Group representations arising from Lorentz conformal geometry

Abstract

It is shown that there exist conformally covariant differential operators D 2 l, k of all even orders 2 l, on differential forms of all orders k, in the double cover ▪ n of the n-dimensional compactified Minkowski space ▪ n. These act as intertwining differential operators for natural representations of O(2, n), the conformal group of ▪ n. For even n, the resulting decompositions of differential form representations of O ↑(2, n), the orthochronous conformal group, produce infinite families of unitary representations, the most interesting of which are carried by “positive mass-squared, positive frequency” quotients for 2l ⩾ ¦n − 2k¦. Physically, these generalize unitary representations of the conformal group associated with the modified wave operator D 2,0 = □ + ( (n − 2) 2 ) 2 , and the Maxwell operator on vector potentials D 2, (n − 2) 2 = δd . All the representation spaces produced, unitary and nonunitary, may be viewed as infinite systems of harmonic oscillators. As a by-product of the spectral resolution of the D 2 l, k , one gets some striking wave propagative properties for all of the equations D 2 l, k Φ = 0, including Huygens' principle in the curved spacetime ▪ n. The operators D 2 l, k have not been seen before except in the special cases k = 0 or n, and k = (n ± 2) 2 , l = 1 (the Maxwell operator). Thus much new information is obtained even in the physical case n = 4.