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). The latter has its origin arising from the perspectives of Planck scale deformations of relativity symmetries. 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 is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
One can introduce \( {P}_{\mu }=\frac{1}{c}{E}_{\mu } \) and \( I=\frac{1}{c}F \), with c being the speed of light, to write the algebra in terms of Jμν, Xμ, Pμ, and I as generators. The latter form would be more familiar looking. Physics at that level would be better described in c = 1 units anyway. Even a simple contraction picture of the Poincaré to Galilean symmetry has the same feature. Interested readers can see Ref. [5], which gives a detailed pedagogical description of the story.
Strictly speaking, we should write the algebra elements with a factor of − i, which we leave out here and below. The true generators of the real Lie algebra are really of the form − iJ, rather than simply J itself. The conventional − i is, of course, to have the ‘generators’ J represented by Hermitian operators in a unitary representation.
Contractions of ISO(m, n) to G(m, n) or HR(m, n) (also commonly denoted by C(m, n)) differ only for one generator of the algebra, which plays the role of the time translation generator in G(3) and the central charge needed for the Heisenberg commutation relation as in HR(3); for G(1, 3), it gives translations of an absolute (proper) time-like coordinate σ in a five dimensional ‘spacetime’ coset picture, besides the then relative coordinate time t, as in the (1 + 3)D Minkowski spacetime.
The standard \( \hslash \) dimension is that of λpc; hence giving the correct choice as \( \lambda =p=\sqrt{\frac{\hslash }{c}} \). Here, we are neglecting c, which should be taken to be trivial at this level. All of the \( \hslash \)’s here should actually be \( \frac{\hslash }{c} \). The exact dimensions – including the c factors – are given in Table 2.
References
Chew, C.S., Kong, O.C.W., Payne, J.: A quantum space behind simple quantum mechanics advances in high energy physics 2017, special issue on planck-scale deformations of relativistic symmetries, Article ID 4395918, 1–9 (2017)
Chew, C.S., Kong, O.C.W., Payne, J.: Observables and dynamics, quantum to classical, from a relativity symmetry perspective, NCU-HEP-k070 (2018) submitted; moreover, see the references therein
Inönü, E., Wigner, E.P.: On the Contraction of Groups and their Representations. Proc. Nat. Acad. Sci. (US) 39, 510–524 (1953). See also and E. Saletan, Contraction of Lie Groups, J. Math. Phys. 2 (1961) 1–21
Gilmore, R.: Lie groups, Lie algebras, and some of their applications, Dover (2005)
Kong, O.C.W., Payne, J.: The Newtonian limit of special relativity: A relativity symmetry contraction perspective, NCU-HEP-k067 (2018)
Kong, O.C.W.: A deformed relativity with the quantum \( \hslash \). Phys. Lett. B 665, 58–61 (2008). see also arXiv:0705.0845 [gr-qc] for an earlier version with some different background discussions
Amelino-Camelia, G.: Testable scenario for relativity with minimum length. Phys. Lett. B 510, 255–263 (2001)
Amelino-Camelia, G.: Relativity in spacetimes with short-distance structure governed by an observer-independent (Plankian) length scale. Int. J. Mod. Phys. D 11, 35–59 (2002)
Magueijo, J., Smolin, L.: Lorentz invariance with an invariant energy scale. Phys. Rev. Lett. 88(190403), 1904031–1904034 (2002)
Magueijo, J., Smolin, L.: Generalized Lorentz invariance with an invariant energy scale. Phys. Rev, D 67, 044017 (2003)
Kowalski-Glikman, J., Smolin, L.: Triply special relativity. Phys. Rev. D 70, 065020 (2004)
Snyder, H.S.: Quantized space-Time. Phys. Rev. 71, 8–41 (1947)
Yang, C.N.: On quantized space-time. Phys. Rev. 72, 874 (1974)
Mendes, R.V.: Deformations, stable theories and fundamental constants. J. Phys. A 27, 8091–8104 (1994)
Chryssomalakos, C., Okon, E.: Generalized quantum relativistic kinematics: A stability point of view. Int. J. Mod. Phys. D 13, 1817–1850 (2004). ibid (2004) 2003–2034
Cho, D.N., Kong, O.C.W.: Relativity symmetries and Lie algebra contractions. Ann. Phys. 351, 275–289 (2014)
Connes, A.: Noncommutative geometry. Academic Press (1994)
Das, A., Kong, O.C.W.: Physics of quantum relativity through a linear realization. Phys. Rev. D 73, 124029 (2006)
Kong, O.C.W., Lee, H.-Y.: Poincaré-Snyder relativity with quantization, NCU-HEP-k036 (2009)
Kong, O.C.W., Lee, H.-Y.: Classical and quantum mechanics with poincaré-snyder relativity, NCU-HEP-k037 (2010)
Kyprianidis, A.: Scalar time parametrization of relativistic quantum mechanics: The covariant Schrödinger formalism. Phys. Rep. 155, 1–27 (1987). and references therein
Fanchi, J.R.: Parametrized relativistic quantum theory, Kluwer Academic Publishers (1993)
Aldaya, V., de Azcárraga, J.A.: Quantization as a consequence of the symmetry group: An approach to geometric quantization. J. Math. Phys. 23, 1297–1305 (1982)
Aldaya, V., de Azcárraga, J.A.: Symmetries of the pre-Klein-Gordon bundle: a Lagrangian analysis of quantum relativistic symmetry. J. Phys. A: Math. Gen. 18, 2639–2646 (1985)
Aldaya, V., de Azcárraga, J.A.: Group manifold analysis of the structure of relativistic quantum dynamics. Ann. Phys. 165, 484–504 (1985)
de Azcárraga, J.A., Izquierdo, J.M.: Lie groups, Lie algebras, cohomology and some applications in physics. Cambridge University Press, Cambridge (1995)
Trump, M.A., Schieve, W.C.: Classical relativistic many-body dynamics, Kluwer Academic Publishers (1999)
Bacry, H., Levy-Leblond, J.M.: Possible Kinematics. J. Math. Phys. 9, 1605–1614 (1968)
Feynman, R.P.: Mathematical formulation of the quantum theory of electromagnetic interaction. Phys. Rev. 80, 440–457 (1950)
Hostler, L.: Quantum field theory of particles of indefinite mass. I. J. Math, Phys. 21, 2461–2467 (1980)
Stückelberg, E.C.G.: La Mecanique du point materiel en theorie de relativite et en theorie des quanta. Helv. Phys. Acta 15, 23–37 (1942)
Feynman, R.P.: A relativistic cut-off for classical electrodynamics. Phys. Rev. 74, 1430–1438 (1948)
Acknowledgments
The authors are partially supported by research grants number 105-2112-M-008-017 and 106-2112-M-008-008 of the MOST of Taiwan.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kong, O.W., Payne, J. The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including all Known Relativity Symmetries, Classical and Quantum. Int J Theor Phys 58, 1803–1827 (2019). https://doi.org/10.1007/s10773-019-04075-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-019-04075-x