popper-logic-scientific-discovery

some observations on quantum theory 215
believe to be incontestable. If a statement concerning the position of an
electron in atomic dimensions is not verifiable then we cannot attribute
any sense to it; it becomes impossible to speak of the “path” of a
particle between two points at which it has been observed.’ 6 (Similar
remarks are to be found in March 7 , Weyl, 8 and others.)
Yet as we have just heard, it is possible to calculate such a ‘senseless’ or
metaphysical path in terms of the new formalism. And this shows that
Heisenberg has failed to carry through his programme. For this state of
affairs only allows of two interpretations. The first would be that the
particle has an exact position and an exact momentum (and therefore
also an exact path) but that it is impossible for us to measure them both
simultaneously. If this is so then nature is still bent on hiding certain
physical magnitudes from our eyes; not indeed the position, nor yet
the momentum, of the particle, but the combination of these two
magnitudes, the ‘position-cum-momentum’, or the ‘path’. This interpretation
regards the uncertainty principle as a limitation of our knowledge;
thus it is subjective. The other possible interpretation, which is an
objective one, asserts that it is inadmissible or incorrect or metaphysical
to attribute to the particle anything like a sharply defined ‘positioncum-momentum’
or ‘path’: it simply has no ‘path’, but only either an
exact position combined with an inexact momentum, or an exact
momentum combined with an inexact position. But if we accept this
interpretation then, again, the formalism of the theory contains metaphysical
elements; for a ‘path’ or ‘position-cum-momentum’ of the
particle, as we have seen, is exactly calculable—for those periods of
time during which it is in principle impossible to test it by
observation.
It is illuminating to see how the champions of the uncertainty relation
vacillate between a subjective and an objective approach. Schlick
for instance writes, immediately after upholding the objective view, as
we have seen: ‘Of natural events themselves it is impossible to assert
6
Schlick, Die Kausalität in der gegenwärtigen Physik, Die Naturwissenschaften 19, 1931, p. 159.
7
March, op. cit. passim (e.g. pp. 1 f. and p. 57).
8
Weyl, Gruppentheorie und Quantenmechanik, 2nd edition 1931, p. 68 (cf. the last quotation in
section 75, below: ‘. . . the meaning of these concepts. . . .’). *The paragraph referred to
seems to have been omitted in the English translation, The Theory of Groups and Quantum
Mechanics, 1931.