[FOM] Kieu's work

[Laura Elena Morales Gro.]

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/quant-ph/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 Aharanov-Bohm
experiment you (LE) made at the end of your first paragraph:

(1) One starts with an electron beam  hitting on a plate with two
pin-holes 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.