Special properties of the irreducible representations of the proper Lorentz group

It is shown that the finite‐ and infinite‐dimensional irreducible representations ( j 0, c) of the proper Lorentz group SO(3,1) may be classified into the two categories, namely, the c o m p l e x‐o r t h o g o n a l and the s y m p l e c t i c representations; while all the integral‐j 0 representations are equivalent to complex‐orthogonal ones, the remaining representations for which j 0 is a half‐odd integer are symplectic in nature. This implies in particular that all the representations belonging to the c o m p l e m e n t a r y s e r i e s and the subclass of integral‐j 0 representations belonging to the p r i n c i p a l s e r i e s are equivalent to r e a l‐o r t h o g o n a l representations. The rest of the principal series of representations for which j 0 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‐j 0 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‐j 0 representations. Also, by introducing a suitable r e a l basis, the finite‐dimensional ( j 0=0, c=n) representations, where n is an integer, are shown to be r e a l‐p s e u d o‐o r t h o g o n a l 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 p s e u d o‐u n i t a r y 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.