Special properties of the irreducible representations of the proper Lorentz group

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.