Distances and topology
• When it is suggested to define a notion of distance by
“counting the intermediate diagrams nodes”, it is evident that a few
precise details are needed.
By thus simply proceeding, in order to describe a (massless) photon
arriving on the Earth while coming from a “far away” galaxy,
one
would be led to a photon having travelled through a null distance
(within a null duration), that is not what is usually taken as
reference.
• One may propose to think by analogy with the
measure of an usual object constituted with atomes : its length may
seem to us well defined at our macroscopic scale, but may seem rather
“fractal” when one looks at a smaller scale. It is therefore necessary
to define how to “get the microscopical limit”.
Particularly, the definition of the “physical” derivatives is obtained
for small variations leading to
zero... without leading to zero : it must be specified
how far the “limit” is considered.
• Coming back to the counts on the diagrams, it is perhaps necessary to
take as well a kind of mean (may be a “lower limit”) within the set of
possible
diagrams connecting two events, but it is probably necessary for this
to define a “weighting” in the mean (or a way to consider the limit),
which
constitutes more or less the definition of a neighbourhoods filter,
that is to say a topology.
The suggested weighting might be based on a measure like Levin's one,
using
Kolmogorov's complexity
[1], or on the methods used to describe the flows in production lines
[2] or the interactions in the networks [3].
The metric then possibly proceeds from the
topology, but less likely the inverse [4, 5, 6]. What I fear then, is
that
the complexity of the mathematical problem would prevent any immediate
solution. Indeed, although my mathematical knowledge is rather
relatively strong for a physicist (when I considered to found a
research work in theoretical physics, I had in anticipation reinforced
my basic mathematical knowledge), I feel that the approach of this
problem is so different from what has been used previously that it
could require either a fantastic mere chance (a physicist who
accidentally encounters the needed mathematical tools), or a solution
by a high level mathematician (knowing a wide set of tools and curious
to take interest in the physical application of more “abstract”
theories). Bad luck, it is rather late for Grothendieck [7]. One might
also think to someone like Gregori Perelman, able to intrust himself
with a complex project without excessive inclination in hope for
rewards [8], but this kind of
persons is by nature little accessible (if you meet him, tell him to
look for possible interest in my divagations).
• Thus, one may consider that the ”neighbourhoods” of a mouving
particule would not be the same that those defined for the
“propagation middle” (this with respect to what one considers that the
particule is mouving) and that this kind of “topological Doppler
effect” would globally appear in the form of Lorentz's transformation
(or its
generalization for an accelerated motion).
• Moreover, as the different kinds of particules
dont interact in the same way, they probably dont
“see” space with the same
topology/metric : only an unified theory of
interactions would lead to a single metric.
Peculiarly, I am rather inclined to think that
gravitation is nothing but what remains, seen from afar, when one
considers the whole others interactions compensating each other by
mean. I am here induced to consider general relativity, finally, as an
approached macroscopic theory whose metric is not at all appropriate to
describe the microscopic level.
• Whether concerning general relativity or the quantum theories
describing interactions within a renormalized “vacuum”, some approaches
indeed seem to indicate that the integrability of the equations derived
from macroscopic theories would not be independent of the conformal
invariance of the underlying micoscopic phenomena [9].
• Finally, it would be perhaps interesting to search for an approach of
the problem founded upon the method of path integrals [10]. Starting
from the action A, the integration K(xfin, tfin
| xin,
tin) = 
over all the
possible paths leads to a difficulty if one
considers theses paths in a space supposed previously existing. It
would be useful to resolve the reverse
problem : to found a space (interactions network ?) where to
“integrate”, such as would disappear the divergences connected to
quantum theories. Perhaps should we have then the surprise to be lead
to an inexpected space, may be of an intermediate nature between the
countable infinity and the continuous one [11].
__________________
References :
1. see as example : “La complexité mesurée...”,
J.P. Delahaye, Pour la Science n° 314, december 2003.
2. see as example : “L'algèbre des sandwichs”, G. Cohen,
S. Gaubert and J.P. Quadrat, Pour la Science n° 328, february 2005.
3. see as example :
“Étude des réseaux par la
physique statistique”, A. Barrat, M. Barthélémy and A.
Vespignani, Bulletin de l'UPS, n° 212, october 2005 ;
“Vivre serein dans un monde cruel”, J.P. Delahaye
and R. Dorat, Pour la Science n° 346, august 2006.
4. if, starting from the set theory, one tries to build a geometry at
the “elementary” level, one is lead to associate a notion of “straight
line” with a propagator connecting two vertices ; the main part of the
reasoning stops as early as one considers the axioms of order on a
“straight line” : there is no other point but the two vertices and the
“geometry” which follows from this is too rudimentary to be really
useful ; it seems therefore that one would rather have to build the
geometry at the macroscopic level on the basis of a new understanding
at the elementary level ; some similar difficulties had been moreover
evoked as early as Euclide, about the a priori that might be understood
by some of it's axioms ; see as example :
“Foundations of geometry for
university students and high-school students”, R. Sharipov,
fr.arxiv.org/abs/math/0702029, february 2007 ;
“Les avatars de la rigueur mathématique”, É. Barbin, Pour
la Science n° 356, juin 2007.
5. this may also involve some less usual structures like the
pretopologies, or even a couple of
pretopological structures respectively connected to the sources and
detectors devices ; see as example : “Les
mathématiques des frontières floues”, S. Dugowson, Pour
la Science n° 350, december 2006.
6. this may moreover put as much critical questions as mathematicians
are previously circumspect about problems such as those connected to
the axiom of choice (independently of the physical use of mathematics)
; concerning the applications, it seems to me that the sole existence
of
Banach-Tarski's paradox involves the redhibitory feature of the strong
form of the axiom of choice in so far as to describe physical space at
usual scales, more especially as the weak form associated with
Lebesgue's measurability leads to mathematical properties which are
sufficient enough to do that (and that it may be moreover considered to
add some more axioms) ; on the other hand, it is not evident that the
judicious set of axioms should be the same for the description at the
“elementary” microscopic scale considered here (from which rises the
interest for the matematicians to continue to work unpre-judicially on
the different possibilities) ; see as example : “Coloriages
irréels”, J.P. Delahaye, Pour la Science n° 328, december
2005.
7. see as example : “Grothendieck : au fond des choses”, A.
Hobeika, Pour la Science n° 334, august 2005.
8. see as example : “Un mystérieux mathématicien pour une
complexe conjecture”, C. Dumas,
sciences.nouvelobs.com/sci_20060818.OBS8690.html, august 2006.
9. see as example : “Les mille et une facettes de
l'intégrabilité”, D. Bernard and P. di Francesco, Pour la
Science n° 336, october 2005.
10. see as example :
wikipedia,
http://fr.wikipedia.org/wiki/Intégrale_de_chemin ;
lessons by C.
Cohen Tannoudji,
http://www.phys.ens.fr/cours/notes-de-cours/cct-dea/index.html.
11. see as example : “Imaginer l'infini, ou le découvrir
?”, J.P. Delahaye, Pour la Science n° 370, august 2008.
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