[FOM] Kieu's work
Laura Elena Morales Gro.
lemg at math.unam.mx
Wed Mar 10 19:31:15 EST 2004
Motivated mainly by the interest of Martin in the
subject I decided to go and have a look at it and
here is my opinion on Kieu's work.
Having gone through the paper shown in:
http://arxiv.org/abs/quantph/0310052
and some others of him, (by the way, he is a bit
uncareful in writing and defining his variables to
my taste) I assure you he has not an algorithm for
Hilbert´s problem. For to have it he needs to go and
interact with the physical quantum world and here is
where the difficulty lies. And this has to do not only
with unsurmountable technical difficulties when settling
an experimental device, different for each Diophantine
equation in question, but with Heisenberg's uncertainty
principle as well. Besides, he's trapped himself in circular
arguments. I'll try to explain in what follows what he's
doing. He pretends to solve a diophantine equation by modelling
(transforming) it into a perturbative hamiltonian and bringing
a physical system to a ground state under its action and then
look at the energy (the eigenvalues of such a hamiltonian) of the
system, in my simplified version. If the energy of the system
happens to be zero, then the equation has a solution in natural
numbers (the occupation numbers, the quantum principal numbers) if
not, then it has not. He makes sure he has a hamiltonian bounded
from below, beginning in zero energy, by squaring such a hamiltonian.
Even though he does not explicitly mention it, he has light (photons)
in mind when considering the physical system.
With this scenario let's go now to actual calculations. To measure
anything in each of your real systems (needed each one to prove the
algorithm for every diophantine equation you can think of and
supposing you can construct them) you will alter the system every
time you go to it and measure its energy states and/or occupation
numbers. This cannot be avoided because it is implicit in the
uncertainty principle which is at the core of the quantum theory.
Also, for the time required to attain a given state there is no an
idea, notwithstanding that time is finite and the estimations he
proposes.
Moreover, every time you (suppose) turn a dial to pass from an inital
hamiltonian state to the diophantine one no matter how careful how
slowly, you'll (among other facts) perturb states which were not taken
into account because of their low probability but that can anyhow be
altered in a manner to influentiate significantly your coherent states.
Then the probability conditions for a ground (final) energy state change.
By the simple approaching to your system to perform the measurements your
body is interacting with it. How will that affect your theoretical set up?
You can not tell anything unless you know the answers you are looking for
and not even so. In physics the hamiltonians are written when one knows
them and how they describe the system. An example of the unpredictability
of the effects and reactions of a quantum system in the presence of 'nothing'
is given in the following experiment (told to me by a talented particle
physicist): You have a beam of light that impinges on a board where there
are four holes in it. You observe light passes through the (say) first
two of them. Then you put aside (next to) your system a (humanly perfect)
tested device for any sort of fields (physical) and nothing is measured
around it so can be taken as a totally inactive neutral object but which
in its inside contains a solenoid. By the mere fact of setting it aside the
system, the beam of light will deviate, now it'll pass through the next two
holes...
There's nothing wrong in his theoretical approach to the problem but he
has only displaced it from the maths in logic to the quantum mechanics
maths without really dealing with the physical world. His is only a
mathematical curiosity. Not to mention proving his algorithm which he has
not done. He makes calculations with ideal simple models (of which he knew
the answer) and numerical maths methods. He calculates probabilities for an
ideal ground state system which can have or not zero energy deciding so if
the diophantine equation has a solution in natural numbers or not,
respectively. Solving ideal Schroedinger's equations with ideal hamiltonians
for states and products of them which are only handled mathematically is not
dealing with physics. His hamiltonians modelling the diophantine equations
have nothing to do with real, fundamental forces in nature. Forces which,
yes, somehow could be coupled and/or implemented to represent a perturbation
(the Diophantine hamiltonian needed in the system to bring it to the
required final state), perhaps, (depending in the DE) but, he has not done
that for one yet.
Suppose as well the Diophantine Equation selected varies smoothly enough
to couple with the adiabatic principle (which is a convergence theorem for
infinite time), a conceptual argument. But, every time you interact with
the possible system (assuming it can be constructed, which is not the case
and will not be) you alter its energy states, as was said, and for to
measure a particular energy, the uncertainty principle says you need an
infinite time. And you accumulate uncertainty every time you go and look
at (measure in) your system and you need to do that several times.
It is not an argument to support his work to say (as he does), for
instance, that there's nothing wrong in principle to prevent us form
approaching, physically, as closely as desired to the absolute zero
even though we cannot attain the absolute zero. Because precisely this
unattainability comes from the quantum world and is explained there
due to the actual impossibility of extracting the energy of all available
energy levels in an atom (a countable but unbounded peculiar set). To
extrapolate to it from the classical physics is forbidden. He has to deal
with the real quantum world to prove his results, even in one, the simplest
case. And be able to repeat his experiment and reproduce his results a
couple of times more. What would mean he has proved (solved for) only one
diophantine equation. But he, definitely, has not an algorithm for Hilbert's
problem. Nor will he. The quantum world, in its way, forbids it.
With best regards,
LE
Note: I acknowledge fruitful discussions with physicist aplorenz.
"Note added in print" I append to my note the remarks made to me,
and on the algorithms subject, by the physicist aplorentz whose clear
explanation on the experiment I oversimplified for my purposes. Here's
the complete correct version:
**************************************************************
First, I (aplorentz) must correct the description of the AharanovBohm
experiment you (LE) made at the end of your first paragraph:
(1) One starts with an electron beam hitting on a plate with two
pinholes on it. At the other side of the plate one places a screen
to see the interference pattern that the passing of electrons through
the holes would form. Same as light would do in a similar setup.
 

 electron beam  X < maximum of the interference patt.

 
Holed Plate Screen
(2) Now one sets a solenoid on one side of the experiment, which has been
tested to not produce any magnetic field outside the solenoid, and thus
hypothetically does not interact with the electron beam. Next, we switch
again the experiment: the pattern on the screen now appears shifted
respect to the one measured before.
 
X < maximum of the interference patt.
 electron beam  

O  
Solenoid Holed Plate Screen
The lesson is that the change on the configuration of the system alters
the whole answer, in this case the spatial probability distribution.
Second, I (aplorentz) would classify my objections to the "algorithm" in
three parts:
1. It is physically very difficult, if not impossible, to build a system
described by an arbitrary squared Diophantine's Hamiltonian. The point is
that the number operator, N, already contains a linear momentum squared.
Any larger power of N (and so on P squared) is just meaningless from the
physical point of view. (Squared P has the meaning of the kinetic energy
and the potential suppose to be only function of X.)
2. The suggested algorithm does not define the time T which is crucial
for two things: (i) it defines the rate at which the coherent state
Hamiltonian has to evolve into the squared Diophantine's Hamiltonian.
(ii) It also defines the time at which the spectrum of the system has to
be measured (the probability distribution of all possible states).
a) Measuring at the precise time T is physically impossible, the
perturbation introduced by such an instant measurement would introduce a
large (infinite) error in the determination of the probability of each
state. It would be necessary that the system stays at the squared
Diophantine's Hamiltonian for some time, as large as needed to make all
the measurements.
b) Moreover, the algorithm proposes to make as many times the experiment
as needed to find the one single state whose probability over passes 1/2.
If the first measurement fails (no such state is found), one has to
increase the time T and do the experiment again, and so on.
That means going back to the start each time the measurement fails:
(i) prepare the initial coherent state as close to the ground state as
possible (there would be always some uncertainty on that), (ii) evolve the
system again and measure the spectrum, and so on. This implies an enormous
technical control on the experiment. there is no guarantee that the
accumulated uncertainty in the process of all measurements would be under
control.
3. More physically important. A measure in physics is always bounded by
uncertainty principle, which means that non absolute numbers can ever be
obtained. Everything is always measure with some uncertainty. That is the
case of the energy of the "would be ground state" of the squared
Diophantine's Hamiltonian. The best we can do is to say that the energy
would be E with some uncertainty dE. That means that the ground energy
can only be determined to be (or not to be) zero within some
approximation!!!... However, to know if there is a solution to the
Diophantone equation one needs to absolutely know weather the ground state
has absolutely zero energy!!!.
(end of aplorentz's remarks)
***********************************************************************
Thanks Abdel for your corrections and contribution. LE.
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