Special properties of the irreducible representations of the proper Lorentz group

Srinivasa Rao, K. N. and Gopala Rao, A. V. and Narahari, B. S. (1983) Special properties of the irreducible representations of the proper Lorentz group. J. Math. Phys., 24 (10). pp. 2397-2403.

[img] Text (Full Text)
JMathPhys_24_2397.pdf - Published Version
Restricted to Registered users only

Download (804Kb) | Request a copy
Official URL: http://dx.doi.org/10.1063/1.525619

Abstract

It is shown that the finite‐ and infinite‐dimensional irreducible representations ( j0, c) of the proper Lorentz group SO(3,1) may be classified into the two categories, namely, the complex‐orthogonal and the symplectic representations; while all the integral‐j0 representations are equivalent to complex‐orthogonal ones, the remaining representations for which j0 is a half‐odd integer are symplectic in nature. This implies in particular that all the representations belonging to the complementary series and the subclass of integral‐j0 representations belonging to the principal series are equivalent to real‐orthogonal representations. The rest of the principal series of representations for which j0 is a half‐odd integer are symplectic in addition to being unitary and this in turn implies that the D j representation of SO(3) with half‐odd integral j is a subgroup of the unitary symplectic group USp(2 j+1). The infinitesimal operators for the integral‐j0 representations are constructed in a suitable basis wherein these are seen to be complex skew‐symmetric in general and real skew‐symmetric in particular for the unitary representations, exhibiting explicitly the aforementioned properties of the integral‐j0 representations. Also, by introducing a suitable real basis, the finite‐dimensional ( j0=0, c=n) representations, where n is an integer, are shown to be real‐pseudo‐orthogonal with the signature (n(n+1)/2, n(n−1)/2). In any general complex basis, these representations (0, n) are also shown to be pseudo‐unitary with the same signature (n(n+1)/2, n(n−1)/2). Further it is shown that no other finite‐dimensional irreducible representation of SO(3,1) possesses either of these two special properties.

Item Type: Article
Subjects: Physical Sciences > Mathematics
Divisions: PG Campuses > Manasagangotri, Mysore > Mathematics
Depositing User: Kodandarama
Date Deposited: 21 May 2013 03:34
Last Modified: 21 May 2013 03:34
URI: http://eprints.uni-mysore.ac.in/id/eprint/9777

Actions (login required)

View Item View Item