From: terry@aslws01.asl.dl.nec.com (Terry Bollinger)
Subject: Re: Uncertainty Principle [T.Bollinger => LONG]
Organization: (the process of converting a metal pipes in to an organ?)
Date: Sat, 5 Sep 1992 07:15:19 GMT

Howdy ya'll,

(A long one again -- so QUICK, hit that K before it's too LATE!!)
(Headers are provided if you want to take a quick topical scan.)
(Not recommended for readers who prefer group theory mathematics.)

In article <1992Sep4.170847.235@prim> prim!dave@germany.eu.net
(Dave Griffiths) writes:

> I just read (Stephen Hawking) a nice explanation of the Uncertainty
> Principle: to measure the position accurately, you need very short
> wavelength photons which will have such a high energy that they will
> bash the particle so hard that it's momentum becomes uncertain...

> ... is this explanation a simplification for laymen, or is that how the
> UP was really derived?  I thought that the UP was somehow more "fundamental"
> and not just related to practical difficulties of measurement...

THE "DISTURBED PARTICLE" INTERPRETATION OF QUANTUM UNCERTAINTY

Historically, some of the earlier books on QM certainly tended to describe
it that way, especially when folks with an essentially classical background
were trying to come to grips with it.  But it has problems, and does in fact
tend to "overtrivialize" the profoundly bizarre underlying principles that
are involved in QM.  E.g., Dr. Richard Feynman found such views misleading
and somewhat offensive, and in his writings he argues against relying too
heavily on such analogies.  In Vol. III of his Lectures he goes to some
lengths to point out that in Stern-Gerlaugh the separation of the base
states does *not* "disturb" the particles in the sense of irreversibly
preventing the original composite state from being recovered.  You can
"rebuild" the state simply by recombining the base state paths -- something
that makes no sense at all from a strict disturb-the-states perspective.

THE FOURIER TRANSFORM INTERPRETATION OF QUANTUM UNCERTAINTY

One of the most beautiful and readable discussions I've seen on this subject
is by Dr. Roger Penrose in his popular book "The Emperor's New Mind."  On
pages 243-250 of the paperback edition you will find two sections entitled
"The quantum state of a particle" and "The uncertainty principle."  What
you will read is a fascinating discussion of the remarkable symmetry that
lies between particles as they are described in "ordinary" space of three
dimensions, and "momentum" space in which they are positions according to
their momentum values instead of their ordinary-space locations.

In Dr. Penrose's discussion, the wavefunction that describes a particle
has the appearance of a corkscrew in both of these spaces.  (See my earlier
discussion of phone cords if you find that interesting.  A phone cord is
just a bound state of the same sort of thing, meaning that when a particle
wavefunction bounces back and forth in a small space it starts looking an
awful lot like a rotating phone cord -- which is itself a superposition of
two opposing corkscrew-like rotating states).

Now what is intriguing about these two ways of representing the same particle
is that they are "mediated" or related to each other by something called the
"Fourier transform."  And amazingly, therein lies the real answer to why
this odd little thing called the uncertainty principle must exist.  Without
beating on the exact definition of the Fourier transform here, suffice it to
say that one of its unavoidable features is that if you squeeze the length
of the corkscrew down in one space (it matters not in the least *which* one),
you will find it that the length of the corkscrew in the other space must
expand proportionally -- e.g., shorten the momentum space corkscrew by half,
and the one in real space will double (or vice-versa).  If you keep that
simple rule in mind you've really pretty much covered Fourier transforms
for corkscrews -- it only gets messy and complex when you start combining
corkscrews to produce more complicated wavefunctions.

When you keep in mind that the wavefunction gives ALL of the information
that can possibly be known about a particle, this kind of behavior becomes
profound indeed -- and not something understood by simple bat-the-critter-
around analogies.  When the corkscrew in ordinary space becomes very large,
you *cannot* know anything more about the particle until you "force" a
decision upon it, such as with some kind of detector.  (Again, see my earlier
email for a discussion of the probability issue -- it carries over exactly
for corkscrews, right down to the enclosed space of with a volume of "one".)

ON JOINT "MEMBERSHIP" OF A PARTICLE IN THE SPACE AND MOMENTUM "CLUBS"

The resulting relationships are like some sort of pair of clubs with a long-
term unresolved dispute about joint membership.  Join one of the clubs 100%
(make your corkscrew so short that you have an EXACT location in either
ordinary space or momentum space), and you are 100% LOST from the other
club.  Your location in it becomes scattered literally over the entire
universe, so that at any one location you're probability of being "found"
by a detector becomes flat-out zero.  A distressing state of affairs!

Conversely, if you are 100% exactly/precisely/on-the-schnoz located in the
the club we call Space by shrinking your little corkscrew wavefunction down
to an infinitely short rotating spike, you become 100% LOST in the Momentum
club.  (Velocity is still limited by good ol' relativity, but relativity does
not limit momentum.)  Your momentum could be of any magnitude at all, in any
direction at all.

THE ENERGY COST OF SPACE/MOMENTUM "MEMBERSHIP"

There is an irony in these membership relationships, because it turns out
that the COST of 100% membership in one or the other clubs is most decidely
NOT symmetrical -- the two spaces are similar in many ways, but by no means
are the identical.  The irony is this:  In terms of energy costs, 100%
membership in an exact location in the Space club is infinite -- it would
take more energy than exists in the universe to accomplish it!  The cause
of the problem is that it takes ENERGY to stretch that corkscrew over in
Momentum space, and stretch it you must if you want an exceptionally exact
location for your particle (corkscrew) back in the Space club.  I might
note that this has some implications for why particle research keeps
building bigger and bigger particle colliders -- they need those higher
energies if they want to explore the structure of the universe at smaller
and smaller granularities.

It's worth keeping in mind the next time you casually write down a diagram
that *assumes* a whole bunch of infinitely precise particle locations.  With
that kind of philosophical underpinning -- i.e., that any real incarnation
of one of those points would require more energy than currently exists in
the universe -- one might argue that it should not be overly surprising that
the necessity for odd little mathematical games such as "renormalization"
might pop up here and there in such analyses.

The flip side of the irony is that it does NOT take infinite energy to achieve
infinitely precise membership in the Momentum club -- although, alas, it does
require an infinitely long length of time!  Compressing the corkscrew in
Momentum space (again) implies an equivalent expansion of the Space club
corkscrew.  Ah, but it does NOT cost energy to spread things out in ordinary
space (at least in the absence of fields), so the energy cost of being very
precisely in the Momentum club is minimal and quite acheivable.

SIZE ASYMMETRY OF MATTER IN SPACE/MOMENTUM UNIVERSES

The two relationships are seen in the size of our universe.  If you truly
tried to build a universe that was "symmetrical" in its distribution of
paticle memberships between Space space and Momentum space, it would
instantly go ka-BLOOEY and expand drastically into the Space side of things.
Why?  Because it's cheaper that way -- the long intial corkscrews in Momentum
space would act like very energetic springs trying to collapse back together,
and in the process of doing so they would cause an explosion in the no-cost-
to-expand corkscrews of regular Space.

The irony in sum is that in some ways the idea of a truly point particle is
MORE real in momentum space than it is in our ordinary space, in that a good
phyisical setup can achieve a given level of "particleness" in Momentum space
at a very much lower cost than a device for measuring comparable levels of
"particleness" in ordinary Space.

HIDDEN VARIABLES IN QUANTUM MECHANICS

> Does it make any sense to talk about a particle having an unknowable, but
> precise, position and momentum (which would be the case if the problem was
> simply one of measurement)?

Good gravy yes it makes sense.  They're called "hidden variable" theories, and
such notables as Dr. Albert Einstein and (more recently) Dr. Bell of CERN
were both notable in their support of at least *thinking* about whether QM
might turn out to be such a thing.  Some of you might find it very surprising
that I mentioned Dr. Bell as a hidden parameter type, given the fact that he
was the person who first pointed out that their were REAL experiments that
could be performed to test the hidden-parameter hypothesis.  But I stick by
my guns -- if you read his collected work, it's hard not to get the impression
that one of the reasons he came up with Bell's Inequality was that deep down
he was sort of hoping that both Dr. Einstein and Dr. David Bohm were right
after all in supporting hidden parameter theories.  He was just very careful
and facts-first in his orientation, a true experimentalist as well as a most
intriguing theorist.

Dr. Richard Feynman was NOT a hidden parameter type, and the recent results
of experiments (e.g., Aspect) designed to test whether hidden variable might
lurk at the heart of QM seem to indicate that the answere is NO -- you cannot
do such things without violating both the rules of QM and explicitly observed
behavior in such experiments.  Indeed, the spin thought experiment I posted
recently rests on exactly that issue.  If the universe is full of hidden
variables, it's easy to show why my thought experiment does NOT cause any
kind of problem.  Alas, if it does not use hidden variables, then such thought
experiments can be very troublesome indeed.

HIDDEN VARIABLES, PILOT WAVES, AND THE DANGERS OF PHILOSOPHICAL EXTREMES

BTW, it's my guess that part of the reason of the reason why Dr. David Bohm,
author of a superb text on QM and a neo-hidden-variable "pilot wave" in his
interpretation of QM, unexpectely went New-Agey on everbody.  If you insist
that particles are real billiard-balls guided by real "pilot wave" fields
and take that view to its philosophical endpoint, Lo!, you do indeed wind
up with some gruesome problems in trying to reconcile the resulting info
links without becoming rather "holistic" in your perspective.  Even Dr. Bell,
who apparently admired Dr. Bohm's work considerably, once made a remark in
an interview that perhaps the reason everything worked out in QM experiments
was that everything was TOTALLY deterministic.  E.g., experimenters would
always look at at the right thing at the right time to get the results
predicted by the implicit action-at-a-distance rules of QM, not because
such action existed, but because the universe was set up in such a fashion
that it would LOOK that way.  The ultimate hidden-variable explanation!

NO HIDDEN VARIBLES OR PILOT WAVES FOR MOI...

For whatever it's worth, I don't buy the hidden variable school mainly
because it seems to cause more problems than it's worth.  And Dr. Feynman's
beautiful and elegant "integral of history" approache certainly doesn't jibe
well with the pilot wave interpretations that often accompany such ideas,
since you have to assume that somehow the pilot wave coincidentally takes
on EXACTLY the same form produced by the more subtle (and strange) integral
of history methods.  To me Dr. Feynman's insistence on taking the peculiar
behavior of QM at face value and to stop trying to apply meaningless big-
world interpretations on everything seems like a pretty darned good idea,
and one that handles an encounter with Occam's razor a lot better than
ideas that try to code data from the universe into every particle in it.

STEPHEN HAWKING AND SHIRLEY MCLANE

Final note, somewhat related:  I bought the latest Stephen Hawking book
was absolutely appalled by the introduction that seemed to imply that
"my MY, the reader had best take note of this physicist because, after all,
SHIRLEY MCLANE thinks he's really neat!"

Dr. Hawkings, if you ever read this net, SHAME on you for allowing yourself
(and those of us buying books) to be exploited in such a fashion.  I have
been appalled to watch the general decline of science in Western education,
and cannot help but thing how such nonsense subtly but definitively continues
that damage by giving readers the impression that Shirley McLane is somehow
"relevant" to becoming a top-notch physicist.

			Cheers,
			Terry Bollinger  (Speaking only for myself)
Newsgroups: sci.physics