Atiyah-Singer Index Theorem in an SO(3) Yang-Mills-Higgs system and derivation of a charge quantization condition

The Atiyah-Singer index theorem is generalized to a two-dimensional SO(3) Yang-Mills-Higgs (YMH) system. The generalized theorem is proven by using the heat kernel method and a nonlinear realization of SU(2) gauge symmetry. This theorem is applied to the problem of deriving a charge quantization condition in the four-dimensional SO(3) YMH system with non-Abelian monopoles. The resulting quantization condition, eg=n (n: integer), for an electric charge e and a magnetic charge g is consistent with that found by Arafune, Freund and Goebel. It is shown that the integer n is half of the index of a Dirac operator.Comment: 18pages, no figures, minor corrections, published versio

Inequivalence of the Massive Vector Meson and Higgs Models on a Manifold with Boundary

The exact quantization of two models, the massive vector meson model and the Higgs model in the London limit, both describing massive photons, is presented. Even though naive arguments (based on gauge-fixing) may indicate the equivalence of these models, it is shown here that this is not true in general when we consider these theories on manifolds with boundaries. We show, in particular, that they are equivalent only for a special choice of the boundary conditions that we are allowed to impose on the fields.Comment: 14 pages, LATEX File (revised with minor corrections

An unknown story: Majorana and the Pauli-Weisskopf scalar electrodynamics

An account is given of an interesting but unknown theory by Majorana regarding scalar quantum electrodynamics, elaborated several years before the known Pauli-Weisskopf theory. Theoretical calculations and their interpretation are given in detail, together with a general historical discussion of the main steps towards the building of a quantum field theory for electrodynamics. A possible peculiar application to nuclear constitution, as conceived around 1930, considered by Majorana is as well discussed.Comment: Latex, amsart, 20 pages, 2 figures; to be published in Annalen der Physi

Experimental status of pionium at CERN

The DIRAC Collaboration presents a first search for "atomic pi(+) pi(-) pairs" from ionization of pionium.Comment: 2 pages, LaTex, 2 figures, talk at Chiral Dynamics 2000, Newport News (USA), July 17-20, 200

Dirac's Constrained Hamiltonian Dynamics from an Unconstrained Dynamics

We derive the Hamilton equations of motion for a constrained system in the form given by Dirac, by a limiting procedure, starting from the Lagrangean for an unconstrained system. We thereby ellucidate the role played by the primary constraints and their persistance in time.Comment: 10 page

Run-away solutions in relativistic spin 1/2 quantum electrodynamics

The existence of run-away solutions in classical and non-relativistic quantum electrodynamics is reviewed. It is shown that the less singular high energy behavior of relativistic spin 1/2 quantum electrodynamics precludes an analogous behavior in that theory. However, a Landau-like anomalous pole in the photon propagation function or in the electron-massive photon foward scattering amplitude would generate a new run-away, characterized by an energy scale omega ~ m_e exp (1/alpha). This contrasts with the energy scale omega ~ (m_e/alpha) associated with the classical and non-relativistic quantum run-aways.Comment: 3 minor changes; 17 pgs, epsf & aps styles,1 eps & 2 embedded ps fig

The 'Square Root' of the Interacting Dirac Equation

The 'square root' of the interacting Dirac equation is constructed. The obtained equations lead to the Yang-Mills superfield with the appropriate equations of motion for the component fields.Comment: 6 page

Coloring Graphs with Forbidden Minors

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gon\c{c}alves and generalize it to the next step by showing that every graph with no $K_t$ minor is $(2t-6)$-colorable, where $t\in\{7,8,9\}$. We then prove that graphs with no $K_8^-$ minor are $9$-colorable and graphs with no $K_8^=$ minor are $8$-colorable. Finally we prove that if Mader's bound for the extremal function for $K_p$ minors is true, then every graph with no $K_p$ minor is $(2t-6)$-colorable for all $p\ge5$. This implies our first result. We believe that the Kempe-chain method we have developed in this paper is of independent interest