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Mathematical Foundations of Quantum Mechanics
John von Neumann
- 01 Jan 1996
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TL;DR: The Mathematical Foundations of Quantum Mechanics as discussed by the authors is a seminal work in theoretical physics that introduced the theory of Hermitean operators and Hilbert spaces and provided a mathematical framework for quantum mechanics.
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Abstract: Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth century, shows that great insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. He begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which von Neumann regards as the definitive form of quantum mechanics. Using this theory, he attacks with mathematical rigor some of the general problems of quantum theory, such as quantum statistical mechanics as well as measurement processes. Regarded as a tour de force at the time of publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.
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Citations
Krein-space formulation of $$\mathcal{P}\mathcal{T}$$ symmetry, $$\mathcal{C}\mathcal{P}\mathcal{T}$$ -inner products, and pseudo-Hermiticity
TL;DR: In this article, a Krein-space derivation of the inner product of the Schrodinger equation has been presented, which is linked with the pseudo-Hermitian formulation of the symmetric quantum mechanics.
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Free Will, Physics, Biology, and the Brain
Christof Koch
- 01 Jan 2009
TL;DR: The traditionally conceived question of free will is reviewed from the point of view of a physicist turned neurobiologist, and several cognitive neuroscience experiments suggesting that in many instances, their brain decides prior to their conscious mind, and that the authors are often ignorant of their brain’s decisions.
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Theory of “Weak Value" and Quantum Mechanical Measurements
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Unifying typical entanglement and coin tossing: on randomization in probabilistic theories
TL;DR: In this article, the expected purity of a subsystem in any probabilistic theory for which this quantity is well-defined is derived, and the proof of the formula generalizes several randomization notions to general probability theories, including coin tossing and randomization in post-quantum theories.
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The measurement problem in artificial world models.
TL;DR: The measurement problem also bears on the questions of the relation of computation to physics and to formal symbol systems and on what sense dissipationless computation is a useful concept.
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