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 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.

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.

Invariance Groups in Classical and Quantum Mechanics

1973 ◽  
Vol 28 (3-4) ◽  
pp. 538-540 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Classical Mechanics ◽  
Symmetry Groups ◽  
Casimir Invariants ◽  
Classical Phase Space ◽  
Invariance Groups ◽  
Classical And Quantum Mechanics

AbstractThis is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

SOME REMARKS ON QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260018
Author(s):  
GIANFAUSTO DELL'ANTONIO
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Cloud Chamber ◽  
Similarities And Differences

We discuss the similarities and differences between the formalism of Hamiltonian Classical Mechanics and of Quantum Mechanics and exemplify the differences through an analysis of tracks in a cloud chamber.

scholarly journals A Model of Causal and Probabilistic Reasoning in Frame Semantics

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Causal Reasoning ◽  
Probabilistic Reasoning ◽  
Classical Mechanics ◽  
Complex Hilbert Space ◽  
Interpretation Of Quantum Mechanics ◽  
Frame Semantics ◽  
Quantum Entity ◽  
Natural Counterpart

Quantum mechanics admits a “linguistic interpretation” if one equates preliminary any quantum state of some whether quantum entity or word, i.e. a wave function interpretable as an element of the separable complex Hilbert space. All possible Feynman pathways can link to each other any two semantic units such as words or term in any theory. Then, the causal reasoning would correspond to the case of classical mechanics (a single trajectory, in which any next point is causally conditioned), and the probabilistic reasoning, to the case of quantum mechanics (many Feynman trajectories). Frame semantics turns out to be the natural counterpart of that linguistic interpretation of quantum mechanics.

scholarly journals Classical Mechanics as a Fundamental Law of Quantum Mechanics

2020 ◽  
Author(s):  
Yehuda Roth
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Second Law ◽  
Operator Form ◽  
Fundamental Law ◽  
Third Law ◽  
Definition Of ◽  
Force Operator ◽  
Physical Quantities

n our previous paper, we showed that the so-called quantum entanglement also exists in classical mechanics. The inability to measure this classical entanglement was rationalized with the definition of a classical observer which collapses all entanglement into distinguishable states. It was shown that evidence for this primary coherence is Newton’s third law. However, in reformulating a "classical entanglement theory" we assumed the existence of Newton’s second law as an operator form where a force operator was introduced through a Hilbert space of force states. In this paper, we derive all related physical quantities and laws from basic quantum principles. We not only define a force operator but also derive the classical mechanic's laws and prove the necessity of entanglement to obtain Newton’s third law.

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.

GEOMETRIC STRUCTURES IN PHYSICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260026 ◽  
Author(s):  
L. J. BOYA
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Mechanics ◽  
Path Integral ◽  
Riemannian Manifolds ◽  
Symplectic Geometry ◽  
Gauge Theories ◽  
Classical Mechanics ◽  
The Past ◽  
Modern Physics

Geometry and Physics developed independently, until the past twentieth century, where physicists realized geometry is rather flexible and can adapt itself to the needs and characteristics of modern physics. Besides the use of Riemannian manifolds to describe General Relativity, classical mechanics encounters symplectic geometry, not to speak of the bundle connection ingredient of modern gauge theories; even Quantum Mechanics, after the initial Hilbert space period, is seeking nowadays to adapt itself better to a geometrical interpretation, by imperatives of the path integral description and also to incorporate more clearly the symplectic aspects of its classical antecedent.

Sign in / Sign up

Export Citation Format

Share Document