The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including all Known Relativity Symmetries, Classical and Quantum

Abstract

In this article, we utilize the insights gleaned from our recent formulation of space(-time), as well as dynamical picture of quantum mechanics and its classical approximation, from the relativity symmetry perspective in order to push further into the realm of the proposed fundamental relativity symmetry SO(2, 4) of our quantum relativity project. We explicitly trace how the diverse actors in this story change through various contraction limits, paying careful attention to the relevant physical units, in order to place all known relativity theories -quantum and classical -within a single framework. More specifically, we explore both of the possible contractions of SO(2, 4) and its coset spaces in order to determine how best to recover the lower-level theories. These include both new models and all familiar theories, as well as quantum and classical dynamics with and without Einsteinian special relativity. Along the way, we also find connections with covariant quantum mechanics. The emphasis of this article rests on the ability of this language to not only encompass all known physical theories, but to also provide a path for extensions. It will serve as the basic background for more detailed formulations of the dynamical theories at each level, as well as the exact connections amongst them.This article is a study within our group's Quantum Relativity Project, the key idea of which is to formulate pictures of quantum spacetime and the related dynamics from a (relativity) symmetry perspective. We expect the models of quantum spacetime to be of a noncommutative nature, seeing them as intrinsically quantum; hence, they may not be fully described by any real number geometric picture of finite dimension. The latter, as classical/commutative geometry, is of course applicable to modeling classical spacetime, as in the Newtonian, Minkowskian, as well as the dynamical Einstein general-relativistic spacetime. What we should bear in mind is that all of these are only theoretical models of the notion of spacetime, and as such are only as good as the corresponding model of physical dynamics. The pursuit of better models of fundamental physics should go hand-in-hand with the pursuit of better models of spacetime. Real number geometry may not maintain its role as a successful, not to mention convenient, tool in this endeavor. The point of view underlying our project highlights the difficulty one encounters in appreciating traditional quantum mechanics. Quantum mechanics indeed sounds strange, or even counter-intuitive, when thinking about it as a theory of mechanics on classical spacetime. What we hope to convince people of, however, is that it is no less intuitive than classical mechanics when one thinks about it with the proper model of a quantum (physical) space in hand [1, 2] (a conceptual discussion has been presented in an article in Chinese [3]). A quantum particle, for example, always has a definite position within the quantum model of physical space, though it notably cannot be modeled or represented by a finite number o...

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

“…The relativity symmetry for the quantum theory is one of H R (1,3), which fits well into the contraction chain, at least at the symmetry and coset space level [16,17]. It has been well known that from a group theoretical perspective, a general overcomplete coherent state basis can naturally be identified with points of the appropriate coset.…”

“…Each group element can be identified with a point in the H R (1, 3)/SO(1, 3) coset space [14,17]. X (ς) andP (ς) are operators on the abstract representation space H ς spanned by the p…”

“…We can see the latter either through the explicit use of Equation ( 17), or directly from the fact that V(p µ , x µ ) in Equation ( 10) is an η-unitary operator. From the definition ofη in Fock basis and Equation (17), we obtain…”

“…A contraction [34,35] of the Lorentz symmetry SO(1, 3), sitting inside the H R (1, 3), to the Galilean ISO(3) has been discussed in Ref. [17], together with the corresponding coset spaces of interest. The full (quantum) relativity symmetry group obtained by contraction is named H GH (3), with commutators among generators essentially given by…”

“…Formulation given there corresponds to writing the time coordinate as ict. Here, the representation is rather given in the form of the |p µ , x µ coherent states, with p µ and x µ being real Minkowski four-vectors, from our group theoretical grand framework [1,16,17].…”