Geometric everything involving a physical field ?

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_ikR = 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

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

Thus, in addition to the fundamental quadratic form ds² = g_ik dxⁱ dx^k, another linear differential form dj = j_i .dxⁱ, 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

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_ikdxⁱ 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 M⁴xS¹, where M⁴ is Minkowski spacetime and S¹ 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

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 x⁵, substituting the metric (B.5) into the action (B.4) after the integration according to x⁵ 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 = xⁱ , x'⁵ = x⁵ + l(xⁱ).
*) 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.

B.1. The process of unification in physics		B.3. Wheeler's geometrodynamics. Gravity and topology.

Vojtech Ullmann