Space-Time Properties as Quantum Effects. Restrictions Imposed by Grothendieck's Scheme Theory

Leonid Lutsev
2019 Journal of Modern Physics  
In this paper we consider properties of the four-dimensional space-time manifold  caused by the proposition that, according to the scheme theory, the manifold  is locally isomorphic to the spectrum of the algebra  , where  is the commutative algebra of distributions of quantum-field densities. Points of the manifold  are defined as maximal ideals of density distributions. In order to determine the algebra  , it is necessary to define multiplication on densities and to eliminate those
more » ... liminate those densities, which cannot be multiplied. This leads to essential restrictions imposed on densities and on space-time properties. It is found that the only possible case, when the commutative algebra  exists, is the case, when the quantum fields are in the space-time manifold  with the structure group ( ) 3,1 SO (Lorentz group). The algebra  consists of distributions of densities with singularities in the closed future light cone subset. On account of the local isomorphism ( ) Spec ≅   , the quantum fields exist only in the space-time manifold with the one-dimensional arrow of time. In the fermion sector the restrictions caused by the possibility to define the multiplication on the densities of spinor fields can explain the chirality violation. It is found that for bosons in the Higgs sector the charge conjugation symmetry violation on the densities of states can be observed. This symmetry violation can explain the matter-antimatter imbalance. It is found that in theoretical models with non-abelian gauge fields instanton distributions are impossible and tunneling effects between different topological vacua | n〉 do not occur. Diagram expansion with respect to the  -algebra variables is considered. Journal of Modern Physics these properties can be determined by the spectrum of the commutative algebra  of distributions of quantum-field densities. Points of the space-time manifold  are defined as maximal ideals of quantum-field density distributions. The scheme theory imposes restrictions on space-time properties. Schemes were introduced by Alexander Grothendieck with the aim of developing the formalism needed to solve deep problems of algebraic geometry [22] . This led to the evolution of the concept of space [23] . The space is associated with a spectrum of a commutative algebra or, in other words, with a set of all prime ideals. In the case of the classical physics, the commutative algebra is the commutative ring of functions. In contrast with the classical physics, quantum fields are determined by equations on functionals [24] [25] [26] [27]. Quantum-field densities are linear functionals of auxiliary fields and, consequently, are distributions. There are many restrictions to construct the commutative algebra of distributions. In the common case, multiplication on distributions cannot be defined and depends on their wavefront sets. In the microlocal analysis the wavefront set ( ) 3,1 SO (Lorentz group) and the time is one-dimensional. The asymmetry of time, the chirality violation of spinor fields, and the charge conjugation symmetry violation in the boson sector are the necessary conditions for the existence of L. Lutsev
doi:10.4236/jmp.2019.107054 fatcat:vf64ejvsjzfdle5gh34bxrs4ae