_F group, which is then completely broken by the same orbifold projection and Scherk-Schwarz twist. The general features of this construction, where ordinary fermions live on the branes defined by the orbifold fixed-points and messenger fermions live in the bulk, are compared to those of ordinary four-dimensional flavour models, and some explicit examples are constructed.Comment: LaTex, 37 pages, 2 figures; some clarifying comments and a few references adde

Estimates on Green functions of second order differential operators with singular coefficients

We investigate the Green functions G(x,x^{\prime}) of some second order differential operators on R^{d+1} with singular coefficients depending only on one coordinate x_{0}. We express the Green functions by means of the Brownian motion. Applying probabilistic methods we prove that when x=(0,{\bf x}) and x^{\prime}=(0,{\bf x}^{\prime}) (here x_{0}=0) lie on the singular hyperplanes then G(0,{\bf x};0,{\bf x}^{\prime}) is more regular than the Green function of operators with regular coefficients.Comment: 16 page

Green functions and dimensional reduction of quantum fields on product manifolds

We discuss Euclidean Green functions on product manifolds P=NxM. We show that if M is compact then the Euclidean field on P can be approximated by its zero mode which is a Euclidean field on N. We estimate the remainder of this approximation. We show that for large distances on N the remainder is small. If P=R^{D-1}xS^{beta}, where S^{beta} is a circle of radius beta, then the result reduces to the well-known approximation of the D dimensional finite temperature quantum field theory to D-1 dimensional one in the high temperature limit. Analytic continuation of Euclidean fields is discussed briefly.Comment: 17 page

Markov quantum fields on a manifold

We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.Comment: 14 pages, 1 figure, Late

Gauge-Fermion Unification and Flavour Symmetry

After we study the 6-dimensional ${\cal N} = (1, 1)$ supersymmetry breaking and $R$ symmetry breaking on $M^4\times T^2/Z_n$, we construct two ${\cal N} = (1, 1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_3$ where $E_6$ is broken down to $SO(10)\times U(1)_X$ by orbifold projection. In Model I, three families of the Standard Model fermions arise from the zero modes of bulk vector multiplet, and the $R$ symmetry $U(1)_F^{I} \times SU(2)_{{\bf 4}_-}$ can be considered as flavour symmetry. This may explain why there are three families of fermions in the nature. In Model II, the first two families come from the zero modes of bulk vector multiplet, and the flavour symmetry is similar. In these models, the anomalies can be cancelled, and we have very good fits to the SM fermion masses and mixings. We also comment on the ${\cal N}=(1, 1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_4$ and $M^4\times T^2/Z_6$, SU(9) models on $M^4\times T^2/Z_3$, and SU(8) models on $T^2$ orbifolds.Comment: Latex, 33 pages, minor change

Green functions and propagation of waves in strongly inhomogeneous media

We show that Green functions of second-order differential operators with singular or unbounded coefficients can have an anomalous behaviour in comparison to the well-known properties of Green functions of operators with bounded coefficients. We discuss some consequences of such an anomalous short or long distance behaviour for a diffusion and wave propagation in an inhomogeneous medium

Linearized Kompaneetz equation as a relativistic diffusion

We show that Kompaneetz equation describing photon diffusion in an environment of an electron gas, when linearized around its equilibrium distribution, coincides with the relativistic diffusion discussed in recent publications. The model of the relativistic diffusion is related to soluble models of imaginary time quantum mechanics. We suggest some non-linear generalizations of the relativistic diffusion equation and their astrophysical applications (in particular to the Sunyaev-Zeldovich effect).Comment: 12 page

Lepton flavor violating Higgs boson decays from massive seesaw neutrinos

Lepton flavor violating Higgs boson decays are studied within the context of seesaw models with Majorana massive neutrinos. Two models are considered: The SM-seesaw, with the Standard Model Particle content plus three right handed neutrinos, and the MSSM-seesaw, with the Minimal Supersymmetric Standard Model particle content plus three right handed neutrinos and their supersymmetric partners. The widths for these decays are derived from a full one-loop diagrammatic computation in both models, and they are analyzed numerically in terms of the seesaw parameters, namely, the Dirac and Majorana mass matrices. Several possible scenarios for these mass matrices that are compatible with neutrino data are considered. In the SM-seesaw case, very small branching ratios are found for all studied scenarios. These ratios are explained as a consequence of the decoupling behaviour of the heavy right handed neutrinos. In contrast, in the MSSM-seesaw case, sizeable branching ratios are found for some of the leptonic flavor violating decays of the MSSM neutral Higgs bosons and for some choices of the seesaw matrices and MSSM parameters. The relevance of the two competing sources of lepton flavor changing interactions in the MSSM-seesaw case is also discussed. The non-decoupling behaviour of the supersymmetric particles contributing in the loop-diagrams is finally shown.Comment: 44pgs. Version to appear in Phys.Rev.