Ts. D. TSVETKOV^{1} and G. PETROV^{2}

^{1}* Institute for Cryobiology and Food Technologies, BG – 1674 Sofia, Bulgaria*

^{2}* Institute for Nuclear Research and Nuclear Energy, BG – 1113 Sofia, Bulgaria*

TSVETKOV, Ts. D. and G. PETROV, 2014. Scaling without conformal invariants and the causality in the non local relativistic quantum systems in living cells. *Bulg. J. Agric. Sci., *20: 993-1007

By the living cells and organisms as an object of the fundamental cryobiological researches i.e. in this case the metabolisms is minimal and fossils e.g. the mystery by the mammoth baby Lyuba it is possible to be taken in the account the problem of a “time’s arrow” at the microscopic level by the help of the axiomatic-physical methods by the relativistic theory of quantum fields systems by the contemporary considerations of the quantum vacuum as a ground state of anyone relativistic quantum fields system. It can be defined by anyone field operator algebra becomes a fixture by the lyophilized elementary living cells and fossils. So the possibility to understand the geometrical quantum functional theory of the indefinite metric for the further considerations i.e. in this case we consider only relativistic quantum system and the word elementary understands a one structure idealization of the living cells and fossils is to be used the Hilbert functional methods for the indefinite functional metrics (Bogolubov et al.,).

Also the many miracle properties of so defined living cells and fossils apparent enchanting by consideration of his functions yet are putting besides in the molecules but in the fundamental quantum field interactions between the quantum vacuum of anyone quantum fields system in the Microsoft matter and the molecules but taken in the Minkowski space-time or in the flat space-time defined by so called oriented in the time global Lorenzian geometry too.

Since the 1948 the mathematical description of the so-called Casimir world as a part of the physical observed space-time i.e. oriented in the relativistic sense in the time is to be considered by the help of the Hamiltonian quantum field’s theory and furthermore even it is based on the fine play between the continuity and the discrete too. The axiomatic-physical methods of the local quantum fields theory has given us the other possibility than the Lagrange quantum field’s theory and precisely on this rigorously mathematical way to understand the singularities and the black holes, also the dark energy and the dark matter from one uniformly point of view.

Aside from this, the essential difference is that external forces other than gravity, e.g. such as Casimir force, play a major role in the phenomena, i.e. remember there is not observed in our seeing world a local classical relativistic electromagnetic field potential Aμ(x) caused this force. And also it is possible to describe the fundamental interactions between anyone concrete fundamental relativistic quantum field system with someone other or with the external and innerness material objects as a additional boundary conditions by the proving the fulfilling of the causality conditions and consider they as an external classical fields, and everyone internal background fields. At the first in his famous work “To the Electrodynamics of moved bodies”, Leipzig, 1905, Einstein has proved the possibility to understand the nature from the relativistic point of view in the classical physics.

Moreover it can be represented the symmetrical selfadjoint Hamiltonian operator Φ taken by as for simplicity for the relativistic quantum scalar fields by definition obtained as virtual (potential) element in the Hilbert functional space with indefinite metric. That is the quantum field operator obtained by everyone wave fields solution at the fixed time known as a virtual or “potential” quantum field operator. This is acting on the virtual vacuum vector valued functional states as a local entities of the Hilbert functional space with indefinite metric, e.g. the Minkowski space-time has a indefinite quadrate of the interval between events points. So we have in this case the vacuum state which has global properties too and also can be understand better by definition in the global Lorenzian geometry for the events points connected in pairs by the seeing time like or may be not seeing non space like geodesic line with a length non less then the length of every other non space like curve. So also it is realizable the possibility to be obtained the local or non local quantum force currents. They interacts minimal local or global by phase integration over the field potential with the field force carrier knowing as the so called virial current (vis via as a quantum source) i.e. that impact near local or global by interactions with the classical neighborhood in the Microsoft matter in the Minkowski space-time at the distance. The probability interpretation of the spectral family give us the physical interpretation of the observed quantum invariant entity by the relativistic quantum systems even for the dynamically (not thermodynamically) fine structure of the ground state as potential state also as virtual vector valued functional state, i.e. as the element of the Hilbert functional space with indefinite metric by the vacuum interactions in the Casimir world. It knows yet the Casimir force today is measured with exactness by 5 %.

Precisely the impact of this force on the molecular biology (genetics) is still not clear, i.e. there is a new situation of the so called quantum cryobiology. The additional boundary conditions must be taken under account, e.g. in the cosmogony models it is not possible to consider additional boundary conditions. So also it is possible to understand better the molecules by the molecular biology as a classical object interacting with the ground state of the every one relativistic quantum field system. So also by definition it is considered the relevant operator valued functional Banach algebra or in the Schrödinger picture the vacuum wave functional as a solution of the wave equation describing the same relativistic quantum system in the Minkowski spacetime or oriented in the time space-time, e.g. the so called global Lorenz geometry. With other words as in the non relativistic case (see B.C. Goodwin, 1963), by the help of the so called S-matrix theory in the quantum mechanics where this theory is very gut proved we hope to understand better the nature under consideration in the relativistic sense of the axiomaticaly S-matrix theory by the quantum field systems in the living cells and the fossils too.

So also, the Casimir vacuum in the asymptotic past at the left of the one not moved perfectly conductor plate contains then from the micro-causal point of view propagation of the virtual particles for the initial observer understanding as referent system (a map). In the asymptotic future at the right of the same plate and the left of the second parallel moved perfectly conductor plate towards the plate at the rest with a constant velocity v contains the propagation of the see massive particles for the latetime observer, e.g. the Maxwell demon for the events point bounded with chronological geodesic time like line. Moreover at the right of the moved plate anew there is a propagation of the relativistic quantum virtual particles system, e.g. the Maxwell demon for the events point bounded with causal geodesic non space like paths. In mathematical sense it is possible to be defined the topology of Aleksandrov on the everyone space-time (M, g) – also a topology, that can be given in M by the choice of them as base of the topologies of the all sets in the form Vκ’x +(κ’xμ ) ∩ Vκx

-(κxμ ) where the non local events points κxμ, κ’xμ ∈ M are defined in the past and future cone of the space-time.

Precisely the scalar massless relativistic quantum field give us then that his local operators algebras are unitary equivalent in the bounded domains of the locally algebras by the matter field and also they have the same structure properties whish is from more great importance for the theory than the definiteness of the metric of the Hilbert functional space. So it is possible to be defined the double singularities which will be given by the ground state of local relativistic quantum scalar field system too. The symmetries and structure properties are mathematical described by the Banach algebra of the field’s operators defined in the Hilbert functional space with indefinite metric.

Farther the ground state is defined over this algebra but it can be negative too as remember from the indefinite metric of the Hilbert functional space. However then there are a number of additional properties generated from the physical distinctions by the massless systems: his scale i.e. the group of the scale transformations represented by the dilatations and special conformal transformations and conformal symmetries also obtained by the group of the conformal transformations give a double singularities of the quantum systems and the vacuum state, but scale invariance does not imply necessary a conformal invariance and as well the infrared effects leaded to manifest the global structure of the relativistic quantum systems and the vacuum state. Quantum Field Theory QFT and the Renormierungs groups theory RG-groups are classified by scale invariant, Infrared IR fixed point (Wilson’s philosophy). In the Doctor paper (Petrov, 1978) it is showed that the scaling behaviors of the some quantum entities are destroyed in longitudinal and conserved in the cross section’s direction by fulfilling the causality condition for non forward deep inelastic scattering of leptons and hadrons. Also the scale invariance is not from the same nature as the conformal invariance by the massless quantum fields and the scale invariance lead yet not necessarily to the conformal invariance. Also it is possible to consider in the double cone with Alexandrov topology in the Lorenz manifold by the help of the mirror reflections and hyperbolical turns between two mirror one at the rest and the second parallel moved with a constant velocity v at the face a domain of the sequence of fixed events points in Minkowski space-time without accumulative point.

So in this case it is remarkable to understand the possibility to distinguish the chronological and the causality bounded events points in the space-time for n → ∞ where n is the number of the mirror reflections at the moved mirror.

Furthermore the Hilbert functional space understands by means of the space of the test functions from his completion by anyone norm the possibility of the definition of the Casimir quantum vacuum state as well a ground state of the relativistic quantum field system in the Schrödinger picture over the involutes Banach algebra of the field operators defined in the Hilbert functional space with indefinite metric. Then so one functional vector valued vacuum state can be negative as remember of the indefinite metric by definition but this is not from anyone significance for the theory. This question precisely spoken is a pure algebraically formulations of anyone relativistic quantum systems out of the Hilbert functional spaces with indefinite metric.

Furthermore the vacuum state in the Schrödinger picture defined over this algebra can be negative too as remembering of the indefinite metric but that is only a one algebraic problem. It can be shown that, on scaling-invariant time like paths of the virtual quantum particles, there is a redeﬁnition of the dilatation current by the virial current that leads to virtual generators of dilatations operators.