A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems

Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM) (developed in paper I) with two extra ingredients : positive observable valued measures (PObVMs) [which serve to connect state-induced expectation values and classical probabilities] and the `CC condition' [which stipulates that the sets of observables and pure states be mutually separating] is proposed as a universal mechanics potentially covering all physical phenomena. It facilitates development of an autonomous formalism for quantum mechanics. Quantum systems, defined algebraically as supmech Hamiltonian systems with non-supercommutative system algebras, are shown to inevitably have Hilbert space based realizations (so as to accommodate rigged Hilbert space based Dirac bra-ket formalism), generally admitting commutative superselection rules. Traditional features of quantum mechanics of finite particle systems appear naturally. A treatment of localizability much simpler and more general than the traditional one is given. Treating massive particles as localizable elementary quantum systems, the Schr$\ddot{o}$dinger wave functions with traditional Born interpretation appear as natural objects for the description of their pure states and the Schr$\ddot{o}$dinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. A provisional set of axioms for the supmech program is given.Comment: 55 pages; some modifications in text; improved treatment of topological aspects and of Noether invariants; results unchange

Quasi-local Update Algorithms for Numerical Simulations of d=3 SU(2) LGT in the Dual Formulation

In the dual formulation of d=3 SU(2) LGT, the link variables are group representations and valid configurations are those satisfying a number of triangle inequalities. In \cite{lat99} algorithms for local updates that automatically respect these constraints were described. It was also pointed out there that these local updates were not ergodic. In this presentation, we describe two different quasi-local updating algorithms which, in conjunction with the local updates, appear to be ergodic.Comment: LATTICE2000(Topics in Gauge Theories),5 pages,5 figures,LaTe

Analytic Variational Investigation of Euclidean SU(3) Gauge Theory

Analytic variational techniques for lattice gauge theories based on the Rayleigh-Ritz(RR) method were previously developed for euclidean SU(2) gauge theories in 3 and 4 dimensions. Their extensions to SU(3) gauge theory including applications to correlation functions and mass gaps are presented here.Comment: 4-pages in 2-column format, typsetted for LaTeX with document style espcrc2.sty (available on request), IMSc/93-2

Current Status of the Numerical Simulations of d=3 SU(2) Lattice Gauge Theory in the Dual Formulation

We have continued our systematic investigations of the numerical simulations of lattice gauge theories in the dual formulation. These include: i) a more practical implementation of the quasi-local updating technique, ii) a thorough investigation of the sign problem, iii) issues related to the ergodicity of the various update algorithms, iv) a novel way of measuring conventional observables like plaquette in the dual formalism and v) investigations of thermalisation. While the dual formulation holds out a lot of promises in principle, there are still some ways to go before it can be made into an attractive alternative lattice formulation.Comment: LATTICE2000(Topics in Gauge Theories),5 pages,3 figures,LaTex(espcrc2.sty