The dynamic equations related to Kepler motion are scale-invariant. This means that the dynamical model is universal: it works on the same principles at the micro level as well as at the macro level. Why then quantization? Is it telling something we couldn't read in the classical physics? The answer is negative: both the microcosm and the macrocosm show the same type of phenomena that could be taken into consideration by the classical theory. The only thing worth considering from the side of quantum revolution is the inspiration it could bring, in astrophysics for instance. This was lost, however, due to the artificial dichotomy of our spirit.
The dynamic equations related to Kepler motion are scale-invariant. This means that the dynamical model is universal: it works on the same principles at the micro level as well as at the macro level. Why then quantization? Is it telling something we couldn't read in the classical physics? The answer is negative: both the microcosm and the macrocosm show the same type of phenomena that could be taken into consideration by the classical theory. The only thing worth considering from the side of quantum revolution is the inspiration it could bring, in astrophysics for instance. This was lost, however, due to the artificial dichotomy of our spirit.
Classical dynamics is usually considered to be the first level in the development of our understanding of the physical laws in Nature, which today is based on quantum theory. Despite the many successes of quantum theory in describing basic physical phenomena, it is continually running into difficulties in some specific physical situations. Thus, all attempts to quantize the theory of gravity have encountered serious difficulties. In order to find a solution to these problems, it seems to be useful to reconsider the fundamental principles of classical dynamics. In this context, the principles of Hamiltonian dynamics are seen to be of great importance not only for a basic understanding of classical theory, but also for its quantization.
Does quantum chaos exist?If it exists, what is it?If it does not, what was it supposed to have been?Why should we associate chaos with a "third revolution in physics" if it fails to be expressible in terms of quantum mechanics, our most fundamental theory of physical reality?Is there something wrong with quantum mechanics?Or is chaos merely a mathematical construct relevant only for models in classical mechanics?Is the human mind doomed to interpret and understand quantum mechanics in classical terms?The list of unanswered questions in quantum chaos research itself shows symptoms of the phenomenon it attempts to grasp. 1 How do we recognize classical Hamiltonian chaos?We have dealt with this question in our two previous columns.2,3 In Part I we discussed the implications of integrability and non-integrability for the phase-space trajectories of classical Hamiltonian systems with two degrees of freedom.We described the method of Poincaré surfaces of section as a convenient and striking discriminant between the two possibilities.In Part II we searched for an alternate mode of representation for classical Hamiltonian systems, a mode that is an equally powerful indicator of chaos, but one that can be more directly translated into quantum mechanics than the Poincaré map.We proposed a representation based on the construction of classical invariants via time averages of dynamical variables, a representation that can be employed under very general circumstances including integrable and nonintegrable models.In this column we shall construct quantum invariants, demonstrate the impact of non-integrability on these quantities, and discuss their properties in relation to their classical counterparts.In a future column we intend to introduce quantum Poincaré surfaces of section for stationary states and bring full circle our survey of quantum and classical Hamiltonian chaos.Consider a quantized integrable model.Its eigenstates are labeled by quantum numbers.There are as many quantum numbers as there are degrees of freedom.Each quantum number represents a quantized action.In the still evolving language of quantum chaos, quantum numbers are said to have been "lost" in the transition from a quantum system whose classical counterpart is integrable to one whose counterpart is chaotic.The broken, or lost, tori of classical chaos become lost quantum numbers of quantum chaos.As we shall see, the loss of quantum numbers is vividly mirrored in the behavior of the quantum invariants.We begin by describing the construction of the quantum invariants for a
It is argued that the world is a dissipative dynamic system, a phase flow of which is formed by conformally-symplectic mapping. The key assumption is that the concept of energy in microcosm makes sense only for the steady motions corresponding to quantum eigenstates. The constant, which determines the exponential phase volume contraction, is supposed to be a new universal constant, in addition to the speed of light and Planck constant. It is shown that statistical treatment of quantum objects as the ensembles concentrated on smooth connected attractors provides a simple explanation of stochastic behaviour of these objects as well as leads to a natural interpretation of the wave function, stationary Schrodinger equation, and scattering matrix. To validate the general hypotheses stated in the work, some physical models are presented. In particular, the models support the view that the inertial motion and quantum properties are basically determined by the vacuum as a dynamic subsystem. The matter-vacuum interaction is described formally by means of multivalued Hamilton function. A drawback of all these models is the non-locality of equations of motion stemming from the non-locality of variation of multivalued Hamilton function. This drawback can be overcome in the field theory using multivalued functionals with a local first variation opened by S. P. Novikov. Hence, a prospect of construction of the unified field theory in Einstein spirit, without a procedure of quantization is opened.
For nearly a century, physics has lived with a dual identity, as if the Uni- verse spoke two different languages. On one side, General Relativity describes the macrocosm with geometric precision: planets, stars, and galaxies. On the other, Quantum Mechanics—with the equations of Heisenberg, Schrodinger, and Dirac—is relegated to the microscopic realm, where reality becomes probabilis- tic, uncertain, and fuzzy. But what if this boundary is merely an optical illusion? The fundamental error of modern physics might not lie in the equations themselves, but in how we have used them. We have treated the pillars of quantum mechanics as microscopes, when in reality they are powerful cosmological lenses not yet properly calibrated for the macrocosm. Whenever we try to describe a large-scale event, these equations seem to break down. But the truth, viewed through the lens of the Dynamic Mass Transduction (DMT) theory, is different: to describe deep reality, these equa- tions must expand in their values. Paradoxically, the higher the degree of the uncertainty principle—without violating the normalization parameters of the equation—the more realistically the mathematical description approaches cosmic events. Fuzziness is not a mea- surement error; it is the survival algorithm of the Universe.
Those defile the purity of mathematical and philosophical truthy who confound real quantities with their relations and sensible measures. —ISAAC NEWTON, Principia Before we say anything about quantum mechanics, let us take a quick look at the Newtonian (or ‘classical’) view of the physical universe. According to that view, nature consists of an enormous number of particles. When Newtonian physics is combined with the theory of the electromagnetic field, it becomes convenient to think of these particles as dimensionless (even if there is a kind of conceptual strain involved in trying to think of something as having a mass but not any size ), and as possessing electrical properties – negative charge, or positive charge, or neutrality. This leads to the well-known ‘solar system’ view of the atom – with the electrons whirling around the nucleons (neutrons and protons) just as the planets whirl around the sun. Out of atoms are built molecules; out of molecules, macroscopic objects, scaling from dust motes to whole planets and stars. These latter also fall into larger groupings – solar systems and galaxies – but these larger structures differ from the ones previously mentioned in being held together exclusively by gravitational forces. At every level, however, one has trajectories (ultimately that means the possibility of continuously tracing the movements of the elementary particles) and one has causality (ultimately that means the possibility of extrapolating from the history of the universe up to a given time to its whole sequence of future states).