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4 only if A is invariant under the flow of X. The unique sel-f- |

o

adjoint extension of such a. Q.(P), Q (P) , the local quantum if

A A

momentum in A, can be given explicitly as, |

Q

(

) = [q%(p)] .

(64)

y

where Q.(P) has differential expression _A :|

I

1

Q^(P) = -ih(D^ + ^ div X) x(A) , (65) J

i

38

DQ^(P) = e |Q^(P)(|; e . (66)

Proof: see appendix 9.

The principal difficulty lies in the restricted sense in which the Q^(P) of theorem 12 may be considered "local". It is, in general, not possible to define a Q^(P) oa an arbitrarily pre-elected region A of M, so that an apparatus whose sensitive volume is A cannot be

associated with a suitable local operator; equally a "local" observable of non-zero domain A cannot, in general, be measured by an apparatus since the extent of the region A may be metrically very large or even infinite. It follows, therefore, that an alternative method of describing the local behaviour of observables must be sought in the general case, but, before proceeding to do so, we establish explicitly the link between the well-formed "local" observables and the global observable from which they are derived.

Theorem 13: On the reconstruction of a global momentum.

Let { (P)} be a family of well-defined local observables a

generated from the global observable Q(P) by Q (P) =

t ^

[tt Q(P)ir ] , such that {A } forms a partition of M; then we_{a a ,} have the identity.

Q(P) = Z Q. (P) . (67)

a a

Proof: see appendix 9.

§6.2: Uncertainty and measurability.

Having ascertained that local momentum observables cannot, with any degree of generality, be defined in accordance with the desiderata [30] and [31] above, without restriction of the class of wave functions upon which local measurements may be performed, we consider in this

I

section some consequences of an attempt to admit local measurement by such a restriction. The elected set of admissable wave functions must both be sufficiently narrow as to admit of localizability and yet

sufficiently broad to allow the accurate measurement of momentum. But it is at the heart of quantum mechanics that these two concepts, know­ ledge of position and knowledge of momentum, are, in a sense, mutually exclusive, as is embodied in the Principle of Uncertainty. It is clear, therefore, that this fundamental uncertainty will be central to the discussion of quantum measurability. It is this relationship which will lead us again to the concept of completeness as a necessary

condition for global measurability.

We first address the problem of defining the form of the

Uncertainty Principle appropriate to a metrically finite, non-zero, region A of M, within which we seek to determine the momentum P. We shall assume that

[32] there exists a local chart {x^|ie[l,n]} covering A on which P has the canonical coordinatization p^;

then, since the pair (x‘ ,p^) is canonically conjugate, it is clear that the desired Uncertainty Principle will be constructed upon the quantum analogues, Q(x^), Q(p^). However neither Q(x*) nor Q(p^) can be simply defined as a self-adjoint operator over M, so that their

definitions, conditions for existence, and meaning need be separately discussed.

The chart {x^|ie[l,n]} is, in general^^ , non-global, so that x^ is not well defined outwith a subregion of M. However we may define a global observable in accordance with the desideratum [30] of §6.1 by the expression,

2 2Where [x^|i e [l,n]} is global, we may also define Q(x^)=x', DQ(x* ) = {i|j c (M) I Q(x^ )t|j e cC^ (M)} . It is clear that Q(x*) as defined in (68),

(69), is the local observable generated by the global operator Q(x^) within the region A.

40

Q x^ ) = x^ x(A)

(68)

(which is to be interpreted as a function on whose expression

[33] Qq(P) may have no self-adjoint extensions, when Q(P)

clearly does not exist. However we may nevertheless define an Uncertainty Principle by restricting the class of admissable wave-functions to a suitable sub­ set of C^(M), when we may interpret Q(P) as being simply the symmetric operator Q^(P).

within A coincides with the coordinate function x^ (x), and outwith A, f X(A) being the characteristic function of A, is zero identically), and

by the domain of definition,

DQ(x‘ ) = {i|; eX^(M) |Q(x^ )?];■ €X^^(M)} . (69)

Q(x^), being self-adjoint, global, and reducing to x' on A, is clearly the natural quantum analogue of x*.

If x^ is not global, then p^ is not defined everywhere on M; how- ;! ever, since p^ is the local representation of the global momentum P, -J the natural quantum analogue Q(p,) of p, is formed by Q(P)^^ whenever_{1 1}

‘

this latter exists. As was discussed in §3, the operator Q(P) is a self-adjoint extension of a uniquely determined operator Q^(P). Three possible situations may arisej

1

[34] Qq(P) may have many self-adjoint extensions of which *1

at most one may be Q(P); (we assume that such a unique Q(P) does indeed correspond to the quantum momentum, although its exact domain of definition is unknown.)

[35] Q^(P) has a unique self-adjoint extension equal to Q(P) and defined by (5) and (6).

In every case, we construct the commutator of Q(x^) and Q(P) to deduce,

[Q(x' ), Q(P)] c -in x(A) . (70)

Hence we deduce, following the standard manipulation^* , the derived 5 form of the Uncertainty Principle,

AQ(x’ )AQ(P) > 1 p(A) , (71)

in which p(A) = < ij; I X(A) | > is the probability that a particle endowed with the momentum Q(P) will be found in A. The domain of functions ip

to which (71) applies is as yet unknown, since the domain of Q(P) is, in general, unknown; however whatever the exact form of DQ(P), we may define an explicitly known subset of the commutator domain by,

y^(A,P) = € C~(M)

I

c“(A)}

The set P^(A,P) consists of, in a sense, "ideal" wave functions; for not only do they obey the Uncertainty Principle above, but also an apparatus may be constructed where sensitive volume A encloses a maxi­ mally connected subregion of their support, and may certainly, there­

fore, be used to measure the local momentum values within A.

Consider now the progress of a momentum measurement conducted j wholely within the set A. Two aspects of the local measurement may j

3

[36] no account is taken of position or momentum values outwith A, and

[37] the measuring device of sensitive volume A will record the momenta only of these particles perceived immediately before the measurement as lying within A.

see, say, R o m an^^^\ and note that all that is required in that Q(P) be symmetric.

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