Date: 26-Apr-98 11:08 PDT
From: Chris Cole > INTERNET: chris@questrel.com
Subj: Re: Chit-chat about Noesis article
Robert Low wrote:
> In your 'objects and interactions' paragraph, you state that
> the properties of liquids and gases are explained in terms
> of a model in terms of small particles that bounce off one
> another. I'd agree about gases---kinetic theory works well
> for them: but the bulk properties of liquids are just
> beginning to be understood in terms of such models.
> Solids and gases seem to be much simpler to understand than
> liquids.
Your level of sophistication is showing. What I am referring to is
"billiard ball" physics. If you take a red liquid and a clear liquid
and mix them together, you get a pink liquid. Why is that? Because the
red atoms are mixed together with the clear atoms, just like when you
mix red sand with clear sand. The interesting thing is how hard it is
for us to even think of another model that explains this behavior. And
yet in Newton's time there were traditions of alchemy that had radically
different explanatory models. I recently read Michael White's
revisionist "The Last Sorcerer" in which White argues that Newton spent
a significant portion of his career doing alchemy, and that this has
been suppressed.*
When you get down to the details and read Newton's replication of
famous alchemical experiments, it is awe-inspiring to see how quickly
Newton realized what was really going on, and how bankrupt the
alchemical program was. It reminds me of Gell-Mann's reaction to silly
books like "The Tao of Physics." Anyway, I guess revisionists have to
make a living too.
I mentioned gases (which are easy) and liquids (which are borderline)
and left out solids (which are not well explained by billiard ball
physics). What holds all those atoms in those funny crystalline
patterns, anyway? And to give the devil his due, many aspects of
non-equilibrium chemistry (such as detailed reaction dynamics) are just
now being studied with ultra-fast laser pulses and the like. And of
course the alchemists based their whole system on combustion, which is a
non-equilibrium phenomenon (usually). When paradigms shift, frequently
the unanswered questions get swept under the rug. So perhaps there was
method in Newton's madness.
Yes, OK, mea culpa. My only defense is that the "interesting" part of
radiation was beta decay, in the sense that it led to the postulation of
the neutrino and the discovery of the weak force. Alpha and gamma decay
are fairly mundane. Besides, if I'd said, "beta decay is caused by the
weak force," most readers would have said "what's beta decay?" Everyone
knows what radiation is. Poetic license.
> From the paragraph on particles: the property you ascribe
> to the digits of the decimal expansion of pi, namely that
> any finite sequence of digits will appear at some point in
> that expansion, is true of the Borel normal numbers. Now,
> as far as I can remember, all but a set of measure zero of
> the reals are Borel normal, but the last I heard it was not
> known whether pi was in fact Borel normal. Am I out of date
> here, or were you assuming the fairly widely accepted
> conjecture that pi is Borel normal?
I was assuming this. I don't recall seeing anyone claim to have proved
it recently, but I think it's generally assumed.
> Now, on to something more constructive (and less petty).
>
> In 'convergence', you comment on the infinities plaguing
> fundamental theory. There are various reasons those
> infinities could crop up in, for example, QED.
>
> 1/ The universe is just like that. (Sufficiently ugly an
> excuse that nobody takes it too seriously.)
>
> 2/ QED is like that. (i.e. the attempt to quantize classical
> electromagnetism is inherently wrong-headed).
>
> 3/ QED is actually fine, but the perturbation analysis used
> to calculate quantities falls down eventually.
>
> I've read fairly recently that there are suspicions in the
> theoretical physics community that although lots of people
> think (3), it may be that (2) is actually the case.
>
> At the moment, fundamental theory is in a ferment. Something
> very good may grow out of the current obsession with
> membranes---I hope so, but I don't understand that stuff
> well enough to make an informed comment. I'll stick to the
> stuff I think I understand.
Well, it can't be JUST QED that is plagued with infinities, because they
show up in non-Abelian gauge theories too (e.g., QCD). However, I like
the idea that the infinities happen because the theory has been
"extracted" from a complete theory that includes gravity. Evidence for
this (hands waving wildly) is that the infinities are associated with
high energy (AKA short distance). This is where you'd expect a theory
to fall down that assumes that spacetime is continuous when it is in
fact quantized. Nature will only allow one theory. False or
approximate theories will be penalized.
> Last, and most interestingly, 'speculations'.
> You've certainly arrived at some thoughts there which are
> currently being looked at very seriously within the
> theoretical physics community---though maybe not in exactly
> the same form you suggest.
>
> For example, I don't know about using octonions as a
> framework for physics, though there have certainly been
> serious attempts to use quaternions, and such higher
> structures have been used in isolated places, though not as
> a fundamental framework (at least, not in the stuff I'm
> familiar with). There is, though, a vociferous school which
> uses Cayley algebra as a language for physics---if you want
> to chase that up, I can send you some references.
>
> There's also a school attempting to make fundamental sense
> of quantum gravity via what are known as spin-networks:
> these are combinatorial structures which take spin states
> as fundamental, and out of which space-time appears in some
> limit. Again, I don't know much more about them than that:
> some of John Baez's material, available through the WWW,
> might be of interest to you, and provide a pointer to the
> more technical literature. (Again, if you don't know about
> his stuff, and you want to, I can get you the URLs.)
>
> Anyway, thanks for an interesting, thought-provoking
> article. (Too much of the other kinds of stuff in
> Noesis :-( )
Octonions and Cayley numbers are the same thing.
Here is how I came to the octonion idea:
Take the fundamental relativistic equation of motion for a boson:
(1) (p^2 - m^2) * psi = 0See if you can find the "square root" of this equation, in the
sense of
finding a linear operator in p that when "squared" equals the operator
on the left side of (1). Trying an arbitrary linear function,
(2) (p[0] - a[1] * p[1] - a[2] * p[2] - a[3] * p[3] - a[4] * m) * psi = 0,
squaring and working out the algebra leads to the conditions:
(3) a[i] * a[j] + a[j] * a[i] = 2 * delta(i,j)
In other words, the a's are quaternions, which is the mathematics's way
of saying that the fermions have spin.
It is not unusual to have more types of numbers show up when you take a
square root. In fact, there is a progression from real -> complex ->
quaternion -> octonion, and it stops there! There is no normed algebra
with an identity element over the real field beyond the octonions. Thus
the "natural" progression seems to point to octonions. If we try to
pull Dirac's trick once again, we write down an operator like:
(4) (d[0] - o[1] * d[1] - o[2] * d[2] - o[3] * d[3] - o[4] * d[4]) * psi = 0
And we "square" it, we get the identities for the the o's that
define
the octonions, as expected. The really cool thing is that the d's are
the square roots of the p's; in other words, they are fractional
derivatives. How's that for exotic!
I invented the Mathematica system (well, its precursor, really) to try
to do calculations with these strange beasties. The problem was there
were too many possible solutions. This same thing has cropped up in
superstring theory. This is nature's way of slapping our hand: we're
really trying to get something for nothing. We're trying to get physics
out of mathematics, which isn't going to work. We need a really good
physical insight, like the equivalence principle, to guide us to the
mathematics that is correct. So I gave up on octonions, although I'm
sure that they'll be in there when the right idea comes along.
* The fact that Newton devoted the same huge energy
to his alchemical (and religious) studies that he
did to physics and mathematics is well-known to
historians of science; it would be fairer to say
that it's been largely ignored. --KL