scholarly journals Euler’s reflection formula, infinite product formulas, and the correspondence principle of quantum mechanics

2021 ◽  
Vol 62 (6) ◽  
pp. 063504
Author(s):  
Tamar Friedmann ◽  
Quincy Webb
Keyword(s):  
Quantum Mechanics ◽  
Correspondence Principle ◽  
Infinite Product ◽  
Reflection Formula ◽  
Product Formulas

Sound as a transverse wave

2017 ◽  
Vol 13 (1) ◽  
pp. 4522-4534
Author(s):  
Armando Tomás Canero
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Waves ◽  
Longitudinal Wave ◽  
Transverse Wave ◽  
Sound Propagation ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Wave Model ◽  
Scientific Experiments

This paper presents sound propagation based on a transverse wave model which does not collide with the interpretation of physical events based on the longitudinal wave model, but responds to the correspondence principle and allows interpreting a significant number of scientific experiments that do not follow the longitudinal wave model. Among the problems that are solved are: the interpretation of the location of nodes and antinodes in a Kundt tube of classical mechanics, the traslation of phonons in the vacuum interparticle of quantum mechanics and gravitational waves in relativistic mechanics.

scholarly journals IMPLICATIONS OF A QUANTUM MECHANICAL TREATMENT OF THE UNIVERSE

1998 ◽  
Vol 13 (05) ◽  
pp. 347-351 ◽  
Author(s):  
MURAT ÖZER
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Early Universe ◽  
Mechanical Treatment ◽  
Correspondence Principle ◽  
Scale Factor ◽  
Quantum Mechanical ◽  
Quantum Mechanical Treatment ◽  
The Standard Model ◽  
The Universe

We attempt to treat the very early Universe according to quantum mechanics. Identifying the scale factor of the Universe with the width of the wave packet associated with it, we show that there cannot be an initial singularity and that the Universe expands. Invoking the correspondence principle, we obtain the scale factor of the Universe and demonstrate that the causality problem of the standard model is solved.

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.

OBSERVABLE TIME IN QUANTUM COSMOLOGY

1995 ◽  
Vol 04 (01) ◽  
pp. 105-113 ◽  
Author(s):  
V. PERVUSHIN ◽  
T. TOWMASJAN
Keyword(s):  
Quantum Mechanics ◽  
First Principles ◽  
Quantum Cosmology ◽  
Correspondence Principle ◽  
Proper Time ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Conformal Time ◽  
Energy Factor ◽  
Definition Of

We show that the first principles of quantization and the experience of relativistic quantum mechanics can lead to the definition of observable time in quantum cosmology as a global quantity which coincides with the constrained action of the reduced theory up to the energy factor. The latter is fixed by the correspondence principle once one considers the limit of the “dust filled” Universe. The “global time” interpolates between the proper time for dust dominance and the conformal time for radiation dominance.

scholarly journals Ensembles and Experiments in Classical and Quantum Physics

2003 ◽  
Vol 17 (16) ◽  
pp. 2937-2980
Author(s):  
Arnold Neumaier
Keyword(s):  
Quantum Mechanics ◽  
Quantum Physics ◽  
Law Of Large Numbers ◽  
Correspondence Principle ◽  
Axiomatic Approach ◽  
Mathematical Concepts ◽  
Large Numbers ◽  
The Universe ◽  
Classical And Quantum Mechanics ◽  
Elegant Solution

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the ''probability via expectation'' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

scholarly journals The correspondence principle in quantum field theory and quantum gravity

2018 ◽  
Author(s):  
Damiano Anselmi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Correspondence Principle ◽  
External World ◽  
Quantum Field ◽  
Human Ability ◽  
Local Symmetries ◽  
Matter Sector

We discuss the fate of the correspondence principle beyond quantum mechanics, specifically in quantum field theory and quantum gravity, in connection with the intrinsic limitations of the human ability to observe the external world. We conclude that the best correspondence principle is made of unitarity, locality, proper renormalizability (a refinement of strict renormalizability), combined with fundamental local symmetries and the requirement of having a finite number of fields. Quantum gravity is identified in an essentially unique way. The gauge interactions are uniquely identified in form. Instead, the matter sector remains basically unrestricted. The major prediction is the violation of causality at small distances.

FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS

2021 ◽  
Vol 9 (1) ◽  
pp. 171-179
Author(s):  
M. Zarichnyi
Keyword(s):  
Game Theory ◽  
Mathematical Biology ◽  
Dynamic Programming ◽  
Quantum Mechanics ◽  
Convex Sets ◽  
Correspondence Principle ◽  
Optimization Problems ◽  
Mathematical Economics ◽  
Idempotent Mathematics ◽  
Idempotent Measure

Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.

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