The Hilbert Space Model for Quantum Mechanics and the EPR Dilemma

1989 ◽  
pp. 85-104
Author(s):  
David W. Cohen
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Space Model

Nonlinear quantum mechanics in a q-deformed Hilbert space

2019 ◽  
Vol 383 (23) ◽  
pp. 2729-2738 ◽  
Author(s):  
Bruno G. da Costa ◽  
Ernesto P. Borges
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

THE UNREASONABLE EFFECTIVENESS OF EXPONENTIALLY SUPPRESSED CORRECTIONS IN PRESERVING INFORMATION

2013 ◽  
Vol 22 (12) ◽  
pp. 1342030 ◽  
Author(s):  
KYRIAKOS PAPADODIMAS ◽  
SUVRAT RAJU
Keyword(s):  
Hilbert Space ◽  
Black Hole ◽  
Quantum Mechanics ◽  
Nonperturbative Effects ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Information Theoretic ◽  
Black Hole Evaporation ◽  
Strong Subadditivity ◽  
Unreasonable Effectiveness

We point out that nonperturbative effects in quantum gravity are sufficient to reconcile the process of black hole evaporation with quantum mechanics. In ordinary processes, these corrections are unimportant because they are suppressed by e-S. However, they gain relevance in information-theoretic considerations because their small size is offset by the corresponding largeness of the Hilbert space. In particular, we show how such corrections can cause the von Neumann entropy of the emitted Hawking quanta to decrease after the Page time, without modifying the thermal nature of each emitted quantum. Second, we show that exponentially suppressed commutators between operators inside and outside the black hole are sufficient to resolve paradoxes associated with the strong subadditivity of entropy without any dramatic modifications of the geometry near the horizon.

scholarly journals Ontology in Quantum Mechanics

2021 ◽  
Author(s):  
Gerard ’t Hooft
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Black Hole ◽  
Quantum Mechanics ◽  
Evolution Equations ◽  
Black Hole Physics ◽  
Special Kind ◽  
Quantum Evolution ◽  
The Standard Model ◽  
Low Energies

It is suspected that the quantum evolution equations describing the micro-world as we know it are of a special kind that allows transformations to a special set of basis states in Hilbert space, such that, in this basis, the evolution is given by elements of the permutation group. This would restore an ontological interpretation. It is shown how, at low energies per particle degree of freedom, almost any quantum system allows for such a transformation. This contradicts Bell’s theorem, and we emphasise why some of the assumptions made by Bell to prove his theorem cannot hold for the models studied here. We speculate how an approach of this kind may become helpful in isolating the most likely version of the Standard Model, combined with General Relativity. A link is suggested with black hole physics.

scholarly journals Nonlinear Generalisation of Quantum Mechanics

2021 ◽  
Author(s):  
Alireza Jamali
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Lorentz Invariance ◽  
Current Theory ◽  
Quantum Mechanical ◽  
Dynamical Condition ◽  
Current Definition ◽  
Klein Gordon ◽  
Definition Of ◽  
Mechanical Momentum

It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.

scholarly journals Necessity of Complex Hilbert Space for Quantum Mechanics

1973 ◽  
Vol 49 (3) ◽  
pp. 707-713 ◽  
Author(s):  
Toshiyuki Toyoda
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Complex Hilbert Space

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

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