The book presents a detailed description of the spacetime model as a causal virtualfield physical structure. Under certain conditions this model leads to fixation of the symmetry group and field representations and also to the representation of reversible local times in the world of virtual events as chains of local Treflections.
The book is intended for experts in quantum field theory and elementary particle physics and for all those interested in the problems of submicroworld physics.
Contents


 Foreword 
Section I 
 Quantum virtual nature of geometric extension 
Section II 
 Lagrangian quantum fields on lightlike hyperplanes as approximations to the fields of the model 
Section III 
 Calculation of the fieldbilinear part (I.6) of the fundamental equation of nearness (I.3) in u and V Lagrangian formalisms 
Section IV 
 Necessity of Tsymmetry of the closed algebra equation (I.3), its realization in the calculation via identification of the fields of u and Vformalisms, connection with the spacetime curvature and "currentfield" 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 nonHermiticity of scalar fields in the model. Explicit notation of the field Umatrix, 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 spacetime in the large as an algebraic causal network and the corresponding continuous approximation 
 Appendix I 
 Appendix II 
 References 
Foreword


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 spacetime "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
"figureofeight" 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 nonstandard representations of
fermion and scalar fields. It is assumed that fields of
geometric type in the model can appear as effective
fieldsconnectednesses owing to the model covariance supergroup
localization in the global algebraic spacetime 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 subPlanck
energies. The degree of model reality could in principle be
verified by calculating the lowenergy 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.
About the Author


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 HighEnergy 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 Kalgebras) and to
construct an example of a simple Lie superalgebra. In 1990 the "Theoretical
and mathematical physics" journal published his first version of the
operatorfield model of spacetime 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 lightlike nearness in spacetime 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.