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QUANTUM GEOMETRY OF THE SPACE-TIME
Model “Dimensional Four” of Matter, Energy and
Vacuum Rafael Javier Martínez Olmo E-mail: Rafael@gravityquantum.com Madrid Spain Madrid, on Wednesday, April 22, 2009
Philosophical prologue. The object of this work is the unification of the General Theory of
the Relativity and the Quantum Mechanics, by means of the elimination of the
randomness in the second one. The motive by which it has been, until now, impossible to achieve this
unification, is in an ERROR in the Logic. No physical object or ideal logical
entity can possess two properties that is incompatible at the same time, to
exist and not to exist, to be continuous and discreet, to be random and to be
causal. This error infected the mathematics by means of the abstraction and
this infection happened to the physics by the mere imitation. Although the math, considering its abstract character, is almost
immune to this virus, it same it do not happen with the Natural Philosophy, which
today, we call Physics, considering its concrete character, being wanted for
to describe the reality of
the laws of the universe, and to submit it to the experimental contrast. Not exist anybody, who could think about any object or quality, so
much if it is ideal as if it is material, which exists and does not exist
simultaneously, except naturally, the ideal concept of mathematical point of
dimension = 0. The point exists, because it has an exact and real position in the
spacetime, which is determined by its four coordinates of spacetime. The point does not exist, because it do not have dimensions
(dimensions = 0). It has no height, it has no length, it has no breadth, and
it does not in the time. This incoherence is subtle and invisible like a virus. We know its
consequences for 2600 years, but its origin has always happened unnoticed. Historically
it has appeared, in front of us, three times. It always conceal in the concept of continuity. The first time that presented its incoherence is in the four sophisms
of Zenón de Elea, 2.600 years ago, and whose logic is perfect, closed, and
conclusive. These sophisms were presented to the local philosophers, in Athens. Perhaps
in front of Sócrates. (We know that Aristóteles and Platón, lightly posterior, knew perfectly the discourse given by Zenón). The second time that appeared again the contradiction is in the
invention of the infinitesimal calculation. In that epoch, many intellectuals
were against the logic of such a theory, since there is no way of explaining
the movement, the speed, or the acceleration, on terms of the point. The mathematical point has neither another point that is its following
one, nor its previous one. The second point, the third point, the point
quarter, etc. Not exist. The motive is that the set of the numbers Real (R) lacks an ordinal
number associated with its elements. In spite of being a completely tidy set,
it lacks good order. This contradiction, implicit in the “calculus”, and
never solved, by being irresolvable, since it take implicitly the incoherence
of the mathematical point (dimensional = zero) with that it is constructed
the mathematical "continuum" (R). However,
since that the calculation works, and the subtle contradiction was not
discovered, the opinions opositoras had to be quiet
unwillingly. By it, the movement entered the Incoherent Point Club. It is
continuous (by definition) but it is not possible of to demonstrate
mathematically in punctual terms, only it is possible of to demonstrate
mathematically in terms of intervals (e> 0). The third time that appeared the incoherence, it is in the birth of
the Quantum Mechanics (Principle of suspense of Heisenberg). The
incoherence of the point, it went unnoticed again, it broke in two to the
Physics, introducing the randomness in the quantum world. Since then the universe possesses simultaneously two qualities
incompatible: it is random and it is causal, depending on the size of the
object of the Nature that we is studying, random if this one belongs to the
microcosm and causal if the object belongs to the macrocosm. . Physical prologue. The universe and all its content has
four dimensions. The Spacetime has four dimensions. The universe, up to where we know
experimentally, also has four dimensions. The matter always has three spatial
dimensions (volume) and exists in the time, the energy always occupies a
volume of space and exists in the time, then both have four dimensions. The absolute vacuum, in the sense of
lacking both of matter and of energy, does not exist. There is no way of
isolating the gravity of a “ volume of Spacetime ”, for what we know
experimentally and of the astronomical remarks. The relative vacuum, which, at least, contains gravity, has therefore,
also four dimensions. Following Euclides, it is not
possible to construct, for mere addition, geometric objects of dimension
different from that of its elements of construction. If in the Nature only elements of four dimensions exist the paltry
and indivisible element of our universe has to be four dimensions, a volume
that exists in the time. Also because there do not exist objects
of less than four dimensions, except in the Platonic world of the ideas like
mathematical abstractions. To calculate its value we use the
fundamental constants of the nature, c, h, y G, like Max Planck was already
doing one century ago, so that they give us a volume - time, that is to say,
a volume during a unit of time.
In
System Cegesimal (centimeter, gram, second) Since we know, the universe is isotropous, that is to say, it do not
have spatial favorite directions, it present in any direction the same aspect
and properties. The distinctions that we do between length, breadth and high
place is merely semantic, since we can exchange to will its names and they
keep on representing the same physical reality. If we associate it the idea of isotropy with the elementary and paltry
objento
of the universe, this can be only, clearly, a sphere. To the being a sphere we can calculate the radius Lmo =
4,897 506 921 037 260 E-19 centimeters that it is the paltry distance and any
measurement of a distance is equal to the product of Lmo for a
Natural number (N). This distance is covered at the speed of the light in, Tmo =
1,633632464842480E-29 seconds. This is the interval of paltry time, and any
measurement of time is equal to the product of Tmo for a
Natural number (N). Since it is the paltry distance, there cannot exist a wavelength
shorter than Lmo, 4,897 506 921 037 260 E-19
centimeters. For being the paltry wavelength the supreme value of energy is
an Emo =
253,177 660 585 902 TeV. Any wavelength is equal to the product of Lmo for a
Natural number (N). Likewise, the value of any measurement of a quantity of energy is
equal to divide Emo by a Natural number (N) For the equivalence between mass and energy E = m • c^2, only we have
to divide for c^2 to obtain the value of the elementary paltry mass Mmo =
4,512 946 783 762 060 E-19 grams, and any measurement of a mass is equal to
the product of Mmo for a Natural number (N). This dimensional procedure was established by Max Planck to
determine the absolute units of the Nature, because it is obtained of the
universal constants, each of which comes expressed in its own dimensional
units, for example the speed of the light is a length divided in a time. Max Planck considered,
already one century ago, the length, the mass, the time, the electrical load
and the temperature as the fundamental elements to describe the Nature, and
using in addition to c, h, G, the constant of Coulomb and
the constant of Boltzmann. These last two constants is not used in this work, for considering them
to be unnecessary for its intention. The difference between the
dimensional procedure of Max Planck,
with him established in this work, is in the response to: “What is
fundamental?” Question that is humanly opinable,
against of the idea established at present, reflected well in the following
paragraph: “The system measures several of the
fundamental magnitudes of the universe: time, length, mass, electrical load
and temperature. The units Planck usually call (joking) for the physicists like the
" God's units ". This eliminates any anthropocentric arbitrariness
of the system of units”. Extracted from Wikipedia, Units of Planck. In
this work it is considered to be elementary, a volume of paltry and
indivisible spacetime of four dimensions. With a unique dimensional formula,
the fundamental units is obtained of the Quantum Geometry of the Nature. [G•
h • c ^-2] In the following table, we see
the differences between both methods.
The magnitudes calculated by Max Planck for
the length and the time, is extremely small and it involve quantities of
inaccessible energy to our technology. The costs necessary for its
construction exceed the imagination, all this, without entering its size that
would be cosmic. A wavelength concerning 10E-33 cm is equivalent to an energy of
3,89626E+11 TeV approximately, one trillion of times the
maximum energy of the current particle accelerators. On the contrary, the
mass is big and accessible to the experimentation; nevertheless, we have not
found any interrelation between its value and any experimental fact. Clearly it seems that the units of Planck is
erroneous and do not have any physical meaning. On the contrary, the magnitudes calculated in this work can be in the
correct way, and it is contrastable experimentally at present with the
particle accelerators which we have. The supreme quantity of energy 253, 177 TeV; it is
to only two orders of magnitude of the last particle accelerators, probably
the quantum geometry has been already photographed, but it have happened
unnoticed because we did not know what it was necessary to search. For the first time, the Quantum Gravitation there is scope of the
experimental physicists, who will be those that they determine if the
conclusions of this work is in accordance with the Nature or it is erroneous. Extract. If we understand the Differential calculus as a Physical Theory
(reality). Not like a mathematical model, which brings us near to this one
infinitely, but without reaching her. Using the mythological set of the Real
numbers (R). We see that it is the logical way that shows us the elementary
and quantum nature of the Geometry of the "Spacetime"; whose
elementary components, in the sense of without parts, is volumes in the time
(VT), of four dimensions (D4), infinitesimal (e> 0) and of curved topology
(p). This route based
on the Differential calculus, is the Quantum Geometry. The writing tries to
sit the bases and theoretical essentials of this way, which unifies the
experimental results of the Quantum Mechanics with the theory of the General
Relativity, by means of the hidden variables (Op). Spacetemporary
curvatures of the quantum events. The model is of relation and independently from fund (Quantum
Relativity), where so much the General Relativity like the Quantum Mechanics it
is modified, the first one is specified in an elliptical geometry of radial dynamics
(D = 4). The second one is completed introducing the causality, replacing the
statistical treatment (chance, random) with a geometric treatment (grounds,
determinist). The Quantum Geometry it give a causal explanation of the Quantum
Mechanics, establishing an one to one connection, between the statistical
results and the elementary geometries (without parts) of the
"Spacetime", which contains the above mentioned quantum event. Likewise,
it determines the geometry and the metrics of the Nature in its elementary
scale, specifying the model of Quantum Relativity. The attempts of unification of both theories, it have always failed
for the appearance of the infinities. The origin of these indeterminations is
in an equivocal reading of the "Calculus". Consequence of the bug
of expressing the continuity departing from points dimensional zero (D = 0;
R). The set of the real numbers (R) have allowed us up to today to
represent the physical reality to our scale. But he sinks of unequivocal form
on having faced the description of the elementary nature of the
"Spacetime"; that can be described only in elementary (indivisible)
terms, for geometric objects of “ Four Dimensions ”, Volumes in the Time
(VT); and represented by the set of the natural numbers. N: 0 ÏN Consequently, the continuous term is put up only to discreet under the
concept of mathematical point (D = 0; R)
(infinite divisibility). Physical singularities do not exist in the Nature, it
is the result of using the set of the real numbers out of context (scale). It
is mathematical singularities (division for zero) and Not physical. This
model is finite, quantitative, and predictive. Contrastable experimentally at
levels of energy between 1 and 8 TeV. It will be irrefutable at a level of energy lower
than 253,177 660 585 902 TeV. (Supreme
value of energy of an Elementary Quantum event). INDEX
a.
The Foundation
b.
The Principle of
“ Spacetime ” paltry
c.
The principle of
“ Paltry information ”
e.
The first Calculations and Results
IV.
The Development
of the Quantum Geometry
b.
We will have walked from R up
to N. The
Art of To Measure
d.
The Sophism.
Parménides and Zenón de Elea
f.
The Implicit Presumptions
g.
The ambiguity of the concept “Point”
h.
The ambiguity of
the Vacuum
i.
The ambiguity
of the “Calculus”
V.
The Hypothesis Basic of the geometry of “ Spacetime
“ a.
The Infinitesimal
Calculation
VI.
The Quantum
Geometry - the first Part a.
The Elementary
Geometry of the Nature. b.
The Paltry "Spacetime", The Dynamics 4D
VII.
The Quantum Geometry - the
second Part a.
A sketch of
Geometry Source b.
The Compatibility
with Newton's Gravity c.
The Modifications
in the “ Calculus” d.
The General
Theory of the Relativity (R.G.) e.
The Quantum Mechanics
(M.C.)
VIII.
The Cosmological Geometry
A.
The Quantum
Geometry or Elementary At present the Physics
it is divided in two; two models for two fundamental theories of all our
physical knowledge of the nature, and both is contradictory between yes. To
admit the essentials of one of them, he means to invalidate other one. We search from This situation is very
unsatisfactory at theoretical level, where both theories is opposite. In the
General Relativity, the constant of Planck is not born in mind, arguing its
smallness and therefore, its insignificance in the calculations and
essentials of the theory. On the other hand, the
Quantum Mechanics he does not bear in mind to the gravity, arguing of equal
form its smallness, in this case, the extreme weakness of its associate
field, which makes her, in fact, undetectable at small distances, that is to
say, in which there demonstrates the whole phenomenology described in the
experiments of the Quantum Mechanics. The essential
difference is in that for the Relativity Special and General, the idea of
causality is essential. On the contrary, the Quantum Mechanics he does not
bear it in mind by no means; it is more, it proclaims the nonexistence of
causality in the quantum world, everything materializes in functions of
probability on events. The experimental
physicist could have verified the validity of the General Relativity up to
scales of the order of the millimeter, but from there and considering its
extreme weakness we do not know anything. Although the scientific community
inclines for thinking, that at the distances at which the particle
accelerators is employed, the component gravitational must be paltry and
certainly without relevancy to quantum effects. He has gone over in the
particle accelerators at distances of the order of 10E- The situation is
disconcerting, both theories is loaded with good arguments, both is supported
by the experimentation and also with amazing precisions, but it is not
possible that the two is true, because they rest on fundamental contradictory
concepts, or one of them or the two is erroneous For the General
Relativity (R.G.) the "Spacetime" is relational with regard to its
content, the matter and the energy, (he influences and is influenced). The
"Spacetime" is curved by the presence of these, and mass and energy
move following these curvatures. For the Quantum Mechanics (M.C.) is only a
frame of reference (absolute or of Newton) and he has a secondary importance,
let's remember that it is a statistics, and in the statistics the variables “
spacetime “ is regarding external, neutral, not related to the events. (They
neither influence they is not even influenced by them.) Both lose its prophetic power, when he
changes the scale of distances between the point A and the point B. what is badly, he
is related to the magnitude distance and the ends of its scale II. “ Spacetime “ and Causality The General Relativity he shows
that the "Spacetime" is not only a neutral stage. That a
"volume - time" it is indistinguishable from any other "volume
- time". To having taught us that the curved matter to this one, and in
turn the "Spacetime". This way curved, governs the movement of the
matter and the energy. With which, it stops being a frame of neutral
reference to describe the physical phenomena, to the being another active
component of the nature, since he influences the mass and the energy and in
turn it is influenced by them. For it is reasonable to think
that the geometry of the "Spacetime", which contains the quantum
event, is the causal person in charge of the quantum phenomena, to the being
the geometry inside which they happen. It must be labeled in terms of
the physics like of “hidden variable”, "relational", “independently
from fund”, “determinist “and "quantum" 1.
When
the Quantum Mechanics this proposed the nonexistence of causality in the
quantum world he supposed a finished revolution in the same base of the
classic physics, where the principle of causality is out of discussion, any
effect has a cause and any cause creates an effect and there did not exist
physical experiment that was demonstrating the opposite. Nevertheless, in the experiments in
quantum mechanics the same cause always produces different effects. The same
ones in quantity and each of them with a probability that is always the same.
Following the lead let's say that the expressed results meanwhile per cent
they is 1st 12 %, 2nd 18 %, 3rd 40 %, 4th 8 %, 5th 3 %, 6th 9 %, 7th 10 %, so
many times as we always repeat the experiment will give themselves 7 results
and each of them in the aforesaid proportion. This with an extreme precision
let us say with not major change of 1 between one trillion 10E+12 of events. This leads us to asking us why
this mistake happens in the law of the causality that he says that the same
cause always produces same effect, imagine Vd. that supports an object with
the hand and releases it, the object will always fall down and also following
the same direction perpendicular to the soil [[1]]. It
always happens thus, as he confirms our daily experience. Now it be imagined that it is the
quantum experiment of the previous example. when we were releasing the object
he would fall down in seven different directions, and only in seven, and if
we were repeating the experiment a much raised number of times, we would
verify that each of these directions recurs in an exact proportion to earlier
described, but before to release the object we will never know that this time
is going to happen in particular: in whom of seven directions he will go? We
do not know it any more than in terms of probability. The first thing that was thought
is that perhaps, the mistake in the causality, owed not to have born in mind
anything, let's call him "x", that is to say a variable, or perhaps
several, and that once found it would know, thanks to its value, which of the
possible events would be produced and everything would be again like he was
always. So many Einstein as many people
others searched and searched which might be this or these “ invisible
variables ” because he is really difficult to admit, since it can be the
nature to small aleatory scale and to our scale to be a determinist, being
our scale a mere sum of the quantum states. However, for much that nothing is
searched he was, and one to an all the proposed ideas, which have been
formulated, they have been dismantled for the Mechanics Quantum, quite which
seems to indicate that God, not only plays the given ones, but it should be a
finished Ludópata. This supposed a rupture of the
theoretical structure of the physics, which gives like result that the
physicists think to the “way relativist” when they treat of the nature of the
macrocosm, and of “quantum way” when they treat of the nature of the
microcosm. Although as they see, they is
completely opposite; and one of the two, or perhaps the two, has to be
erroneous, since the nature cannot be simultaneously two mutually exclusive things,
aleatory and determinist. 2.
"Relational"
model. With this adjective I want to indicate that the model shares the idea
of the general relativity of that the "spacetime" is intimately related
to the mass and the energy and vice versa, being influenced mutually. 3.
Model
“independent background”. With this expression, he wants to indicate that the
model does not depart from any preconceived idea of the geometric structure
of the "spacetime". In the sense from if it is a space of Hilbert,
Mikonski, Riemann, hyperbolic, elliptical, Euclidean, “almost-Euclidean” or
from any other name and on this mathematical base to construct the theory,
much on the contrary it is the theory the one that generates in a simple and
natural way the quantum or elementary geometry of the universe. 4.
“Determinist”
shaped. I believe that I have already been sufficiently explicit, he consists
of the fact that the model considers fundamental and I validate the principle
of cause and effect or of causality, as the Relativity or the classic
physics. 5.
“Quantum”
model. With it I want to express that there is essential for the theory the
constant of Planck and the principle of suspense of Heisenberg. Moreover, the
central idea of its construction is based on the constants of the nature and
in the way of operating of dimensional
formwith them that Planck used.
III.
The Last one Route Bases of the model: 1.
The Infinitesimal calculation as
physical theory “Natural Calculation” is a hypothesis, very guessed right, on
the infinitesimal and discreet nature of the quantum geometry of the
"Spacetime". It is the base of this model of Quantum Geometry. 2.
The “Natural Calculation” confirms the
physical hypothesis of which it is possible to describe with major great
precision and mathematical generality, the curved and volumetric nature of
the geometry of the "Spacetime", if we always 3.
Planck operating in a dimensional way
with the fundamental constants of the nature; speed of the light in the
vacuum (c), gravitational constant (G), and of Planck (h); calculated a series of units of measurement
that take its name Length, Mass, Time of Planck. Whose values we understand
like absolute, in the sense of generated by the proper Nature, and that today
we use, for example, to predict at the distances and energies where the
quantum gravitation must show its effects. None of these units of Planck is a
volume time, it turns out to be clear that Planck did not think that a
"volume - time" had a fundamental and elementary character, yes on
the contrary, he considered as such: the length, the mass, and the time. Following the same dimensional process
of Planck, we calculate the natural unit of "Volume - time". It is
necessary to multiply the gravitational constant G by the constant of Planck h and to divide by the square of c (speed of the light) and we obtain, if
it is allowed me, the "volume - time" of Planck With only this universal The
existence of this VT of paltry value, he supposes likewise, for the size of
the dimensions of the "spacetime" a paltry and elementary value [[2]]. 4.
We have the physical evidence of which
the geometry of the nature always appears to “Dimensional Four” and us finite
(D = 4). The universe possesses four dimensions, both the energy, and the
matter, like the proper "Spacetime" that contains them there is
“Four Dimensional ones” (D = 4). Nobody knows any object or physical entity,
which should possess minor number of dimensions. Nobody knows any form to
change the number of dimensions of the physical previously mentioned objects
not also. Yes we can on the other hand, diminish the value (length) of any of
the dimensions of the "spacetime", but for the previous reflection
we cannot equal them to zero, since he would eliminate the dimension. 5.
Since the universe does not present
favorite directions, of a paltry, indivisible value 6.
The Nature he detests the mathematical
concepts zero and infinity, to the effect that they lack physical reality.
They do not exist in the Nature. 7.
If finally the Nature it is constituted
only by volumes - times (VT), and we reason with Euclides who is not possible
to construct geometric objects of dimension different from that of its
constituent elements, the elementary brick of our universe has to be also,
with all certainty, a VT (4D). Thanks to its
spherical form we can calculate the radio, obtain 4,897 506 921 037 470 And- Finally the supreme curvature (Cma) is the inverse one of the radio of
the sphere 4,897 506 921 037 470 E+19 cm-1. This
is the maximum value of curvature that a VT has, and it is the supreme level of
energy of a quantum elementary event 253,177660585902 TeV Mmo of 4,512 946 783 762 060 E-19 corresponds with the elementary mass gr. Notice that this geometric property of the
"Spacetime", has to take the matter as an origin, since it is the
matter the one that generates the "Spacetime" for radiation and on
account of its mass, hence the principle of minimal distance only affects to
the volumes time occupied by elementary material particles (with mass). In physical terms we can define this property of the
geometry of the "Spacetime", saying that the elementary masses
(Mmo), Lmo believe a "volume - time" of radio around him that he
prevents no other Mmo from being able to be over a distance lower of her. 8.
Any elementary mass (Mmo), it is a
geometric and energy origin of the geometric spherical creation, with radial
dynamics of the "Spacetime", to the scale of Planck,
Being
the geometry of any "volume - time" VT.
Where, Every elementary VT with mass believes
in the unitary time (Tmo), a spherical and concentric field around him that
he determines a "volume - time" in the shape of spherical ring [[3]] of section Lmo, which is a place of
the "Spacetime" where no other particle with mass can be, since he
would infringe the principle of paltry "Spacetime" (ETmo) The only way of achieving this, it is
radiating energy to a very short and energy wavelength 253,177 660 585 902
TeV to support its "private" inviolable "volume - time". The wavelength
is of 4,897506921037260E- This spherical radiation that comes from
every elementary particle with mass (Mmo) expands around him, of spherical
form, at the speed of the light, creating and giving form to the
"Spacetime", in the sense of dividing this one in volumes time
elementary (spherical rings) that Mmo can be occupied by other masses.
Without break the principle ETmo. b.
The principle of “Spacetime paltry” Every elementary VT with mass believe around him for emission of
gravitational radiation and on account of its mass, a spherical field,
volumetric and concentric of radio the paltry length (Lmo)
4,897506921037260E- c.
The principle of “ Paltry information ” Every elementary VT only possesses paltry
information of the Nature. N=1 Every elementary
VT only possesses a paltry instruction to process the information of the
nature. N=1 1.
The Quantum Mechanics he describes the universe
when the radio of the volumes - times, which contain the event, tends to
4,897506921037260 And- 2.
The hidden variables of the Quantum Mechanics
they is in the geometry, to paltry scale, of the "volume - time"
that contains the quantum event. Up to today, we have thought that the
curvature spacetime to this scale was almost flat, therefore, of almost void
influence. For against,
the quantum geometry, he places the origin of the curvature and therefore of
the geometry, in any particle with mass, therefore the supreme curvature or
maximum energy is over a distance paltry Lmo, of any "volume -
time" occupied by an elementary mass (Mmo). The supreme curvature is
1/Lmo. = 4,897506921037260E+ 3.
Since we know for Feynman, the quantum
electrodynamics (Quantum Electrodynamics Q.E.D.) were suffering in its
beginning of that all the results that he was giving were infinite. The
motive is that the sum of histories must bear in mind all the possible, and
these ways depend on the distance between particles, this distance becomes
zero, the zero introduces the infinity in the results. Feynman determined not to take the calculations
up to zero to avoid the indetermination, replacing it with a very small
number 10E- These problems eliminate, if instead of using a
small arbitrary distance we use the paltry Natural distance,
4,897506921037470 And- 4.
The geometry “ Dimensional Four ” at quantum
level behaves as if there was two-dimensional, since the radio determines
three spatial dimensions and the time the dynamic evolution, also the radio,
he has the same ordinal that the time, (there coincides the tag of the
spherical ring and the time since it was radiated). c.
The first Calculations and Results
IV.
The development of Quantum Geometry In the essentials of the theory
these affirmations have done: 1)
The existence of a paltry VT like natural
principle.(OLmo) (not reducible, not divisible). 2)
That the infinities that is obtained
when we work with R.G. and M.C., their have its origin in the use, in both,
of the set 3)
In the Nature not existed (absence of
reality) of the mathematical concepts, zero and infinity. These
three questions is more that related between if, and since we will see, its
clarification happens for understanding and fixing some ambiguous concepts of
an unequivocal way. For it we go away to support like essentials in the
logic, the theory of numbers and the infinitesimal calculation and like
example the sophism of Parmenides and Zenón. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
(quantum geometry).
