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arXiv:2410.14711 · Chinese Journal of Physics

A problem with the conservation law observed in macroscopic quantum phenomena is a consequence of violation of the correspondence principle

А. В. Никулов

Published Sep 20, 2024• 4 citations
arXivHTMLPDF
DOI
10.1016/j.cjph.2024.09.023

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One of the unsolved problem in theoretical physics during some decades remains a construction of the complete and closed theory in which quantum mechanics and special relativity would be consistent without divergences and renormalization (Dirac, 1978). It may be assumed that divergences show conservation laws violation in the conventional theories, and a cause for it may be in turn violation of the group-theoretic principles in these theories, in accordance with the Noether theorems. A success of renormalization allows one to believe that the theory without divergences is possible. This paper is devoted to consideration of possibility to develop the consistent group-theoretic scheme of the quantum mechanics merely. It consists of Introduction, three parts, and Conclusion. The requirements which allow one to consider the quantum mechanics as a consistent group-theoretic theory are formulated in Introduction. The Noether theorems set one-to-one correspondence between conservation laws of the variables to be measured, i.e. observables (Dirac, 1958), and groups of symmetries of the solutions transformations of equations for complex wave functions, spinors, matrices and so on in which the space-time properties appear (Olver, 1986). These solutions do not obey to be an observables but the last ones may be constructed as the Hermitian forms corresponding to these observables on their basis. The mathematical tool to express the space-time symmetry properties is the group theory. Two circumstances connected with the stated above attract attention in the generally accepted schemes of the quantum mechanics. The exact conservation laws fulfilment is inconceivable in any theoretical scheme under absence of the complete set of the Hermitian forms, based on the main equations solutions and its derivatives, each of them would be corresponded to the observables. Some of these Hermitian forms have to be conserved, another have to be changed but all of them have to satisfy to some completeness condition expressed mathematically. The last subject has exceptional significance since if only some part of the unknown complete set of observables really existing is included into the theory, then both physical interpretation and conservation laws would be dependent on the Hermitian forms which are excluded from the theory. Of course, such theory can not be recognized to be the consecutive, complete and closed theory. One of the impressive consequence of the observables complete set and corresponding completeness condition absence is the well known question on the hidden parameters 13
Our environment and nature as a whole are fundamentally not in equilibrium. The existence of life on earth is only possible for this reason. Everyday examples of non-equilibrium processes are various weather phenomena driven by air and heat flows, traffic jams on motorways or the swarm behavior of animals in groups. These are just a few examples of many more phenomena and motivate a better understanding of non-equilibrium processes in classical as well as quantum systems. Non-equilibrium means that the known concepts, which are valid in equilibrium, are generally not or only approximately applicable. In equilibrium, all macroscopic currents add on average up to zero and the configuration of a system is solely determined by the fundamental principle of entropy maximization and the symmetries present in the system. According to the Noether theorem, which represents one of the most fundamental relations in physics, each continuous symmetry is associated with a conservation law. In this thesis, we consider quantum systems that have a set of conservation laws of which some are weakly violated by an external perturbation. We show that under these circumstances highly non-equilibrium states can be reached which are characterized by large currents. The phenomenon that a small perturbation can have a very large effect if it breaks a conservation law can, for example, be illustrated with a greenhouse. Inside the greenhouse the energy is approximately conserved due to the good insulation. As a consequence, the interior can be heated up to very high temperatures by even weak sunlight.
It is shown that violation of the Born Rule leads to a breakdown of the correspondence between the quantum electromagnetic field and its classical counterpart. Specifically, the relationship of the quantum coherent state to the classical electromagnetic field turns out to imply that if the Born Rule were violated, this could result in apparent deviations from the energy conservation law applying to the field and its sources (Poynting's Theorem). The result, which is fully general and independent of interpretations of quantum theory, suggests that the Born Rule is just as fundamental a law of Nature as are the field conservation laws.