Positive Definite Distributions Research Articles

The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by C p( G)(0< p⩽2), (ii) the K-biinvariant elements in C p(G) by I p(G), (iii) the positive definite (zonal) spherical functions by P , and (iv) the spherical transform on C p( G) by ϑ → \\ ̂ gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto C l( G), (ii) there exists a unique measure μ of polynomial growth on P such that T[ ψ]=∫ pψdμ for all ψϵI 1( G) (iii) all measures μ of polynomial growth on P are obtained in this way, and (iv) T may be extended to a particular I p(G) space (1 ⩽ p ⩽ 2) if and only if the support of μ lies in a certain easily defined subset of P . These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.

Positive-definite distributions and intertwining operators

An example is given of a positive-definite measure μ on the group SL(2, R) which is extremal in the cone of positive- definite measures, but the corresponding unitary representation Lμ is reducible. By considering positive-definite distributions this anomaly disappears, and for an arbitrary Lie group G and positive-definite distribution μ on G a, bijection is estab- lished between positive-definite distributions on G bounded by μ and positive-definite intertwining operators for the repre- sentation Lμ. As an application, cyclic vectors for Lμ are obtained by a simple explicit construction. Introduction* The use of positive-defi niteness as a tool in abstract harmonic analysis has a long history, the most striking early instance being the Gelfand-Raikov proof via positive-definite functions of the completeness of the set of irreducible unitary representations of a locally compact group (5). More recently, it was observed by R. J. Blattner (1) that the systematic use of positive-defi nite measures gives very simple proofs of the basic properties of induced representations, and the cone of positive-defi nite measures on a group was subsequently studied by Effros and Hahn (4). The purpose of this paper is two-fold. First, we give an example to show that positive-definite measures do not suffice for the study of intertwining operators and irreducibility of induced representations, despite the claim to the contrary in (4). Specifically, we exhibit a positive-definite measure μ on G — SL(2,1?) such that μ lies on an extremal ray in the cone of positive-defi nite measures on G, but the associated unitary representation Lμ is reducible, contradicting Lemma 4.16 of (4). Our second aim is to show that when G is any Lie group, then the correspondence between intertwining operators and positive func- tionals on G asserted by Effros and Hahn does hold, provided one deals throughout with positive-definite distributions instead of just measures. The essential point is the validity of the Schwartz Kernel Theorem for the space C?(G), together with a result of Bruhat (3) about distributions on GxG, invariant under the diagonal action of G. Using this correspondence, we obtain cyclic vectors for representations defined by positive-defi nite distributions, using a modification of the construction in (7) (The proof of cyclicity given in (7) is invalid, since it assumes the existence of a measure on G corresponding to

Tensors in Hilbert space

If one multiplies the Schrödinger wave function in the coordinate representation by the corresponding function in the momentum representation one gets a wave function which can be regarded as a mixed tensor in Hilbert space. This leads to a positive-definite distribution function for the coordinate and the momentum. One can form a linear combination of such state tensors corresponding to a new kind of state. One can also form tensors describing non-observable states providing more information about the system than is permitted by quantum mechanics. Some generalizations are suggested, such as having the coordinate and momentum functions depend on different times.

On quadratic lagrangians in General Relativity

Theories of gravitation similar to General Relativity but with an additionalR2 term in the Lagrangian are explored. The Schwarzschild metric is not the exterior solution that can be continued to the interior of the body to give a positive definite mass distribution. The experimental consequences ofR2 terms are investigated. Furthermore, it is shown that a theory with anR2 term only possesses an interesting singular dependence on the coupling constant.