It is possible to centre a standard apparatus of characteristic dimension d^ f 0 at every point of the configuration manifold M only if'^ M is geodetically complete.
Proof: see appendix 7.
This result is intuitively clear; for suppose that M is not geodeti cally complete, then it possesses a "boundary" and points may be chosen sufficiently close to this "boundary" that a region of radius d^ cannot be constructed around them. It is important to observe that, if the space is not geodetically complete, then it is impossible, solely in terms of Hamilton’s equations of motion derived from the Hamiltonian
(51), to describe correctly the motions of test and reference particles during collisions sufficiently close to the spatial "boundaries" of the manifold, due to the presence of position-dependent constraints
analogous to those discussed in §5.1. The measuring procedure is therefore intrinsically suspect when employed to measure any momentum P within a region sufficiently close to the "boundary" of M. Henceforth, therefore, we shall confine our attention exclusively to manifolds which are geodetically complete.
'^Observe that [20], which here assures [19], requires that within each local set A the departure from Euclidean space must be negligible. It may, in general, be impossible to assure this for the class of all sets of characteristic dimension d ' f , o 0, regardless of how small do is chosen.
30 -
We next consider, abstracting from our analysis of local measure
ment only the recoil sets B, the global behaviour of the characteristic
errors A^p which arise from an impulsive measurement of a momentum P, so as to introduce a concept of classical global measurability. The
essential idea underlying this concept is that, for a suitably chosen s reference class of local recoil sets B, the corresponding class of
intrinsic uncertainties A^p in the locally assigned values of the momentum P must be bounded above by some finite global tolerance or
standard of accuracy A^. For, unless such a global standard A^ > A^p exists, the accuracy of the local values of P decreases without limit as B is varied within its reference class; so that a fixedly reliable momentum value is not everywhere assignable, or, as we may say, P is not (classically) globally measurable. Turning our attention to more detailed considerations, we note that to be directly comparable with completeness the concept of classical global measurability must
necessarily depend only upon properties of the momentum P or equivalently of the associated vector field X, so that the various local errors A^p within an elected reference class must 'first be standardized to eliminate dependence upon the properties of the particles under test. This is readily achieved by requiring
[24] that the alligned Cartesian momentum be taken in the calculation of A^p to be some pre-assigned universally applicable value P^ ^ 0.
The exact character of the reference class of the local sets B is again inspired by comparability with completeness. We elect
[25] that the reference class of local recoil sets is, for each maximal integral curve ^ of the vector field X associated with the momentum P, the set of all intervals
of ^ of fixed'^ length d(^). . j';
The following precise formulation of classical global measurability can now be given.
[26] A momentum P is classically globally measurable if and only if, for all maximal integral curves of the associated vector field X, it is possible to elect a finite upper bound to the set of characteristic uncertainties A^p, standardized as in [24], and
generated by the reference class [25] of local recoil sets B.
This is as illustrated in figure 2 overleaf. It will be noted that this final statement of classical global measurability is removed by a considerable margin from its original intuitive motivation in the idea of standard measurement, the reasons for this abstraction being largely connected with the need to admit as globally measurable as many momenta "having critical points" as possible. We shall defer the discussion of the link between local measurement and classical global measurability until §7, when we shall have in our possession the parallel concept of quantum global measurability.
Let us, for the present therefore, confine our attention to momenta P without critical points on M, and derive in this case the
connection between completeness and global measurability. To this ] end we find it convenient to group together several properties of a | maximal but incomplete integral curve of an incomplete vector field j into the following theorem, the notation and results of which will be î central in the subsequent discussion.
'G Note especially that the length d(0) can vary between integral curves and in particular need not exceed on every curve any fixed non-zero value. This variability is necessary to admit as globally measurable a large class of momenta whose vector fields possess
critical points, and whose maximal integral curves can be of arbitrarily short length.
32
Figure 2: To illustrate the local recoil sets B within the reference
— 2--- a
class defined on the integral curves of a vector field.
S.O-.'ï
The underlying graph is a family of maximal integral curves of the vector field X = x(-^ + y 3y), which has a set of critical points x = 0 which separates the flow into two distinct regions x ^ 0. We show
along typical flow-lines of the vector field a representative group of local recoil sets together with the local Cartesian coordinate directions (y^ ,y^).
Theorem 10: On the properties of an incomplete maximal integral curve