The existence of a topological double-covering for the GL(n, R) and diffeomorphism groups is reviewed. These groups do not have finite-dimensional faithful representations. An explicit construction and the classification of all\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {SL} $$ \end{document}(n, R), n=3,4 unitary irreducible representations is presented. Infinite-component spinorial and tensorial\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {SL} $$ \end{document} fields, “manifields”, are introduced. Particle content of the ladder manifields, as given by (...) the\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {SL} $$ \end{document}(3, R) “little” group, is determined. The manifields are lifted to the corresponding world spinorial and tensorial manifields by making use of generalized infinite-component frame fields. World manifields transform w.r.t. corresponding\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {Diff} $$ \end{document} representations, which are constructed explicitly. (shrink)

Einstein's general relativity theory describes very well the gravitational phenomena in themacroscopic world. In themicroscopic domain of elementary particles, however, it does not exhibit gauge invariance or approximate Bjorken type scaling, properties which are believed to be indispensible for arenormalizable field theory. We argue that thelocal extension of space-time symmetries, such as of Lorentz and scale invariance, provides the clue for improvement. Eventually, this leads to aGL(4, R)-gauge approach to gravity in which the metric and the affine connection acquire the (...) status ofindependent fields. The Yang-Mills type field equations, the Noether identities, and conformal models of gravity are discussed within this framework. After symmetry breaking, Einstein's GR surfaces as an effective “low-energy” theory. (shrink)

“CHN≓ (1966)was an algebraic algorithm which reproduced and extended the predictions of the “non-interacting≓ quark model in the asymptotic high-energy region. It wus formulated within the conceptual framework of on- mass- shell physics and of the complex angular-momentum plane. Prior to the advent of the standard model, it was reinterpreted in terms of the Melosh transformation relating “current≓ to “constituent≓ quarks. It is now lied up to the QCD paradigm.

We relate personal encounters of three kinds with geometrical approaches in the development of a relativistic quantum field theory of the fundamental interactions—including interactions with Nathan Rosen. We characterize the geometrical structures involved and discuss the more recent attempts to develop a unified theory based on a Klein-Kaluza contraction of the eightfold extended supergravity.

Since 1961, the experimental exploration at the fundamental level of physical reality has surprised physists by revealing to them a highly geometric scenery. Like Einstein's (classical) theory of gravity, the “standard model,” describing the strong, weak, and electromagnetic interaction, testifies in favor of Plato's reported allegation.

We review the mathematical theory ofSL(n, R) and its double-covering group $\overline {SL} (n,R)$ , especially forn = 2, 3, 4. After discussing a variety of physical applications, we show that $\overline {SL} (3,R)$ provides holonomic curved space (“world”) spinors with an infinite number of components. We construct the relevant holonomic “manifield” and discuss the gravitational interaction of a proton as an example.

We review the issues of nonseparability and seemingly acausal propagation of information in EPR, as displayed by experiments and the failure of Bell's inequalities. We show that global effects are in the very nature of the geometric structure of modern physical theories, occurring even at the classical level. The Aharonov-Bohm effect, magnetic monopoles, instantons, etc. result from the topology and homotopy features of the fiber bundle manifolds of gauge theories. The conservation of probabilities, a supposedly highly quantum effect, is also (...) achieved through global geometry equations. The EPR observables all fit in such geometries, and space-time is a truncated representation and is not the correct arena for their understanding. Relativistic quantum field theory represents the global action of the measurement operators as the zero-momentum (and therefore spatially infinitely spread) limit of their wave functions (form factors). We also analyze the collapse of the state vector as a case of spontaneous symmetry breakdown in the apparatus-observed state interaction. (shrink)

Schrödinger's approach was analytical, but it is equivalent to an algebraic treatment. We review the evolution of group theory as a physical tool and its application to the Hilbert space of Schrödinger's eigenstates. Special emphasis is put on recent results relating to the relativistic quantized string.