Application of the double numbers in the representation of the Lorentz transformations

Authors

DOI:

https://doi.org/10.31349/RevMexFisE.20.010204

Keywords:

Lorentz transformations, double numbers, Wigner angle, Thomas precession

Abstract

We show that the orthochronous proper Lorentz transformations that leave one of the Cartesian coordinates fixed can be represented by $2 \times 2$ unitary matrices with determinant equal to 1, whose entries are double numbers. This representation is employed in the calculation of the Wigner angle, which arises in the composition of two boosts in arbitrary directions.

References

G.F. Torres del Castillo, 3-D Spinors, Spin-Weighted Functions and their Applications (Springer, New York, 2003), Sec. 1.1. https://doi.org/10.1007/978-0-8176-8146-3

G.F. Torres del Castillo, Some applications in classical mechanics of the double and the dual numbers, Rev. Mex. F´ıs. E 65 (2019) 152. https://rmf.smf.mx/ojs/rmf-e/author/submission/550

G.F. Torres del Castillo, Applications of the double and the dual numbers. The Bianchi models, Rev. Mex. F´ıs. E 17 (2020) 146. https://doi.org/10.31349/RevMexFisE.17.146

G.F. Torres del Castillo and K.C. Gutiérrez-Herrera, Double and dual numbers. SU(2) groups, two-component spinors and

generating functions, Rev. Mex. F´ıs. 66 (2020) 418. https://doi.org/10.31349/RevMexFis.66.418

W. Rindler, Relativity, Special, General, and Cosmological, 2nd ed. (Oxford University Press, Oxford, 2006).

J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998), Sec. 11.8.

T. Padmanabhan, Sleeping Beauties in Theoretical Physics (Springer, Cham, 2015), Chap. 13. https://doi.org/10.1007/978-3-319-13443-7

C.B. Van Wyk, Rotation associated with the product of two Lorentz transformations, Am. J. Phys. 52 (1984) 853. https://doi.org/10.1119/1.13538

A. Ben-Menachem, Wigner’s rotation revisited, Am. J. Phys. 53 (1985) 62. https://doi.org/10.1119/1.13953

M.W.P. Strandberg, Abstract group theoretical reduction of products of Lorentz-group representations, Phys. Rev. A 34 (1986) 2458. https://doi.org/10.1103/PhysRevA.34.2458

P.K. Aravind, The Wigner angle as an anholonomy in rapidity space, Am. J. Phys. 65 (1997) 634. https://doi.org/10.1119/1.18620

J.A. Rhodes, M.D. Semon, Relativistic velocity space, Wigner rotation and Thomas precession, Am. J. Phys. 72 (2004) 943. https://doi.org/10.1119/1.1652040

G.P. Fisher, The Thomas precession, Am. J. Phys. 40 (1972) 1772. https://doi.org/10.1119/1.1987061

A.A. Ungar, Thomas precession and its associated grouplike structure, Am. J. Phys. 59 (1991) 824. https://doi.org/10.1119/1.16730

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Published

2023-01-13

How to Cite

[1]
G. F. Torres del Castillo, “Application of the double numbers in the representation of the Lorentz transformations”, Rev. Mex. Fis. E, vol. 20, no. 1 Jan-Jun, pp. 010204 1–, Jan. 2023.