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 andfurthermore even it is based on the fine play between the continuity and the discrete too.Following the classical Einstein’s gravitational theory Weyl in 1918 attempt to incorporate electromagnetism into thetheory by gauging the metric tensor i.e.by letting: gμν = exp(-γ∫dxμWμ(x))gμν, where γ was a constant and the vector field Wμ was to be identified with the electromagnetic vector potential.Although this idea was attractive, following Einstein, it was physically untenable because it would imply that the spacing of spectral lines would depend on the history of the emitting atoms, in manifest disagreement with experiment to this time by the quantum understandings of the nature.However, after the advent of Wave Mechanics in 1926, the idea was resurrected by application to other physical situations.This new observation that the usual electromagnetic differential minimal principle was equivalent to the integral minimal principle and that this was the correct version of Weyl’s proposal in which the constant was chosen pure imaginary γ = i/ћ, where ћ is the Planck constant h divided by 2π and the electromagnetic factor was chosen to multiply more the Schrödinger wave functional Ψ*ακ”(φκ’, t) by t ∈ (t-2(n - ½j), t2(n - ½j)], n = 0,1,2,…, j ≤ 2n, understanding mathematical as an operator acting on everyone function which described the relativistic quantum systems considered for simplicity by us only for relativistic scalars particles systems: (Formula presented), where j is the number of the virtual (“potential”) scalar Dirac particles called by us scalarino fulfilled anticommutative relations and occupied the local place localized (micro causality in the Casimir world) by the neighbourhood of the 4-ponts yμ-2(n-½j) in the coordinate Minkowski space-time.Moreover they have a helicity (projection of the spin on the longitudinal direction of the moving) also at the left at the mirror at the rest act the helicity operator on the vacuum state λ|ℓ0> = |-½> and at the right λ|r0> = |½> and the same is for the moved mirror.Between the mirrors is the helicity zero.Further for ϰ → ϰ’ or by localization ϰ, ϰ’→ 0 is defined: ∫Dφακ(Ψφακ(ακ, t-2(n-½j))δ(ακ‘-φακ)Ψ*ακ“(φακ’, t-2(n-½j) + ε½))Dακ‘= 1, for the Dirac δ-functional by definition δ(ακ‘-φακ) = <φακ|ακ‘> and then is (Formula presented).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 better the Whitney’s singularities theory applied on the vacuum and the black holes, also the dark energy and the dark matter from one uniformly point of view.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 in the relativistic theory of quantum wave fields by the contemporary considerations of the quantum vacuum as a ground state of anyone relativistic quantum fields e.g.j = 0 and n →∞.This 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 by Hilbert space 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 of the indefinite functional metrics (Bogolubov at al., 1987).Also the many miracle properties of so defined living cells and fossils apparent enchanting by consideration of hisfunctions yet are putting besides in the molecules but in the fundamental quantum field interactions between the quantum vacuum of anyone quantum fields 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.So also it is possible to be solving the many body problems by our so called Gedanken experiment with hyperbolic turns and reflections for Casimir world defined by two mirrors moving parallel to another i.e.the one can be at the rest and the other move with a constant velocity v.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 with anyone other or with the external and innerness material objects as a additional boundary conditions (localizing) by the proving of 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 field operator Φ taken for simplicity for the relativistic quantum scalar fields by definition obtained as functional virtual (potential) vector valued state in the functional state Hilbert space with indefinite metric.That is the quantum field operator obtained by anyone wave fields solution at the fixed time known as a virtual or “potential” quantum field operator.This is acting on the vector valued functional virtual vacuum 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 and is Lorenz invariant.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 at least one seeing non space like geodesic line with a length non less as 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 by the help of the ensembles of the so called virtual current particles e.g.scalars and his scalarino.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 point source in three dimensional space or quantum sink in two dimensional space at a given time) i.e.that impact near local or global by interactions at the distance with the classical local neighborhood in the Microsoft matter in the Minkowski space-time.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 state 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 (localization) 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.So also by definition it is considered the relevant operator valued functional Banach algebra or in the Schrodinger picture the vacuum wave functional as a solution of the wave equation describing the same relativistic quantum system in the Minkowski space-time or oriented in the time space-time, e.g.the so called global Lorenz geometry.With other words as in the non-relativistic case (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 axiomatically S-matrix theory by the relativistic quantum systems in the living cells and the fossils too.So also the Casimir vacuum in the asymptotic past at the left “ℓ” side of the one perfectly conductor plate at the rest 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 “r“side of the same plate and the left side of the second parallel moved perfectly conductor plate towards the plate at the rest with a constant velocity v contains the propagation of some see massive particles for the late-time observer, e.g.the Maxwell demon for the events point bounded with time or non-space like geodesic line.Moreover at the right side 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 time or non-space like causal geodesic 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 (Formula presented) - where the non-local events points kxm, k’xm ∈ M are defined in the past and future cone of the space-time.Precisely the mass less 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 which is from more great importance for the theory than the definiteness of the metric of the Hilbert or Banach functional vector valued state space.So it is possible to be defined the double singularities which will be given by the ground state of local relativistic quantum system too.The symmetries and structure properties are mathematical described by the Banach algebra of the operators valued field’s defined in the Hilbert functional vector valued state space with indefinite metric.Farther the ground state is defined over the Banach algebra but it can be negative too as remember from the indefinite metric of the Hilbert functional state 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 relativistic quantum systems and the vacuum vector valued 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 (G.Petrov, 1978) (the Thesis) it is showed by the help of the mathematical generalized Fourier analysis 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.It is possible to consider in the double cone with Alexandrov topology in the Lorenz manifold of the Casimir world 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 chronology and the causality by the ensemble from assembling and folding surfaces of bounded events points in the space-time for n → ∞ where n is the number of the mirror reflections at the moved mirror.Furthermore by means of the space of the test functions from his completion by anyone norm the Hilbert functional space understands the possibility of the definition of the Casimir quantum vacuum state as well a ground state of the relativistic quantum field system in the Schrodinger picture over the involutes Banach algebra of the operators valued fields defined in the Hilbert functional state space with indefinite metric.Then so one virtual (“potential”) 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 in the Hilbert functional state spaces with indefinite metric.It can be shown that, on scaling-invariant time like or causally non space like paths of the virtual quantum point or sink sources, e.g.current particles, there is a redefinition of the dilatation current by the virial current that leads to virtual (“potential”) generators of dilatations operators. © 2016, National Centre for Agrarian Sciences.All rights reserved.