Dirac, Von Neumann, and the Derivation of the Quantum Formalism

2020 ◽  
pp. 203-218
Author(s):  
Jim Baggott
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Critical Stage ◽  
Von Neumann ◽  
Quantum Formalism ◽  
Conceptual Problems ◽  
Underlying Reality

The evolution of quantum mechanics through the 1920s was profoundly messy. Some physicists believed that it was necessary to throw out much of the conceptual baggage that early quantum mechanics tended to carry around with it and re-establish the theory on much firmer ground. It was at this critical stage that the search for deeper insights into the underlying reality was set aside in favour of mathematical expediency. All the conceptual problems appeared to be coming from the wavefunctions. But whatever was to replace them needed to retain all the properties and relationships that had so far been discovered. Dirac and von Neumann chose to derive a new quantum formalism by replacing the wavefunctions with state vectors operating in an abstract Hilbert space, and formally embedding all the most important definitions and relations within a system of axioms.

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 Measurement theory in classical mechanics

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
So Katagiri
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Measurement Theory ◽  
Classical Mechanics ◽  
The State ◽  
Measurement Device ◽  
Von Neumann ◽  
Relative Interpretation ◽  
Classical And Quantum Mechanics

Abstract We investigate measurement theory in classical mechanics in the formulation of classical mechanics by Koopman and von Neumann (KvN), which uses Hilbert space. We show a difference between classical and quantum mechanics in the “relative interpretation” of the state of the target of measurement and the state of the measurement device. We also derive the uncertainty relation in classical mechanics.

scholarly journals ON KOOPMAN–VON NEUMANN WAVES

2002 ◽  
Vol 17 (09) ◽  
pp. 1301-1325 ◽  
Author(s):  
D. MAURO
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Wave Functions ◽  
Quantum Case ◽  
Von Neumann ◽  
Interference Experiment ◽  
Classical Wave

In this paper we study the classical Hilbert space introduced by Koopman and von Neumann in their operatorial formulation of classical mechanics. In particular we show that the states of this Hilbert space do not spread, differently from what happens in quantum mechanics. The role of the phases associated to these classical "wave functions" is analyzed in detail. In this framework we also perform the analog of the two-slit interference experiment and compare it with the quantum case.

scholarly journals Quantum Mechanics and Its Evolving Formulations

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 124
Author(s):  
Jean-Pierre Antoine
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Time Evolution ◽  
Von Neumann ◽  
Rigged Hilbert Space ◽  
Space Formulation ◽  
Space Approach

In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.

scholarly journals ON KOOPMAN–VON NEUMANN WAVES II

2004 ◽  
Vol 19 (09) ◽  
pp. 1475-1493 ◽  
Author(s):  
E. GOZZI ◽  
D. MAURO
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Wave Functions ◽  
Von Neumann ◽  
Similarities And Differences ◽  
Classical And Quantum Mechanics

In this paper, we continue the study started in Ref. 1, of the operatorial formulation of classical mechanics given by Koopman and von Neumann (KvN) in the 1930s. In particular, we show that the introduction of the KvN Hilbert space of complex and square integrable "wave functions" requires an enlargement of the set of the observables of ordinary classical mechanics. The possible role and the meaning of these extra observables is briefly indicated in this work. We also analyze the similarities and differences between non-selective measurements and two-slit experiments in classical and quantum mechanics.

scholarly journals Quantum logic as motivated by quantum computing

2005 ◽  
Vol 70 (2) ◽  
pp. 353-359 ◽  
Author(s):  
J. Michael Dunn ◽  
Tobias J. Hagge ◽  
Lawrence S. Moss ◽  
Zhenghan Wang
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Computer Science ◽  
Quantum Logic ◽  
Theoretical Computer Science ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Definition Of

§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection.The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.

Conceptual problems in quantum mechanics

1992 ◽  
Vol 162 (10) ◽  
pp. 93
Author(s):  
V.P. Demutskii ◽  
R.V. Polovin
Keyword(s):  
Quantum Mechanics ◽  
Conceptual Problems

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
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