Apropos Rukhsan's remarks on QM and geometry, I offer the interesting
remarks of Ed Dellian on the topic:
*****************************
Dear Bob,
thank you very much for recalling my work in the geometry of quantum
theory (QM). I think that the mysteries and paradoxes of QM are rooted
in its mathematical foundation by Planck, Heisenberg and Schrödinger.
Unfortunately this foundation is not disputed. But it implies a basic
mathematical mistake. just note that Planck's E/f = h = constant shows a
/linear/ relation of energy, E, to the concept of momentum, p. This
"linear relation" is nothing other than /a geometric proportion/, E ~ p.
If you put E over p, reminding that p = h/lambda, and lambda times f = c
= constant, you finally get E/p = c = constant (also known as Poynting's
vector, and in special relativity!). This I call /the "linear" concept
/of energy E. But from classical mechanics we know a /different concept
/of E. It stands in a /squared /relation to momentum p; it is the
concept E = p^2/2m; I call it/the "squared" energy //concept/. Both
concepts are used and confused in the mathematical foundation of QM.
Heisenberg introduced the "linear" relation E/p = c, while Schrödinger
based his equation on the traditional "squared" concept. Nobody realized
an inconsistency, because both men asserted their formalisms to be
equivalent. This seems indeed to be the case, since both concepts seem
to bear identical dimensions. But actually, even though the dimensions
are seemingly identical, the concepts themselves are not, as you can
prove by relating them to each other according to "E (Heisenberg) over E
(Schrödinger) = pc/p^2/2m". See that E (H) /E (S) results not in "1",
as it would be the case if the concepts were equivalent; rather it
results in E (H)/ E (S) = 2mc/p, or, with p = mv, E (H) / E (S) = 2c/v.
This confusion of incommensurable mathematical concepts is responsible
for a lot of mathematical complexity of modern QM, as I could show you
in detail.
Now, even though modern QM mostly uses the Schrödinger equation
only, confusion remains, because the "squared" classical energy concept
(wich is the basis of Schrödinger's equation) is itself a very
problematic one. It was conceived by Leibniz in 1686, when he proposed a
measure of "force" which he developed in considering the "force" of a
falling body in proportion to space, h. Now, since h is proportional to
the square of velocity, as has been shown by Galileo, Leibniz conceived
the "force" as being also proportional to the square of velocity. So he
gained the formula "force" = mv^2", which he called "vis viva", which
was later on called "energy". Somewhat later Coriolis (?) completed it
by the factor "1/2" at will, and so classical mechanics was based on
this Leibnizian concept in the "squared" form E = mv^2/2 = p^2/2m
(present also with the Hamiltonian H). Unfortunately, this concept bears
a congenital defect. Galileo has shown (Discorsi, 1638) that to put the
velocity of a falling body proportional to space leads into the
absurdity of the body to cover different places in space at the same
time. For this reason, Galileo developed the realistic geometric
proportion of velocity not to space /but to time/. Leibniz, however,
took the wrong proportion "velocity to space"; and this mistake is still
present with the classical "squared" concept of energy, E = p^2/2m. As
this concept lies at the basis of Schrödinger's QM, it is no wonder that
the absurd consequence of things appearing at different places in space
at the same time characterizes QM as well (entanglement, non-locality).
Note that this mistake of Leibniz was well-known to Newton who called it
a "wonderfully philosophical error" - but to no effect, as the error
survived, and infects modern QM, being responsible for the "mysteries"
and "paradoxes" of a theory of mechanics which would be clear and simple
had it not been led into the absurd by a mistaken mathematical concept.
Easter was fine, yes. Time was filled considering the fact that
according to experience of the balance, the center of rotation of a
many-body system (a galaxis, for example) _can never be a massive
object_. Just think of the center of a hurricane, or of the water vortex
at the outlet of your bath tube. Rather it must be a mere immaterial
point in space. Note that even the center of gravity of our system is
not the center of the sun but rather an immaterial point in space; note
that the sun itself rotates around that point (which was already known
to Copernicus, Galileo, and Newton). Therefore, "black holes" (massive
objewcts) at the rotation centers of galaxies are impossible.
Best wishes,
Ed.
*********************************
It has always bothered me that QM, as commonly conceived, has nothing at
all to do with mechanics. (I always refer to it as "quantum physics".
Perhaps Ed has made a connection?
The best to all,
Bob
> Dear All
> As Prof Kauffman has pointed out that there are many mysteries in
quantum
> theory which need to be decoded. The measurement problem being the
central
> one. And I agree with Prof Kauffman that taking the eigenvalue aspect of
quantum theory seriously and relating to Lambda calculus can help us to
understand its deeper aspects.
> However I would like to point to yet another related aspect. Spin is called
> essentially quantum mechanical property which has no classical analogue.
Yet when one does construct the formalism to treat spin we just use
SU(2)
> group which provides the double cover for SO((3) group and all of it was
known before quantum theory as well. Similarly fermions are also very
quantum objects but their algebra was once again developed by Grassmann
in
> an entirely different context. It begs the question how does the
Grassmann
> algebra which was developed to understand geometry is exactly the same
for
> building blocks of matter. Is somehow quantum
> properties of matter coming from geometry. You will be surprised that in
recent developments in quantum theory(Berry phases) it has been found
that
> important physical properties of matter are related to geometry and
topology of space of quantum states.
> So all of it suggests that we have a long way to go before we resolve
the
> paradoxes of quantum theory. Geometry and topology are going to be
beacon
> lights in this endeavor. I am not forgetting algebra and logic which are
already there in the quantum theory itself,Heisenberg commutation
relations
> are algebraic and logical expressions of underlying quantum world. Rukhsan
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