Noncompact Gauge-Invariant Simulations of U(1), SU(2), and SU(3)

We have applied a new gauge-invariant, noncompact, Monte Carlo method to simulate the $U(1)$, $SU(2)$, and $SU(3)$ gauge theories on $8^4$ and $12^4$ lattices. The Creutz ratios of the Wilson loops agree with the exact results for $U(1)$ for $\beta \ge 0.5$ apart from a renormalization of the charge. The $SU(2)$ and $SU(3)$ Creutz ratios robustly display quark confinement at $\beta = 0.5$ and $\beta = 2$, respectively. At much weaker coupling, the $SU(2)$ and $SU(3)$ Creutz ratios agree with perturbation theory after a renormalization of the coupling constant. For $SU(3)$ the scaling window is near $\beta = 2$, and the relation between the string tension $\sigma$ and our lattice QCD parameter $\Lambda_L$ is $\sqrt{\sigma} \approx 5 \Lambda_L$.Comment: For U(1), we switched from beta = 2 / g^2 to beta = 1 / g^2; 3 pages; latex and espcrc2.sty; one figure generated by PiCTeX; our contribution to Lattice '9

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We explain why compact U(1) confines and how to fix it. We show that plaquettes of negative trace carry most of the confinement signal in compact SU(2). We show how to perform noncompact gauge-invariant simulations without auxiliary fields. We suggest a way to simulate fermions without doublers.Comment: Talk presented at LATTICE96(algorithms), 4 pages, Latex, espcrc2, picte

Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

In this paper, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in [Dykema and Strawn, 2003]. Our central technique involves continuous liftings of paths from the polytope of eigensteps to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense, and also that generic FUNTFs are full spark.Comment: 33 pages, 4 figure

Synthetic Quantum Systems

So far proposed quantum computers use fragile and environmentally sensitive natural quantum systems. Here we explore the new notion that synthetic quantum systems suitable for quantum computation may be fabricated from smart nanostructures using topological excitations of a stochastic neural-type network that can mimic natural quantum systems. These developments are a technological application of process physics which is an information theory of reality in which space and quantum phenomena are emergent, and so indicates the deep origins of quantum phenomena. Analogous complex stochastic dynamical systems have recently been proposed within neurobiology to deal with the emergent complexity of biosystems, particularly the biodynamics of higher brain function. The reasons for analogous discoveries in fundamental physics and neurobiology are discussed.Comment: 16 pages, Latex, 1 eps figure fil