DIFFEOMORPHISM GROUPS AND CURRENT ALGEBRAS: CONFIGURATION SPACE ANALYSIS IN QUANTUM THEORY

1999 ◽  
Vol 11 (01) ◽  
pp. 1-23 ◽  
Author(s):  
S. ALBEVERIO ◽  
YU. G. KONDRATIEV ◽  
M. RÖCKNER
Keyword(s):  
Differential Geometry ◽  
Quantum Theory ◽  
Configuration Space ◽  
Current Algebra ◽  
Relativistic Quantum ◽  
Unitary Representations ◽  
Poisson Measure ◽  
Diffeomorphism Groups ◽  
Dirichlet Operator ◽  
Space Analysis

The constuction of models of non-relativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space Γ of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on Γ, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on ℝd.

scholarly journals Relativistic quantum mechanics

1932 ◽  
Vol 136 (829) ◽  
pp. 453-464 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Correspondence Principle ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Relativistic Quantum Mechanics ◽  
Classical Electrodynamics ◽  
Quantum Phenomena

The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approxi­mation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.

Configuration space analysis of the rheumatoid wrist

1997 ◽  
Vol 16 (2-3) ◽  
pp. 309-322 ◽  
Author(s):  
C.F. Small ◽  
R.E. Ellis ◽  
J.T. Bryant ◽  
I.L. Dwosh ◽  
D.R. Pichora ◽  
...  
Keyword(s):  
Configuration Space ◽  
Space Analysis ◽  
Rheumatoid Wrist

Quantum Theory for Chemical Applications

2020 ◽  
Author(s):  
Jochen Autschbach
Keyword(s):  
Quantum Theory ◽  
Computational Chemistry ◽  
Chemical Engineering ◽  
Relativistic Quantum ◽  
Point Group ◽  
Theoretical Background ◽  
Group Symmetry ◽  
Structure Theory ◽  
Chemistry Laboratory ◽  
Mathematical Background

‘Quantum Theory for Chemical Applications (QTCA): From basic concepts to advanced topics’ is an introduction to quantum theory for students and practicing researchers in chemistry, chemical engineering, or materials chemistry. The text is self-contained such that only knowledge of high school physics, college introductory calculus, and college general chemistry is required, and it features many worked-out exercises. QTCA places special emphasis on the orbital models that are central to chemical applications of quantum theory. QTCA treats the important basic topics that a quantum theory text for chemistry must cover, and less-often treated models, such as the postulates of quantum theory and the mathematical background, the particle in a box, in a cylinder, and in a sphere, the harmonic oscillator and molecular vibrations, atomic and molecular orbitals, electron correlation, perturbation theory, and the basic aspects of various spectroscopies. Additional basic and advanced topics advanced topics that are covered in QTCA are band structure theory, relativistic quantum theory and its relevance to chemistry, the interactions of atoms and molecules with electromagnetic fields, and response theory. Finally, while it is not primarily a guide to computational chemistry, QTCA provides a solid theoretical background for many of the quantum chemistry methods used in contemporary research and in undergraduate computational chemistry laboratory courses. The text includes several appendices with important mathematical background, such as linear algebra and point group symmetry.

Losing Sight of the Forest for the ψ‎

2020 ◽  
pp. 185-211
Author(s):  
Alisa Bokulich
Keyword(s):  
Quantum Theory ◽  
Quantum State ◽  
Configuration Space ◽  
Mathematical Formalism ◽  
Alternative Formulation ◽  
Quantum Hydrodynamics ◽  
Many Body ◽  
Quantum Realism ◽  
The World ◽  
The Many

Traditionally \1 is used to stand for both the mathematical wavefunction (the representation) and the quantum state (thing in the world). This elision has been elevated to a metaphysical thesis by advocates of wavefunction realism. The aim of Chapter 10 is to challenge the hegemony of the wavefunction by calling attention to a littleknown formulation of quantum theory that does not make use of the wavefunction in representing the quantum state. This approach, called Lagrangian quantum hydrodynamics (LQH), is a full alternative formulation, not an approximation scheme. A consideration of alternative formalisms is essential for any realist project that attempts to read the ontology of a theory off the mathematical formalism. The chapter shows that LQH falsifies the claim that one must represent the many-body quantum state as living in 3n-dimensional configuration space. When exploring quantum realism, regaining sight of the proverbial forest of quantum representations beyond the \1 is just the beginning.

scholarly journals Renormalizable and Unitary Model of Quantum Gravity

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1334
Author(s):  
S. A. Larin
Keyword(s):  
Quantum Theory ◽  
Quantum Gravity ◽  
Curvature Tensor ◽  
Relativistic Quantum ◽  
Dimensional Regularization ◽  
Graviton Propagator ◽  
Unitary Model ◽  
Theory Of Gravity ◽  
Made In

We consider R + R 2 relativistic quantum gravity with the action where all possible terms quadratic in the curvature tensor are added to the Einstein-Hilbert term. This model was shown to be renormalizable in the work by K.S. Stelle. In this paper, we demonstrate that the R + R 2 model is also unitary contrary to the statements made in the literature, in particular in the work by Stelle. New expressions for the R + R 2 Lagrangian within dimensional regularization and the graviton propagator are derived. We demonstrate that the R + R 2 model is a good candidate for the fundamental quantum theory of gravity.

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