Judith R. Goodstein: Einstein’s Italian Mathematicians – Ricci,Levi-Civita,and the Birth of General Relativity

Mathematische Semesterberichte -     

Titu Andreescu und Branislav Kisačanin: Math Leads for Mathletes 1 & 2 – A rich resource for young math enthusiasts,parents, teachers,and mentors

Mathematische Semesterberichte -     

Regular connections among generalized connections

The properties of the space of regular connections as a subset of the space of generalized connections in the Ashtekar framework are studied. For every choice of compact structure group and smoothness category for the paths, it is determined whether is dense in or not. Moreover, it is proven that has Ashtekar–Lewandowski measure zero for every non-trivial structure group and every smoothness category. The analogous results hold for gauge orbits instead of connections.     

On the Gribov Problem for Generalized Connections

 The bundle structure of the space of Ashtekar's generalized connections is investigated in the compact case. It is proven that every stratum is a locally trivial fibre bundle. The only stratum being a principal fibre bundle is the generic stratum. Its structure group equals the space of all generalized gauge transforms modulo the constant center-valued gauge transforms. For abelian gauge theories the generic stratum is globally trivial and equals the total space . However, for a certain class of non-abelian gauge theories – e.g., all SU(N) theories – the generic stratum is nontrivial. This means, there are no global gauge fixings – the so-called Gribov problem. Nevertheless, for many physical measures there is a covering of the generic stratum by trivializations each having total measure 1. Finally, possible physical consequences and the relation between fundamental modular domains and Gribov horizons are discussed. Received: 4 March 2002 / Accepted: 20 August 2002 Published online: 30 January 2003 Communicated by H. Nicolai     

Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

Let ${\cal A}$ be the affine space of all connections in an SU(2) principal fibre bundle over ?³. The set of homogeneous isotropic connections forms a line l in ${\cal A}$ . We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding $l \hookrightarrow {\cal A}$ does not continuously extend to an embedding $\overline{l} \hookrightarrow \overline{\cal A}$ of the respective compactifications. Here, the Bohr compactification $\overline{l}$ corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and $\overline{\cal A}$ to that of loop quantum gravity. Analogous results are given for the anisotropic case.     

Representations of the Weyl Algebra in Quantum Geometry

The Weyl algebra of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied. It is shown that, in the piecewise analytic category, every regular representation of having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional assumptions concern the dimension of the underlying analytic manifold (at least three), the finite wide triangulizability of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms – but neither any domain properties of the represented Weyl operators nor the requirement that the diffeomorphisms act by pull-backs. For this, the general behaviour of C*-algebras generated by continuous functions and pull-backs of homeomorphisms, as well as the properties of stratified analytic diffeomorphisms are studied. Additionally, the paper includes also a short and direct proof of the irreducibility of .     

Irreducibility of the Weyl algebra in loop quantum gravity

The irreducibility of the standard Weyl algebra representation in loop quantum gravity is proven using a very short and direct argument.     

Asymptotic positivity of Hurwitz product traces: Two proofs

Consider the polynomial tr(A+tB)^m in t for positive hermitian matrices A and B with m∈N. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of AB≠0. More precisely, we show—once complex-analytically, once combinatorially—that the k-th coefficient is positive for all integer m?m₀, where m₀ depends on A, B and k.     

Stratification of the Generalized Gauge Orbit Space

Different versions for defining Ashtekar's generalized connections are investigated depending on the chosen smoothness category for the paths and graphs – the label set for the projective limit. Our definition covers the analytic case as well as the case of webs. Then the action of Ashtekar's generalized gauge group on the space of generalized connections is investigated for compact structure groups G. Here, first, the orbit types of the generalized connections are determined. The stabilizer of a connection is homeomorphic to the holonomy centralizer, i.e. the centralizer of its holonomy group. It is proven that the gauge orbit type of a connection can be defined by the G-conjugacy class of its holonomy centralizer equivalently to the standard definition via -stabilizers. The connections of one and the same gauge orbit type form a so-called stratum. As the main result of this article a slice theorem is proven on . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, is topologically regularly stratified by . These results coincide with those of Kondracki and Rogulski for Sobolev connections. Furthermore, the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of G. Finally, it is shown that the set of all gauge orbits with maximal type has the full induced Haar measure 1. Received: 12 January 2000 / Accepted: 8 May 2000