Current commutation relations, continuous tensor products, and infinitely divisible group representations

by R. F. Streater, in "LOCAL QUANTUM THEORY", Ed. R. Jost, Academic Press, 1969.

ABSTRACT

This is an account of a lecture given in August 1968 at the Varenna Summer School in Theoretical Physics.

We reformulate and solve the problem posed by Guichardet (Commun. Math. Phys., 5, 262-, 1967) and give the necessary and sufficient conditions for a continuous tensor product to exist, in the case of cyclic group representations. Dubin and I had already shown that in general, continuous tensor products of cyclic group representations do not exist. A preliminary account of this work was presented to the Winter School at Karpacz, 17/2/68-2/3/68. In the present paper I define the concept of an infinitely divisible cyclic representation of a group, and show (theorem 4) that the necessary and sufficient condition for a continuous tensor product $\otimes_x^\Omega U(g)$ of a group representation g\mapsto U(g) to exist is that (U,Omega) should be infinitely divisible; here Omega is a cyclic vector for the representation U. In the trivial case when the group is R, then the generator of the group defines a random variable, and the cyclic vector defines a probability measure on the line; then the theory reduces to the concept of infinite-divisibility of this measure. This has been further studied in my lecture at MaPhySto, Aarhus.

Another main result of the paper is the embedding theorem, which shows that any infinitely divisible group representation can be embedded as an action on a certain Fock space. This I call the Araki-Woods embedding theorem, by analogy with a similar-looking result (H. Araki and E. J. Woods, Publ. Res. Inst. Math. Sci., Kyoto, 2, 157-, 1967).

The results are used to show how to construct representations of current algebra, as defined in an earlier paper

The theory can be easily reformulated in terms of infinitely divisible positive-definite functions on groups, namely, the expectation of the unitary operators representing the group in the cyclic vector (known as the characteristic function of the cyclic representation), which is always a positive semi-definite continuous function on the group. In the paper, this function is shown to be the exponential of a continuous conditionally positive function on the group, if and only if the cyclic representation is infinitely divisible. The commutative case is well known in probability. The concept of infinite divisibility in probability was introduced by di Finetti, and the classification of these was achieved by Lévy.

H. Araki was in the audience at this lecture, and pointed out that the work overlapped with some of his work in progress, which duly appeared in: Factorisable Representations of Current Algebra, Publ. Rre. Inst. Math. Sci., Kyoto, 5, 361-422 (1970-71). He avoided the use of the term infinite divisibility, and analysed the more general problem of all factorisable representations of current algebra. He remarked that the condition of being conditionally positive can be expressed as a cocycle condition, and found all the conditionally positive continuous functions on a compact group, by showing that all cocycles are coboundaries in that case.

The details of a proof for finite permutation groups can be found in A Continuum Analogue of the Lattice Gas.

In 1970, this work was extended to Lie algebras and with Daniela Mathon to Clifford algebras.

The case of SL(2,R) is particularly interesting, because there are nontrivial cocycles for some representations of this group. See "Representations of the group SL(2,R), where R is a ring", by A. M. Vershik, I. M. Gelfand, and M. I. Graev, Uspehi-Mat-Nauk, 28, 83-128, 1973.

A very nice review, with complete proofs and lots of new results, can be found in Guichardet's book "Symmetric Hilbert Spaces and Related Topics", Lecture Notes in Mathematics, 261, Springer-Verlag, 1972, and in that of K. R. Parthasarathy and K. Schmidt, "Positive-definite kernels, continuous tensor products, and central limit theorems of probability theory", Lecture Notes in Mathematics, 272, Springer-Verlag, 1972.

An extensive review of work which is derived from or related to this paper see Compound Poisson Processes and Levy Processes in Groups and Symmetric Spaces, by David Applebaum.

Generalizations of the work in this paper have been given by Robin Hudson and Sylvia Pulmannova in their paper Chaotic expansion of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus.