Group representations arising from Lorentz conformal geometry

Abstract

It is shown that there exist conformally covariant differential operators D2l,k of all even orders 2l, on differential forms of all orders k, in the double cover {A figure is presented}n of the n-dimensional compactified Minkowski space {A figure is presented}n. These act as intertwining differential operators for natural representations of O(2, n), the conformal group of {A figure is presented}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 D2,0 = □ + ( (n - 2) 2)2, and the Maxwell operator on vector potentials D2, (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 D2l,k, one gets some striking wave propagative properties for all of the equations D2l,k Φ = 0, including Huygens' principle in the curved spacetime {A figure is presented}n. The operators D2l,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. © 1987.