Symmetries in Foundation of Quantum Theory and Mathematics

Abstract

In standard quantum theory, symmetry is defined in the spirit of Klein's Erlangen Program: the background space has a symmetry group, and the basic operators should commute according to the Lie algebra of that group. We argue that the definition should be the opposite: background space has a direct physical meaning only on classical level while on quantum level symmetry should be defined by a Lie algebra of basic operators. Then the fact that de Sitter symmetry is more general than Poincare one can be proved mathematically. The problem of explaining cosmological acceleration is very difficult but, as follows from our results, there exists a scenario that the phenomenon of cosmological acceleration can be explained proceeding from basic principles of quantum theory. The explanation has nothing to do with existence or nonexistence of dark energy and therefore the cosmological constant problem and the dark energy problem do not arise. We consider finite quantum theory (FQT) where states are elements of a space over a finite ring or field with characteristic p and operators of physical quantities act in this space. We prove that, with the same approach to symmetry, FQT and finite mathematics are more general than standard quantum theory and classical mathematics, respectively: the latter theories are special degenerated cases of the former ones in the formal limit p → ∞.

Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts

“…which follows from C 2 = n 2 − 1 and the definition of C 2 , Equation (81). Substituting in the differential expressions Equations (134) and (141) for a and L we find that…”

“…The 2005 book by Gilmore on Lie algebras has 4 pages of homework problems on the H atom to duplicate results in early papers [78]. The last papers I am aware of that used SO (4,2) were applications in molecular physics [79,80] and more general in scope [81]. Carl Wulfman published a book on dynamical symmetries in 2011, which provides a helpful discussion of dynamical symmetries for the hydrogen atom [82].…”

The Nakano–Nishijima–Gell-Mann Formula from Discrete Galois Fields

“…However, these approaches are incomplete and a unified formulation is unknown. Here, we assert that a different mathematical structure, namely a world with a finite [4] and discrete space-time [5][6][7][8][9][10][11][12][13][14], is needed for the following reasons.…”

“…For instance, fractional numbers such as 1/3 are not defined. What we would consider in this paper is called a Galois field (F p = Z/pZ), that is, the set of integers where two numbers differing by a multiple of a prime number p are equivalent [5][6][7][8][9][10][11][12][13][14]. F p has a finite, periodic, and discrete structure.…”

“…Third, the theory of quantum gravity needs to be expanded because of the difficulty of renormalizability. This problem would not appear in F p in the first place [5,6,13,14], so the expansion is expected to be achieved by using a discrete system. In standard particle physics theory, the existence of antiparticles depends on additional assumptions, while in the Galois field such strange assumptions are not necessary [13].…”