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Appendix B
UNITARY FIELD THEORY
AND QUANTUM GRAVITY
B.1. The process of unification in physics
B.2. Einstein's
visions of geometric unitary field theory
B.3. Wheeler's
geometrodynamics. Gravity and
topology.
B.4. Quantum geometrodynamics
B.5. Gravitational
field quantization
B.6. Unification
of fundamental interactions. Supergravity.
Superstrings.
B.7. General
principles and perspectives of unitary field theory
B.4. Quantum geometrodynamics
To recap, the geometrodynamic view of the world of classical (non-quantum) physics is admirably uniform: there is nothing but empty space-time, all matter, fields, and charges are a manifestation of the geometric and topological properties of empty space. Classical physics could thus be fully unitarianized.
However, we know that
nature is much more diverse - more than 300 types of
"elementary" particles are known, the charge is
quantized, etc. Geons with their large mass and overall classical
structure cannot be directly related to elementary particles. Similar to
geometrodynamic electric charges generated in a curved a multiply connected
space are not directly related to quantized charges of elementary particles.
From this point of view, the whole classical geometrodynamics
appears only as an interesting physical-mathematical prank...
^{ }To harmonize geometrodynamics
with the world of elementary
particles it
would be possible to try to implant appropriate quantum fields and particles (meson, neutrino, etc.) into
"pure" geometrodynamics. Then it would be necessary to
introduce appropriate phenomenological coupling constants and
would appear other shortcomings of the existing apparatus of
quantum field theory (difficulty renormalizations), so that in
fact the appealing features of
geometrodynamics as a
perfectly unitary theory woul be
disapperared.
^{ }Wheeler went a different way: he
tried a quantum formulation of his classical geometrodynamics in
the hope that the resulting quantum
geometrodynamics
could explain the properties of elementary particles. In other
words, he investigated the consequences of quantum
uncertainty relations in geometrodynamics.
Quantum
field fluctuations and space-time geometry
The basic postulate of quantum mechanics is the well-known
Heisenberg uncertainty principle Dx. Dp ³ h , where h º h/2 p @ 1,05.10^{-27 }g cm^{3}/s is the Planck constant. The
relation of uncertainty DA. DB ³ h holds between every two dynamically coupled quantities A and B; for example
quantity A: | quantity B: | |
position x | ç-----c | momentum p |
time t | ç-----c | energy E |
electric field intensity E |
ç-----c | magnetic field strength B |
internal curvature of space | ç-----c | external curvature of space |
If we observe, for example, a magnetic field in a small spatial region characterized by the dimension L, there will contain the energy proportional to B^{2} .L^{3} and the time required to measure the field will be L/c; uncertainty relation DE. Dt ³ h then gives (DB)^{2 }.L^{4} ³ h .c, or DB ³ h.c/L^{2}. It can therefore be said, that the quantum fluctuations of the electromagnetic field in size L, are of the order of magnitude
D E ~ D B ~ D A ^{i} ~ Ö ( h .c) / L ^{2} . | (B.8) |
Thus, the field is constantly "oscillating" between configurations whose fluctuation range is greater the smaller the spatial areas we observe. Effect of these quantum fluctuation of the motion of the electron around the atomic nucleus is shown schematically in Fig.B.5 (these quantum fluctuation "overlap" Broglie waves through the Bohr model of the atom - see "Bohr model of the atom" in §1.1 "Atoms and atomic nuclei" monograph "Nuclear physics and physics of ionizing radiation", Fig.1.1.6) .
Fig.B.5. Schematic drawing of the motion of an electron around an atomic nucleus. A closer look at the Kepler trajectory of the electron would reveal the chaotic irregularities caused by quantum fluctuations in the electric field. The mean deviation from the global trajectory is zero, but the mean square deviation leads to a small shift in the energy level. This shift was actually measured as part of the Lamb-Rutheford shift. |
Assuming the universal validity of the quantum uncertainty principle, a similar situation must occur even in geometrodynamics: quantum fluctuations in the space-time geometry will be expected. In the spatial region with characteristic L dimensions (investigated in a locally inertial system with diagonal metric coefficients -1,1,1,1), the fluctuations of the space-time geometry, ie metrics, connections and curvatures, will be of the order of equal
D g _{ik} ~ L * / L , DG ^{l }_{ik} ~ L * / L ^{2} , D R ^{i }_{klm} ~ L * / L ^{3} , | (B.9) |
where
L * = Ö ( h .G / c ^{3} ) @ 1.6. 10 ^{-33} cm | (B.10) |
is the so-called Planck-Wheeler length, which was formally introduced as early as 1899 by M.Planck and whose fundamental significance was clarified by J.A.Wheeler in 1955.
Fluctuations
in geometry and topology of spacetime
In the scales L » 10^{-8 }cm, with which atomic physics
works, Dg ~ 10^{-25}; even for the scales L » 10^{-13 }cm of nuclear physics, the
quantum fluctuations of the metric Dg » 10^{-20 }are
completely negligible. Therefore, in all the situations we face
so far, we can rightly consider space-time as a smooth continuum. The basic postulate of classical
(non-quantum) physics - special and general theories of
relativity, that space is locally Euclidean, is very
well fulfilled.
Fig.B.6. At very small scales, the spontaneous quantum
fluctuations of space metrics increase (a, b). In the area of Planck lengths, these
fluctuations may eventually increase to such an extent that the
space becomes multiple continuous
(c) - they
grow into bizarre fluctuations in the topology (enlarged sections
in Fig.d).
However, if we go to ever smaller scales (Fig.B.6a,b), quantum fluctuations gradually increase, when in areas of size L » 10^{-33 }cm, where according to (B.9) Dg ~ 1, the fluctuations of the metric already so strong that the overgrowth melts in topology fluctuations - Fig.B.6c,d. The dynamic evolution of curved empty space in connection with the quantum principle thus leads to specific laws at very small distances: in microscales of the order of ~10^{-33 }cm not only the geometry, but also the topology of space fluctuates very strongly (Dg » 1).
From a normal point of
view, spacetime appears to us as a continuous smooth continuum.
Similarly, when we look at the surface of the ocean from a
high-flying plane, we see a completely smooth surface, only
slightly globally curved (in the shape of a globe) - Fig.B.7a. If
the observer jumps with a parachute and gradually approaches the
surface, he can see more and more clearly that it is rippled
(Fig.B.7b). When he finally lands on the water with a rubber
dinghy, he realizes how far the surface is to an perfectly flat
and smooth surface - the surface ripples sharply, spraying foam.
^{ }In meter scales, the local
curvature of the surface (waves) fluctuates strongly, in
centimeter and millimeter scales even the topological
structure of the surface fluctuates - drops separate, foam bubbles are formed
(Fig.B.7c).
^{ }
Fig.B.7. The analogy between the geometric-topological structure
of spacetime and the structure of the sea surface .
a) When viewed from
a height of several kilometers, the sea level appears to be an
ideally smooth surface.
b) From a height of
several tens of meters, the surface appears wavy, but otherwise
smooth.
c) It can be
seen from the immediate vicinity that not only the curvature of
the surface, but also its topological structure (bubbles, drops)
fluctuates strongly.
Similarly, in our
spatiotemporal "continuum" the smaller the microareas we observe, the more pronounced the quantum fluctuations
of the geometry will be manifested, until finally the space
topology itself will fluctuate strongly in the Planck length l_{p} » 10^{-33 }cm. For example, topological
tunnels will be formed and destroyed, closed areas will be
created (virtual "black
micro-holes" that will evaporate immediately quantum - see
§4.7 "Quantum radiation and thermodynamics of black
holes"), even new "micro-univers"
("bubbles" may separate which
then disappear; but with randomly generated fluctuations of
sufficient magnitude, they are likely to expand in inflation
and create a "new" macroscopic "universe" -
§5.5, part "Chaotic inflation and quantum cosmology", passage "The
emergence of more universes"). According to quantum
geometrodynamics, therefore, this seemingly empty vacuum is the
scene of the most turbulent microevents - spacetime has a kind of
"foamy" constantly spontaneously
fluctuating microstructure.
^{ }If an idealized point *) test
particle enters such a place, it will be mercilessly thrown
left-right, up-down, and chaotically here and there in all
directions, until it loses any idea of temporal sequence and
spatial proportions — space and time cease to exist for
it.
*) If the test particle had non-zero
dimensions, it would be immediately torn apart by fluctuating
tidal forces.
Quantum fluctuations of
spacetime: reality, or maybe forever just a hypothesis ?
To some extent, by formally combining the laws of quantum physics
and general theory of relativity, it can be concluded that in
areas around the Planck's length » 10^{-33 }cm, quantum vacuum fluctuations are so
large that not only the geometry fluctuates, but also spacetime
topology - spacetime here "boils" like a bubbling
"quantum foam". In these Planck-Wheeler
microscales, quantum foam is ubiquitous : it is not only in
singularities inside black holes, but also in interstellar space,
around us, inside the cells of our brain, inside atomic nuclei
and particles. Under normal conditions, however, the
quantum fluctuations of the spacetime metric are so slight, that no experiment has yet revealed them.
To "see" them, we would have to have a hypothetical
"supermicroscope" providing a magnification
of »10^{32 }times and higher *).
*) A more high-quality optical microscope
is able to provide a maximum magnification of about 3.10^{3 }times. The
fundamental limitation here is given by the wavelength of visible
light. The electron microscope can give a magnification of up to » 10^{5}-x. Tunneling
microscopes and electrostatic microscopes are under development,
which will make it possible to image even heavier atoms. However,
it is unimaginably far from Planck's details!
^{ }If we on the imaginary virtual
experiments on this "supermikroskopu" gradually
increased magnification on »10^{8 }x, we observed atoms
at a magnification »10^{13 }x would be able to "see" the atomic nuclei, the
nucleons and quarks inside them. But even with further growth of
magnification, the space would remain completely smooth, with
only a slight continuous curvature given the gravitational mass.
Only if we increased the magnification by another incredible 20
orders of magnitude to see the dimensions »10^{-32 }cm, we would first observe small, but
gradually larger and larger fluctuations in the curvature of
space. And at a magnification of about 10^{33} times, the space would resemble bubbling
stochastic-quantum "foam" (Fig.B.6d).
^{ }Directly achieving such an
magnification is of course impossible not only
for technical, but mainly for principal (quantum-physical)
reasons. Even complex and ingenious indirect experiments will not
be able to prove quantum fluctuations of space-time in the
foreseeable future (but see the following
passage on the possible influence of quantum fluctuations of
space on the speed of high-energy photons...). They will remain for a long time (maybe forever?) only
at the level of an interesting hypothesis ..!..
Possible practical
consequence :
Is
high-energy g- radiation moving slower than light
?
All electromagnetic radiation propagates in a vacuum at exactly
the speed of light c , independent of the movement of the
source and the observer. This is a basic finding, firmly rooted
in the special theory of relativity. Regardless of the wavelength
- speed c propagates radio waves, visible light *), X and
gamma radiation.
*) The classical dispersion, observed with light in the matter's
optical environment, originates in the (collective) interactions
of the electromagnetic wave with the atoms of matter; does not
occur in a vacuum.
Fig.B.7d) Influence of quantum micro-fluctuations of space-time
geometry on the speed of motion of high-energy photons of gamma
radiation.
However, in connection with the mentioned quantum-gravitational
fluctuations of spacetime, there may be phenomena which
may, in certain circumstances, call this basic initial statement
of theory of relativity into question somewhat *). Fig.B.7.d
shows a situation where two photons are emitted from a certain
source at the same time: one photon with a lower energy, ie a
longer wavelength, the other photon of high-energy gamma
radiation with a very short wavelength. For radiation with a
longer wavelength, the quantum fluctuations of the metrics are
averaged and completely smoothed out in the respective longer
scale, so that this radiation in the classical vacuum will move
exactly at the speed of light v = c. However, photons of
high-energy radiation g with a very short wavelength will be "more
sensitive" to fluctuations in the space-time metric on a
fine scale than low-energy photons. Such waves will travel along
a slightly undulating geodetic path, photons will in a sense
"intertwine" the unevenness of the path, caused by
subtle metric perturbations, and their effective velocity v_{ef} will be slightly less
than c . Locally, such a photon travels a slightly longer
trajectory than would correspond to a smooth space. We can
compare this to the movement of a car with small wheels and large
wheels on a bumpy road: when driving the wheels at the same
circumferential speed, a car with small wheels (copying a bumpy surface) will
drive a little slower than a car with a large wheel diameter (whose circumference extends beyond small depressions).
* ) This
phenomenon cannot be considered as a violation
or failure of a special theory of relativity, which is exactly
valid in flat spacetime without metric defects (or in a locally
inertial systems in GTR).
^{ }These
differences in propagation speed are reflected only at the very high
energy radiation g, in the field of GeV and TeV. Here, too, the
differences in speed are very small (of the order of 10^{-20}), without the
possibility of laboratory measurements. In the future, they could
only be demonstrated by a temporal comparison of the detection of
light and flashes of hard g-radiation from catastrophic processes in outer space.
At cosmological distances of billions of light-years, even these
slight differences in speed could "accumulate" and have
measurable effects (the problem, however, is to distinguish these
differences from the differences in emission times in the sources
themselves ...).
^{ }Interactions with
quantum-gravitational fluctuations of space can lead to
dissipative phenomena and a slight modification of the
kinematics not only of hard photon radiation, but also
of high-energy particles in space.
Huge quantum
microfluctuations
Quantum fluctuations cause the space to have, in addition to
macroscopic (gravitational, cosmic) curvature, also
"microcurvature" of radius of the order of L*»10^{-33 }cm and that the necks of
topological tunnels are formed everywhere, whose dimensions and
mutual distances are also of the order of ~L*. If we have a
topological tunnel of size L (and thus areas ~L^{2}),
there will be quantum fluctuations of electric field intensity of
the order of Ö(h.c)/L^{2},
so that the total field strength flow indicating the effective
electric charge will be of order of q ~ Ö(h.c) » 10.e,
regardless of the dimensions of the tunnel. The typical charge in
geometrodynamice but not directly related to
the charge of elementary
particles, because it is an order of magnitude larger than
the elementary quantum of charge e and is not quantized. The energy density
E^{2}/8p field in a typical tunnel reaches
enormous values ~ h.c/L*^{4} » 5.10^{93 }g/cm^{3} and characterized by mass-energy
per one topological tunnel makes m ~ Ö(h.c/G) @
2.2.10^{-5 }g » 10^{19 }GeV, which is at least 20 orders of
magnitude more than the rest mass of elementary particles (and
about 9 orders of magnitude more than the maximum energy
registered in cosmic rays).
^{ }These huge values are obviously at odds with the very low mean energy density
that we observe in space. However, if we take into account the
contribution of gravity to the density of energy and mass, then
two typical tunnel mouths with masses m_{1} » m_{2} ~ Ö(h.c/G) = ~10^{-5
}g, spaced r_{1,2} » L* @ 10^{-33 }cm, will have a binding energy E_{gr} = -G.m_{1}.m_{2}/r_{1,2} » - c^{2}Ö(h.c/G) during mutual gravitational
interaction. Mass defect of two adjacent mouths of topological
tunnels Dm_{gr} = E_{g} /c^{2} ~ -Ö(h.c/G) @ -10^{-5 }g, which is negative and of the same order
as the (positive) electromagnetic mass of both structures, can
therefore locally to compensate for
the energies of the
respective fluctuations. Such locally compensated fluctuations no
longer show gravitational attraction with more distant
concentrations of matter and energy. After such a total
compensation of the huge picofluctuations, the vacuum may look as
if we observe it.
^{ }However, the observed elementary
particles, which are probably not elementary
at all, have
dimensions of the order of 10^{20} times larger than the Planck
length, and thus could perhaps be a kind of "collective
excitations" (involving a large number of elementary
fluctuations ) in a sea of strong microgeometry
fluctuations, which are canceles everywhere else on average and
macroscopically form the usual "vacuum". Whether
this is the case and how it is going, no one knows yet ...
Micro
fundamental values of physical quantities
Therefore, even though the quantum geometrodynamice far failed to
explain the structure of elementary particles, provides important limit fundamental values of some basic physical quantities :
l _{p} º L * = Ö ( h .G / c ^{3} ) @ 1.6. 10 ^{-33} cm | - Planck's length | (B.11) |
t _{p} º t * = L * / c @ 10 ^{-43} s | - Planck's time | |
m _{p} º m * = Ö ( h .c / G) @ 2.2. 10 ^{-5} g | - Planck's mass |
Quantum spacetime
structure here shows, that a smaller distance than L* » 10^{-33 }cm and shorter time intervals than t* » 10^{-43 }s are irrelevant, because the spatial relations
and time relations lose their meaning here due to quantum fluctuation in topology. So from a
quantum point of view, space and time are not
a smooth
infinitely divisible continuum, but effectively have a
"grainy" discrete
structure of
some "atoms" or "elementary particles" of
space and time - it decays into elementary Planck lengths ("quantum of space") and Planck
times ("quantum of time", also sometimes
called chronons).
^{ }From the point of
view of nuclear physics, the issues of continuous or discrete
internal structure of space and time are discussed in the passage
"Is the world
continuous or discrete at the deepest level?" in §1.1 of the book "Nuclear
Physics and Physics of Ionizing Radiation".
^{ }We can imagine the following illustrative
justification for the smallest possible length: Visual
resolution of two nearby points in space requires the use of a
wavelength of light shorter than the distance of the points. The
energy of photons is inversely proportional to the wavelength, so
the closer the points are, the more photon energy we need.
According to GTR, this photon energy causes the curvature of
spacetime. At Planck's distance »10^{-33 }cm the curvature of spacetime (caused by the necessary
high-energy photons) would increase so much, that the points
would be inside the horizon of a (virtual) black hole - measuring
such a small distance becomes fundamentally impossible,
spacetime is no longer continuous here. At the Planck length, the
GTR loses validity, and the physical processes are controlled by
the quantum theory of gravity (§B.5 "Quantization of the gravitational
field"). Quantum geometrodynamics suggests that in the smallest
microscales there is a kind of "quantum of space"
on the background of a general manifold without a metric
structure, space can be likened to "quantum foam". This
idea could be important for quantum field theory
in general, since the energy of the quanta is inversely
proportional to the wavelength of the respective "wave
ball". If the wavelengths cannot be less than a certain
lower limit (in our case »10^{-33 }cm), because a shorter length
simply does not exist, then the energy
of the quanta is limited from above - quanta with infinite energy, which
causes such problems in quantum field theories, are in advance
excluded...
^{ }Planck-Wheeler units are natural
units for the description and modeling of natural laws based on
the properties of spacetime. The fact that there is the smallest length and shortest
time interval in space and time, about which it still
makes sense to talk about, is a knowledge that is beyond the
scope of geometrodynamics and perhaps even the whole of physics..!..
^{ }
Gravity, black holes and space-time physics : | ||
Gravity in physics | General theory of relativity | Geometry and topology |
Black holes | Relativistic cosmology | Unitary field theory |
Anthropic principle or cosmic God | ||
Nuclear physics and physics of ionizing radiation | ||
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