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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.2. Einstein's visions of geometric unitary field theory
A. Einstein firmly
believed that nature, although literally bursting with the
diversity of various structures and phenomena, is very economical on basic principles. In the spirit of this vision,
Einstein worked on unitary field theories, after the creation of the general
theory of relativity, until the last days of his life. The idea of unitary field theory is extremely deep and beautiful:
according to it, there should be a single,
completely basic and all-encompassing physical field, the special manifestation of which would be all observed fields in
nature (gravitational, electromagnetic,
fields of strong and weak interactions and their manifestations
in subnuclear physics, for example); and thus all
events happenings
in nature. In fact, there is nothing in the world but this field,
of which everything is composed - even material formations (eg
particles) are a kind of local "densification" of this
field.
*) Speculations
on unitary field theories^{ }
Einstein continuously published these unitary theories [78,
volume 2]. Nevertheless, there were reports (or speculations and
assumptions) that Einstein had succeeded at the end of his life
in creating a truly perfect and "functioning" unitary
field theory, which he refused to publish for fear that
the world was not yet ready for its consequences, which could be misused.
As tempting as it may sound, and many people believe it, these
reports are unfounded. In the light of the
further development of unitary theories of the field, it turned
out that despite all the erudition and thought mastery of
A.Einstein, the path he had taken could not lead
to such a result.
^{ }In this context, the so-called Philadelphia
experiment, which was allegedly carried out by the
US Navy in October 1943 using an electric generator developed by
Nikola Tesla, is often mentioned. Around
the deck of the large warship Eldridge, moored in the port of
Philadelphia, coils were placed, powered by a generator on board
with a strong electric current. The aim of the experiment was to
"make invisible" the ship due to a strong
electromagnetic field, which should allegedly "deflect"
light..?.. Instead, under the influence of this strong
electromagnetic field, the ship Eldridge allegedly disappeared,
shrouded in fog *), appeared shortly in the 400km distant port of
Norfolk and returned to its original location after a few hours (it was allegedly accompanied by various catastrophic
phenomena, including the disappearance of sailors).
*) If there is any truth to this whole story - as
it probably is not - the fog could have been
caused by smoke from the burned windings of electromagnets
overloaded by a strong electric current ..?..
^{ }^{ }It was supposed to be a consequence
of Einstein's unitary field theory, when strong the
electromagnetic field "opened a new dimension in space -
time" through which "teleportation"
took place. Here, too, these are false
unsubstantiated legends and the mystification of
the character of science fiction or UFO, spread by sensation-seekings
conspiracy authors. The electromagnetic field used, although it
might have seemed fantastically strong at the time, was in fact insignificantly
weak to produce any observable relativistic effect (after all, even the strongest field would not
lead to the stated effects according to the unitary
field theories..). At present, much stronger
electromagnetic fields are commonly used in many applications (eg accelerators or nuclear magnetic resonance, often
with the aid of superconducting electromagnets), and of
course there are no indications of similar
effects ..!..
So far, we have stood firm in the position of the source ® field : there is a source (which is in a sense "primary"), that excites the field around it, and the task of physics is to determine the laws by which the source creates this field. The source is something different from the field, it is a kind of "substance" - an foreign-born element to the theory of the field itself. Looking at the Maxwell equations F^{ik}_{;k} = 4p j^{ i} or in Einstein equation R_{ik }- ^{1}/_{2} g_{ik}R = 8pT_{ik}, we see that on the left side there is an expression describing the field and on the right side a quantity describing the source. If we compare the character both quanties, we can say, along with Einstein, that "phenomenological" source on the right side (tensor of energy-momentum T_{ik} or four-stream j^{ i}) acts in comparison with concise expression describing the field on the left side, as "wooden hut next to a palace of glass and aluminum". In perfect field theory there should be no such dualism, a source different from the field should not exist; the source should also be "composed" of an field.
Thus, the unitary theory of the field the classic problem "Which way the source excite the field around it?" he turns completely to the head and asks: "How is what we consider to be a source, composed from its field?". The problem of field excitation, as well as the problem of the interactions with other particles with this field, then automatic disappears - everyting is a field, that develops in a certain way (according to its internal laws) in space and time. Only in our observation do some areas of the field appear to us as "sources" and others as excited or acting "fields".
In the 1920s, physics
knew only two types of forces: electromagnetic and gravitational. Both of these forces decrease
with the square of the distance from the (point) source, and the
speed of gravity according to the GTR is the same as the speed of
light - so the idea was suggested that they were somehow
interconnected.
^{ }^{ }After
creating a general theory of relativity - which is actually a geometrization of
gravity - A.Einstein worked hard for almost 40 years to create a unitary
theory of the gravitational and
electromagnetic field and thus complete his impressive life's
work. This is because the electromagnetic field has many similar
properties to the gravitational field (§1.4-1.5), so it
naturally offered itself as the most suitable
"candidate" for geometrization and thus for unification
with an already geometrized gravitational field. And the most
natural way to include electromagnetism in gravity seemed to be
to generalize the geometric properties of Riemann's curved
spacetime GTR, so that the newly formed
geometric structures somehow describe the electromagnetic
field.
^{ }^{ }The
unitary theories of the gravitational-electromagnetic field,
developed in the 1920s and 1940s by Einstein and other
physicists, did not lead to the desired result, and therefore we
will make only a very brief mention of them (for
more details see eg [169], [78], [146]). These theories can be roughly divided
into two groups :
a)
Generalization of geometric properties of 4-dimensional spacetime
The first attempt in this direction belongs to H. Weyl, who in
1917-19 generalized Riemann's geometry in the sense that with the
parallel transfer of a vector around a closed curve, not only the
direction but also the size of the vector can change. In this
Weyl (conformal) geometry, the general covariance group (used in GTR) is
extended by calibration transformations of the metric g_{ik}
g ' _{ik} = l (x) . g _{ik} , | (B.1) |
in which the lengths of all vectors at a given point are multiplied by the same arbitrary coefficient l, which may vary from point to point. The length of the vector l then changes according to the law at infinitesimal parallel transmission
d l = - l. j _{i} dx ^{i} . | (B.2) |
Thus, in addition to the fundamental quadratic form ds^{2} = g_{ik} dx^{i} dx^{k}, another linear differential form dj = j_{i} .dx^{i}, describing the nonintegrability of the length of the vectors, arises in Weyl geometry. The quantities j_{i} are components of the four-vector and during the calibration transformations (B.1) they are transformed according to the law
j ' _{i} = j _{i} - ¶ ln l (xi) / ¶ x ^{i} . | (B.3) |
The resulting four-vector Weyl interpreted as an electromagnetic 4-potential and a four-dimensional rotation F_{ik} = j_{k; i} - j_{i; k} this field, which is calibration invariant, as a tensor of the electromagnetic field F_{ik}. The equations of the electromagnetic and gravitational fields should then arise from a single variational principle, invariant with respect to both general coordinate transformations and calibration transformations (B.1); this led to the quadratic Lagrangian and thus to the 4th order differential equations.
Another method of generalizing the axiomatics of Riemann geometry for the purposes of unitarization was designed and elaborated by A.Einstein in 1946-53. The generalization consists in the fact, that instead of a symmetric tensor g_{ik} in basic form g_{ik}dx^{i }dx^{k} is allowed even assymmetric metric tensor g_{ik} as well as asymmetrical coefficients of an affine connection G_{ik}^{l}. It was the antisymmetric part of the metric that Einstein tried to interpret as an electromagnetic field, while the symmetrical part described gravity similarly as in GTR.
b) Five-dimensional unitary theories
Another approach
to the problem of the unification of gravitational and
electromagnetic field developed in the year 1921-25 T.Kaluza and O.Klein, who for a
general description of physical properties suggested using the five-dimensional manifold (in which spacetime
GTR is some
4-dimensional subspace) in hope that the fifth dimension could
express the electromagnetic field. Kaluza and Klein were
apparently inspired by the way in which Minkowski unified the
three-dimensional separate electric and magnetic fields by moving
to fours-dimensional spacetime (§1.5). From the generalized
theory of gravity (curved spacetime) in five dimensions, we could
then get the theory of electromagnetism and gravity in four
dimensions.
We observe physical spacetime as four-dimensional, so the "excess" fifth dimension (which has no direct physical meaning) must be disposed of by placing a suitable condition on the five-dimensional geometry. Kaluza originally introduced a relatively artificial requirement of "cylindricality", according to which there should be a one-dimensional group of isometric transformations in a five-dimensional manifold; this creates a Killing vector field, which leads to the fact that the 5-dimensional geometric structure can be fully described by the geometry of a four-dimensional hypersurface. Later, Einstein, Bergmann and Bargmann [17] proposed another geometric condition: the closedness (compactness) of a five-dimensional manifold in the fifth dimension. The five-dimensional manifold would then have the topological structure M^{4}xS^{1}, where M^{4} is Minkowski spacetime and S^{1} is a circle, ie the manifold would have the shape of a thin tube. If the radius of this tube (compaction radius) is small enough (subatomic dimensions), no macroscopic object can move in the fifth dimension and spacetime effectively appears to be four-dimensional.
The integral of the action of the general theory of relativity in five-dimensional space is considered in the form
(B.4) |
where g_{AB} is a five-dimensional metric, g^{(5)} = det (g_{AB}) and R^{(5) }= g^{AB}.R_{AB} is the scalar curvature of five-dimensional space. The metrics of five-dimensional space are chosen in the form
g _{AB} = j ^{-1/3} . | | | g _{ik} + A _{i} A _{k} j | A _{i }j | | | , | A, B = 0,1,2,3,5 | (B.5) |
| | A _{k }j | j | | | i, k = 0,1,2,3 |
where g_{ik} is the usual metric tensor of 4-dimensional spacetime, the 5th component of g_{5k} is identified (except for the scalar factor j) with the four-potential of the electromagnetic field *). Assuming that the metric g_{AB} does not depend on the coordinate x^{5}, substituting the metric (B.5) into the action (B.4) after the integration according to x^{5} we get
(B.6) |
If we omit the scalar field j
(see note *), the
integral of the action in Kaluz-Klein theory is equal to the sum
of Einstein's gravitational term and U(1) -calibration term given
by the tensor F_{ik} = A_{i; k} - A_{k; i} , which can be
interpret as Maxwell's electromagnetic field. The calibration transformation A_{i} ® A_{i} + ¶l/¶x^{ i} is generated by a
special coordinate transformation in 5-dimensional space: x'^{i}
= x^{i} , x'^{5} = x^{5} + l(x^{i}).
*) In (B.5) the
parameterization of the metric g_{AB} and the designation of quantities are
chosen without prejudice to generality so as to obtain Einstein's
and Maxwell's equations in the usual form. The fifth field
variable - the scalar quantity j - is
superfluous in Kaluzov-Klein theory, and Kaluza ruled it out by
simply placing it exactly one. Attempts were later made to
understand the significance of this scalar field and give
it a cosmological significance; Brans and Dicke related this
field to a long-range scalar field in
their so-called scalar-tensor theory of gravity (outlined in
§A.2).
For some time, Einstein and Bergman held the hope, that
the periodicity of the fields
with respect to the fifth compactifed coordinate (along which would
field can be varied with a period equal to the length of the
circle compactification or its integer multiples) could explain the quantum phenomena and possible to create classic
models of elementary particles. However, this resemblance to
Bohr-Broglie's quantization proved to be superficial, and the
corresponding hopes did not materialize.
^{ }^{ }One of the objections to Kaluz-Klein's
theory is that this theory is not really uniform in the true
sense of the word: gravity and electromagnetism are separated
here in an invariant way - like "oil and water"....
Multidimensional
unitary theories
Kaluza-Klein theory did not lead to the desired
results and for a long time it fell into virtually oblivion. In
recent years, however, an unexpected "renaissance" of
the Kaluz-Klein concept has taken place in connection with
efforts to geometrically formulate supergravity theories. These
are generalized Kaluz-Klein theories
built in
superspaces, where additional dimensions of a
spinor character expressing the internal properties of
interactions are introduced; it turns out that, for example, the
11-dimensional Kaluza-Klein theory could unify all known
particle interactions - see §B.6. Kaluz-Klein theories also
provide interesting possibilities for models of the universe with
a higher number of dimensions, as mentioned in §5.7.
^{ }^{ }The
basic idea of multidimensional unitary theories with compacted
dimensions is that the physical laws we observe depend on the
geometric properties of other, hidden dimensions. There are many
solutions in multidimensional theories, differing, for example,
in the metric size of compactifications. The compacted dimensions
are too small to be observed or detected in any way. However,
different geometries of additional dimensions imply different
kinds of particles and forces, which causes different physical
phenomena even in the macroscopic world.
So much so briefly about the proposals of the unitary field theory of the 20s-40s. These theories have never crossed the speculative level. In the meantime, research in the field of atomic physics has revealed new laws that govern the microworld - the laws of quantum physics. And two other types of fundamental interactions have been discovered - strong and weak interactions, which have completely different properties than gravitational and electromagnetic forces. This pushed earlier unitary theories into the background.
We will now describe in a more detailed way in §B.3 another approach, the so-called geometrodynamics of J.A.Wheeler [180], [275], [277]. The name was created by analogy with electrodynamics. Just as electrodynamics describes the dynamics of an electromagnetic field, geometrodynamics examines the dynamics of space-time geometry, ie, according to the GTR, the behavior of a gravitational field. "Geometrodynamics" is often understood to mean the whole general theory of relativity; here, however, it will be about geometrodynamics in the narrower sense.
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 | ||
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