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Principles of a Unified Theory of Spacetime and Physical Interactions

Principles of a new approach (binary geometrophysics) are presented to construct the unified theory of spacetime and the familiar kinds of physical interactions. Physically, the approach is a modified S-matrix theory involving ideas of the multidimensional geometric models of physical interactions of Kaluza-Klein's type as well as Fokker-Feynman's action-at-a-distance theory. Mathematically, this is a peculiar binary geometry being described in algebraic terms. In the present approach the binary geometry volume is a prototype of three related notions: the S-matrix, the physical action (Lagrangian) of both strong and electroweak interactions, and the multidimensional metric. A transition from microworld geometrophysics to the conventional physical theory in classical spacetime are characterized.


Introduction
In attempts to construct microworld physics for the last half of a century hopes have been pinned successively on a few key ideas and principles. So, in the fifties those were the principles of quantum field theory, in the sixties attempts were made to construct the S-matrix approach and quantum theory axiomatics, in the seventies and early eighties a primary emphasis was placed upon group methods and gauge theory of physical interactions, at the very end of the 20th century hopes were pinned first on supesymmetric theories and then on superstrings and p-branes. At the same time, other (side) concepts: Fokker-Feynman's action-at-a-distance theory, Penrose's twistor theory, multidimensional geometrical models of Kaluza-Klein's type etc. were investigated.
In all these investigations a relation to the coordinate spacetime was formulated in any event. The latter was a priori assumed four-dimensional, either was generalized to superspaces or multidimensional manifolds, or it was considered that for constructing microworld physics needless to proceed from the coordinate spacetime, and the momentum space should be used as the basis for constructing a theory (as was declared by Chew in the S-matrix approach [1]), either it was proposed to start from complex notions of another kind, e.g. from twistors in Penrose's program [2,3], then arriving at both quantum theory and the classical spacetime.
In the present paper a new approach is proposed to construct microworld physics, involving a number of ideas and methods of the previous research and allowing one to consider the essence of spacetime and physical interactions from another viewpoint. The theory being put forward is called binary geometrophysics. Its premises have somewhat in common with Penrose's twistor program, however our starting points are more abstract and richer in their consequences. Binary geometrophysics is closest to the S-matrix theory. The word "binary" reflects two sets of states of the system, i.e. the initial (i) and the final (out) ones, forming the basis of the theory. The theory itself is a peculiar binary geometry describing microworld physics.
As known, the idea of S-matrix approach was put forward by J. Wheeler [4] and W. Heisenberg [5] and was developed into the S-matrix in the sixties in the works by a number of authors (e.g., see [1,6]). Its basis constituted principles of Lorentz invariance, analyticity, causality etc. Particles and their characteristics were related to poles in the complex plane.
Binary geometrophysics and the S-matrix theory share, apart from the two kinds of states, a possibility of constructing a theory "independently of whether microscopic spatial-temporal continuum does exist or not" [1, p.17]. In binary geometrophysics there is no a priori spacetime for microworld. It arises only in the relations between macro-objects (according to the idea of a macroscopic nature of the classical spascetime [7]). On the very elementary level there becomes invalid the differential and integral calculus alongside spatial-temporal continuum. There remain only algebraic methods.
To construct binary geometrophysics, the mathematical (algebraic) formalism of Yu.I. Kulakov's binary physical structures [8,9,10] has been used. (The latter was previously developed for other purposes.) As in the S-matrix theory, of importance here is a symmetry, however, in our theory there emerges a more general than the Lorentzian symmetry from which one may turn to Lie groups, in particular, to the Lorentz group and internal symmetries being used in gauge theories.
In the present approach the function continuity condition is used only at the first stage to find algebraic laws of structures (relation systems), and then from merely algebraic considerations one may arrive at a definition of particles and a relationship between constants and charges in strong and electroweak intractions. But above all, in the framework of binary geometrophysics, from the unified expressions prototypes of three related notions: the S-matrix, the action (Lagrangian) for strong and electroweak interactions of elementary particles and the multidimensional metric are derived.
It should be emphasized that binary geometrophysics has a relational pattern. Relations between elements, i.e. primary notions of the theory, play the key part in it. Specifically, the microanalogue of reference frames in Relativity [18], called a binary system of complex relations (BSCR), is put in the forefront. If micro-oobjects are denoted by the symbol µ, and macro-objects by the symbol m, then the starting points of the relational theory should be defined by the symbol R µ (µ), where µ in parentheses means that micro-objects are considered in it, and the subscript µ means that this is done with respect to micro-objects as well. In such terms classical mechanics, describing macro-objects with respect to macroinstruments, is denoted as R m (m), and quantum mechanics -as R m (µ).
The first two sections present principal ideas and principles of binary geometrophysics at the level R µ (µ) in the absence of notions of the classical spacetime, and in the fourth section a transition to the conventional theory R m (m) in the presence of these notions is discussed.

Binary Geometry of the Microworld
Briefly outline the principal notions of BSCR forming the basis of the theory R µ (µ) and give their physical interpretation.

Two Sets of Elements
It is postulated that there are two sets of elements. Denote the first set by a symbol M, and the second one by N . Elements of the first set are denoted by Latin letters (i, j, k, . . .), and the elements of the second one -by Greek letters (α, β, γ, . . .). Between any pair of the elements of different sets a pair relation -some complex number u iα is given (see Fig. 1). The elements of the two sets have the following physical meaning. The elements of the first set M characterize initial states of particles, and the elements of the second one Nfinal states. Thus, an idea of the S-matrix approach (more exactly, an elementary chain of any evolution), i.e an initial-to-final transition, proves to be laid in the very fundamental notions of BSCR. The complex relations between elements in the two states are a prototype of both the amplitudes of a transition between states and the notions of momenta, and finally particle coordinates.

Law and Fundamental Symmetry
It is postulated that there exists some algebraic law, connecting all possible relations between any r elements of the set M and s elements of the set N : The integers r and s characterizes rank (r, s) of BSCR. The essential point of the theory is a requirement of fundamental symmetry lying in the law (1) to be valid, while substituting the given set of elements by any others in the corresponding sets. The fundamental symmetry and the continuity condition allow functional-differential equations to be derived from them and the form of both pair relations u iα as well as the function Φ (r,s) itself to be found (see [8,9,10]). In binary geometrophysics BSCR of symmetric ranks (r, r) are used, with nondegenerate and degenerate systems of relations being distinguished. For the nongeranerate BSCR the law is written via a determinant of pair relations: Φ (r,r) (u iα , u iβ , . . .) = u iα u iβ · · · u iγ u kα u kβ · · · u kγ · · · · · · · · · · · · u jα u jβ · · · u jγ = 0, where the pair relations are represented in the form Here i 1 , . . . , i r−1 -(r − 1) parameters of the element i, and α 1 , . . ., α r−1 -(r − 1) parameters of the element α.

Elementary Basis
The origin of the element parameters should be especially dwelled on. They are analogues of the notions of coordinates in the conventional geomentry. To arrive at them, in the law (1) one should separate r − 1 elements of the set M and s − 1 elements of the set N and consider them to be standard. Fig. 1 denotes these elements by letters m, n, µ, ν. Then this law may be interpreted as a relationship determining a pair relation between two nonstandard elements (say, the elements i and α) in terms of their relations to standard elements (see Fig. 1). Relations between the standard elements themselves may be considered given forever. Then the pair relation u iα proves to be characterized by s − 1 parameters of the element i (its relations to s − 1 standard elements of the set N ) and similar r − 1 parameters of the element α. The system of the standard elements comprises elementary basis of BSCR.

Fundamental Relations
In the BSCR of rank (r, r) an important part is played by fundamental (r − 1) × (r − 1)-relations, being nonzero minors of the maximal order 1 in the determinant of the law (2). For BSCR of any rank (r, r) the fundamenta relations are written in terms of a product of two determinants composed of parameters of one sort: (4)

Linear Transformations
Transitions from one elementary basis to another may be shown (see [11]) to be described by linear transformations of element parameters of two sets: where C s r C ⋆s r are the coefficients defining a class of binary systems (standard elements) being used. Restrict ourselves to the case of complex conjugate coefficients.

Two-Component and Finsler Spinors
Low-rank BSCR notions are, in fact, used in modern theoretical physics. In In particular, the two-component spinor theory [19] naturally arises in the framework of the minimamal nondegenerate rank (3,3) BSCR. Really, according to this theory, the elements are characterized by pairs of complex parameters i → (i 1 , i 2 ), α → (α 1 , α 2 ), i.e. are vectors of the two-dimensional complex space where the linear transformations (5) are defined. Restrict ourselves to the liner transformations leaving invariant each of the 2 × 2 determinants on the left in (4). Since each of such determinants is an antisymmetric bilinear form: i 1 k 2 − i 2 k 1 = Inv; α 1 β 2 − α 2 β 1 = Inv, the definition of two-component spinors becomes evident. For a selected class of transformations (5) the coefficients satisfy the condition C 1 1 C 2 2 − C 1 2 C 2 1 = 1, i.e. these transformations belong to a six-parameter group SL(2, C). Such transformations single out the privileged class of standard elements corresponding to inertial frames of reference in General Relativity.
In the framework of rank (3,3) BSCR, the transformations keeping invariant both the antisymmetric forms on the right in (4) and the pair relations u iα form the three-parameter group SU (2) corresponding to rotations in three-dimensional space. As follows from the foregoing, one may state that four-dimensionality of the classical spacetime and the signature (+ − −−) are due to the rank 3,3) of the first nondegenerate BSCR.
At this level of theory development in the framework of rank (3,3) BSCR we, in fact, arrive at the notions comprising Penrose's twistor theory. However in binary geometrophysics, relation systems of a higher rank (r, r) are considered. In them the elements are described by r − 1-dimensional vectors.
Requiring the corresponding antisymmetric forms in (4) under the transformations (5), we arrive at a nonstandard generalization 2 of the two-component spinors which are naturally called Finsler (r − 1)-component spinors [21]. These transformations form a group SL(r − 1, C).

Description of Elementary Particles
In binary geometrophysics the elementary particles are described by a few elements in each of the two sets. In the framework of a simplified model based on the rank (3,3) BSCR, massive leptons (electrons) are described by pairs of elements, and neutrinos -by one element in each of the sets. Let the electron (e) be described by two pairs of elements: i, k and α, β, then it may be characterized in terms close to the conventional ones, i.e. by the four-component column and line: where the quantities with subscripts denote covariant components of spinors.
On the elements describing the leptons imposed are constraints along the horizontal (in one set of elements) and along the vertical (in two sets). Thus, for free leptons the vertical constraints mean that the parameters of two pairs of elements in the two sets are complex conjugate to one another, following the rules of quantum mechanics. For the electron in (6) this means that i s = * α s , k s = * β s . The horizontal constraints connect the elements i and k. They correspond to Dirac's equations for free particles in a momentum space [11].

Momentum Space (Velocity Space)
In the rank (3,3) BSCR, of element parameters -two-component spinors -the four-vectors physically interpreted as velocity (or momentum) components of particles are conventionally constructed. Introducing Dirac's four-row matrices in the corresponding representation, the electron four-velocity components may be presented as follows: where e = e † γ 0 . Defining the matrix γ 5 , the left and right components of the electron may be conventionally introduced, then the left component of the electron is described by the pair of elements i, α, and the right component is described by the pair k, β. It is evident that there is only one component for the neutrino. In the general case, when the particles are described by a triple of elements, a generalization of the above rules is necessary. The above-mentioned transition from parameters to velocities may be intrepreted as a transition from a binary geometry to a unary one by a specific glueing of two pairs of elements of the two BSCR sets into new elements of one set being described by Lobachevsky's geometry. It is easy to verify that for one particle being characterized by the expression (6) we have For a free particle the four-dimensional velocities defined in such a way possess the well-known property u µ u µ = Const = 1. Introducing the second particle e 2 being described by the parameters: j, s, γ, δ, the scalar product of velocities (momenta) of two particles may be defined as where the expression in square brackets denote the fundamental 2 × 2-rank (3,3) BSCR relations defined in (4).
For interacting particles the conditions of complex conjugation of elements of the two sets are not satisfied any more. Introducing for particles in initial and final states the complex conjugate quantities and constructing from them, using the well-known formulas, four-velocities (momenta), we arrive at the characteristics of particles before and after interaction typical for the S-matrix.
9. Choice of BSCR Rank In our works (see [11,12]) properties of physical theories based on the BSCR of ranks (2,2), (3,3), (4,4), (5,5), (6,6) and higher have been analyzed step by step. The analysis has shown that the prototype of the known kinds of physical interactions is constructed in the framework of rank (6,6) BSCR. Therewith individual elementary particles (fermions) should be described by triples of elements in each of two sets M and N . This corresponds to the familiar notions of the structure of baryons consisting of three quarks. In this approach these notions are extended to leptons as well.
10. Base 6 × 6-Relations As a prototype of such related notions as the S-matrix or the action (Lagrangian) for two interacting particles there should be some expression containing two triples, i.e. six elements of one set M and six elements of the set N . Such is the base 6 × 6-relations being written in the framework of the rank (6,6) BSCR as follows where the vertical lines underline the fact that the first particle is described by the elements i, k, j, α, β, γ, and the second one is described by the elements s, l, r, δ, λ, ρ. This expression is invariant under the transformations of parameters of the group SL(5, C) and is a specific volume in the binary geometry of rank (6,6). The physical interpretation of the base 6 × 6-relation is illustrated by the digrams of Fig. 2, where on the left the 12-tail structure of binary geometrophysics is depicted. Two triples of the lower lines describe initial states of two particles, and two upper triples -their final states. A generalization of Feynman's type diagram is presented in the middle, and a standard diagram of particle scattering is given on the right.

Transition from the Bionary Volume to the S-Matrix Prototype or the Action (Lagrangian)
To pass from the base 6 × 6-relation to prototypes of the S-matrix or the action (lagrangian) of two particles, the following set of approaches, procedures and principles should be used.

Splitting Procedures
First of all, it is necessary to perform a procedure of splitting (or reduction) corresponding to the 4+ s-splitting procedure into four-dimensional spacetime and additional dimensions in multidimensional P ′µ  (7), and from the internal ones charges of elementary particles are constructed. In fact, this procedure means splitting the initial rank (6,6) BSCR into two subsystems: a rank (3,3) BSCR (with two parameters) and a rank (4,4) BSCR (with three parameters). As a result of splitting, the initial transformation group SL(5, C) is narrowed down up to two subgroups: SL(2, C) -for external parameters and SL(3, C) (or a narrower one SU (3)) -for internal parameters.
The left lower or right upper 3 × 3-subblocks M [(2), (2)] contain terms of the form · · · · , where the horizontal line divides the parameters characterizing two particles. Such terms describe scalar interactions of two particles. They correspond to Higgs' scalar bosons. These subblocks will be called mass ones.
The diagonal left upper M (4, 0) and right lower M (0, 4) subblocks contain external parameters of only one of the particles. They correspond to the action (Lagrangian) of "free" particles. Such submartices will be called free for the first and second particles.
The rest four extreme 3 × 9-9 × 3-subblocks contain three parameters of one particle and one parameter of another particle. There are criteria allowing contributions of such terms to be excluded.

Unified Principle of Describing Baryons and Leptons
The theory being presented here permits interactions (processes involving) baryons, massive leptons and neutrinos to be described in the same way. All particles mentioned are described by triples of elements in each of the two sets (states). The baryons are generally characterized by all three twocomponent columns of the external parameters containing nonzero components. Thus, e.g., in the set M the baryon (b) is characterized by a rectangular 3 × 5-matrix of the parameters of its consistuents: where two upper lines correspond to external two-component spinors, and the three rest -to additional parameters (three-component Finsler spinors) describing internal degrees of freedom (charges) of an interacting particle. For massive leptons (electrons (e)) the matrix has the same form (14), however, one of the upper two-component columns consists of zero external parameters. Let it be the third column: j 1 = j 2 = 0. For neutrinos (ν), being also described by ×5- (14), two two-component columns of the external parameters are zero. Let it be the first two columns: i 1 = i 2 = 0, k 1 = k 2 = 0. For the leptons defined in such a way the number nonzero terms in the base 6 × 6-relation (13) reduces considerably. It should be noted that the number zero columns of the external parameters is an invariant property of particles with respect to distinguished parameter transfomation groups. The particle definitions by external parameters given here are in agreement with those presented in paragraph 7 of the previous section.

Exchange Character of Physical Interactions
If one write the base 6 × 6-relation for two particles with the same internal parameters, then it vanishes due to antisymmetry of determinant columns. A nonzero result arises in using the exchange mechanism of physical interactions. It is based on the postulate that, according to values of the external parameters, the particles may be in two kinds of states: in the U -state ("normal") or in one of the X-states ("excited"). The interaction process consists in an exchange of states between particles.
The "normal" or U -state is characterized by a nonzero 3×3-determinant of the internal parameters. The analysis of possible simplest kinds of X-states (taking account of the principle of correspondence to the conventional theory) shows that the X-states may be of two kinds. One type (X C ) is also characterized by a nonzero determinant of three columns of the external parameters, and for another type (X N ) such a deteminant is zero.
4a) X C -states (charged) The simplest variant of determining X C states leading to nonzero base 6 × 6-relations lies in changing the sign of of the three columns of the internal parameters of the U -state. In all there are three such possibilities (channels) which, according to the column number with changed sign, are called X X -, X Y -and X Z -states. It can be shown that for these X C -states only three pairs of terms in the M (2, 2)-matrix prove to be nonzero. They are located in its 3 × 3-subblocks similarly to nonzero elements in six nondiagonal Gell-Mann's matrices λ n . Such terms correspond to interactions via charged vector bosons in the conventional theory.
4b) X N -states (neutral) In X N -states a pair or all three columns (three-dimensional vectors) of the internal parameters are collinear. In the simplest case on may assume that all three vectors (of the column) c ′ (s) , where s = 1, 2, 3, are collinear, i.e. representable as c ′ (s) = C ′ s c ′ , where c ′ is a three vector, and C ′ s are three coefficients. The coefficients C ′ s in diagonal summands of the submatrix M (2, 2) may be shown to enter only as differences, i.e. only two combinations of them are independent. One may choose the combinations as follows: Two particular cases: ) C ′ = 0;C ′ = 0; ) C ′ = 0;C ′ = 0 define two channels: an -channel with the corresponding X A -state and ) a channel with the corresponding X B -state, which should be compared to two channels of interactions via neutral vector bosons in the conventional theory. The two combinations of the coefficients in (15) correspond to two Gell-Mann's diagonal matrices: λ 3 and λ 8 in the conventional representation. The above considerations characterize, so far only qualitatively, an essence of five channels (three charged and two neutral) in the known types of physical interactions. Details will be presented in the next article.

Intermediate Vector Boson Treatment
In the given approach, corresponding to (Fokker-Planck's) action-at-a-distance concept, there are not intermediate carriers of interactions (vector bosons). To them correspond the above channels of interactions (types of X-states of elementary particles). For strong interactions eight gluons correspond to these channels: three pairs of charged gluons corresponding to the X X -, X Y -and X Z -states, and two neutral --gluons corresponding to the states X A and X B . Similarly "intermediate" vector bosons "carrying" electroweak interactions.
6. "Matreshka" Principle In the conventional gauge field theory three types of spaces are, in fact, used to describe interactions: 1) an external one corresponding to the classical four-dimensional spacetime with the Lorentz transformation group (SL(2, C) group), 2) an internal space of electroweak interactions in which there takes place the group SU (2) × U (1) and 3) an inernal (chromatic) space of stong interactions with the group SU (3). The theory is being constructed as a composition of these spaces, which is called the "bricks" principle Binary geometrophysics assumes a more economical approach of electroweak interactions being considered as truncated strong interactions, which is achieved by fixing one of the lines of the internal parameters in (14) (let it will be a line of the parameters with index 3), when the group SL(3, C) (or SU (3)) of admissible transformations of the internal parameters is narrowed down up to the subgroup SL(2, C) (or SU (2)) of the internal parameters with indices 4 and 5. It turns out that in such a truncated theory there remain valid considerations presented above of two types of interactionsanalogues of those via charged and neutral vector bosons, with the difference that for one particle generation only one channel (of three X C ) survives, which corresponds to interactions via one pair of charged vector bosons.
In such a theory, electromagnetic interactions may be shown to correspond to strong ones via neutral A-gluons, weak interactions via neutral vector Z-bosons to strong ones via B-gluons, and electroweak interactions via charged vector W ± -bosons to stong ones via one of the pairs of charged vector gluons (say, X ± -gluons).
Such a principle of actual embedding of one type of physical interation to another may be called the "matreshka" princile, an alternative to the conventional "bricks" principle.

Physical Interaction Channel Symmetry Principle
The above procedures and principles allow prototypes of the action (Lagrangian) of strong and electroweak interaction of baryons in terms of quarks (their consistuents) as well as electroweak interaction of leptons (massive ones and neutrinos) between one another and with baryons (quarks) to be written down in the same way. These prototypes are constructed as combinations of products of the four-velocities (of the external parameters) of the particle (quark) components with the corresponding coefficients of the internal parameters having a physical meaning of charges in strong and electroweak interactions. However, therewith there remain uncertain quantities and values of the independent constants and charges in the corresponding interactions. This gap is eliminated by using the physical interaction channel symmetry.
It turns out that for obtaining the familiar relations between charges it suffices to know, first, the fact of an existence of the above channels, second, the character of the presence of charges in the corresponding interactions (multiplicativity in charges in the interactions via charged vector bosons) and, third, the conditions of total symmetry of the above channels for all quarks in strong interactions or for left components of particles (quarks or leptons) in electroweak interactions. Therewith strong interactions are unambigously shown to be characterized by only one constant corresponding to the constant g 0 in chromodynamics, and electroweak ones -by two constants corresponding to the familiar charges g 1 and g 2 in Weinberg-Salam's model.
In so doing it is natural that one relationship for two interaction constants is written down, which permits Weiberg's angle to be calculated theoretically.

Description of Elementary Particle Generations
The approach proposed (in the framework of binary geometrophysics) allows one to resolve (or opens a new avenue of attack on) a number of problems of the modern theory of physical interactions. For example, there opens a possibility of theoretical justification of the existence of just three generations of elementary particles. Simpifying the situation, one may argue that the availabitity of three generations of elementary particles (quarks and leptons) is closely related to the existence of three channels of strong interactions of quarks via charged vector bosons. As mentioned here, in electroweak interactions of the particles of the same generation an analogue of only one pair of charged gluons in the form of W-bosons; in interactions of the particles of two other generations there appear analogues of two other pairs of charged gluons. For leptons, in particular, the definitions of three generations are also related to arrangement of zero columns of the external parameters.
It should be particularly emphasized that the approach proposed corresponds basically to the conventional gauge models of physical interactions (e.g., see [22,23]). In its framework notions of antiquarks and antileptons are easily defined, besides introduced are mesons as well as particles being described by pairs of elements in each of two sets of the rank (6,6) BSCR, with, to fit the conventional approach, on of these elements corresponding to the definition of a quark, and another -of an antiquark. A number of other problems will be elucidated in a succession of articles.

Architectonics of Binary Geometrophysics
The aforesaid relates only to the fundamentals of binary geometrophysics. The simplest act of evolution -a transition of two interacting particles from one state to another has been discussed, with elementary particles being as elementary bases for the latter. This always meant three specific subsets of elements comprising the first and second interacting particles and elementary basis.
An importamt problem of binary geometrophysics program lies in constructing a transition from an algebraic theory of the level R µ (µ) to the conventional spatial-temporal and other attendant notions of modern physics. The most essential points here are as follows.
First of all, to the three specific susets of elements mentioned the fourth subset, formed by all other elements that did not enter in these three subsets, should be added. The first three of these subsets may be simple (individual particles) or rather complex up to macro-objects, whereas the added fourth subset is extremely complex by definition.
To turn to the conventional physical theories, it is necessary to take the steps describable as the following chain: