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Group Theoretical Foundations of Quantum Mechanics

Ronald Mirman

Published Nov 1, 1995• 11 citations
OpenAlex

Abstract

Contents: Foundations Why geometry, so physics, require complex numbers Properties of state functions The foundations of coherent superposition geometry, transformations groups and observers The Poincare group and its implications The dimension of space Bosons, fermions, spinors and orthogonal groups The complete reasonableness of quantum mechanics Terminology and conventions The Einstein Podolsky Rosen paradox Experimental meaning of the concept of identical particles Nonexistence of superselection rules Complex groups, quantum mechanics and the dimension and reality of space The reality and dimension of space and the complexity of quantum mechanics.

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Semiotic Interpretation of Probability in Quantum Mechanics

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The presented dissertation deals with how to interpret the conception of reality within both the assumed metaphysical frameworks of the proponents of the Einstein-Podolsky-Rosen (EPR) paradox and the Copenhagen interpretation of quantum mechanics. The hypothesis of the dissertation is based on assumption that EPR operated with presuppositions stemming from classical-physical determinism (e.g., the concept of particle independence) that, at the time of their paradox's formulation, no longer corresponded to the contemporary level of knowledge. In contrast, the Copenhagen interpretation operates with a concept of reality that significantly differs from the metaphysics of classical physics. This dissertation identifies the EPR proponents' approach to reality as philosophical foundationalism and interprets it using the philosophy and semiotics of Charles Sanders Peirce as a dualism that does not respect the essential role of representation in development of scientific theories. The aim is to present a metaphysical and semiotic framework that would be complementary to the interpretation of quantum mechanics as an indeterministic theory which understands the representation of probability and uncertainty inherent in quantum mechanical descriptions as intrinsic parts of the conception of reality, rather...

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Steering paradox for Einstein–Podolsky–Rosen argument and its extended inequality

Tianfeng Feng, Changliang Ren, Qin Feng +4 more

The Einstein–Podolsky–Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of a local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of “quantum nonlocality,” which manifests itself in three types: quantum entanglement, quantum steering, and Bell’s nonlocality. Although Bell’s nonlocality is more often used to show “quantum nonlocality,” the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality, thus making it amenable to experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is friendlier for experimental implementation, i.e., one may measure the observables only in the x , y , or z axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.

Photonics Research•Mar 19, 2021•4 citations

A Steering Paradox for Einstein-Podolsky-Rosen Argument and its Extended Inequality

Tianfeng Feng, Changliang Ren, Qin Feng +4 more

The Einstein-Podolsky-Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of "quantum nonlocality", which manifests itself by three different types: quantum entanglement, quantum steering, and Bell nonlocality. Although Bell nonlocality is more often used to show the "quantum nonlocality", the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality,thus making it amenable to an experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is more friendly for experimental implementation, i.e., one may only measure the observables in $x$-, $y$-, or $z$-axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with the theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.

arXiv:2004.04352•Apr 9, 2020•0 citations
Contents: Foundations Why geometry, so physics, require complex numbers Properties of state functions The foundations of coherent superposition geometry, transformations groups and observers The Poincare group and its implications The dimension of space Bosons, fermions, spinors and orthogonal groups The complete reasonableness of quantum mechanics Terminology and conventions The Einstein Podolsky Rosen paradox Experimental meaning of the concept of identical particles Nonexistence of superselection rules Complex groups, quantum mechanics and the dimension and reality of space The reality and dimension of space and the complexity of quantum mechanics.
The presented dissertation deals with how to interpret the conception of reality within both the assumed metaphysical frameworks of the proponents of the Einstein-Podolsky-Rosen (EPR) paradox and the Copenhagen interpretation of quantum mechanics. The hypothesis of the dissertation is based on assumption that EPR operated with presuppositions stemming from classical-physical determinism (e.g., the concept of particle independence) that, at the time of their paradox's formulation, no longer corresponded to the contemporary level of knowledge. In contrast, the Copenhagen interpretation operates with a concept of reality that significantly differs from the metaphysics of classical physics. This dissertation identifies the EPR proponents' approach to reality as philosophical foundationalism and interprets it using the philosophy and semiotics of Charles Sanders Peirce as a dualism that does not respect the essential role of representation in development of scientific theories. The aim is to present a metaphysical and semiotic framework that would be complementary to the interpretation of quantum mechanics as an indeterministic theory which understands the representation of probability and uncertainty inherent in quantum mechanical descriptions as intrinsic parts of the conception of reality, rather...
This article surveys a selection of key conceptual and interpretational developments in quantum mechanics, tracing the theory from its foundational postulates to contemporary discussions of measurement, nonlocality, and the emergence of classicality. Beginning with the structure of Hilbert space and the postulates governing state evolution and measurement, the epistemic stance of the Copenhagen interpretation and its modern reformulations are examined. The Einstein-Podolsky-Rosen argument, Bell's theorem, and Hardy's paradox are then discussed as probes of locality and realism, alongside the deterministic but explicitly nonlocal de Broglie-Bohm theory. The measurement problem and the implications of contextuality are analyzed in relation to objective collapse models, which introduce new physical dynamics to account for definite outcomes. Finally, the role of decoherence in the suppression of interference and the emergence of classical behavior is explored, together with the interpretational frameworks of many-worlds and consistent histories. This material aims to provide a coherent introductory overview of how several of the most prominent interpretations address the central concern of what quantum mechanics tells us about the nature of physical reality.
The Postulates and Principles of Quantum Mechanics Commutability and Compatibility of Quantum Observables Non-Commuting Operators and Uncertainty Principle Quantum Dynamics and Schrodinger Equation Time Evolution Operator and Propagator Symmetry, Transformation and Continuous Groups Lie Groups and Lie Algebras O(3) Group, SU(2) Group and Angular Momentum Dynamical Symmetry and Hydrogen Atom Lorentz Group and Group Representation Chiral Transformation and Spinor Algebra Gamma Matrix Algebra and Discrete Symmetry in Dirac Spinor System.
The Einstein–Podolsky–Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of a local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of “quantum nonlocality,” which manifests itself in three types: quantum entanglement, quantum steering, and Bell’s nonlocality. Although Bell’s nonlocality is more often used to show “quantum nonlocality,” the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality, thus making it amenable to experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is friendlier for experimental implementation, i.e., one may measure the observables only in the x , y , or z axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.
The Einstein-Podolsky-Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of "quantum nonlocality", which manifests itself by three different types: quantum entanglement, quantum steering, and Bell nonlocality. Although Bell nonlocality is more often used to show the "quantum nonlocality", the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality,thus making it amenable to an experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is more friendly for experimental implementation, i.e., one may only measure the observables in xxx-, yyy-, or zzz-axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with the theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.