In this chapter we assume that H is a Gaussian Hilbert space. We will use H to define and study several operators on L2(Ω, F(H), P). These operators are important in quantum physics, but we will not go into any such applications here; cf. for example Segal (1956), Glimm and Jaffe (1981, Chapter 6) and Meyer (1993).

There are also other applications of these operators, and we will use some of the results below in Chapter 15.

Some of the operators will be studied again in Chapters 15 and 16, where we also consider actions on Lp for p ≠ 2.

The reader may note that the operators treated here can be defined for any abstract Fock space based on an arbitrary Hilbert space, and that many of the results make sense in this generality, cf. for example Baez, Segal and Zhou (1992), Meyer (1993) and Parthasaraty (1992). Nevertheless, we will exclusively consider the Gaussian case here, where the extra structure is both helpful and, we hope, illuminating. (Of course, any result that can be stated for an abstract Fock space is valid in general as long as it is valid for the concrete realization treated here for Gaussian spaces.)

We warn the reader that several of the operators defined below will be unbounded and defined only on a dense subset of L2.