#8
Message
par Gatti » 29 juin 2005, 10:01
Je N'ai pas l'intention d e me laisser ecraser aussi legerement par une mal baisée!
Voici le debut de la demonstration en question sous sa forme mathematique la plus academique. Je Ne publierai pas les quarante pages qui lui sont consacrée pour des questions de copyreigt mais je les tiens a la disposition du president des sceptiques du QUebec QUE NOUS INTERPELLONS UNE NOUVELLE FOIS .Il ferait une tres bonne affaire publicitaire (vis a vis de toute la communauté scientifique travallant en recherche fondaentale) en faisant etudier de tres pres les equatons d ePhilippe VIOLA .
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BIOQUANTUM THEORY
I
THEORY OF PSI FIELDS
Table of contents
Introduction
I - THE TOTAL UNIVERSE
I.1 - The external world
I.2 - The internal world
I.3 - The quantum world
II - PSI FIELDS AND WAVES
II.1 - PSI sources : charges and currents
II.2 - PSI fields
II.3 - PSI waves
III - PSI PROCESSES
III.1 - PSI interactions
III.2 - Propagation of PSI waves
III.3 - Emission of waves by PSI charges
III.4 - PSI radiation
III.5 - PSI scattering
IV - GRAVITATIONAL PSI FIELDS
IV.1 - Quantum gravity
IV.2 - Quantum biogravity
V - ELECTROMAGNETIC PSI FIELDS
V.1 - Quantum electrodynamics (QED)
V.2 - Quantum bioelectrodynamics (QBED)
VI - PSI-FLUIDS
VI.1 - Quantum hydrodynamics (QHD)
VI.2 - Quantum magnetohydrodynamics (QMHD)
VI.3 - Quantum biomagnetohydrodynamics (QBMHD)
VII - SUPERSYMETRIC PSI FIELDS
VII.1 - Bosonic and fermionic PSI fields. Supersymetrization
VII.2 - Supergravity and superbiogravity
VII.3 - Super-PSI-fluids
I
THE TOTAL UNIVERSE
I.1
The external world
In bioquantum theory, the elementary (i.e. non structured) external world is a complex Einstein manifold Z of complex dimension 4 endowed with a hermitian metric which locally writes :
(I.1) ds2 = gij*(z,z*)dzidz*j
where i,j run from 0 to 3 and * is the complex conjugaison, and which is solution of the Einstein equations on Z :
(I.2) Rij* - (1/2)R.gij* = (8G/c4)Tij* , DextiTij* = 0 , Dexti*Tij* = 0
where Dext , G and c are the covariant derivative on Z, Newton's gravitational constant and the velocity of light in the vaccum, respectively. Since :
(I.3) gij* = (gi*j)* (i,j = 0,1,2,3)
there exists a discrete unitary symmetry S, called the mirror symmetry, the left action of which exchanges Z and its complex conjugate Z* :
(I.4) S.Z = Z* , S.Z* = Z , S2 = 1
Thus, if one writes Z = X iY, X is the symmetric part of Z and Y, its skew-symmetric part. Indeed, from (I.4), one can easily see that :
(I.5) S.X = X , S.Y = -Y
Z has therefore torsion and its local invariance group is the unitary Lie group SU(3,1) isomorphic toC(3,1)/Z2, where C(3,1) is the real conformal group. This shows that, up to the choice of sign, Z geometrically behaves as a real conformal Einstein manifold of dimension 4. However, if there exists a scalar function K(z,z*) such as :
(I.6) gij* = 2K/ziz*j (i,j = 0,1,2,3)
then Z is Kähler.If this is so, then Z is analytic and torsion-free.
The choice of Z as the elementary external world is much more convenient to treat matter and antimatter on a equal footing and to perform quantization with respect to the de Broglie duality, as we will see further.
For the time being, let us introduce structure into the geometry of Z. Let e be a real scale parameter and consider the probabilistic space B = (Z,T,P), where T is a Borel tribe and P is a probability law with parameter e. The space B can be viewed as a dense family of external worlds {Z(e)}e in R such that :
(I.7) Z(0) = Z , Z(e) (included in or equal to)Z(e') (0 e e')
This amounts to go from a deterministic Einstein universe Z to a fractal one into which structures at all scales organize through a stochastic process with probability law P. So, at scale e, each point z(e) of the universe Z(e) writes :
(I.8) z(e) = <z(e)> + , ||2 = i*i = e2
where <.> stands for the mean value and is the fluctuating part or correlation length. Since the mean value is scale-independent, one has :
(I.9) <z(e)> = <z(0)> = z in Z for all e in R
which shows that <Z(e)> identifies with the elementary external world Z. However, the situation here is much more complicated than in conventional 3-space, for the value e = 0 now corresponds to a "light-like" fluctuation, as comes out from (I.8) and from the hyperbolic nature of the metric on Z. This type of fluctuation is singular in the sense that, despite that the length of the 4D vector i is indeed zero, that of the 3D spatial vector a (a = 1,2,3) is not, but equal to the length of the time-like component o :
(I.10) ||2 = a*a = |o|2 = o*o ≥ 0
So, there still remains spatial and time fluctuations, of the same amplitude.
The structure rule governing the coherent organization of sub-universes Z(e) from e = 0 to infinity is given as the general solution of a stochastic differential system of the form :
(I.11) dzi(e)/de = eFi(z,z*) , dz*i(e)/de = eF*i(z,z*) (i = 0,1,2,3)
For e = 0, this gives :
(I.12) dzi/de = 0 , dz*i/de = 0 (i = 0,1,2,3)
which indeed shows that z is scale-independent. For e 0, assuming the Ito conditions are satisfied, the universe Z(e) deduces from Z by means of the evolution group G(e) :
(I.13) Z(e) = G(e).Z , G(0) = 1
and the fractal nature of the structured external world is revealed by the scale invariance of G(e), which writes :
(I.14) dG(e)/de = 0
When non-reversibility occurs, this evolution group splitts into two semi-groups : G+(e), only defined for e ≥0 and G-(e), only defined for e 0 with a common connection at e = 0 (branching condition).
On the fractal external world Z(e), a trajectory is a non-regular curve (e) with the property of self-similarity (fractal curve). Assuming the curve (0) is C1 and smooth, that is, analytic on Z, the curve (e) can be built step by step applying the group action G(e) on the left, as in (I.13) :
(I.15) (e) = G(e).(0)
So, if one writes G(e) = exp(eH), where H is a stochastic operator, one gets the evolution equation :
(I.16) d(e)/de = - [H, (e)]
with [.,.] the usual Lie bracket, while eq. (I.14) becomes :
(I.17) [H, G(e)] = 0 for all e in R
When the probability law P is gaussian, its momenta are non zero up to the second order only, which corresponds to the fractal dimension 2. In this case, the total derivative d/de writes :
(I.18) d/de = /e + Re{(zi/e).(Dext)i} + e(Dext)i(Dext)*i
So, when P is gaussian, its density e(z,z*) = exp(|z|2/2e2) / e4 (with - if |z|2 is time-like and + if it is space-like) satifies the equation :
(I.19) e/e = e(Dext)i(Dext)*ie
However, at light-like points z of Z, that is, on the complex light cone, one finds e(z,z*) = 1/ e4. Such a singular situation does not appear in purely riemannian space for which the metric is elliptic. In the hyperbolic case, on the contrary, the probability density is everywhere the same on the whole surface of the light cone. At e = 0, one has a single light cone associated to all point of Z that may show light-like fluctuations and when e -> infinity , one gets a fractal complex light cone in place of the preceding one.
More generally, a classical field on Z(e) is a non-regular application :
(I.20) : Z(e) -> Rn or Cn (n ≥ 1)
where z(e) is a stochastic variable. The total covariant derivative of is d/de, where the operator d/de is given by (I.18) in fractal dimension 2. In the general case, depends both on z(e) and on z*(e).
All results extend to the discrete case replacing differentials by finite differences, continuous curves and fields by their discrete counterparts and the Poisson bracket {.,.} on the cotangent vector bundle T*Z of Z by the Lie bracket [.,.]. Continuous spectrum of propagation and scattering are replaced by discrete spectrum of interactions and functions by their corresponding operators. So, one goes from commutative spaces and algebras to non-commutative ones. A typical and simple example of a discrete structured external world is the well-known Cantor set in which the elementary component (i.e. that at scale e = 0) is a full universe Z and the large-scale structure of the fractal external universe Z(e) when e -> infinitylooks like a discrete infinity of isomorphic non-interacting elementary external worlds of characteristic diameter zero, infinitely close to each other.
Let us say a word about gauge transformations in the structured external world. Such transformations are to be considered when studying couplings of a mechanical complex system with an external field.
Let us introduce the non-linear scale-dependent covariant derivative of z(e) :
(I.21) Dz/de = v + A(z,z*) , v = dz/de , v* = dz*/de
where the external field A may be complex-valued and consider the hermitian kinematical function :
(I.22) L(z,z*,v,v*,e) = (Dz/de)(Dz/de)*
= |v|2 + v.A*(z,z*) + v*.A(z,z*) + |A(z,z*)|2
If one introduces a scale transformation of the first kind of the form :
(I.23) z -> z' = z.ei(e) , real
The expression (I.21) above becomes :
(I.24) Dz'/de = (v + iD/de).ei + A{z.eiz*.e-i}
and, by complex conjugaison, one also gets :
(I.25) (Dz'/de)* = (v* - iD/de).e-i + A*{z.eiz*.e-i}
while the modulus of z remains unchanged. The L-function thus becomes :
(I.26) L(z',z'*,v',v'*,e) = |v|2 - i(v-v*)(D/de) + (D/de)2 +
+ A{z.eiz*.e-i}(v* - iD/de).e-i +
+ A*{z.eiz*.e-i}(v + iD/de).ei +
+ |A{z.eiz*.e-i}|2
This expression has to be equal to (I.22) in order to have scale-invariance. This implies a transformation of the second kind on the field A :
(I.27) A{z.eiz*.e-i} = ei.{A(z,z*) - iD/de}
to be performed simultaneously with (I.23).
Finally, let us examine the fractal geometry of the structured external world.