Progress in Applied Mathematics

Since the 1948 the mathematical description of the so-called Casimir world as a part of the physical observed space-time in the relativistic sense 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. there is not observed in our seeing world a local classical relativistic electromagnetic field potential Aμ(x). 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.
By the living cells as an object of the fundamental cryobiological researches i.e. in this case the metabolisms is minimal so that by the help of the axiomatic-physical methods by the relativistic theory of quantum fields systems considering as a Micro fields it is possible to be taken in the account the problem of a “time’s arrow” at the microscopic level by the contemporary considerations of the quantum vacuum in the Casimir world as a ground state of anyone relativistic quantum system becomes a fixture by the lyophilized elementary living cells. So the possibility to understand the Hilbert space with indefinite metric for the further considerations the word elementary understands a one structure idealization of the living cells. Also the many miracle properties of so defined living cells 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 Minkowsi space-time too. 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 acting on the virtual vacuum vector valued functional states of the Hilbert functional space with indefinite metric. So also it is realizable the possibility to be obtained the local or non local quantum force currents, i.e. whish interact minimal local or global by phase integration over the field potential with the field force carrier knowing as the so called virial current i.e. that impact near local or global at the distance of interactions with the classical 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 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, t.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 impulse wave equation describing the some relativistic quantum system in the Minkowski space-time. With other words by the help of the so called S-matrix theory as in the non relativistic case where this theory is very gut proved we hope to understand better the nature under consideration.It knows the following fact that it is potentially force with a long-range action at the distance or with other words asserted every experiment in this genetics domain without clearness of the role of his impact on the neighborhoods in the living cell. Just therefore this is to be taken very good under account from the point of view of the nanophysics too. May be he is the cause for not observing of the so-called Goldstein massless bosons or the quarks as it is the case by the Coulomb force between the charged particles.Moreover the Casimir vacuum state of the relativistic quantum scalar field system may be not belonging in the operator definition’s functional domain of the field’s operators, but fulfill the additional causal and boundary conditions by the solving the boundary value problem for the carrier of the interaction force, the virtual fundamental scalar particles called by us scalars belonging to the domain structure of the Casimir world. 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 the propagation of the see massive particles for the late-time observer, e.g. the Maxwell demon, and moreover at the right of the moved plate anew a propagation of the virtual relativistic quantum particles system. Precisely the scalar massless relativistic quantum field give us then that his local 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 dies 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. 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 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. The theoretical underpinnings of scale without the conformal invariance in relativistic quantum physics are given in the light of the results of the non local operator’s expansion on the light cone. Then the Casimir vacuum state of a given relativistic scalar quantum fields systems, precisely due to deep connectionsbetween scale-invariant theories and the recurrent scaling behaviors of the quantum entities in the Casimir world can be defined over the involutes Banach algebra of the field operators acting on the virtual vector valued state defined in the Hilbert functional space 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. Also just that lead to the generations of the virtual vacuum fluctuations described by the relativistic quantum fields operators created an involutes Banach algebra of the quantum field operators out of the Hilbert space with indefinite metric. Finally, it can be develop a systematic algorithm by the Casimir world for the research of scaling-invariant non space like paths of virtual particles caused by virtual fluctuations of the vacuum with a zero point energy ZPE and broken scaling-invariant time like and non space like paths of see massive particles.

Belaus, A. M., & Tsvetkov, Ts. D. (1985). Nauchnie ocnovi tehnology sublimatsionova konservanija (p.208). Kiev, Naukova Dumka, (Ru).

Bogolubov, N. N., Logunov, A. A., Oksak, A. I., Todorov, I. T. (1987). Obshie prinzipi kvantovoi teorii polya (p.615). Kiev, Naukova Dumka, (Ru).

Bogolubov, P. N., Matveev, V. A., Petrov, G., Robaschik, D. (1976). Causality properties of form factors, Rept. Math. Phys, 10, 195-202.

Bordag, M., L., Kaschluhn, G. P., Robaschik, D. (1983). On nonlocal light-cone expansion in QCD. Yd. Fiz., 37(1), 193 - 201.

Bordag, M., Petrov, G., Robaschik, D. (1984). Calculation of the casimir effect for scalar fields with the simplest non-stationary boundary conditions. Yd. Fiz., 39, 1315-1320.

Casimir, H. B. G. (1948). On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Akad. Wet., 60, 793.

Fortin, J. F., Grinstein, B., & Stergiou, A. (2011). Scale without conformal invariance: An Example . arXiv:1106.2540.

Geyer, B., Petrov, G., & Robaschik, D. (1977). Kausalitaetseigenschaften von formfaktoren und kinematische bedingungen. Wiss. Z. Karl-Marx-Univ., Leipzig, Math.-Naturwiss. R., 26, 101-122 (DE).

Goodwin, B. C. (1963). Temporal organisation in cells, (A dynamik theory of cellular control processes). Academic Press London and New York, 251.

Dodonov, V. V. (2001). Nonstationary casimir ẽect and analytical solutions for quantum fields in cavities with moving boundaries. Retrieved from http://arxiv.org/PS_cache/quant-ph/pdf/0106/0106081v1.pdf

Kaempffer, F. A. (1965). Concepts in quantum mechanics (pp.392). Academic Press, New York and London.

Mitter, H., & Robaschik, D. (1999). Eur. Phys. J. B., 13, 335-340.

Petrov, G. (1974, May). The causality of the form factors of the virtually compton scattering amplitude. I. Scientific Symposium. of Bulg. students, Berlin, 10-12, (DE).

Petrov, G. (1978). Kausalität von formfaktoren, kinematische bedingungen und folgerungen für das Scaling-Verhalten im Nichtvorwärtsfall der tief-inelastischen Lepton-Hadron-Streuung, Dissertation zur Promotion A, Karl-Marx-Universität, Leipzig.

Petrov, G. (1978). Restrictions on the light-cone singularity imposed by causality. Karl-Marx-Univ., Leipzig, L. 856/78 III/8/1767.

Petrov, G. (1985). Calculation of casimir effect for the electromagnetic field with the non-stationary boundary conditions. Bulg. J. Phys., 12(4), 355-363.

Petrov, G. (1989). Casimir relativistic effect for massless scalar field with uniformly moving mirrors. Rev. Roum. Phys., 34(5), 453 – 460.

Petrov, G. (1992). Impulsive moving mirror model in a schrödinger picture with impulse effect in a banach space. Dubna, JINR E2-92-272.

Petrov, G. (1992). Impulsive moving mirror model and impulse differential equations in Banach space. Dubna, JINR E2-92-276.

Petrov, G. ( 2010). Non equilibrium thermodynamics by the concrete quantum field system with a impulse effect in the nanophysics. Agriculture and Biology Journal of North America ISSN Print: 2151-7517, ISSN Online: 2151-7525.

Tsvetkov, Ts. D., & Petrov, G. (2004). Non equilibrium thermodynamic and fields effects in the cellular cryobiology and anhydrobiology. Bulgarian Journal of Agricultural Science, 10, 527-538.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2005). Non equilibrium thermodynamics by the concrete quantum field systems in the cellular cryobiology an anhydrobiology. Bulgarian Journal of Agricultural Science, 6, 387-403.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2005). Non equilibrium thermodynamics by the vacuum waves in the Banach space for the see scalar’s systems in the cellular cryobiology an anhydrobiology. Bulgarian Journal of Agricultural Science, 6, 645–6602.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2006). Non equilibrium thermodynamics for the interactions of the see quantum scalar’s system with the classical bio fields in the cellular cryobiology and anhydrobiology. Bulgarian Journal of Agricultural Science, 12, 621 – 628.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2007). Non equilibrium thermodynamics of the cryobiology and quantum field theoretical methods in the temporal organization in cells. Bulgarien Journal of Agricultural Science, 13, 379-386.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2007). Causality implies the Lorenz group for cryobiological “freezing-drying” by the vacuum sublimation of living cells. Bulgarian Journal of Agricultural Science, 13, 627-634.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2008). Non equilibrium thermodynamics by vacuum energy momentum tensor of interacting quantum fields and living cells. Bulgarien Journal of Agricultural Science, 14, 351-356.

Tsvetkov, Ts. D., Petrov, G., & Tsvetkova, P. (2009). Causality properties of the energy-momentum tensor of the living cells by the low temperatures. Bulgarian Journal of Agricultural Science, 15, 487-493.

Tsvetkov, Ts. D., Petrov, G., & Hadzhy, A. (2011). Theoretical field analysis of the concrete quantum field system with AN impulse effect in the elementary living cells. Bulgarian Journal of Agricultural Science, 17, 1-10.

Tsvetkov, Ts. D., Petrov, G., & Hadzhy, A. (2012). Boundary conditions on the quantum scalar field system with a fluctuation’s impulse operator of the vacuum state in living cells. Bulgarian Journal of Agricultural Science, 18, 147-152.

Tsvetkov, Ts. D., & Petrov, G. (2012). The scalar particles in Casimir vacuum state of the relativistic quantum field system in living cells. Bulgarian Journal of Agricultural Science, 18, 819-826.

Tsvetkov, Ts. D., & Petrov, G. (2013). Casimir vacuum state of the relativistic quantum scalar field system in living cells. Bulgarian Journal of Agricultural Science, 19(6), 387-396.

Tsvetkov, Ts. D., & Petrov, G. (2013). Scale without conformal invariants in the non local relativistic quantum field systems in living cells. Bulgarian Journal of Agricultural Science, 19(6), 1171-1182.

Zwetkow, Zw. D. (1985). Vakuumgefriertrocknung (p.245). VEB Fachbuchverlag Leipzig.