ideally simple form
c(a^3) = 0 . (53)
Turning now to compare the above modes of quantization, we
immmediately perceive that, whereas each procedure is based upon a 1 natural correspondence, each is nevertheless inconsistent with all the
others. There would seem, at present, to be no overwhelming reason i to prefer any one of the above procedures to any other. However, so
as to be definite in the sequel, we shall assume that the choice [26] is the correct one, since this results in the ideally simple canonical decomposition
QgCa^^PiPj) = -o(a^^PiPj) »
(54)
and, as has been demonstrated by Kimura^^^), in the attractive equivalent expression
Qo(^^^PiPj) = 8 *Qo(Pi)g*a^jg4QQ(pj)g * . (55)
in which as usual g denotes the determinant of the metric tensor j •
§4.2: Some remarks upon the general problem of determining the
quantities ^ [o,[n/2]].
Let us address the problem of determining in the general case the quantities shall outline two possible methods of
solution, discuss their limitations and advantages, and finally
'T tentatively find in favour of the second means of quantization.
165
Consider firstly the following general axiom:
[27] The formal quantum operators Q^(A) corresponding to the multilinear momenta A are such that
Qq((g^^P^Pj,A>) = (+ih)~*[Q^(g^jp^Pj), Qq(A)] . (56)
Whilst this is a natural and physically appealing quantization rule, it is nevertheless, as was demonstrated by Bloore, Assimakopoulos, and G h o b r i a l , in general inconsistent, the only cases when the above system (56) uniquely determines the operators Q^(A) (of at most second order) corresponding to reducible configuration manifolds, either of one-dimension, or of vanishing Ricci tensor, or of constant curvature. We may conclude, therefore, that this schema cannot have any general applicability in the problem of the quantization of multilinear momentum observables.
We may alternatively be less ambitious in our aims and demand instead that the system (56) above be valid only for those multilinear momenta which are constants of the free motion. The disadvantage of
this reduced scheme lies in the extreme computational difficulty
involved in the explicit calculation even of the lowest order observables Qo(A).
Restricting for the moment our attention to one-dimensional mani folds, we propose as our second axiom of quantization the following:
[28] The formal quantization of the multilinear momenta is such that the class of quantizable momenta is, in a sense to be made explicit below, "maximally large", the quantities Bn_2k(A) being assumed to have the general form
B„_,^(A) = , k e to,[n/2]] , (57)
' V 2 k + 1 ’" n
in which the quantities are real constants.
Recalling the necessary and sufficient condition's for the quantiza bility of a general local observable Q°(A) of theorem 7, and substi tuting the expressions (57) thereinto, we deduce, for the local quantizability of the observable A on the interval f = (a,b), the conditions
a j} ^ . (c) = 0, c £ {a,b}_, p £ [o,[n/2-|]] . (58) 2k lin-Zk-p+l'-'^n
These may be illustrated for the special case of n = 5, say, by the explicit system alllll(c) ^ 0 , =2*jll^^(c) = 0 ' =4*^llll(c) = 0 > (59) lllll/_\ _ _5_11111
1
a?: (c) = 0 , ^2^1111 (c) = 0 ,= 0 .
in which c £ {a,b) and where we have noted, in accordance with (8), that £^ = 1. It is now-immediate from the general pattern illustrated in (59) above, and the general independence of the various derivative conditions above, that, independent of the choice of f , the number of equations of constraint will be minimum provided only that
62^ = 0 , k £ Co,[n/2-|]] . , (60)
This set of equations uniquely determines the observables of odd order, and determines the even ordered observables to within a,scalar function £^a,^ n 11 « • • • X^ . Preservation of positivity under quantization ([25]) is
1 n
then sufficient® to set e^ = 0, so that, in the case of a one dimensional manifold, [28] leads to the quantization rule
Qo(A) = 5^(A) . (61)
®We omit the demonstration which is achieved by means of examples on the manifold (M,G) = ((R,l).
167
It is now natural to suppose, at least for the purpose of the sequel, that the above equation (61) holds quite generally for all manifolds, and all multilinear momentum observables. Indeed we
conjecture, though cannot prove, that this generalisation is a conse quence of axiom [28] above.
It will be well to note in closing that the above discussion of possible means- of quantization does not claim to be exhaustive; for example we have not, due to technical problems in effecting a general
(21-22)
comparison, included mention of We y l ’s rule or its generalis ations, though this is in fact (for see footnote 6) inconsistent with the methods of quantization [24] and [26] above.
§5: An Illustration of the Proposed Quantization Scheme.
We develop in this section, by way of an illustration of the above proposed quantization scheme, the explicit form of certain quantum observables defined on the configuration manifold (R of
coordinatization {x|x e (R} and with the usual metric. More precisely we shall, for a representative selection of C real-valued functions f
> . . . f
of the (complete) momentum xp, seek to determine the coefficients of the expansion
in accordance with the requirements.
[29] that the Taylor series decomposition of f(xp) enables the reduction of the quantization problem to that of the
observables Q^(x^p^) by means of the axiom 00 (k)
Q (f(xp)) = 2 ri--- Q (x^p^) , (63)
°
k=o
[30] that the quantization of the component observables k k
Q^(x p ) is prescribed by the equations
.
(64)
k k Let us, therefore, seek a decomposition of the operators Q^(x p ) in terms of the powers of the fundamental quantum momentum Q^(xp), so as to obtain the following theorem:
Theorem 8: On the expansion of the operators Q^(x^p^), k e N.
Ic Ic « «
The operators Q^(x p ) have a decomposition of the form