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Thesis (Ph. D.)--University of Wisconsin--Madison, 1941. / Typescript. Includes abstract and vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 44).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1981. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 139-148).

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2002-02Bitstream added on 2014-06-13T18:48:13Z : No. of bitstreams: 1 melo_cam_me_ift.pdf: 748344 bytes, checksum: 4ea018751b23b3395fccb8e9c49dc573 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho apresentamos um estudo detalhado da abordagem de Schwinger para a Mecânica Quântica, fazendo sua generalização para sistemas com escalares pertencentes ao conjunto dos quatérnions. Analisamos, em especial, a estrutura da Álgebra de Medida e sua relação com as propriedades físicas observáveis. Estudamos ainda o problema da liberdade de gauge relacionado à quantização do campo eletromagnético livre, e implementamos uma solução alternativa para este problema utilizando o Princípio Variacional de Schwinger, e o campo auxiliar B(x) introduzido por Nakanishi. / In this work we have presented a detailed study of the Schwinger s approach to the Quantum Mechanics, making its generalization for systems with scalars which belong to the quaternion set. In particular, we have analysed the structure of the Algebra of Measurement and its relation with the observable physical properties. We have also studied the problem of the gauge freedom related to the quantization of the free electromagnetic eld and implemented an alternative solution to this one employing the Schwinger s Variational Principle and the B- eld introduced by Nakanishi.

Bell's Theorem demonstrates that factorizable theories for the EPR experiment (EPR) cannot reproduce the quantum correlations. Factorizability is motivated by various locality conditions. So to understand the nature of nonlocality in EPR, we first need to understand the conceptual relations between factorizability, these various locality conditions and the nature of the quantum correlations. That is the main focus of my thesis. My main conclusion is that these conceptual relations are more subtle than the literature has usually suggested. Chapter 1 is a general introduction. In Chapter 2, I review the general framework of factorizable stochastic theories for EPR. I argue that factorizability can be motivated by various locality conditions, even in theories that admit time-dependent states and take the measurement interactions to be neither instantaneous, nor simultaneous. In Chapter 3, I focus on Cartwright' s (1989) and Humphreys' (1989) theories of probabilistic causation for singular events, which are based on modifications of traditional causal linear modelling. I argue (against Cartwright) that local models for EPR in the framework of these theories are committed to factorizability; and so they cannot reproduce the EPR correlations. In Chapter 4, I turn to Stochastic-Einstein Locality (SEL). Hellman (1982) proposed that SEL with some provisos characterizes the No-Superluminal-Action (NSA) of the Special Theory of Relativity (STR), and he argued that SEL is not violated in EPR. Butterfield (1994) proposed that SEL (without Hellman's provisos) characterizes the lack of superluminal Lewisian causation, and he argued that SEL is violated in EPR. I argue that SEL (without Hellman's provisos) is motivated by NSA and spatiotemporal separability. Thus, the violation of SEL might arise from the failure of spatiotemporal separability. And this failure is compatible with NSA and superluminal Lewisian causation. Accordingly, Hellman's and Butterfield's views need not be in tension. In Chapter 5, I focus on the implications for nonlocality of Jarrett's (1984) analysis of factorizability into "locality" and "completeness". I argue that although this analysis cannot distinguish between failures of factorizability which are compatible with STR and those which are not, it is significant for clarifying the implications of Bell's theorem for nonlocality. In Chapter 6, I qmsider three arguments that are intended to deny superluminal causal propagation in EPR, and a fourth argument that is intended to establish the opposite conclusion. I argue that in various ways these arguments have gone wrong. Three of these argument rely on implicit assumptions that have been overlooked. Accordingly, two of them reached a reasonable conclusion, i.e. the failure of separability, for the wrong reasons; whereas the third reached a wrong conclusion, i.e. that the failure of the contiguity of causal processes explains the failure of factorizability in EPR. The fourth relies on a wrong assumption, and thus it reaches a too strong conclusion, that the EPR correlations require superluminal action. My main conclusion is that the quantum correlations require nonseparability . In Chapter 7, I focus on decision theory in the context of EPR. In both EPR and the famous Newcomb's problem (NewProb), there are unscreenable-off correlations. I argue that NewProb can be related to EPR, in the sense that a NewProb can be realized by that experiment.

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The difficulties associated with the quantization of the gravitational field suggests a modification of space-time is needed. For example at suffici~ly small length scales the geometry of space-time might better discussed in terms of a noncommutative algebra. In this thesis we discuss a particular example of a noncommutative algebra, namely the symplectic Schonberg algebra, which we treat as a geometric algebra. Thus our investigation has some features in common with recent work that explores how geometry can be formulated in terms of noncommutative structures. The symplectic Schonberg algebra is a geometric algebra associated with the covariant and the contravariant vectors of a general affine space. The "embedding" of this space in a noncommutative algebra leads us to a structure which we regard as a noncommutative affine geometry. The theory in question takes us naturally to stochastic elements without the usual ad-hoc assumptions concerning measurements in physical ensembles that are made in the usual interpretation of quantum mechanics. The probabilistic nature of space is obtained purely from the structure of this algebra. As a consequence, geometric objects like points, lines and etc acquire a kind of fuzzy character. This allowed us to construct the space of physical states within the algebra in terms of its minimum left-ideals as was proposed by Hiley and Frescura [1J. The elements of these ideals replace the ordinary point in the Cartesian geometry. The study of the main inner-automorphisms of the algebra gives rise to the representation of the symplectic group of linear classical canonical transformations. We show that this group acts on the minimum left-ideal of the algebra and in this case manifests itself as the metaplectic group, i.e the double covering of the symplectic group. Thus we are lead to the theory of symplectic spinors as minimum left-ideals in exactly the same way as the orthogonal spinors can be formulated in terms of minimum left-ideals in the Clifford algebra .. The theory of the automorphisms of the symplectic Schonberg algebra allows us to give a geometrical meaning to integral transforms such as: the Fourier transform, the real and complex Gauss Weierstrass transform, the Bargmann (3) transform and the Bilateral Laplace transform. We construct a technique for obtaining a realization of these algebraic transformations in terms of integral kernels. This gives immediately the Feynmann propagators of conventional non-relativistic quantum mechanics for Hamiltonians quadratic in momentum and position. This then links our approach to those used in quantum mechanics and optics. The link between the theory of this noncommutative geometric algebra and the theory of vector bundles is also discussed.

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