  Boris Tsirelson constructed an uncountable family of type III product systems
of Hilbert spaces through the theory of Gausian spaces, measure type spaces and
`slightly coloured noises', using techniques from probability theory. Here we
take a purely functional analytic approach and try to have a better
understanding of Tsireleson's construction and his examples.
  We prove an extension of Shale's theorem connecting symplectic group and Weyl
representation. We show that the `Shale map' respects compositions (This
settles an old conjecture of K. R. Parthasarathy). Using this we associate a
product system to a sum system. This construction includes the exponential
product system of Arveson, as a trivial case, and the type III examples of
Tsirelson.
  By associating a von Neumann algebra to every `elementary set' in [0,1], in a
much simpler and direct way, we arrive at the invariants of the product system
introduced by Tsirelson, given in terms of the sum system. Then we introduce a
notion of divisibility for a sum system, and prove that the examples of
Tsirelson are divisible. It is shown that only type I and type III product
systems arise out of divisible sum systems. Finally, we give a sufficient
condition for a divisible sum system to give rise to a unitless (type III)
product system.
