Symplectic Field Quantization within Sensosphere

Maybe Sensosphere or mkptmk sensorial networks are interpretable by the dynamic functions within a transfinite HyperComplex Hilbert space?

As sounds are actually sonars, sonarrrrrs, sonarrrrrrrrs (this effect is specialy easy to perceiving within an old railway tunnel, as in Rincón de la Victoria Cantal), we haven't a lineal sound travelling as a plane rightly from New Your to Málaga, no!, what you have there are (imagine a cloudy day when flying) a succesive series of resounds (magnified by the clouds!), going as if dense smoke in a whet forest, difficult fire, in its beginnings...

You have a sucession of sound clouds, whenever you hear something around you. As more or less giant worms whose suden deformations in their body are so quick and noticeable, under your macroscopic vision.

What about three dimensional fractals-inthe-air.

Air, by means of thinking and feeling on-and-within echolocation in your everyday life, seem more and more visible for you, as a so very dynamic system, or ecosystem, air show a constant dynamic presence around your everyday life. One flying bird means a sample of turbulences that you know very well, each time these birds passing so near of your hears...

No matter if in this puntual observation, you are not hearing, and sensing in your skin, the fly of thi pigeon playing with the air, far from you. As pigeon is a common bird, another citizen, with you, you know them very well, your shared experience along your life, and the life of millions and millions people, are universal transconnectors, within that great neurosemantic network we call Noosphere, or Sensosphere. They are constancies among very much humankind. If you add in a list all the foods and life sharing significant part of humankind, under the neurosemantic coherence principle, this species act as sensorial constants, as a significant part of our neurosemantic networks that are shared by more people. This result means an objective way to measure all that join us Humans, as a superorganism, within Gaya...

Sensosphere is you because of sensorial equipment, or co-equipment, As oil for cars, "your" sensorial, quantum, equipment, are not only developped by "you" and isolated being, taken for him or her, the information they need...

Sensorial equipments exist as a symbiotic partenariat among a transfinite fuzzy, and quantum set, call Sensosphere.

To ignore sensorial systems (co-systems) as an essential core of Neurosciences, until now, has means a great cognitive dissonance, juts because our brains, scientist brains, after so long duration of scientific brain-solipsism, had forgotten that here in front of you you have "The World", the physical reality so active with you and within their rich interactions, as you are also an active participant, coparticipant, within the flowing of a transfinite dance of quantum tridimensional air bubles, where quantum transfigural geometry permits us to represent all that, symplecticaly, as the nice clouds, of diferent forms and speed of change, you may have seen in all your life...

Quantum Science is so much intuitive. Contraintuitive when you situate out of dynamic quantum thinking. When you situate your self out from shamanism, or the multiple ways other old cultures have learn to see the World...

Shamanism would be the ("human") (co-human) biotechnology, the scientific basis would come from many disciplines, symplecticaly from Quantum Sciences (Qu. Physics, Qu. Biology...)...

Sensosphere is very easy to approach for you if you choose some "observational-set", or observation.

Mindfulness is learn across many multicultural disciplines. Maybe the less disciplined among them would be Shamanism, that in last years have covered a significant critical mass within european-origin people, and within academias.

Mindfulness make available for you the possibility of increasing, with time, the spam of time you could concentrate within a certain experience, or "observational-set". I prefer to say "observational set" to not forget the always quantum, fractal, and compositional nature of any observation.

In that way you, an intuiscientist, are walking-and-observing. Intuition is the result of caring observations. Caring observations are like pamperings, or caresses, we call "technically, "kissels".

A kissel would be any observational-set or any observational subset we chose as, maybe, the sensorial effects resulting in you, after the quiet process of observating, has make an important, or significant impact in your neurosemantic networks, so a LTP (long Term potentiation of your sensorial capacities, or your sensibility, or row tropisms, as remarked by Margulis, Lynn...

So from a transfinite set of sensations all them dancing around, on, towards, and from, you, the sensosphere (ecobrain would be an arbitrary subset of sensosphere, as an example the sensosphere within a classroom would be a "ecobrain"), we take a sub sample, maybe the site to where our eyes-and-senses decided to focus in that moment, maybe for the originality, maybe because the efficiency in recompensation for the highest neurosemantic diversity. For example, looking in Paseo Marítimo de La Cala, one day, a struendous flying group of parrots, following a small dog, keeping in her mouth one parrot, was a very sygnificant observation which is very difficult to forget in a complete fashion...

In that way we chose arbitrary partitions within this splendidous "new" hard disk, sensosphere, or mkptmk, for a certain time, maybe some minutes. After that, we follow walking with our senses open, trying to hunting new significant observations, or "observation-set"...

After significant observations, it is common your mind will receive a "conceptual recompensation"; for me is the case, actualy, as my endeavours these years was reconnecting and recomposing, a "harappa" called "Global Science", using as superglue all the results of integration we meet within internet across the search engines. This process of abduction (new thinking) is how science or popular knowledge growths. The new things, and thoughts, in a rural, preliterate society, are sumbiotically interconnected, as in such society all the material world is in permanent conversation within community, taken as such all both parts, men, women and plants and all life in the biorregion...

If all the conversation you have in a restaurant, is complemented, synchronically, by the birds, other people, and the sea and their waves, your "human" conversations where formaly you separate all that evident conversations from your "special" (from an anthropocentric point of view) human conversation of all the conversations within the same ecosystem you are.

Is as if you are so slept, walking within a very reach forest, that you sudenly discover all the trees that before were invisible for you. As perceiving a forest as a desert.... Fractaly sahara haas the greter geodiversity, and biogeodiversity in the world, just connecting visualy, with the greates library, the universe, for more number of people.... The implications of so clear universal semiotics on a place on Earth are significant ones...

Essential unitarization of symplectics and applications to field quantization

Stephen M. Paneitz^*

Department of Mathematics, University of California, Berkeley, California 94720, U.S.A.

Received 1 September 1981.

Communicated by the Editors

Available online 20 July 2004.

Abstract

Symplectic operators satisfying generic and group-invariant (spectral) positivity conditions are studied; the theory developed is applied and illustrated to determine the unique invariant frequency decomposition (equivalently, linear quantization with invariant vacuum state) of the Klein-Gordon equation in non-static spacetimes. Let (H, Ω) be any linear topological symplectic space such that there exists a real-linear and topological isomorphism of H with some complex Hilbert space carrying Ω into the imaginary part of the scalar product. Then any bounded invertible symplectic S Sp(H) (resp. bounded infinitesimally symplecticA sp(H)) which satisfies Ω(Sv, v) > 0 (resp. Ω(Av, v) > 0) for all nonzero v ω H, where S + I is invertible, is realized uniquely and constructively as a unitary (resp. skewadjoint) operator in a complex Hilbert space which depends in general on the operator and typically only densely intersects H. The essentially unique weakly and uniformly closed invariant convex cones in sp(H) are determined, extending previously known results in the finite-dimensional case. A notion of “skew-adjoint extension” of a closed semi-bounded infinitesimally symplectic operator is defined, strictly including the usual notion of positive self-adjoint extension in a complex Hilbert space; all such skew-adjoint extensions are parametrized, as in the von Neumann or Birman-Krein-Vishik theories. Finally, the unique complex Hilbertian structure—formulated on the space of solutions of the covariant Klein-Gordon equation in generic conformal perturbations of flat space—is uniquely determined by invariance under the scattering operator. The invariant Hilbert structure is explicitly calculated to first order for an infinite-dimensional class of purely time-dependent metric perturbations, and higher-order contributions are rigorously estimated.

References

1. Bateman Manuscript Project (32rd ed.),Higher Transcendental Functions Vol. 2, McGraw-Hill, New York (1953).

2. M.S. Birman, On the self-adjoint extensions of positive definite operators. Math. Sb. 38 (1956), pp. 431–450 [Russian] .

3. Standard Mathematical Tables (32rd ed.), CRC Press, Cleveland (1975).

4. P. Chernoff and J. Marsden, Properties of Infinite Dimensional Hamiltonian Systems. In: Lecture Notes in Mathematics No. 425, Springer-Verlag, Berlin/Heidelberg/New York (1974).

5. J.M. Cook, The mathematics of second quantization. Trans. Amer. Math. Soc. 74 (1953), pp. 222–245.Full Text via CrossRef

6. J.M. Cook, Complex Hilbertian structures on stable linear dynamical systems. J. Math. Mech. 16 No. 4 (1966), pp. 339–349. Full Text via CrossRef

7. J.L. Daleckiï and M.G. Krein, Stability of Solutions of Differential Equations in Banach Space. In:Translations of Math. Monographs Vol. 43, Amer. Math. Soc., Providence, R.I. (1974).

8. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. , Academic Press, New York (1978).

9. A. Horn, Doubly stochastic matices and the diagonal of a rotation matrix. Amer. J. Math. 76 (1954), pp. 620–630. Full Text via CrossRef

10. , Quantum field theory in curved space-times: a general mathematical framework, in“Differential Geometrical Methods in Mathematical Physics II,” Lecture Notes in Mathematics No. 676, pp. 459–512, Springer-Verlag, Berlin/Heidelberg/New York.

11. T. Kato, Perturbation Theory for Linear Operators. , Springer-Verlag, Berlin/Heidelberg/New York (1966).
T. Kato, Perturbation Theory for Linear Operators. (2nd ed.), Springer-Verlag, Berlin/Heidelberg/New York (1976).

12. B. Kay, Linear spin-zero quantum fields in external gravitational and scalar fields I: A one particle structure for the stationary case. Comm. Math. Phys. 62 No. 1 (1978), pp. 55–70. Full Text via CrossRef |View Record in Scopus | Cited By in Scopus (32)

13. M.G. Krein, The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications I. Math. Sb. 20 62 (1947), pp. 431–495 [Russian] .

14. N.N. Lebedev, Special Functions and their Applications. , Dover, New York (1972).

15. A. Lichnerowicz, Propagateurs et Commutateurs en Relativité Générale. Inst. Hautes Études Sci. Publ. Math. No. 10 (1961).

16. C. Moreno, Spaces of positive and negative frequency solutions of field equations in curved space times I, II. J. Math. Phys. 18 (1977), p. 2153. Full Text via CrossRef
C. Moreno, Spaces of positive and negative frequency solutions of field equations in curved space times I, II. J. Math. Phys. 19 (1978), p. 92. Full Text via CrossRef

17. J. Palmer, Symplectic groups and the Klein-Gordon field. J. Func. Anal. 27 (1978), pp. 308–336.Article | PDF (1444 K) | View Record in Scopus | Cited By in Scopus (3)

18. J. Palmer, Complex structures and external fields. Arch. Rat. Mech. Anal. 75 (1980), pp. 31–49. View Record in Scopus | Cited By in Scopus (0)

19. S. Paneitz. In: Doctoral thesis, Massachusetts Institute of Technology (June 1980).

20. S. Paneitz, Unitarization of symplectics and stability for causal differential equations in Hilbert space.J. Func. Anal. 41 (1981), pp. 315–326. Article | PDF (584 K) | View Record in Scopus | Cited By in Scopus (1)

21. S. Paneitz, Invariant convex cones and causality in semisimple Lie algebras and groups. J. Func. Anal. 43 (1981), pp. 313–359. Article | PDF (2205 K) | View Record in Scopus | Cited By in Scopus (8)

22. S. Paneitz and I.E. Segal, Quantization of wave equations and hermitian structures in partial differential varieties. In: Proc. Nat. Acad. Sci. USA 77 (December 1980), pp. 6943–6947 No. 12 . Full Textvia CrossRef | View Record in Scopus | Cited By in Scopus (1)

23. L. Parker. In: F.P. Esposito and L. Witten, Editors, Asymptotic Structure of Space-Time, Plenum, New York (1977).

24. I.E. Segal, Foundations of the theory of dynamical systems of infinitely many degrees of freedom I.Mat. Fys. Medd. Danske Vid. Selsk 31 No. 12 (1959).

25. I.E. Segal, Mathematical Cosmology and Extragalactic Astronomy. , Academic Press, New York (1976).

26. I.E. Segal, Quantization of symplectic transformations. Math. Anal. Appl. B Advan. Math. Suppl. Stud.7B (1981), pp. 749–758.

27. I.E. Segal, Field quantization. Phys. Scripta 24 No. 5 (1981), p. 827. Full Text via CrossRef

28. E. Vinberg, Invariant convex cones and orderings in Lie groups. Func. Anal. Appl. 14 No. 1 (1980), pp. 1–10. View Record in Scopus | Cited By in Scopus (8)

29. M. Vishik, On general boundary conditions for elliptic differential equations. Tr. Mosc. Math. Obsc. 1(1952), pp. 187–246 [Russian] .
M. Vishik. Amer. Math. Soc. Transl. (2) 24 (1963), pp. 107–172.

30. J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Mat. Ann. 102(1929), pp. 49–131.

31. M. Weinless, Existence and uniqueness of the vacuum for linear quantized fields. J. Func. Anal. 4(1969), pp. 350–379. Article | PDF (1473 K) | View Record in Scopus | Cited By in Scopus (7)

32. A. Weinstein, The invariance of Poincaré's generating function for canonical transformations. Inv. Math. 16 (1972), pp. 202–213. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (2)

33. A. Weinstein, Symplectic geometry. Bull. Amer. Math. Soc. 5 No. 1 (July 1981), pp. 1–13.

34. V.A. Yakubovich and V.M. Starzhinskii. In: Linear Differential Equations with Periodic Coefficients Vol. II, Wiley, New York/Toronto (1975).

35. I.E. Segal, Mathematical characterization of the physical vacuum for a linear Bose-Einstein field.Illinois Math. J. 6 (1962), pp. 500–523.

36. A. Alonso and B. Simon, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators. J. Oper. Theory 4 (1980), pp. 251–270.

37. P. De la Harpe, Classical Banach-Lie Algebras and Banach-Lie groups of Operators in Hilbert Space. In: Lecture Notes in Mathematics No. 285, Springer-Verlag, Berlin/Heidelberg/New York (1972).

^* Miller Research Fellow. Current address: Massachusetts Institute of Technology, Cambridge, MA 02139.