Harmonic Analysis as the Exploitation of Symmetry- A Historical Survey

Paper by George W. Macke

The Topos-theoretical Approach to Quantum Physics

I oppgaven tas det sikte på å anvende begreper og metoder fra kategoriteori og især toposteori innen kvantefysikken. I den resulterende teorien, "toposfysikk", brukes toposteori (teorien om generaliserte mengdeuniverser og generaliserte rom) som et verktøy for å konstruere kvantefysikken ved å "lime sammen" klassiske perspektiver eller "snapshots". Første kapitel gir den nødvendige bakgrunn for å forstå den fysiske motivasjonen bak konstruksjonene i de påfølgende kapitler, med særlig oppmerksomhet viet emnene logikk, kvantisering og rom. I kapitel 2 presenteres først elementær teori om kategorier og topoi. Det gis deretter en gjennomgang av toposfysikkens sentrale trekk: konstruksjonen av et tilstandsrom for kvantemekanikken ved hjelp av (kovariante eller kontravariante) funktorer over en kategori av kommutative operatoralgebraer. To ulike tilnærmingsmåter, Andreas Döring og Chris Ishams "neorealisme" og Chris Heunen, Nicolas P. Landsmaan og Bas Spitters "Bohrifikasjon" presenteres i detalj. I kapitel 3 anvendes den sistnevnte tilnærmingen på teorien om "loop quantum gravity" (LQG). Kapitelet har derfor en kort oppsummering av hovedresultatene innen LQG. Det undersøkes hvordan LQG kan interpreteres innen toposfysikk ved å ta i bruk Christian Fleischhacks formulering av LQG som en Weylalgebra. De topologiske egenskapene til tilstandsrommet i LQG innen toposmodellen undersøkes, og det vises hvordan kravene til gauge- og diffeomorfiinvarians kan interpreteres i teorien. Endelig, ved hjelp av Ishams teknikk for å kvantisere generelle strukturer, utvides den toposfysiske modellen til å inkludere et bredere kategoriteoretisk rammeverk. Vi definerer en målteori for kategorier og undersøker teoriens byggesteiner, pilfeltene over en kategori, i kategorien av deres representasjoner. Vi antyder hvordan modellen kan anvendes innen Sorkins teori om kausale mengder, og som basis for en teori om kvantisert logikk

New Foundation in the Sciences: Physics without sweeping infinities under the rug

It is widely known among the Frontiers of physics, that “sweeping under the rug” practice has been quite the norm rather than exception. In other words, the leading paradigms have strong tendency to be hailed as the only game in town. For example, renormalization group theory was hailed as cure in order to solve infinity problem in QED theory. For instance, a quote from Richard Feynman goes as follows: “What the three Nobel Prize winners did, in the words of Feynman, was to get rid of the infinities in the calculations. The infinities are still there, but now they can be skirted around . . . We have designed a method for sweeping them under the rug. [1] And Paul Dirac himself also wrote with similar tune: “Hence most physicists are very satisfied with the situation. They say: Quantum electrodynamics is a good theory, and we do not have to worry about it any more. I must say that I am very dissatisfied with the situation, because this so-called good theory does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small—not neglecting it just because it is infinitely great and you do not want it!”[2] Similarly, dark matter and dark energy were elevated as plausible way to solve the crisis in prevalent Big Bang cosmology. That is why we choose a theme here: New Foundations in the Sciences, in order to emphasize the necessity to introduce a new set of approaches in the Sciences, be it Physics, Cosmology, Consciousness etc

The pursuit of locality in quantum mechanics

The rampant success of quantum theory is the result of applications of the 'new' quantum mechanics of Schrödinger and Heisenberg (1926-7), the Feynman-Schwinger-Tomonaga Quantum Electrodynamics (1946-51), the electro-weak theory of Salaam, Weinberg, and Glashow (1967-9), and Quantum Chromodynamics (1973-); in fact, this success of `the' quantum theory has depended on a continuous stream of brilliant and quite disparate mathematical formulations. In this carefully concealed ferment there lie plenty of unresolved difficulties, simply because in churning out fabulously accurate calculational tools there has been no sensible explanation of all that is going on. It is even argued that such an understanding is nothing to do with physics. A long-standing and famous illustration of this is the paradoxical thought-experiment of Einstein, Podolsky and Rosen (1935). Fundamental to all quantum theories, and also their paradoxes, is the location of sub-microscopic objects; or, rather, that the specification of such a location is fraught with mathematical inconsistency. This project encompasses a detailed, critical survey of the tangled history of Position within quantum theories. The first step is to show that, contrary to appearances, canonical quantum mechanics has only a vague notion of locality. After analysing a number of previous attempts at a `relativistic quantum mechanics', two lines of thought are considered in detail. The first is the work of Wan and students, which is shown to be no real improvement on the usual `nonrelativistic' theory. The second is based on an idea of Dirac's - using backwards-in-time light-cones as the hypersurface in space-time. There remain considerable difficulties in the way of producing a consistent scheme here. To keep things nicely stirred up, the author then proposes his own approach - an adaptation of Feynman's QED propagators. This new approach is distinguished from Feynman's since the propagator or Green's function is not obtained by Feynman's rule. The type of equation solved is also different: instead of an initial-value problem, a solution that obeys a time-symmetric causality criterion is found for an inhomogeneous partial differential equation with homogeneous boundary conditions. To make the consideration of locality more precise, some results of Fourier transform theory are presented in a form that is directly applicable. Somewhat away from the main thrust of the thesis, there is also an attempt to explain the manner in which quantum effects disappear as the number of particles increases in such things as experimental realisations of the EPR and de Broglie thought experiments