scholarly journals Koopman wavefunctions and Clebsch variables in Vlasov–Maxwell kinetic theory

2021 ◽  
Vol 87 (4) ◽  
Author(s):  
Cesare Tronci ◽  
Ilon Joseph
Keyword(s):  
Kinetic Theory ◽  
Classical Mechanics ◽  
Canonical Formalism ◽  
Canonical Structure ◽  
Hamiltonian Structures ◽  
Von Neumann ◽  
Gauge Transformations ◽  
Phase Dynamics ◽  
Space Formulation

Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here, we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov–Maxwell kinetic theory. The celebrated Koopman–von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is non-canonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant of the Koopman–von Neumann construction that carries a different phase dynamics. Going back to van Hove's prequantum theory, the corresponding Koopman–van Hove equation provides an alternative Clebsch representation which is then coupled to the electromagnetic fields. Finally, the role of gauge transformations in the new context is discussed in detail.

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.

Subsequent and Subsidiary? Rethinking the Role of Applications in Establishing Quantum Mechanics

2015 ◽  
Vol 45 (5) ◽  
pp. 641-702 ◽  
Author(s):  
Jeremiah James ◽  
Christian Joas
Keyword(s):  
Molecular Structure ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Mechanics ◽  
Theory Construction ◽  
Science Pedagogy ◽  
Old Quantum Theory ◽  
The Common ◽  
Complete Quantum

As part of an attempt to establish a new understanding of the earliest applications of quantum mechanics and their importance to the overall development of quantum theory, this paper reexamines the role of research on molecular structure in the transition from the so-called old quantum theory to quantum mechanics and in the two years immediately following this shift (1926–1928). We argue on two bases against the common tendency to marginalize the contribution of these researches. First, because these applications addressed issues of longstanding interest to physicists, which they hoped, if not expected, a complete quantum theory to address, and for which they had already developed methods under the old quantum theory that would remain valid under the new mechanics. Second, because generating these applications was one of, if not the, principal means by which physicists clarified the unity, generality, and physical meaning of quantum mechanics, thereby reworking the theory into its now commonly recognized form, as well as developing an understanding of the kinds of predictions it generated and the ways in which these differed from those of the earlier classical mechanics. More broadly, we hope with this article to provide a new viewpoint on the importance of problem solving to scientific research and theory construction, one that might complement recent work on its role in science pedagogy.

Vortex Pinning and Dynamics in HTS films: Role of Extended Linear Defects

2011 ◽  
Vol 1367 ◽  
Author(s):  
Alexander L. Kasatkin ◽  
Constantin G. Tretiatchenko ◽  
Volodymyr M. Pan
Keyword(s):  
Lorentz Force ◽  
Classical Mechanics ◽  
Vortex Pinning ◽  
Dislocation Model ◽  
Linear Defect ◽  
Single Vortex ◽  
Linear Defects ◽  
Anisotropic Superconductor ◽  
Hts Films

ABSTRACTThe model of single vortex escape from extended linear defect and subsequent vortex dynamics under the Lorentz force action in a rather thick (d > 2λ) 3D anisotropic superconductor is developed. We consider the case of parallel c-oriented linear defects as well as the case of equidistant linear row of such kind of defects, which represents the dislocation model of low-angle [001] tilt grain boundary in HTS films and bicrystals. The suggested model based on the classical mechanics approach allows to describe behavior of an elastic vortex string in the potential well of linear defect and under the action of Lorentz force on its end within the Meissner current carrying layer and to determine the depinning critical current density at low magnetic fields and temperatures.

scholarly journals CANONICAL APPROACH TO 2D SUPERSYMMETRIC WZNW MODEL COUPLED TO SUPERGRAVITY

2002 ◽  
Vol 17 (29) ◽  
pp. 1923-1936 ◽  
Author(s):  
OLIVERA MIŠKOVIĆ ◽  
BRANISLAV SAZDOVIĆ
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Moody Algebra ◽  
Gauge Transformations ◽  
Local Gauge ◽  
Gauge Symmetries ◽  
Wznw Model ◽  
Canonical Approach ◽  
Explicit Expressions

Starting from the known representation of the Kac–Moody algebra in terms of the coordinates and momenta, we extend it to the representation of the super Kac–Moody and super Virasoro algebras. Then we use general canonical method to construct an action invariant under local gauge symmetries, where components of the super energy–momentum tensor L± and G± play the role of the diffeomorphisms and supersymmetry generators respectively. We obtain covariant extension of WZNW theory with respect to local supersymmetry as well as explicit expressions for gauge transformations.

scholarly journals On deformations of classical mechanics due to Planck-scale physics

2020 ◽  
Vol 29 (10) ◽  
pp. 2050070
Author(s):  
Olga I. Chashchina ◽  
Abhijit Sen ◽  
Zurab K. Silagadze
Keyword(s):  
Quantum System ◽  
Evolution Equations ◽  
Hidden Variables ◽  
Classical Mechanics ◽  
Planck Scale ◽  
Leibniz Rule ◽  
Uncertainty Relations ◽  
Interesting Possibility ◽  
Heisenberg Uncertainty ◽  
Space Formulation

Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

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.

scholarly journals Super Quantum Airy Structures

2020 ◽  
Vol 380 (1) ◽  
pp. 449-522
Author(s):  
Vincent Bouchard ◽  
Paweł Ciosmak ◽  
Leszek Hadasz ◽  
Kento Osuga ◽  
Błażej Ruba ◽  
...  
Keyword(s):  
Classification Scheme ◽  
Graphical Representation ◽  
Free Energies ◽  
Recursion Relations ◽  
Gauge Transformations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Topological Recursion

Abstract We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Furthermore, we present various examples of super quantum Airy structures, both finite-dimensional—which include well known superalgebras and super Frobenius algebras, and whose classification scheme we also discuss—as well as infinite-dimensional, that arise in the realm of vertex operator super algebras.

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 the relativistic micro-canonical ensemble and relativistic kinetic theory for N relativistic particles in inertial and non-inertial rest frames

2015 ◽  
Vol 12 (04) ◽  
pp. 1550049 ◽  
Author(s):  
David Alba ◽  
Horace W. Crater ◽  
Luca Lusanna
Keyword(s):  
Kinetic Theory ◽  
Canonical Ensemble ◽  
Particle System ◽  
Classical Mechanics ◽  
Canonical Partition Function ◽  
Relativistic Limit ◽  
Relativistic Kinetic Theory ◽  
Definition Of ◽  
New Formulation ◽  
Relativistic Particles

A new formulation of relativistic classical mechanics allows a reconsideration of old unsolved problems in relativistic kinetic theory and in relativistic statistical mechanics. In particular a definition of the relativistic micro-canonical partition function is given strictly in terms of the Poincaré generators of an interacting N-particle system both in the inertial and non-inertial rest frames. The non-relativistic limit allows a definition of both the inertial and non-inertial micro-canonical ensemble in terms of the Galilei generators.

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