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- Larsson, Jan-Åke, 1969-
(author)
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Quantum paradoxes, probability theory, and change of ensemble
- 2000
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Doctoral thesis (other academic/artistic)abstract
- In this thesis, the question "What kind of models can be used to describe microcosmos?" will be discussed. Being difficult and very large in scope, the question has here been restricted to whether or not Local Realistic models can be used to describe Quantum-Mechanical processes, one of a collection of questions often referred to as Quantum Paradoxes. Two such paradoxes will be investigated using techniques from probability theory: the Bell inequality and the Greenberger-Horne-Zeilinger (GHZ) paradox.A problem with the two mentioned paradoxes is that they are only valid when the detectors are 100% efficient, whereas present experimental efficiency is much lower than that. Here, an approach is presented which enables a generalization of both the Bell inequality and the GHZ paradox to the inefficient case. This is done by introducing the concept of change of ensemble, which provides both qualitative and quantitative information on the nature of the "loophole" in the 100% efficiency prerequisite, and is more fundamental in this regard than the efficiency concept. Efficiency estimates are presented which are easy to obtain from experimental coincidence data, and a connection is established between these estimates and the concept of change of ensemble.The concept is also studied in the context of Franson interferometry, where the Bell inequality cannot immediately be used. Unexpected subtleties occur when trying to establish whether or not a Local Realistic model of the data is possible even in the ideal case. A Local Realistic model of the experiment is presented, but nevertheless, by introducing an additional requirement on the experimental setup it is possible to refute the mentioned model and show that no other Local Realistic model exists.
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| 4. |
- Persson, Tomas, et al.
(author)
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Commuting elements in non-commutative algebras associated to dynamical systems
- 2003
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In: Series: Mathematical Modelling in Physics, Engineering and Cognitive Science. - 9176363864 ; 6, s. 145-172
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Conference paper (peer-reviewed)abstract
- In this article the problem of explicit description of commuting functions of noncommuting elements satisfying commutation relation of the form AB = BF(A) is considered and connection to periodic points of corresponding dynamical system is established.
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| 5. |
- Persson, Tomas, et al.
(author)
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From dynamical systems to commutativity in non-commutative operator algebras
- 2003
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In: Series: Mathematical Modelling in Physics, Engineering and Cognitive Science.. - 1651-0267. - 9176363864 ; 6, s. 109-143
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Conference paper (peer-reviewed)abstract
- This article is devoted to investigation of connection of operator representations of commutation relations XX*=F(X*X) and AB = BF(A) to periodic points and orbits of the dynamical system generated by the function F. Conditions on the general function F for two monomials in operators A and B to commute are derived. These conditions are further studied for dynamical systems generated by affine and q-deformed power functions, and for the beta-shift dynamical system.
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| 6. |
- Petersson, Henrik, 1973-
(author)
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Infinite dimensional holomorphy in the ring of formal power series : partial differential operators
- 2001
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Doctoral thesis (other academic/artistic)abstract
- We study holomorphy in the ring of formal power series in an infinite number of variables. Thus we restrict our study to (infinite dimensional) holomorphy on sequence spaces and we show that we obtain a rich theory without requiring any topological structure on the domain space. We make a comprehensive PDO-study for the spaces under consideration.As a basis we establish the Martineau duality, described by the Fourier-Borel transform, between spaces of entire and of exponential type functions in an infinite number of variables. In order to study PDO:s and PDE:s in spaces in the ring of formal power series, such an established duality is a useful tool for the transpose of a differential operator become the operator of multiplication by the corresponding symbol. The second part of the thesis is devoted to applications of the Martineau duality for various PDO-related problems. The following topics are considered: Existence theorems, Approximation theorems, Fischer decompositions, Cauchy problems, PDE-preserving projectors (Kergin projector) and Pseudo-differential operators.Some of the main results are the following. We prove Malgrange type existence theorems for infinite dimensional differential operators on A, Exp and F and we show that homogenous solutions can be approximated by such solutions consisting of exponential (finitely supported) polynomials. Here A and Exp are the spaces of entire respectively exponential type functions and F is the Fischer-Fock Hilbert space (in an infinite number of variables). We extend the notions of Fischer decomposition and Fischer pair, studied by H. Shapiro, J. Aniansson etc. A Fischer pair for a space is a pair of maps (here differential operators) whose kernel respective transpose image decompose the space into a direct sum. We establish some necessary and sufficient conditions for that a given pair of maps will make up a Fischer pair. By generalizing a known result for finite dimensional domain spaces, we show the existence of non-trivial Fischer pairs for A and Exp.Moreover, we prove some infinite dimensional generalizations of results obtained by H. Shapiro. In particular we show that densely defined differential operators together with their adjoints constitute Fischer pairs for F. We show that Cauchy-and dual Cauchy problems, w.r.t differential operators, in Exp respective in Exp'=A are well-posed. We prove that the infinite dimensional Kergin operator has interpolating and PDE-preserving properties and that it is uniquely determined by these properties. A PDE-preserving projector is a projector that preserves homogeneous solutions to differential equations.
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