|AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy||Gravity, black holes and physics|
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.6. Unification of fundamental interactions. Supergravity. Superstrings.
An objective reason why A.Einstein and his followers, despite
all erudition and effort, failed to develop a satisfactory
unitary field theory, was mainly that they did not have
sufficient experimental data on the laws
of the microworld - the properties of elementary particles.
They tried, so to speak, to "invent
nature from your office desk". This is partly true of Wheeler's
quantum geometrodynamics described in §B.4, which although introduces
the quantum principle, but otherwise it
also considers the
properties of only two types of fields - gravitational and
Now the situation is diametrically different. There is a huge amount of experimental data on the interactions of many types of elementary particles at different energies. From these experimental data, important general principles were derived that must be taken into account when compiling any unitary theory that seeks to claim an adequate description of reality (§B.7). Besides the gravitational and electromagnetic interactions appear two other types of interactions that play a fundamental role in the microworld: a strong interaction permitting the existence of atomic nuclei and a weak interactions causing e.g. radioactive b -decay.
Unification of 4 kinds of interactions
Extensive research in the field of nuclear and particle physics has resulted in the so-called standard model of particle physics, in which 25 fundamental particles and 4 forces (interactions) give a perfectly functioning explanation for everything in nature and space - see §1.5, passage "Standard model - uniform understanding of elementary particles" in the monograph "Nuclear physics and ionizing radiation physics". Nevertheless, even with this great success, fundamental physics is not satisfied. Firstly, from a fundamental point of view 4 different interactions are bit "too much", there should be only one! - to achieve the so-called "theory of everything" TOE (Theory Of Everything). And another problem: one of the fundamental forces - gravity - is essentially different from the others in that it has not yet succeeded in creating a completely consistent quantum theory (discussed above in §B5 "Gravitational field quantization")...
Strange properties of gravity
The non-linearity of gravity in GTR, "self-gravity", the curvature of space-time, are properties very different from other kinds of interactions. This is probably the main gnoseolological obstacle to complete unitarization. Some experts, especially in quantum physics, emphasize here (probably unjustified) even the so-called information paradox in black holes (this issue is discussed in the passage "Return of Matter by Quantum Evaporation"). This contradiction of quantum physics and general relativity here is (in my opinion) artificial and unfounded, it is just the subjective opinion of some experts...
Modern unitarization efforts take place on the ground of quantum field theory and their goal is to unify the fundamental interactions between elementary particles - strong, weak, electromagnetic and gravitational interactions. Because this is a very large area, mostly outside the focus of this book, we summarize only the basic principles and knowledge of unitarization in the physics of elementary particles - see Fig.B.8, which is a continuation of the basic unitarization scheme of Fig.B.1. Before discussing specific methods of unitarization, however, we will briefly approach the basic ideas of unification of fields and the important role played by symmetry and conservation laws.
Fig.B.8. Schematic representation of the basic structure of unification of fundamental interactions.
field « physical space ® unification
Already in §B.2 we outlined a very deep and beautiful idea of unitary field theory: according to it there should be a single, completely basic and all-encompassing physical field, the manifestation of which would then be all observed fields in nature - gravitational, electromagnetic, fields of strong and weak interactions (and all their manifestations, even in subnuclear physics). Then there is nothing in the world but this field, of which everything is composed - even material formations (eg particles) are a kind of local "condensation" of this field.
Physical field is a space in which a certain quantity is distributed in a certain way - at each point a certain scalar, vector or tensor describing a given field (potential or force action) is defined. The time evolution (variability) of the field is expressed in classical physics by the functional dependence of the field quantities on time, in the theory of relativity by the introduction of 4-dimensional spacetime. Several types of fields can be present in a given space, which in the classical approach is expressed using several vectors or tensors at each point in space.
If we want to unify these fields into one unitary field, we can proceed in terms of the relationship between space and field in basically two ways :
Both of these methods may be equivalent to some extent, but the existence of a very sophisticated mathematical apparatus of differential geometry and manifolds topology favors the second approach, which will be applied below to geometric formulations of supergravity and superstring theory.
The term symmetry in physics evolved from symmetries in geometry (and includes these geometric symmetries as an important special case). In geometry, symmetries express the regularity of the shape of certain geometric formations. We have here the central symmetry of a circle or sphere, the axial symmetry of a cylinder, chiral symmetry - mirror left-right symmetry. The symmetry of a geometric shape is manifested by the fact that with certain changes in the position - the so-called transformations, the properties of the studied geometric object do not change - they are invariant with respect to these transformations.
In physics, it does not have to be such a simple and illustrative symmetry, but it can also be a certain abstract symmetry at the level of mathematical equations.
Symmetry and its disruption
At the level of basic regularities, nature appears to be essentially symmetrical, but at the level of practical phenomena and events, symmetry tends to be more or less broken. We can illustrate this with a simple example from everyday life: if we stand a pencil on the table exactly perpendicular in the tip, is equally likely to fall to either side due to rotational (cylindrical) symmetry. If it fell preferentially to one side, it would mean that the symmetry is disturbed (this could be caused by inhomogeneity of the pencil material, a slight deviation from the cylindrical shape, asymmetrical tip trimming; possibly by external influences such as the presence of an electric field, air flow in the room, etc.).
When such symmetry disturbances occur in the region of particle interactions, it indicates the presence of a certain field causing this asymmetry and also the existence of some new particle which is the quantum of that field; this field can also be understood as a "compensating" field needed to maintain the basic symmetry, which seems to be disturbed at the level of the phenomenon.
Symmetry and conservation laws
In physics, the symmetry of a particular physical system is understood constancy (invariance) important feature of this system for transformations of variables that describe it. The symmetries of the equations of motion, describing the dynamics of the system, with respect to the transformations of their variables, play a very important role here. Thus, such transformations of quantities describing a given physical system that leave the shape of the equations of motion of this system unchanged. These symmetries can (but do not have to *) respect the solution of these equations.
*) Equations describing the dynamics of the system have certain symmetries, but the resulting state of the system, which is the solution of these equations, does not respect this symmetry - for example, due to asymmetric initial conditions.
According to E.Noeter's theorem, the invariance of equations of motion with respect to certain transformations leads to laws of conservation of certain physical quantities. In classical mechanics, the law of conservation of energy can be considered as a consequence of homogeneity of time (independence of time shift), the law of conservation of momentum as a consequence of space homogeneity (invariance to spatial translations) and the law of conservation of angular momentum as a consequence of space isotropy (symmetry with spatial rotations).
The first important principle of symmetry in modern physics was the Lorentz invariance, originally discovered as a (more or less random) mathematical property of Maxwell's equations derived from the experimentally observed laws of electromagnetism. The course of physical reasoning at the time was roughly as follows :
Thanks to A.Einstein and his theory of relativity, physicists realized that the principles of symmetry can be a powerful gnoseological tool; let us just remember that it was from the requirement of symmetry to general transformations of space-time coordinates, together with the principle of equivalence, that the general theory of relativity emerged. Now the method in theoretical physics is more of a scheme :
|principles of symmetry||®||lagrangian||®||field equation .|
A physically justified requirement of a certain symmetry can serve as a certain "design principle" or aid in the creation of physical theories and models. And the presence of symmetries in physical models makes it possible to make some important theoretical predictions without having to know specific detailed (and often quite complex) solutions of equations of motion.
Calibration - gauge -
transformations; calibration - gauge - field
The term calibration - gauge - *) in physical, technical and laboratory fields, means a procedure in which the exact setting of the relationship between the measured quantities on measuring instruments or tools is performed or verified. A typical simple example is the calibration of a measuring cylinder for liquids, which is provided with marks (dashes, lines) with a marked volume [milliliters]. Or calibration of a voltmeter, which is provided with a scale with marked divisions of voltage values [V, mV]. Calibration is usually performed by comparison with values from other (more accurate) measuring instruments and standards. A special field of metrology deals with precise calibration.
*) The word "calibration, caliber" comes from French caliper = inner diameter of mainly rifles or cannons, drilling, caliber of firearms. Portable - something that has a certain size, diameter, precise settings, calibration.
The english word "gauge" comes from old French "jauge" - graduated rod used for measuring.
Here in field theory, calibration - gauge is meant as a suitable setting - transformation, "calibration, gauge" - of certain quantities (coordinates, field potentials), which preserves the values of field intensity, but at the same time allows a more suitable mathematical notation and easier specific solution in field theory. This certain "freedom" in the choice of potentials allows you to choose the shape of the potentials - to perform their "gauge" so that it is as advantageous as possible for the given problem. The invariance of the fields under gauge transformation is called gauge invariance.
We have already encountered this procedure in our book, for example, in §1.5, passage "Properties of Maxwell's equations" - "Field potentials", §2.5 "Einstein's equations of gravitational field", passage "Linearized theory of gravity" and §2.7 "Gravitational waves", passage "Plane gravitational waves in linearized gravity".
This classical concept of invariance to gauge transformations [Lagrangian -> field equations -> potential transformations preserving field equations], the current methods of building unitary field theories are reversed: the requirement of invariance to a certain gauge transformation is declared as primary and from its fulfillment the properties of field are derived. When a local gauge invariance with respect to specific (physically more or less justified) symmetries is required, certain additional terms containing derivatives of transformation parameters appear in the field equations, to eliminate which it is necessary to introduce corresponding compensating terms, which can be interpreted as a some field - this creates a new gauge field that could describe a new further interaction..?.. It will be discussed in more detail below. From a physical-philosophical point of view, this may be a debatable approach, but it "works"...
The composition of two transformations, in which the system does not change, also results in an operation that keeps the system the same - from a mathematical point of view, the set of all symmetries of a given system is a group :
For a better understanding of some of the terms and designations used below, typical of unitary field theories, it will be useful to insert a short mathematical nipple outlining the description of transformations using group theory.
A group is a (non-empty) set G, between the elements of which a binary operation "l" is defined, assigning to each two elements a, b Î G a new element c = a lb Î G, which is also an element G. This binary transformation is associative : (a lb) l c = a l (b l c), has a unit element iÎG: ali = ila = a for each element aÎG, and for each element aÎG there exists an element inverted a-1ÎG: ala-1=a-1la= i. The most common example of a group is the set of all positive rational numbers in a normal multiplication operation ("l" = " . "). If the binary operation "l" is commutative, ie a l b = b l a for each element a, b ÎG, is called G the Abel group. The number of elements g of a group G is called the order of the group. If g is infinite but countable, G is called an infinite discrete group.
If the elements of the group form a continuous set, the order of the group is no longer applicable. On the other hand, it is possible to introduce certain topological properties defining a manifold into a continuous set of group elements (for more information on topology, see §3.1 "Geometric-topological properties"), or even metrics. The above-introduced binary operation c = a l b, defining a group, can then be written as a functional relation c = f(a, b). If all these group operations (inducing the display of the group G on itself) are continuous, the set G forms a topological group. The topological group that is a manifold is called the Lie group *). A typical example of a Lie group is the Euclidean space Rn. Also, the set of continuous transformations forms a Lie group. It is the groups of transformations in which certain quantities are preserved that play an important role in the physics of fields and particles. Since physical transformations are mathematically expressed by matrices of transformation coefficients (Aij,Aab), the respective transformation groups are matrix groups.
*) Sets of this kind, which are both groups and varieties, were first introduced in the 70s of the 19th century Norwegian mathematician Sophus Lie (1842-1899) in the analysis of the properties of solutions of differential equations. Lie groups combine structures from three basic mathematical areas: algebra, analysis, and geometry. They are used in solving differential equations, in differential geometry and algebraic topology, in quantum mechanics, in the theory of relativity, in describing particle interactions, unitary field theories (supergravity, superstrings - see below). In physical applications, they usually appear as Lie groups of symmetries of the studied system.
The unitary group U(N) is defined as the group of all transformations x'a = Aabxb (a,b=1,2,....,N), which preserves the scalar product and the invariance of the unit length of the vector |x| = x*axa - ie for the transformation matrix the relation A*abAba = 1 applies (asterisk * denotes a component complex asociation). If another restriction Det A = 1 applies, it is a so-called unimodular subgroup SU(N) of group U(N). .
Groups in physics
In physics, groups have found their first application in crystallography, where they express the symmetry properties of the crystal lattice of solids. In relativistic physics, groups first appeared in the work of H.Poincaré, who showed that Lorentz transformations of spatial and temporal coordinates between inertial frames of reference form a (Lie) group; this group of general Lorentz transformations (inhomogeneous, including translations) is called the Poincaré group. However, in the further development of the special and especially the general theory of relativity, we can encounter the use of groups only sporadically and marginally. From the end of the 1920s and the beginning of the 1930s, group theory began to be applied more in quantum mechanics in the analysis of multielectron configurations of atoms and in quantum chemistry.
Unitary symmetry groups
New horizons for the application of groups have opened up in nuclear physics since the 1940s and 1950s in describing the properties of elementary particles. The large number of elementary particles that were discovered in high-energy interactions, naturally led to efforts to systematize them and to introduce unitarization schemes. In particular, each baryon and lepton is assigned a baryon number B and a lepton number L (particle +1, antiparticle -1), which are retained in all interactions. Significant similarities and symmetries between some elementary particles, especially hadrons, were found.
If we look away from the electric charge, protons and neutrons, for example, can be considered as two states (doublets) of one particle - a nucleon. Similarly, the pions p+, po, p- form a triplet of similar particles. When studying the strong interactions themselves, which are charge-independent, we can disregard the charge. To describe these similarities and symmetries, a new quantity isotopic spin or isospin T *) was introduced. Nucleons have an isospin T = 1/2, with the projection of the isospin T = +1/2 corresponding to a proton and a T = -1/2 neutron. Pions were assigned isospin T = 1, with projections -1, 0, +1 for p-, po, p+. In the system of interacting nucleons and pions, the law of conservation of isospin applies. To express these symmetries, the SU(2) group was created - a special, unitary (unimodular) group in a complex 2-dimensional space; this group is a locally isomorphic group of rotations O(3) in 3-space, expressing the isotropy of space - symmetry to spatial rotations, leading to the law of conservation of angular momentum.
*) It was based on a formal analogy with ordinary spin, where a particle with spin 1/2 occurs in two states with spin projection -1/2, +1/2 and a particle with spin 1 in three states with spin projections -1, 0 , +1. Isospin T is a vector in thought (auxiliary) "isotopic space". In general, a particle with isospin T can occur in (2T + 1) states with isospin projections on the reference axis: -T, (-T + 1), (-T + 2), ..., -1, 0, 1, ..., (T-2), (T-1), T.
Another important step was the discovery of some "strange" (unexpected) properties of the interactions of mesons K and hyperons in their combined pair production, which led to the introduction of the concept of strangeness described by the quantum number S ("Strange"). Later we were introduced general quantum number called hypercharge Y = B + S, the sum of baryon number B and strangeness S. It turned out that with strong interactions, both isospin T and hypercharge Y are preserved. This led to the search for the group SU(2) ´Y, describing the extended symmetry properties of hadroms. In 1964, M.Gellman and E.Neeman proposed to use a minimal Lie group, containing SU(2) ´ Y as a subgroup - the group of unitary symmetry SU(3). This extended symmetry led to the construction of a baryon multiplet - a decouplet (3/2+), in which, however, one place was missing at that time; the hyperon W was thus predicted, which was soon actually discovered.
The hadron symmetry group is a 4-parameter isospin and hypercharge conservation group. Further analysis of unitary symmetry showed that the system of hadrons can be very well explained by the hypothesis that hadrons are composed of subparticles - a triplet of quarks. This created quark chromodynamics as a theory of strong interactions ... (is described in "Quark structure of hadrons").
......... symmetry groups U (1), SU (2), SU (3), ....., SO (...), ... ., Lie algebras ...
In the terminology of the theory of groups of unitary symmetries it can be said that particles are representations of a group of symmetries. More precisely, we identify (interpret, assign) the components of the basis of the irreducible representation of the symmetry group with a set of physical states - particles (or their ecxitated states, resonances).
local symmetry; Gauge field
When studying physical systems, the corresponding symmetries can be divided into four categories. Depending on the relationship [system] - [environment] it can be :
¨ External symmetry - invariance to changes in the "position" of the system (or its parts) in space and time (in addition to the usual transformations of space-time coordinates, it is eg reversal of time "T" or spatial inversion "P") ;
¨ Internal symmetry - invariance to transformations of internal characteristics of the system (eg exchange of particles for antiparticles, charge association "C") .
In terms of space-time "range", constant or variability of transformations, symmetries of two kinds are applied in physics :
The basic starting point of gauge
is the thesis that all four basic interactions in nature are a
consequence of the requirement of the invariance of the theory to
the respective gauge transformations. Within the gauge
theory it is possible to formulate quantum electrodynamics (where
the electromagnetic field is obtained as a gauge field at the requirement of
lagrangian invariance of free spinor field to local phase
transformations from group U(1)) and Einstein's gravitational
theory (gravitational field arises from
gauge transformations of spacetime - Poincaré group).
Gauge fields in calibration theories are primarily "intangible" (their quantum has zero rest mass), which is adequate for electromagnetic and gravitational fields. Development of theory e.g. weak interactions in gauge theories but it causes some problems stemming from the fact that these interactions are mediated intermediate bosons (W+, W-, Z°), which, due to short range interaction considerably heavy weight (tens of GeV/c2). This problem was overcome by the mechanism of the so-called spontaneous symmetry breaking , , which is a modification of the Lagrangian, in which both the Lagrangian and the equations of motion still have the original given symmetry, but their own physical states no longer have this symmetry (there is no contradiction - for example, motion in a centrally symmetric field under asymmetrical initial conditions may not be at all symmetrical). This spontaneous symmetry breaking then causes, that the respective gauge field to effectively act as a non-zero mass field without violating the gauge invariance.
Fig.B.9. Illustration of the mechanism of spontaneous symmetry breaking in gauge theories.
a) For the effective potential of the shape of a simple symmetrical pit with a single minimum, the ground state is also symmetrical.
b) For such a form of symmetric effective potential, the ground state of the field j no longer has symmetry.
c) The movement of a ball released exactly along the axis into a glass with a dented bottom illustrates the case when, despite the equation of movement of the ball, initial conditions and shape of the glass are symmetrical, the final state does not have this symmetry: the ball always rolls off into the wall recess - previous symmetry is spontaneously broken.
The essence of the
mechanism of spontaneous symmetry breaking is roughly outlined in
Fig.B.9. On Fig.B.9a shows potential energy (effective
potential) scalar field j of mass m and
the coupling constant l with isngle
(model) Lagrangian L = (1/2) (j,i)2 - (m2/2) j2 - (l/4) j4. The effective potential V(j) = (m2/2)
j2 - (l/4) j4 has (for m2 > 0)
the shape of a symmetrical potential well, in which the most
advantageous energy state corresponds to the field j = 0. If the effective
potential would have
the form V(j) = - (m2/2) j2 - (l/4) j4 (corresponding to the case m2 <0), the
potential well will have the form according to Fig.B .9b, so that the
minimum V(j) will no longer correspond to
the state j = 0, but the field j = j o
= ± m/Öl. Although the potential of V(j) is still symmetrical with respect to the
change of the sign j ® -j, the
basic state of the field j no longer respects this symmetry
(the ball symbolically representing the state of the field always
rolls to one of the minima - Fig.B.9c).
After breaking the symmetry, the spectrum of particles (mass of excitations) changes. In this simple case, j = 0, m 2 <0 would be a theory of tachyons with an imaginary mass m2(j=0) = d2V/dj2|j=jo= - m2 < 0, while after symmetry breaking the square of the mass becomes positive by excitation of the scalar field: m2(j=jo) = d2V/dj2|j=jo= 2.m2.
Thus, the basic idea of the Higgs-Kibble mechanism *) is to introduce an auxiliary scalar field (Higgs field) with such an interaction potential into the Lagrangian of the gauge theory, but the Lagrangian as a whole would remain gauge invariant. Then the gauge fields will effectively behave as a field with non-zero mass about you. In addition, so-called Higgs bosons - scalar particles with non-zero rest mass, as a quantum of these auxiliary scalar fields, also appear in the theory.
*) This hypothesis was first introduced in 1964 by the authors P.Higgs, F.Englert and R.Brout, G.Guralnik, C.Hagen and T.Kibble. The Higgs field in 1967 was used by S.Weiberg, A.Salam and S.Glasshow to build the theory of electroweak interaction with heavy intermediate bosons W±, Z° (mentioned below).
Thus, it turns out that theories of all fundamental interactions can be uniformly created within gauge theories differing mainly by the calibration group. Gauge theory thus also forms a suitable basis for unifying interactions: two types of interactions with the calibration group G1 and G2 can be united to create the gauge theory with gauge group G obs and exceeding group G1xG2 as its subgroups. In constructing unified theories of weak, strong, and electromagnetic interactions, this basic idea is supplemented by the assumption that before the symmetry was broken, all vector bosons mediating interactions were intangible. However, after spontaneous symmetry breaking (due to the formation of constant scalar fields in the whole space), some of the vector bosons gain mass and the corresponding interactions become short-range - the symmetry between the different types of interactions is broken.
electromagnetic and weak interactions
The first significant success on this path was recorded in the unification of electromagnetic interaction and weak interaction in the so-called electroweak interaction - this is the Weinberg-Salam-Glashow theory. Before the formation of a constant scalar Higgs field H, this theory has a gauge symmetry SU(2)xU(1) and describes the electroweak interactions of particles caused by exchanges of immaterial vector bosons. After the formation of the scalar field H, the symmetry is spontaneously broken up to the subgroup U(1), the corresponding part of the vector bosons (W+, W-, Z°) acquires a mass (of the order ~ eH » 102 GeV), the respective interactions become short-range ® weak interactions, while the other field Ai remains an intangible ® electromagnetic field. It was thus possible to unite weak and electromagnetic interactions into one theory, in which they appear as two different aspects of the same phenomenon.
Weinberg-Salam's theory of electroweak interaction can now be considered experimentally practically verified, because in 1973 the existence of so-called weak "neutral currents" (causing reactions of type nm + e ® nm + e was proved at CERN), and especially in 1983, intermediate bosons W±, Z°, whose masses (mW @ 82 GeV, mZ @ 93 GeV) and the decay methods agree very well with the predictions of the Weinberg-Salam model.
Electro-weak interaction with intermediate bosons W± very elegantly explains the nature of beta-radioactivity by transmutation of quarks inside neutrons or protons - it is explained in more detail in §1.2 "Radioactivity", part "Radioactivity beta" in monograph "Nuclear physics and ionizing radiation physics" :
Schematic representation of the mechanism of b- neutron decay (top) and b+ -proton transformation (bottom) within the standard model of elementary particles.
Strong interactions and
Before you briefly talk about the next stage of unification - grandunified theories, mention a few words about the specific properties of the strong interaction (a more detailed explanation in §1.5 "The Elementary Particles", passage "Quark structure of hadrons" in book "Nuclear Physics, ionizing radiation"). On the basis of extensive experimental material, obtained mainly in the 50s and 60s in the search for new elementary particles, significant symmetries were observed in the properties of elementary particles, which in 1964 resulted in the formulation of a quark model of hadrons, according to which all hadrons are composed of still "more elementary" quarks (this name was taken from the literary work of James Joyce with a significant dose of recession). Quarks are fermions with a spin of 1/2 and a third electric charge. To explain the system of hadrons within the additive quark model, 6 species (metaphorically used the word flavor - "smell") of quarks were gradually introduced, symbolically marked "u" (up), "d" (down), "s" (strange), "c"(charm), "b" (bottom)," t "(top - the existence of a t-quark is indicated by a striking symmetry in the system of leptons and quarks). For the same reason, it was necessary to assign quarks a new internal quantum number - "color", which takes on three discrete values called "red", "blue", "yellow"; while baryons are "colorless" ("white") combinations of three colored quarks, mesons then a combination of quarks and antiquarks. However, the main difficulty of the quark hypothesis is that no free particles with quark properties have ever been observed. Therefore, if quarks exist at all, they must be very strongly bound in hadrons.
In the late 1960s, the quark model was to some extent supported by the results of experiments with high-energy electron scattering on nucleons (deeply inelastic scattering) showing that in such "hard bombardment", the nucleon does not behave as a compact particle, but as a cluster of several (three) more or less free scattering centers - so-called partons. The quantum numbers of the partons (charge, spin, isospin) corresponded to the values expected for quarks. However, the direct identification of quarks and partons was hindered by a contradiction: on the one hand, partons in nucleons behaved as free in experiments, on the other hand, quarks are so strongly bound that they cannot be released from nucleons.
Significant progress in understanding the properties of strong interaction was achieved in the 1970s, when the so-called quantum chromodynamics (QCD, Greek chromos = color ) was formulated and developed ,  as a theory of strong interaction; the same right can be called "quark chromodynamics". This theory is constructed in a similar manner as quantum electrodynamics (QED), but is based on non-abel gauge symmetry physically associated with the color quark. A significal paradoxical feature of QCD is asymptotic freedom: the effective coupling constant of the interaction between quark approaches zero during shrinking distances, but rapidly increases with increasing distance. Asymptotic freedom allows naturally understand the seemingly incompatible characteristics quark as partons: quarks at small distances inside nucleons hardly interact, while in terms of greater distances are bonded very strongly. Closely related to asymptotic freedom is the hypothesis of perfect "imprisonment" of quarks, according to which quarks cannot exist as free particles (infinitely large energy needed for release), but only bound in hadrons.
The strong interaction between quarks in QCD is mediated by a vector gauge field, whose zero rest mass quantum, called gluons, play a similar role here as photons in QED. Unlike quantum electrodynamics, gluons have a "color" charge and interact with each other (they can "emit" each other); due to this nonlinearity, the vacuum in QCD has a complex structure, especially in the region of "infrared" (low-energy) vacuum fluctuations.
Jets - traces of hadronized quarks
At very high energies, during hard and deeply inelastic collisions of electrons with protons, a number of secondary particles are formed, which fly out unisotropically in some kind of directed "jets". A detailed analysis of the angular distribution and energy of particles in jets showed the following mechanism of interaction, which can be divided into two stages:
During the 1st stage, the high-energy electron, when interacting with the proton, transfers part of its kinetic energy to one of the quarks, which after this scattering moves for a short time practically freely (asymptotic freedom) inside the proton; similarly, the remainder of the proton formed by the two remaining quarks. However, the quarks will not be released from the proton. As soon as the distance between the radiated quark and the rest of the proton exceeds about 1 fm (= 10-15 m), the 2nd stage occurs: the forces between them begin to increase sharply and in the quark-gluon field the quarks and antiquarks are produced, which are formed into mesons and baryons - the so-called "hadronization" of the quark-gluon plasma. The result is the emission of two angularly collimated sprays of particles - jets, which fly out approximately in the directions of flight of the quark and the rest of the proton in the first stage. These jets are actually the traces of quarks.
This mechanism is simply illustrated in the figure, which comes from §1.5 "Elementary Particles" of the book "Nuclear Physics and Physics of Ionizing Radiation" :
In quantum chromodynamics, there is a problem of CP-disruption of the combination of charge symmetry and parity in quark theory, which is solved by introducing particles called axions - they are light (rest mass about 10-5 eV) hypothetical particles with spin 0, which are sometimes discussed as possible part of the so-called hidden (dark, non-radiant) matter in the universe (see §5.6 "The future of the universe. The arrow of time. Hidden matter.").
Further details on the properties of elementary particles and their interactions are given in §1.5 "Elementary particles" of the book "Nuclear physics and physics of ionizing radiation".
If we have the theory of strong interactions (QCD) and the theory of electroweak interactions (Weinberg-Salam model), which are all gauge theories, there is naturally an attempt to combine these theories into one even more general theory of interactions. This next stage of unitarization is sometimes refereed to as Grand Unification (GUT - Grand Unification Theory). The group of gauge symmetry G in this large unification must contain subgroups SU(3)colorx[SU(2)xU(1)]electroweak Ì G; the simplest group of this type is SU(5), but models with gauge groups SO(10), E6 and others are also used .
In grandunification theories, vector bosons X and Y (also called leptoquarks because they cause transitions between quarks and leptons) that are intangible before breaking the symmetry - like all other vector particles; leptons can easily be converted into quarks and vice versa *). First Higgs field disturbs the initial symmetry SU(5), into SU(3)xSU(2)xU(1) - strong interaction described SU(3) are separated from electroweak described by the group SU(2)xU(1). The X- and Y-mesons gain a large mass (of the order of mX,Y ~ 1015 GeV), which strongly suppresses the transformation of quarks into leptons and makes the proton practically stable. Another Higgs field then disrupts the symmetry between weak and electromagnetic interractions as in the Weinberg-Salam model.
*) This circumstance could have been of great importance in the formation of baryon asymmetry in the very early stages of the evolution of the universe (it is discussed in §5.4 "Standard cosmological model. The Big Bang. Forming the structure of the universe." and also in §5.5).
One of the main predictions of grandunification theories is the instability of a proton, which should decay into muons or positrons and into one neutral or two charged pions [p ® (m+ or e+) + (po or p++p-)] with lifetime of the order of tp » 1030 -1033 years. This decay would be caused by the conversion of a quark to a lepton via the X boson, and due to the enormous mass of the X boson, its probability is extremely small. However, observing proton decay would be very important, as it would decisively show that grandunification theory is on the right track. Experiments *) so far give estimates tp > 1030 years.
*) These attempts to observe proton decay are made deep underground (due to cosmic ray shielding), where large water tanks are located, equipped with many photomultipliers that could detect faint flashes caused by the passage of fast particles formed as proton decay products. The most perfect device of this kind is Superkamioka-NDE in Japan, which did not detect any proton decay, but was very successful in the detection and spectrometry of neutrinos (see the "Neutrinos" section in §1.2 of the book "Nuclear Physics and Ionizing Radiation Physics").
The idea of grand unification is certainly very attractive and promising. However, there are still many unresolved issues and problems, e.g. mass hierarchy problem generated by the mechanism of spontaneous symmetry breaking in the scalar portion of theory - emerges here too many free parameters (more than 20), it is not clear how to choose between them to several alternative models and others; GUT is too phenomenological. In addition, grandunification theories do not include gravity. Thus, theories attempting great unification are not yet in such a state that they can be considered "finite" theories of interactions. However, their use, for example, in cosmology already leads to new interesting concepts beneficial for both cosmology and elementary particle physics - see §5.5.
Opinions on the role of gravity in the structure of elementary particles vary widely; they extend between two extreme positions :
If the universality of
gravity can be extrapolated down to the microscale of elementary
(subnuclear) particles, at least the first part of the second
extreme view b) would certainly apply. The local
densities of matter and energy here reach such values, that the
gravitational interaction would become strong. The view is
growing that at present it is no longer possible to separate the
physics of elementary particles and the physics of gravity; it
even seems that without the inclusion of gravity, a consistent
and uniform theory of the particles that make up matter cannot be
established. It is therefore a natural effort to complete the
unitarization of interactions in quantum field theory by
including the gravitational interaction, its unification with the
other three types of interactions. This ambitious unitarization
program is called superunification
To unite gravity with other types of interactions in the spirit of the above-mentioned scheme of unitarization of gauge theories means to combine internal symmetries with geometrical ones, ie to find a common group including both space-time transformation group (eg Poincaré group) characterizing gravity in GTR and internal (not space) symmetry of weak, strong and electromagnetic interactions. It turned out that such a unification (in a non-trivial way, ie not as a mere direct product) was not possible within Lie groups, but it was necessary to use new algebraic structures in a generalized group, often called Lie superalgebras or graduated Lie algebras. In generalized groups, the respective algebras are determined by both commutation and anticommutation relations between individual generators. Those Lie superalgebras which contain as their subalgebras a goup of space-time transformations (e.g. Poincaré groups) are called supersymmetric.
The algebra of supersymmetry is designed to contain, in addition to ordinary Poincaré group generators (space-time shifts Pk and rotations Mkj), also spinor generators Qi with suitable commutation relations. If such an algebra is realized in the field space, the generators transform Qi tensor fields to spinor fields and vice versa. Because the quantum theory of tensor fields describing bosons with integer spin (governed by Bose-Einstein statistics) and spinor fields describe fermions with half-integer spin (Fermi-Dirac statistics), operators Q i actually generates the transformation for transferring fermions bosons, and vice versa. In supergravity , the sharp boundary between fermions and bosons in current physics is thus removed. Another characteristic feature of supergravity is that in addition to the gravitational field, which is a gauge field against local transformations of spacetime, it also contains a spinor field - a gauge field with respect to local supersymmetric transformations generated by Q i; such a field is denoted as Rarit-Schwinger and its quantum is called gravitino (it can have spin 3/2 or 5/2 *).
*) In supersymmetric unitary theories of elementary particles, each particle is assigned its so-called superpartner - each boson has its fermion superpartner and fermion, on the other hand, has its boson counterpart. The most frequently discussed supersymmetric particles are the mentioned gravitin and also photin - weakly interacting mass particles with spin 1/2, introduced as a supersymmetric partner of the photon. Supersymmetric particles to fermions are sometimes discussed: s-leptons as superpatters to leptons, eg s-electron , s-muon , s-neutrino (also called neutralino - it should have a high weight of dozens of GeV), or quarks - s-quark. Elementary particles are discussed in more detail in the book "Nuclear Physics and Physics of Ionizing Radiation" , §5.5 "Elementary Particles".
Supersymmetry means that "force" and "matter" particles (ie, field and matter) are two aspects of the same reality. In principle, supersymmetry makes it possible to solve the infinity problem in such a way that the closed-loop contributions in Feynman diagrams for virtual bosons lead to positive infinities and for virtual fermions to negative infinities, so that they could optimally cancel each other out.
The simplest supergravity theory - so-called simple supergravity created in 1976 , , was more of a model experiment because it contains a minimal number of fields; it also excludes quarks and leptons. Physically more realistic variants of supergravity theories try to expand the number of spinor generators and also introduce internal symmetry generators. This creates an extended supergravity, which contains 4N spinor generators Qa i (a = 1,2, ..., N) carrying the internal symmetry index a. If we limit ourselves to particles (fields) with a spin not exceeding the value 2, in the spacetime of dimension d = 4, the N-extended supergravity theories with N = 1,2, ..., 8 are possible. The simplest extended supergravity theory is the N = 2- supergravity, unifying Maxwell's and Einstein's theory; two gravitons are assigned to photons and gravitons here. Maximum extended N = 8 -supergravity contains: one graviton field, 8 Rarit -Schwinger fields (gravitin), 28 vector fields (bosons) with spin 1, 56 spinor fields (fermions) with spin 1/2 and 70 scalar fields. Thus, multiplets of extended supergravity theories have a much richer structure than in simple supergravity. However, although they contain an excessive number of fields, they do not contain the fields of some known particles, e.g. the m- meson...
From the unitarization scheme in Fig.8, we see two seemingly diametrically different paths: Einstein's geometric path ending with Wheeler's geometrodynamics and the path of quantum field gauge theories leading to supergravity, which has nothing to do with geometric character. Because Einstein's concept of gravity as a geometric structure of spacetime is based on very deep and illustrative principles, the question naturlly arises as to whether the geometric means you can not construct even a theory and supergravity. Physically, it would mean that the "charges" in supergravity theories should have their origin in the generalized geometric structure of spacetime, similar to the gravitational "charge" in the GTR has its origins in the curvature of spacetime *).
*) An interesting variant of multidimensional unitary theory, which has emerged recently, is the theory of so-called superstrings. In this theory, particles and quantum fields are interpreted as excited states of oscillations (one-dimensional) relativistic strings in multidimensional space (most often d = 10). These superstrings with the characteristic length in the order Planck »10-33 cm can be both open (free ends) or closed, wherein the interaction string consists either at the connection of the ends of two strings (a third string is formed) or in the rupture of one strings into two parts. The main advantage of string theory is considered its better renormalization properties - there are no "ultraviolet" divegens. Superstring theory is briefly discussed below in a separate passage at the end of this chapter.
formulation of supergravity. Multidimensional unitary theories.
Indeed, it has been shown that supergravity can be formulated as a geometric theory in superspace (the superspace created by the extension of Minkowski spacetime is generally curved and has other spinor dimensions) using a differential geometry apparatus generalized to the situation where some of the coordinates anticommutate. It is thus a space with torsion, and it has been shown that all components of curvature can be expressed using torsion and its covariant derivatives. Torsion thus becomes an important geometric object in supergravity.
Recent attempts at a geometric formulation of supergravity thus lead to a certain "renaissance" of Kaluz-Klein's theory (see §B.2): theories are constructed in multidimensional (d> 4) "spacetime", which with the help of spontaneous compactification could give a realistic theory in spacetime of the effective dimension d = 4. The mechanism of spontaneous compaction consists in finding a special vacuum solution of generalized Einstein's equations in d-dimensional spacetime, corresponding to the representation of a d-dimensional manifold in the form ed = e4 ´ Bd-4, where e4 is four-dimensional spacetime (mostly Minkowski is considered) and Bd-4 is a compact "inner" space. Excess d-4 dimensions ("extra-dimensions") are "rolled" on sufficiently small scales (mostly Planck scales of 10-33 cm are considered), as discussed above in the passage "Physical field « physical space ® unification".
The total (resulting) d-space in multidimensional unitary field theories is formed by the outer 3 + 1 dimensional spacetime (generally curved) and other d-4 extradimensions (usually 5-7) of the inner space. These extradimensions form a special manifold, whose geometric properties, especially holonomy and connections, suitably model the (unitary) symmetries of the interactions of elementary particles - thus unifying them with geometric gravity 3 + 1 dimensional spacetime.
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 extra-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 types of particles and forces, which causes different physical phenomena in the macroscopic world.
Generalized Kaluz-Klein unitary theories for various dimensions d> 4 have been studied. In order for such a theory to be complete and realistic, ie to unify all known particle interactions, it must contain a phenomenological group of internal symmetry SU(3)xSU(2)xU(1). As Witten  recently showed, in order for the "inner" space Bn to have SU(3)xSU(2)xU(1) - a group of isometries, its minimum dimension must be equal to n = 7, ie the dimension of the initial manifold the Kaluz-Klein theory must be d = 11, which coincides with the result for the maximum N = 8- supergravity in (d = 4) -space-time obtained in .
In Chapter 5 (§5.5 "Microphysics and cosmology. Inflationary universe.") the cosmological consequences of grandunification and supergravity theories were discussed. In the earliest stages of the universe's evolution at high temperatures, when spontaneous compaction has not yet taken place, spacetime could have all its 11 dimensions. The spontaneous compaction that then occurred could, in principle, lead to all possible vacuum solutions, so that "islands" could be created in which spacetime could have different topologies, number of dimensions and metric signatures. The earliest universe could thus be a kind of "window" into the higher dimensions of the generalized Kaluz-Klein unitary theory.
Although supergravity is not yet complete, it is undoubtedly a very promising unitarization concept. To verify the correctness of the path taken by supergravity, it would be essential if we could experimentally prove the existence of gravitins, which are characteristic of supergravity theories. So far, however, the only "laboratory" for the indirect verification of supergravity theories is cosmology - the consequences of phenomena in the very early universe.
One of the basic concepts of physics is the concept of a material point - an idealized object, whose mass (and other parameters) are concentrated in a single geometric point of space. The trajectory through which a material point in space runs is a curve, each point of which can be characterized by spatial coordinates and time. The dynamics of a material point in classical mechanics is given by Newton's equations (§1.2); in quantum mechanics, particle dynamics is described by the Schrödinger equation; trajectories connecting the initial and final state of a particle in space are the starting point for quantization using Feynman integrals over trajectories (§B5).
In classical mechanics, the notion of a material point was merely an idealization of real bodies, useful for the analysis of their motion. However, the special theory of relativity has reinforced the importance of the notion of a material point: no elementary (fundamental) object can have finite spatial dimensions, because no signal or interaction can propagate at superluminal speeds. When two bodies of non-zero dimensions collide, not all parts can react immediately, which means that the body is composed of more elementary objects: Þ the elementary object must be a point object.
However, the point nature of fundamental objects - field sources - leads to serious problems in field theory: at limit transitions to zero dimensions, mathematically divergent expressions leading to infinite values. It is necessary to get rid of these divergences (basically ad hoc) by methods of renormalization - to perform a suitable gauge transformation so that the results of the calculation match the experimental values.
However, it was possible to find a way to systematically avoid these unfavorable mathematical divergences - these are theories in which, instead of points, the elementary objects are one-dimensional lines or loops of non-zero length - the so-called strings. If these strings were small enough (microscopic), they might not be observable - they would look "from a distance" like points. Thus, the basic building blocks of nature would not be particles with zero dimensions, but one-dimensional strings that vibrate different ways, corresponding to different types of particles. And the particle interactions would correspond to the joining and uncoupling of the strings. The strings are basically the same, but they differ in the degree (mode) of their vibrations - according to which the string can be, simply put, for example, an electron or a quark.
According to this concept, everything in the universe - all forces and all matter - is made up of small vibrating energy lines called superstrings. The different ways in which the string vibrates resonantly represent different types of particles. Different types of forces and particles can come from different vibrations of the same string. One of the main pitfalls of superstring theory is the question of their experimental verification. Superstrings, if they exist, are extremely small (Planck dimensions). Therefore, there is no hope for their direct experimental demonstration ...
Description of the
motion of a free string
A free (relativistic) particle of rest mass mo in spacetime (d = 4) is described by the integral of the action (see §1.6) S0 = mo. n ds = mo.nÖ[(dxi/dt)(dxi/dt)] dt , where s is the space-time interval and t is the proper time of the particle. This action S0 (index " 0 " here indicates that it is a point, ie 0-dimensional particle) is proportional to the length of the worldline particles (relativistic interval s ) - Fig.B.10 left. The variational principle of the smallest action dS = 0 then leads to Lagrange's equations, from which follow the equations of motion of relativistic mechanics in STR (1.100), resp. (2.5b) in GTR. This procedure can be generalized to a different number of dimensions than d = 4.
Left: The trajectory of a "0-dimensional" free particle in space-time is a 1-dimensional worldline that can be parameterized by the length of the interval s or by the eigenvalue t .
Right: The trajectory through which a 1-dimensional string runs in space-time is a 2-dimensional world surface, which can be parameterized by its own time t and another parameter s , characterizing the position of a point on the curve representing the string.
The natural generalization of the integral of
the action from a material point to a string leads to the fact
that the action of the string will be proportional to the size
of the world surface , which the string
"sweeps" during its movement (evolution) in space-time
- Fig.B.10 right: S1 = T. n Ö[det(hab)] ds dt, where hab (a, b = 1,2) is a two-dimensional metric on the world surface;
T describes the "tension" of
the string, given by the weight of the string
per unit length.
.................. ..... add ....... relativistic and quantum description of the string ....... ......
String Theory in Strong
String theory has a complex history. The idea of one-dimensional objects - strings - was born in the late 60's during one of the attempts to describe strong interactions. The study of hadron collisions (especially p- mesons) at high energies led to the so-called Venezian model, which quantifies the amplitudes of effective cross sections using products and proportions of G- functions whose argument is squares of the sums of 4-momentums of interacting particles and resulting particles. It turned out that the spectrum of Venezian's model is identical with the spectrum of normal modes of "vibration" of a one-dimensional quantized object - relativistic strings (in 1968, M.Virasoro and J.Shapiro noticed this). And Feynman diagrams describing the interactions of two particles can be consolidated into one diagram, in which 4 interacting particles (2 incoming and 2 outgoing) are shown as open strings (linear shapes of a topologically equivalent line); interchangeable particles mediating the interaction can also be shown. Each string can "vibrate" in different ways and accordingly appear as particles of a certain type (electron, photon, ...) - particles are excited states of "vibration" of the string. More specifically, different vibrations of the string model different parent particles.
Note: The size of the superstrings was considered here in the order of 10-13 cm, corresponding to the characteristic range of strong interaction.
Detailed mathematical analysis has shown that the quantum theory of the boson string is consistent (eg in terms of conformal invariance) only if the dimension of spacetime is d = 26. This dramatically exceeds the observed number of dimensions d = 4 of our spacetime. This discrepancy can be resolved by the hypothesis of "coiling" or compacting excess dimensions into small closed (compact) manifolds, as mentioned above in connection with the generalized Kaluzo-Klein unitary theories, or in §5.7 in connection with the quantum cosmology of the very early universe.
Another disadvantage of the original string theory is that in the spectrum of a free boson string (which contains only transverse modes) the ground state corresponds to a particle with a negative square of mass, ie a particle with imaginary mass - tachyon (for fundamental reasons, especially in terms of causality, we have already ruled out the possibility of the existence of tachyons in §1.6). The second excited state is already more favorable - it corresponds to a quantum with zero rest mass and spin 2, which can be identified with a graviton, see below.
In the mid-1970s, quantum chromodynamics was created (it was briefly mentioned above), which interprets strong interactions using quarks and gluons, which act on each other through the so-called "color charge". The great success of quantum chromodynamics has pushed existing string models into the background for more than 10 years.
Note: However, some physicists at the time imagined that the quarks in hadrons were connected by strings (gluon tubes) that held them together as "rubber fibers" (H.B.Nielsen, Y. Nambu, L.Suskind).
As outlined above in the section on supergravity, attempts to unify the gravitational interaction with other types of interactions within gauge quantum field theories have led to the notion of supersymmetry. This theory connects bosons and fermions: for each boson he predicts a "superpartner" who is a fermion, and vice versa. The application of these new symmetries, expressed geometrically (by commutation and anticommutation relations in spacetime) to string theory, led to a reduction in the required number of dimensions of spacetime from the original d = 26 to d = 10 (and no longer contained any tachyon). This created a supersymmetric string theory, or superstring theory. In addition to the boson string, a fermion string, or superstring, which has another spinor variable, appears here as its partner.
In the spectrum of excitations of a relativistic quantized string, there is a particle with zero rest mass and spin s = 2, which can be identified with a graviton - a quantum of gravitational waves. This led J.Sherk and J.Schwarz in 1974 to the idea that although string theory was not suitable for describing strong interactions, it could become a suitable tool for building a quantum theory of gravity. However, the size of these hypothetical strings must be radically reduced from the originally considered 10-13 cm to the dimensions of 10-33 cm of Planck-Wheeler length, characteristic of quantum gravity (introduced in §B.4).
Strings, or superstrings, are elementary one-dimensional structures that can - as resonators - vibrate in different frequency modes. The vibrations, which are determined by the dimensions of the string and its tension, are quantized, the corresponding energy takes on discrete values. The frequency of these vibrations and the number of waves determine the basic properties of the particles (eg mass or charge). Since the string has small dimensions, it can oscillate in other independent "directions", given by extra-dimensions.
Superstring excitations can be "vibrational", "rotational", and excitations of "inner degrees of freedom" - internal symmetry, supersymmetry. Different quantum excitations (normal superstring modes) are interpreted as a spectrum of elementary particles. This spectrum proves to be so rich, that it can model not only all the building blocks of a standard model of elementary particles, but also quantum gravity. Successful completion of the superstring concept would thus represent a unified approach to the diverse world of elementary particles and all their interactions - the so-called "theory of everything" TOE (Theory Of Everything) could be achieved.
A truly perfect unitary theory of "everything" should also explain the origin and concrete values of basic natural constants, resp. ratios of these constants. From a cosmological point of view, this question is briefly discussed in §5.5, passage "Origin of natural constants".
Another dimension, M-theory, 11-dimensional
The further development of superstring theory continued the research of M.Gren, J.Schwarz and E. Witten, who found such gauge groups that the theory of superstrings was fully covariant in space-time (in the spirit of GTR). Five such models of superstring theory were found, the most interesting of which were two so-called heterotonic theories with gauge groups SO(32) and S8xS8.
An important role in the theory of superstrings in recent years has been played by the analysis of mathematical (and consequently physical) equivalence or duality between different superstring models. These dualities represent new types of symmetries, unifying different models, which may take different forms at first glance, but lead to equivalent physical results. Two types of dualities have been found between existing superstring models. S-duality is manifested by the equivalence of two superstring models, in which we replace the coupling constant g with its inverse value: g ® 1/g. T-duality has a geometric character: a model with a certain coordinate, compacted on a circle of radius R, is equivalent to another superstring model with compactification on a circle ~1/R (more precisely Lstr2/R, where Lstr is the length of the superstring). Sometimes the so-called U-duality, created by a combination of S and T-duality, is also discussed.
Another consequence of dualities and unification of superstring models is the extension of the proper dimension of strings from the original d = 1 to objects with a different (higher) number p of spatial dimensions, eg 2-dimensional objects - membranes. From this is derived the abbreviated name "brane". Such multidimensional objects are no longer called superstrings, but p-branes: for p = 0 it is a point, for p = 1 it is a string, for p = 2 a membrane, etc.
The study of string dualities has shown that all existing superstring theories can be combined into a more general theory, called M-theory (E.Witten, 1995; the designation "M" comes from the name membrane, some authors link it to the attributes mystery, magic, etc.) *). Such a unified M-theory can be realized by increasing the dimension of the manifold to d = 11. If we compare this with the concepts of supergravity above, we see that the number of dimensions coincides with 11-dimensional supergravity; close connections between this two unitary theories, at least in the low-energy limit case, were also analytically proven.
*) Another variant was also proposed, the so-called F-theory (C.Vafa, 1996), using primarily 12 dimensions, but two of which are immanently twisted (to 2-toroid). It provides a large number of solutions, potentially usable in models of particle physics ...
Allegorically, the designation "M" is sometimes associated with "Mother" and "F" with "Father"...
The six extra-dimensions of the general space are compacted into an internal so-called Calabi-Yau manifold, whose geometric properties of the SU(n) holonomy suitably model the symmetries of the interactions and elementary particles. Elementary superstrings can oscillate in different dimensions. The geometric structure of inner space determines the laws of individual interactions between elementary particles and the values of physical constants (such as charges and masses of particles) that characterize individual particles - the "obvious" laws of nature in "outer" 3-dimensional space. However, these basic physical laws, which we have observed in our nature and the universe, are here a consequence of the more fundamental internal laws of the unitary theory of superstrings *).
*) These unitary theories allow many solutions using different actions (Lagrangians) and also depending on how the internal space is compacted. There can be a plethora of these solutions. We can also interpret it asdifferent universes with different apparent laws in 3 + 1-dimensional spacetime, which again leads us to think about the "multiversion" in §5.5 "Microphysics and cosmology. Inflation universe" and in §5.7 "Anthropic principle and the existence of multiple universes" ...
Astrophysical and cosmological
consequences of superstring theories
As with earlier quantum field theories and multidimensional unitary theories, interesting hypotheses of astrophysical and cosmological consequences *) of superstring theory are offered here.
*) The discussion of some cosmological consequences of quantum and multidimensional theories has already been outlined, for example, in §5.5 "Microphysics and cosmology. Inflationary universe" and in §5.7 "Anthropic principle and existence of multiple universes". Astrophysical implications for black hole physics in §4.7 "Quantum radiation and thermodynamics of black holes".
Different solutions of superstring
theory, as well as other unitary field theories, can predict different
universes with different properties (dimensions, values
of physical constants or mass spectra of elementary particles);
the anthropic principle may also say its own to
the reflection of these possibilities and their selection - see
"Anthropic Principle or the Cosmic God".
It is very difficult to decide between different versions of the origin of the universe based on astronomical observations. The only way to test the initial phases of the universe is to detect relict gravitational waves (mentioned in §5.5), which as the only type of radiation could pass through a dense and ionized substance filling the early universe. These primary (relict) gravitational waves would have a different spectrum for different scenarios of initial phases of the evolution of the universe (eg for the inflation model, the amplitude of the waves would increase towards long wavelengths, for the ecpyrotic model, on the contrary, towards short wavelengths). However, their measurement will only be possible in the future using large detectors located in space, such as the forthcoming LISA (§2.7, section "Detection of gravitational waves", final passage). Another possibility is a detailed analysis of fluctuations and polarizations of relic microwave radiation, which could be "modulated" by primordial gravitational waves.
Superstring theory is currently in the stage of intensive development. In addition to the pioneers J.Schwarz, M.Green, E.Witten, several hundred physicists (especially the younger generation) and a number of research groups work on it.
Superstring theory is considered by many physicists to be the most promising current candidate for a complete unitary field theory, unifying all 4 types of interactions and also quantum physics with the general theory of relativity, to the long-awaited "theory of everything". However, many physicists are skeptical of superstring theory. They point out the ambiguity of its conclusions, the opacity and excessive mathematical complexity, especially the difficulty and even the impossibility of experimental verification in the foreseeable future. Certain possibilities of indirect verification could result from experimental verification of electric and gravitational force action at microscopically short distances, where the usual law of inverted squares ~1/r2 could be modified by the dependence of ~1/r2 + d, in which the number of additional (hidden) dimensions d would appear..?..
|B.5. Gravitational field quantization||B.7. General principles and perspectives of unitary field theory|
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