Estimate of the number of one-parameter families of modules over a tame algebra

The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of dimension at most d divides into a finite number f(d,A) of modules and one-parameter series of modules. We prove that the number of m-by-n canonical parametric block matrices with a given partition into blocks is bounded by 4^s, where s is the number of free entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page

Estimate of the number of one-parameter families of modules over a tame algebra

The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of dimension at most d divides into a finite number f(d,A) of modules and one-parameter series of modules. We prove that the number of m-by-n canonical parametric block matrices with a given partition into blocks is bounded by 4^s, where s is the number of free entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page

Quantifier elimination in C*-algebras

The only C*-algebras that admit elimination of quantifiers in continuous logic are $\mathbb{C}, \mathbb{C}^2$, $C($Cantor space$)$ and $M_2(\mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory of $M_n(\mathcal {O_{n+1}})$ is not $\forall\exists$-axiomatizable for any $n\geq 2$.Comment: More improvements and bug fixes. To appear in IMR

Covariance and Time Regained in Canonical General Relativity

Canonical vacuum gravity is expressed in generally-covariant form in order that spacetime diffeomorphisms be represented within its equal-time phase space. In accordance with the principle of general covariance, the time mapping {\T}: {\yman} \to {\rman} and the space mapping {\X}: {\yman} \to {\xman} that define the Dirac-ADM foliation are incorporated into the framework of the Hilbert variational principle. The resulting canonical action encompasses all individual Dirac-ADM actions, corresponding to different choices of foliating vacuum spacetimes by spacelike hypersurfaces. In this framework, spacetime observables, namely, dynamical variables that are invariant under spacetime diffeomorphisms, are not necessarily invariant under the deformations of the mappings \T and \X, nor are they constants of the motion. Dirac observables form only a subset of spacetime observables that are invariant under the transformations of \T and \X and do not evolve in time. The conventional interpretation of the canonical theory, due to Bergmann and Dirac, can be recovered only by postulating that the transformations of the reference system ({\T},{\X}) have no measurable consequences. If this postulate is not deemed necessary, covariant canonical gravity admits no classical problem of time.Comment: 41 pages, no figure

Peculiarities of massive vectormesons and their zero mass limits

Massive QED, in contrast with its massless counterpart, possesses two conserved charges; one is a screened (vanishing) Maxwell charge which is directly associated with the massive vector mesons through the identically conserved Maxwell current. A somewhat peculiar situation arises for couplings of Hermitian matter fields to massive vector potentials; in that case the only current is the screened Maxwell current and the coupling disappears in the massless limit. In case of selfinteracting massive vector mesons the situation becomes even more peculiar in that the usually renormalizability guaranteeing validity of the first order power-counting criterion breaks down in second order and requires the compensatory presence of additional Hermitian H-fields. Some aspect of these observation have already been noticed in the BRST gauge theoretic formulation, but here we use a new setting based on string-local vector mesons which is required by Hilbert space positivity. The coupling to H-fields induces Mexican hat like selfinteractions; they are not imposed and bear no relation with spontaneous symmetry breaking; they are rather consequences of the foundational causal localization properties realized in a Hilbert space setting. In case of selfinteracting massive vectormesons their presence is required in order to maintain the first order power-counting restriction of renormalizability also in second order. The presentation of the new Hilbert space setting for vector mesons which replaces gauge theory and extends on-shell unitarity to its off-shell counterpart is the main motivation for this work. The new Hilbert space setting also shows that the second order Lie-algebra structure of selfinteracting vector mesons is a consequence of the principles of QFT and promises a deeper understanding of the origin of confinement.Comment: 34 pages Latex, several additional remarks and citations, improved formulations, same as published versio

The Structure of GUT Breaking by Orbifolding

Recently, an attractive model of GUT breaking has been proposed in which a 5 dimensional supersymmetric SU(5) gauge theory on an S^1/(Z_2\times Z_2') orbifold is broken down to the 4d MSSM by SU(5)-violating boundary conditions. Motivated by this construction and several related realistic models, we investigate the general structure of orbifolds in the effective field theory context, and of this orbifold symmetry breaking mechanism in particular. An analysis of the group theoretic structure of orbifold breaking is performed. This depends upon the existence of appropriate inner and outer automorphisms of the Lie algebra, and we show that a reduction of the rank of the GUT group is possible. Some aspects of larger GUT theories based on SO(10) and E_6 are discussed. We explore the possibilities of defining the theory directly on a space with boundaries and breaking the gauge symmetry by more general consistently chosen boundary conditions for the fields. Furthermore, we derive the relation of orbifold breaking with the familiar mechanism of Wilson line breaking, finding a one-to-one correspondence, both conceptually and technically. Finally, we analyse the consistency of orbifold models in the effective field theory context, emphasizing the necessity for self-adjoint extensions of the Hamiltonian and other conserved operators, and especially the highly restrictive anomaly cancellation conditions that apply if the bulk theory lives in more than 5 dimensions.Comment: Latex, 23 pages, version to be published in Nucl.Phys.