Abstract
Recent work on Lie’s method of extended groups to obtain symmetry groups and invariants of differential equations of mathematical physics is surveyed. As an essentially new contribution one-parameter Lie groups admitted by three-dimensional harmonic oscillator, three-dimensional wave equation, Klein-Gordon equation, two-component Weyl’s equation for neutrino and four-component Dirac equation for Fermions are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bargmann V 1936Z. Phys. 99 576
Bateman H 1910Proc. Lond. Math. Soc. 8 223
Boyer C P, Kalinis E G and Miller Jr W 1975J. Math. Phys. 16 499
Boyer C P, Sharp R T and Winternitz P 1976J. Math. Phys. 17 1439
Cisneros A and McIntosh H V 1970J. Math. Phys. 11 870
Dickson L E 1924Ann. Math. S 2,25 287
Eilenberger G 1981Solitons, mathematical methods for physicists (Berlin: Springer-Verlag)
Eisenhart L P 1961Continuous groups of transformations (New York: Dover Publ.)
Elliott J P and Dawber P G 1979Symmetry in Physics (New York: Oxford University Press) Vol II
Fock V 1935Z. Phys. 98 145
Fulton T, Rohrlich F and Witten L 1962Rev. Mod. Phys. 34 442
González-Gascon F 1977J. Math. Phys. 18 1763
González-Gascon F and González-López A 1983J. Math. Phys. 24 2006
Guenther N J and Leach P G L 1977J. Math. Phys. 18 572
Hamermesh M 1983 Symmetry groups of a differential equation, Lecture atXII Int. Colloq. in Group Theoret. Methods in Phys. Trieste: ICTP (private communication)
Harnard J and Winternitz P 1980 inGroup theoretical methods in Physics (ed) K B Wolf, Lecture Notes in Physics no. 135 (Berlin: Springer-Verlag) p. 170
Jauch J M and Hill E L 1940Phys. Rev. 57 641
Kalinis E G and Miller Jr W 1974J. Math. Phys. 15 1728
Leach P G L 1978J. Math. Phys. 19 446
Leach P G L 1980J. Math. Phys. 21 32
Leach P G L 1981aJ. Aust. Math. Soc. B23 173
Leach P G L 1981bJ. Math. Phys. 22 679
Levy-Leblond J M 1971Am. J. Phys. 39 502
Lie S and Scheffers G 1891 Vorlesungen uber Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig
Lutzky M 1978J. Phys. A11 249
Makarov A A, Smorodinsky J A, Valiev Kh and Winternitz P 1967Nuovo Cimento A52 1061
McIntosh H V 1971 inGroup theory and its applications (ed.) E M Loebl (New York: Academic Press) Vol II
Noether E 1918Nachr. Ges. Wiss. Goettingen. Math. Phys. K1 235
Olver P J 1976Symmetry groups of partial differential equations, Ph.D. thesis, Harvard University
Patera J, Sharp R T, Winternitz P and Zassenhaus H 1976aJ. Math. Phys. 17 977
Patera J, Sharp R T, Winternitz P and Zassenhaus H 1976bJ. Math. Phys. 17 986
Prince G 1983aBull. Aust. Math. Soc. 27 53
Prince G 198bJ. Phys. A16 L105
Prince G E and Eliezer C J 1980J. Phys. A13 815
Prince G E and Eliezer C J 1981J. Phys. A14 587
Prince G E and Leach P G L 1980Hardronic J. 3 941
Racah G 1951Group theory and spectroscopy, Lecture Notes, Princeton University (mimeographed)
Sattinger D H 1977Group theoretical methods in bifurcation theory (Berlin: Springer-Verlag)
Vinet L 1980 inGroup theoretical methods in physics (ed.) K B Wolf, Lecture Notes in Physics no. 135 (Berlin: Springer-Verlag) p. 191
Wess J 1960Nuovo Cimento 18 1086
Weyl H 1925Math. Zeit. 23 271
Weyl H 1926Math. Zeit. 24 328
Wigner E P 1931Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg, Braunschweig
Wulfman C E and Wybourne B G 1976J. Phys. A9 507
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rudra, P. Symmetry groups of mathematical physics. Pramana - J Phys 23, 445–457 (1984). https://doi.org/10.1007/BF02846620
Issue Date:
DOI: https://doi.org/10.1007/BF02846620
Keywords
- Mathematical physics
- symmetry groups
- differential equations
- Lie groups
- harmonic oscillator
- wave equation
- Klein-Gordon equation
- Weyl equation
- Dirac equation