| Foreword |

Section I |

| Quantum virtual nature of geometric extension |

Section II |

| Lagrangian quantum fields on light-like hyperplanes as approximations to the fields of the model |

Section III |

| Calculation of the field-bilinear part (I.6) of the fundamental equation of nearness (I.3) in *u*- and *V*- Lagrangian formalisms |

Section IV |

| Necessity of *T*-symmetry of the closed algebra equation (I.3), its realization in the calculation via identification of the fields of *u*- and *V*-formalisms, connection with the space-time curvature and "current-field" duality; explicit formulas of the corresponding approximation |

Section V |

| Derivation of equations for unimatrix parameters from the closedness condition of algebra (I.3) within the adopted approximation |

Section VI |

| Calculation of parameters of the unimatrix, the matrices *S*_{1}, S_{2} and the eigenvalues of the Casimir operator for field representations in the charge group *omega* from the closedness condition of algebra (I.3). Fixation thereby of the charge group as group *E*_{6} together with the corresponding representations for fields. Necessity of non-Hermiticity of scalar fields in the model. Explicit notation of the field *U*-matrix, abstracted from the correspondence principle, with calculated parameters and of the exact algebraic equation defining it at two points |

Section VII |

| Possible approach to the description of space-time in the large as an algebraic causal network and the corresponding continuous approximation |

| Appendix I |

| Appendix II |

| References |

The present study is an attempt to formulate the notion of
quantum causal nearness of possible local events in the virtual
word described by a specially organized complex of physical
quantum fields assumed to be primary. The fundamental element of
extension, or the space-time "cell", is described in the model
by the equation of a commutator algebra closed at two such
"nearest" local complexes. The corresponding causal
relationship is visually interpreted as a light line closed as a
"figure-of-eight" loop. The symmetry of this local
construction under -reflection allows representation of a
-reversible flow of local time by a discrete chain of local
-reflections instead of a continuous time shift. The fundamental
causal interval thus constructed is then an indivisible time
step of such symmetric time.

The structural charge symmetry group in such a model turns out
to be fixed as group with non-standard representations of
fermion and scalar fields. It is assumed that fields of
geometric type in the model can appear as effective
fields-connectednesses owing to the model covariance supergroup
localization in the global algebraic space-time network of
"linked" chains.

Under certain conditions this argument allows us to interpret
the superinvariant expression, quadratic in the found complex of
physical fields, as a renormalized approximate Lagrangian
averaged over geometric fields and corresponding to sub-Planck
energies. The degree of model reality could in principle be
verified by calculating the low-energy limit of such a
Lagrangian scheme.

The work was concluded thanks to the boundless devotion and
tolerance of my wife, I. V. Karpenko. I would like to express
sincere gratitude to my friends and colleagues P. Singh,
R. N. Faustov, V. O. Galkin, R. F. Polishchuk, and
G. A. Vilkovyskii for their permanent support.

I am grateful to M. Tsaplina for the English translation of the
book.

*G. Stavraki*

Moscow, 8 June 2008.

**George L. STAVRAKI**
The author is a theoretical physicist, a research worker of the Computation
Center of the Russian Academy of Sciences. In 1966, in his talk at the
International High-Energy Physics School (Yalta) he suggested the possibility
of a joint description of boson and fermion degrees of freedom of quantum
fields within the framework of a unified closed algebra and was the first to
define the concept of Lie superalgebras (referred to as K-algebras) and to
construct an example of a simple Lie superalgebra. In 1990 the "Theoretical
and mathematical physics" journal published his first version of the
operator-field model of space-time as a virtual causal structure. In 2003, in
the note in "Concise encyclopedia of supersymmetry" (Cluwer Academic
Publishers, Dordrecht) he suggested the introduction of the concept of
quantum causal light-like nearness in space-time and in 2006 realized it (in
his work published in "Gravitation and Cosmology") using a special
construction within the framework of quantum field theory. The present book
offers the extended version of the construction.

Stavraki George L.

Theoretical physicist, currently researcher, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences. In 1966 his report on the International school on high energy physics (Yalta) suggested the possibility of local descriptions of a system of interacting boson and fermion fields within the framework of a unified closed algebra that generalizes the canonical commutation relations, and was the first to define the concept of Lie superalgebras (referred to as K-algebras) and to construct an example of a simple Lie superalgebra. In 1990 the “Theoretical and mathematical physics” journal published his first version of the operator-field model of space-time as a virtual causal structure. In 2009 he published a book, “Model of Space-Time as a Field Noncommutative Causal Structure” (M.: URSS) detailing the construction of the model and giving consequences that determine the characteristics of the group charge of its basic fields.