Introduction

The conditions which define a consistent family of histories were stated in Ch. 10. The sample space must consist of a collection of mutually orthogonal projectors that add up to the history identity, and the chain operators for different members of the sample space must be mutually orthogonal, (10.20). Checking these conditions is in principle straightforward. In practice it can be rather tedious. Thus if there are n histories in the sample space, checking orthogonality involves computing n chain operators and then taking n(n − 1)/2 operator inner products to check that they are mutually orthogonal. There are a number of simple observations, some definitions, and several “tricks” which can simplify the task of constructing a sample space of a consistent family, or checking that a given sample space is consistent. These form the subject matter of the present chapter. It is probably not worthwhile trying to read through this chapter as a unit. The reader will find it easier to learn the tricks by working through examples in Ch. 12 and later chapters, and referring back to this chapter as needed.

The discussion is limited to families in which all the histories in the sample space are of the product form, that is, represented by a projector on the history space which is a tensor product of quantum properties at different times, as in (8.7). As in the remainder of this book, the “strong” consistency conditions (10.20) are used rather than the weaker (10.25).