From Osher Doctorow

In his 2003 paper, "Ensembles and experiments in classical and quantum
physics," J. Mod. Phys. B 17 (2003) 2937-2980, and quant-ph/0303047,
Arnold Neumaier continues his somewhat refreshing march down the
borderline between Nonconformity and Conformity in physics including a
revised analysis of Heisenberg's HUP in terms of Bohr's
Complementarity. Unfortunately, his section 3 (Complementarity) is
largely a defense of the existence of complementary quantities.

Max Jammer's 1974 volume (cited fairly often in my previous threads and
posts) actually anticipated Neumaier's 2003 paper, and Neumaier
conveniently even cites Jammer positively, but Jammer came to quite a
different conclusion from Neumaier in his volume, namely that the whole
questions of Indeterminacy (HUP) and Complementarity (HUP) are in my
language up in the air with little agreement on them by theorists.

The Complementarity people at least in their original and typical form
maintain that the space-time descriptions and the energy and momentum
conservation descriptions are "complementary" in the sense of not
simultaneously usable. Even Heisenberg was less extreme than this
version of Complementarity in a sense, since he didn't insist on this
simultaneity in practice while the Bohr people did.

It is instructive to examine Neumaier's criterion for complementary
Hermitian quantities f, g:

1) (f - x)^2 + (g - y)^2 = gamma^2, all real x, y

where gamma is any fixed real number 0. Neumaier tells us that this
captures phenomena where two quantities don't have simultaneous sharp
classical "values".

Neumaier retains his Nonconformity because, despite (1) or his form of
(1), he argues for the parallelism between classical and quantum
concepts. He does this through ensembles or (at least finitely
repeated) aggregates, which is another story (and which Probable
Influence (PI) deemphasizes).

Inequality (1) is actually quite interesting because it somewhat
facetiously resembles what a very advanced future computer might do if
asked for the meaning of life as in Hitchhiker's Guide To the Galaxy:
it would come up with a (almost meaningless) number like gamma, and in
fact it did in the movie (although not gamma).

Why does Neumaier think that he is accomplishing anything by inequality
(1) in his notation? I think that it is largely because he has had
more exposure to mathematics than to physics, especially insufficiently
to physics in the sense that he doesn't realize that the imaginary
scale is as much a scale as the real line and has a phase sense to it
(phase as fundamental physical type of state as with liquid, solid,
gas, plasma, superfluid, superconductor, Bose-Einstein condensate, and
arguably black hole).

He is also apparently entranced by noncommutativity of "complementary"
quantities as opposed to commutativity of non-complementary quantities.
He points out that in the algebra of all linear operators on the
Schwartz space S(R), position q and momentum p are complementary - in
fact, they are Hermitian and this follows from the canonical
commutation relation [q, p] = ih and Proposition 3.4 that he proves
which in addition to saying that the Pauli matrices satisfy (o1 - s1)^2
+ (o3 - s3)^2 = 1 for all s1, s3 in R (the o1 and o3 are two of the
Pauli matrices), p and q satisfying the above canonical commutation
relation also satisfy:

2) (p - k)^2/(delta(p))^2 + (q - x)^2/(delta(q))^2 = h/(d(p)d(q))
where d(p) is short for delta(p), any x, y in R, any positive delta(p),
delta(q) in R.

So we're back at square 1.

What does PI say about (1), (2), complementarity, and indeterminacy
(HUP)? These (1), (2), complementarity, and indeterminacy (HUP) have
no physical significance unless one were living in a complex non-real
phase! Even there, they seem to have all the significance of a
million-place approximation of pi, which is almost none. This is
because inequalities are almost meaningless in complex numbers and
complex variables, which differ from real variables in the absence of
ordering and a different type of multiplication! If ordering is
irrelevant in complex variables and relevant in real variables, what
good is finding inequalities in complex variables even if they reduce
to real number or real variables?

To be very, very concrete, let's say that real variances V(X), V(Y) are
really related by:

3) V(X)V(Y) = 0

while variances of quantum operators such as x and p replace 0 by
h/2pi. It is quite arguable that all this represents is a "scale
difference". We've already seen another example of this in [X, Y] = 0
vs [q, p] = ih. Does anybody really think that ih versus 0 in these
contexts are "profound fundamental variables" or "profound fundamental
constants" having deep explanatory power? Probably the same people who
believe that the speed/velocity of light c has deep explanatory power.
Sure, it looks like c demarcates a phase change and so does h. But if
one cannot provide equations that cross that phases or that tend toward
the phases from both sides (possibly differently from each side, and
maybe even different equations) without using the phase change
constants (and we can include 0 in [X, Y] = 0 above among these
constants), then statistical cross-validation has arguably not been
achieved by coming up with a constant. One might as well claim to
explain psychology by Freud's original idea of anal, oral, and genital
fixations (as constants!). He may have isolated phase boundaries, but
where are the explanations and (one-sided) limits and continuity? Gone
with the wind, perhaps because the physician is as fixated as the
patient.

Osher Doctorow