Role of certain gradient vector fields in quantum mechanics

1983 ◽  
Vol 77 (1) ◽  
pp. 19-30 ◽  
Author(s):  
M. Nishioka
Keyword(s):  
Quantum Mechanics ◽  
Vector Fields ◽  
Gradient Vector

scholarly journals LIOUVILLE THEORY AND SPECIAL COADJOINT VIRASORO ORBITS

1995 ◽  
Vol 10 (06) ◽  
pp. 785-799 ◽  
Author(s):  
A. GORSKY ◽  
A. JOHANSEN
Keyword(s):  
Quantum Mechanics ◽  
Type Theory ◽  
Vector Fields ◽  
Liouville Theory ◽  
Hamiltonian Reduction ◽  
Coadjoint Orbits ◽  
Point Particles ◽  
Witten Theory ◽  
Wrong Sign

We describe the Hamiltonian reduction of the coadjoint Kac–Moody orbits to the Virasoro coadjoint orbits explicitly in terms of the Lagrangian approach for the Wess–Zumino–Novikov–Witten theory. While a relation of the coadjoint Virasoro orbit Diff S1/ SL (2, R) to the Liouville theory has already been studied, we analyze the role of special coadjoint Virasoro orbits Diff [Formula: see text]corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros xi of the vector field of stabilizer [Formula: see text] and additional parameters qi, i = 1, …, n, in terms of quantum mechanics for n-point particles on the circle. We argue that the special orbits are generated by insertions of "wrong sign" Liouville exponential into the path integral. The additional parmeters qi are naturally interpreted as accessory parameters for the uniformization map. Summing up the contributions of the special Virasoro orbits we get the integrable sinh–Gordon type theory.

Randomness and Complexity

2018 ◽  
Author(s):  
Steven E. Vigdor
Keyword(s):  
Quantum Mechanics ◽  
Natural Selection ◽  
Rna World ◽  
Biological Evolution ◽  
Variable Theory ◽  
Cosmological Natural Selection ◽  
Einstein Podolsky Rosen ◽  
World Hypothesis ◽  
Rna World Hypothesis

Chapter 7 describes the fundamental role of randomness in quantum mechanics, in generating the first biomolecules, and in biological evolution. Experiments testing the Einstein–Podolsky–Rosen paradox have demonstrated, via Bell’s inequalities, that no local hidden variable theory can provide a viable alternative to quantum mechanics, with its fundamental randomness built in. Randomness presumably plays an equally important role in the chemical assembly of a wide array of polymer molecules to be sampled for their ability to store genetic information and self-replicate, fueling the sort of abiogenesis assumed in the RNA world hypothesis of life’s beginnings. Evidence for random mutations in biological evolution, microevolution of both bacteria and antibodies and macroevolution of the species, is briefly reviewed. The importance of natural selection in guiding the adaptation of species to changing environments is emphasized. A speculative role of cosmological natural selection for black-hole fecundity in the evolution of universes is discussed.

Symmetries from the solution manifold

2015 ◽  
Vol 12 (08) ◽  
pp. 1560016 ◽  
Author(s):  
Víctor Aldaya ◽  
Julio Guerrero ◽  
Francisco F. Lopez-Ruiz ◽  
Francisco Cossío
Keyword(s):  
Symplectic Structure ◽  
Vector Fields ◽  
Sigma Model ◽  
Canonical Quantization ◽  
General Concept ◽  
Noether Theorem ◽  
Preliminary Step ◽  
Hamiltonian Vector Fields ◽  
Solution Manifold

We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré–Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton–Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.

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.

scholarly journals Gradient Vector Fields Do Not Generate Twister Dynamics

2001 ◽  
Vol 174 (1) ◽  
pp. 91-100 ◽  
Author(s):  
P. Fortuny ◽  
F. Sanz
Keyword(s):  
Vector Fields ◽  
Gradient Vector

Semilinear PDEs in the Heisenberg group: The role of the right invariant vector fields

2010 ◽  
Vol 72 (2) ◽  
pp. 987-997 ◽  
Author(s):  
Isabeau Birindelli ◽  
Fausto Ferrari ◽  
Enrico Valdinoci
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Invariant Vector ◽  
Semilinear Pdes ◽  
The Right ◽  
Invariant Vector Fields

scholarly journals Counting and Discrete Morse Theory

2019 ◽  
Vol 21 (1) ◽  
Author(s):  
Andrew Sack
Keyword(s):  
Morse Theory ◽  
Vector Fields ◽  
Discrete Morse Theory ◽  
Gradient Vector ◽  
Morse Functions

We examine enumerating discrete Morse functions on graphs up to equivalence by gradient vector fields and by restrictions on the codomain.  We give formulae for the number of discrete Morse functions on specific classes of graphs (line, cycle, and bouquet of circles).

scholarly journals Connected components of the space of proper gradient vector fields

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Maciej Starostka
Keyword(s):  
Vector Fields ◽  
Connected Components ◽  
Gradient Vector ◽  
Proper Maps

AbstractWe show that there exist two proper gradient vector fields on $$\mathbb {R}^n$$ R n which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

scholarly journals On Trajectories of Analytic Gradient Vector Fields

2002 ◽  
Vol 184 (1) ◽  
pp. 215-223 ◽  
Author(s):  
Aleksandra Nowel ◽  
Zbigniew Szafraniec
Keyword(s):  
Vector Fields ◽  
Gradient Vector
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