## Descripción

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

**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.

GEORGE L. Stavraki