Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities

Abstract

Abstract. We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold M . On one hand there is a canonical lift of projective structures on M to affine structures on extended manifold M . On the other hand the restriction of algebra of all functions on extended manifold to this special subalgebra of functions implies the canonical scalar product. This leads in particular to classification of second order operators with use of Kaluza-Klein-like mechanisms. Algebra of densitiesIn mathematical physics it is very useful to consider differential operators acting on densities of various weights on a manifold M (see [1] and citations there). We say that s = s(x)|Dx| λ is a density of weight λ on M if under change of local coordinates(λ is an arbitrary real number). We denote by F λ (M ) the space of densities of weight λ on manifold M . (The space of functions on M is F 0 (M ), densities of weight λ = 0.) Densities can be multiplied. If s 1 = s 1 (x)|Dx| λ 1 and s 2 = s 2 (x)|Dx| λ 2 are densities of weights λ 1 , λ 2 respectively then s = s 1 · s 2 = s 1 (x)s 2 (x)|Dx| λ 1 +λ 2 is a density of weight λ 1 + λ 2 . We come to the algebra F(M ) = ⊕ λ F λ (M ) of finite linear combinations of densities of arbitrary weights. Use a formal variable t instead volume form |Dx|. Thus an arbitrary density s = s 1 (x)|Dx| λ 1 + · · · + s k |Dx| λ k can be written as a function on x, t which is quasipolynomial on t, s(x, t) = s 1 (x)t λ 1 + · · · + s k (x)t λ k . An arbitrary density s ∈ F(M ) can be identified with function s r (x)t λr on the extended manifold M , which is quasipolynomial on 'vertical' variable t. There is a natural fibre bundle structure M → M . Extended manifold M is the frame bundle of the determinant bundle of M , (x i , t) are local coordinates on M . Changing of local coordinates is:The fibre bundle M → M can be used for studying projective geometry on M since there is a canonical construction which assigns to an arbitrary projective connection on manifold M an

Operator pencils on the algebra of densities

“…Performing these Kaluza-Klein like considerations (see also [5]) we see that B i (x) − S ik γ k (x) is a vector field, i.e. B i (x) is an upper connection (see [6], [7] for details).…”