Classical measurement of four-vectors

In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time, and temperature, and a variational approach to finding a single-component PDF law p(x) of a single variable x. However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x, since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario. The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device.

The “intrinsic” measurement scenario

In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985), or Frieden (2001). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept of the prior scenario to define our unknown information expression.

For this purpose we proceed to analyze a particular prior scenario. This is an ideal, 4N-dimensional measurement scenario – the vector counterpart of the scalar parameter problem of Sec. 1.2.1.