On Hexagonal Structures in Higher Dimensional Theories

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2),SU(3), G_2, Spin(7), SO(8) as well as E_8 and SO(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.Comment: 9 pages, Latex, 3 figures. Accepted for publication in International Journal of Theoretical Physic

One-parameter Darboux-transformed quantum actions in Thermodynamics

We use nonrelativistic supersymmetry, mainly Darboux transformations of the general (one-parameter) type, for the quantum oscillator thermodynamic actions. Interesting Darboux generalizations of the fundamental Planck and pure vacuum cases are discussed in some detail with relevant plots. It is shown that the one-parameter Darboux-transformed Thermodynamics refers to superpositions of boson and fermion excitations of positive and negative absolute temperature, respectively. Recent results of Arnaud, Chusseau, and Philippe physics/0105048 regarding a single mode oscillator Carnot cycle are extended in the same Darboux perspective. We also conjecture a Darboux generalization of the fluctuation-dissipation theoremComment: 14 pages, 13 figures, correction of the formula in the text after Eq. 7, accepted at Physica Script

Point interactions from flux conservation

We show that the physical requirement of flux conservation can substitute for the usual matching conditions in point interactions. The study covers an arbitrary superposition of δ and δ' potentials on the real line and can be easily applied to higher dimensions. Our procedure can be seen as a physical interpretation of the deficiency index of some symmetric, but not self-adjoint operators

The Spin-Statistics Theorem in Arbitrary Dimensions

We investigate the spin-statistics connection in arbitrary dimensions for hermitian spinor or tensor quantum fields with a rotationally invariant bilinear Lagrangian density. We use essentially the same simple method as for space dimension D = 3. We find the usual connection (tensors as bosons and spinors as fermions) for D = 8n + 3; 8n + 4; 8n + 5, but only bosons for spinors and tensors in dimensions 8n +/- 1 and 8n. In dimensions 4n + 2 the spinors may be chosen as bosons or fermions. The argument hinges on finding the identity representation of the rotation group either on the symmetric or the antisymmetric part of the square of the field representation.Comment: 14 pages. To be published in International Journal of Theoretical Physic

On F-theory Quiver Models and Kac-Moody Algebras

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four dimensional base space. We focus on the base geometry which consists of intersecting F0=CP1xCP1 Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac-Moody (KM) algebras: ordinary, i.e finite dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N =1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine A base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.Comment: 13 pages; new equations added in section 3, one reference added and typos correcte

Note on the natural system of units

We propose to substitute Newton's constant GN for another constant G2, as if the gravitational force would fall off with the 1/r law, instead of the 1/r2; so we describe a system of natural units with G2, c and ħ. We adjust the value of G2 so that the fundamental length L=LPl is still the Planck's length and so GN=L×G2. We argue for this system as (1) it would express longitude, time and mass without square roots; (2) G2 is in principle disentangled from gravitation, as in (2+1) dimensions there is no field outside the sources. So G2 would be truly universal; (3) modern physics is not necessarily tied up to (3+1)-dim. scenarios and (4) extended objects with p=2 (membranes) play an important role both in M-theory and in F-theory, which distinguishes three (2, 1) dimensions. As an alternative we consider also the clash between gravitation and quantum theory; the suggestion is that non-commutative geometry [xi, xj]=Λ2θij would cure some infinities and improve black hole evaporation. Then the new length Λ shall determine, among other things, the gravitational constant GN