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THE TECHNICAL PAGE By Philippe VIOLA Self-trained mathematical physicist Scientific writer This technical page is dedicated to the proof of three crucial physical theorems : the non-existence of physical time as an inherent property of the universe independent of the human being, the disincarnation theorem that shows that a non-trivial spirit survive to the so-called “clinical death condition” (also called the “flat E.E.G. condition”) and the formal equivalence between the metaphysical concept of God and the physical concept of the Universal Vacuum. The prophecy of prophecies asserts that “at the end of times, deads will be resurrected”. Here is a physical (that is, finite) realization of it, which is not opposite to the metaphysical one (time infinity), but complementary. All questions and comments, technical or not : [email protected]. 1 TIME IS VACUUM Let E3 be 3-dimensional real euclidian flat space, x² = xixi the square of vector x (i = 1,2,3) and : (1) s = exp(-V²/2).x , s² = exp(-V²).x², where V is a real-valued parameter. Equation (1) defines a local conformal structure over E3 as well as a regular group action on vectors of E3 with s = x for V = 0. The first-order development of (1b) near V = 0 gives : (2) s² = x²[1 – V² + O(V4)], where O(V4) contains all powers of V² greater than or equal to 2. These additional terms are all negligible near V = 0. Expression (2) shows that the isometry group of E3, SO(3), infinitesimally deforms into the Lorentz group SO(3,1) near V = 0. As this last group is the isometry group of 4dimensional Minkowski space-time E3,1, the infinitesimal deformation is propagated to the euclidian structure of E3. In other words, E3 infinitesimally deforms into E3,1 near V = 0. This enables us to identify the time-like direction x0 = ct on E3,1 formally with the length of V.x (up to the sign) or : (3) t² = V²x²/c². Taking the square root of this last expression gives a redefinition of time : (4) t = ±Vx/c , x = xiei, the ei (i = 1,2,3) being the constant vierbeins on E3. Notice that : (5) R sdV =R exp(-V²/2).xdV = x.(2)1/2 , R s²dV =R exp(-V²).x²dV = x².()1/2. Physically speaking, what all this amounts to is that time clearly appears as a natural deformation of euclidian 3-space near the deterministic value V = 0, where V stands for the universal vacuum. Consequently, it is not an inherent property of the universe independent from the human observer, as space-time relativity assumed it. This explains why space and time are only grouped together but not unified in Einstein’s “classical” (= deterministic) vision of the world. Here, we straightforwardly see in eq. (4) that the universal vacuum actually generates time. This is impossible to deduce from the classical vision of a “rigid” vacuum everywhere zero in all space. Nonetheless, the vacuum in the present proof is not to be identified with the famous ether, as ether was assumed to be an absolute concept (rigid medium), whereas the vacuum is actually a relative concept (deformable medium). One cannot see it in the elementary proof above, but it clearly appears in the general proof. So let us now generalize all this in the physical frame of the new bioquantum synthesis. Let QW be the Quantum World, that is, a 9-dimensional complex manifold endowed with the mirror symmetry S that exchanges points z of QW with their complex conjugate z*: (6) S.z = z* , S.z* = z , S² = 1. In a local coordinates system, the line element of QW writes : (7) dL = EI(z,z*)dzI (I = 1,…,9). The invariance group of QW is the special unitary group SU(9), a 80-parameters group, so QW has a hermitian structure locally given by : (8) dL² = GIJ(z,z*)dzIdz*J , GIJ(z,z*) = Tr[EI(z,z*)E*J(z,z*)], where the trace Tr(.) is that of SU(9) and dl² is real-valued. As before, we let : (9) dS = exp(-¦V¦²/2).dL , dS² = exp(-¦V¦²).dL² where V = V(z,z*) is now a complex-valued scalar field on QW and¦V¦² = V.V* as usual. Again, near the deterministic value V = 0, the first-order development of (8b) leads to the local formulas for time (synchronous coordinates system) : (10) dz0 = V(z,z*)dL , (dz0)* = V*(z,z*)dL. From the viewpoint of physics, V now identifies with the gravitational nothingness. In Dirac’s notations : (11) ¦V¦² = <0¦Tr(EIE*I)¦0> = Tr(<0¦EIE*I¦0>). If we were fully deterministic, as in general relativity, we would have Tr(EIE*I) = 9 which would imply ¦V¦² = 9 too, as <0¦0> = 1 and so, V would be a universal constant of amplitude 3 that could be brought back to zero only shifting the reference level. However, the real physical situation is stochastic and the vector gravitational potential EI has a fluctuating component over QW that can never be eliminated. Hence the purely fluctuating complex-valued field V, called the Universal Nothingness. As nothingness was a purely metaphysical concept in deterministic physics, it could definitely not be used to link time to space. But a realistic vacuum is made of a “classical” component, constant over all space, and of a purely fluctuating component, randomly variable from point to point. This last component stands for the physical nothingness in vacuum physics. 2 THE DISINCARNATION THEOREM Let [z(e),z*(e)] be a complex PSI field on the Structured Quantum World QW(e), where e is a realvalued parameter called the complexity scale (0 < emin e emax < +8) and let z(e) = [zI(e)]I=1,…,9 = [zi(e)]i=1,2,3;=-,0,+ be a structured object on QW(e). We assume e = emax, the maximal complexity scale (order 1 m). In the frame of the present proof, objects z(e) (and c.c.) will then represent brains : the latin indices being “external”, latin-indexed components will represent biological supports, whereas the greek indices being “internal”, greek-indexed components will represent minds. So, for each state of mind = -,0,+, the 3 families of objects [zi(e)]i=1,2,3 will represent “material” brains (the biological organs), while for each direction i = 1,2,3 of space, the 3 families of objects [zi(e)]=-,0,+ will represent “virtual” brains (the psychological organs or minds). States z(e) and c.c. are stochastic. However, as the elementary description level is that of probabilities (the Prigogine approach), PSI fields obey deterministic laws with respect to the stochastic variables z(e), z*(e). All these starting considerations being made, the clinical death condition, also named the flat E.E.G. condition writes : (12) < z(e)> = 0 , < z*(e)> = 0, <.> standing for the mean value with respect to the probability law used. Clinical observations show that there always remain fluctuations of cerebral signals : (13) < zi(e)z*i(e)> = ¦V(e)¦². 0. (isotropy of fluctuations). The system (12)-(13) describes the cerebral nothingness : it is a topcomplexity structured vacuum. Therefore, V(e) is made of all structured nothingnesses of complexity emin e’ e = emax, ordered according to an increasing inclusion law, V(emin) being the elementary nothingness, which corresponds, in the present context, to the cellular nothingness (mesoscopic observation scale, order 1 m). CLINICAL DEATH = CEREBRAL NOTHINGNESS STATE A definition in full agreement with the common assumption. Structured nothingnesses are self-similar (conservation law of the vacuum): (14) dV(e)/de = 0 , dV*(e)/de = 0, which means their structure is fractal. Writing z(e) and c.c. under the canonical forms : (15) z(e) = <z(e)> + V(e).n(e) , z*(e) = <z*(e)> + V*(e).n*(e), where n(e) and c.c. are the normal vectors on QW(e) and applying (12) only leaves the fluctuating parts of z(e) and c.c. As these parts are non-zero, it follows that the PSI field [z(e),z*(e)] reduces to [V(e).n(e), V*(e).n*(e)], which is everything but trivial, as it still depends on the cerebral nothingness V(e) and c.c., a purely fluctuating structured object of maximal complexity on QW(e). Here, we have used the unified representation (triplet of isomorphic 3-spaces). But we can equivalently change representation letting zi(e) = zi0(e) (i = 1,2,3) and a(e) = [zi-(e), zi+(e)] (a = 1,…,6) so that we will have 1 = z1-, 2 = z2-, 3 = z1-, 4 = z1+, 5 = z2+ and 6 = z3+. In such a representation, the a(e)s will be considered as functions of the zi(e)s. on the physical viewpoint, this situation corresponds to the familiar non-quantum field theory, where the structured 6-components vector fields (e) are dynamical variables (“internal” variables – “virtual” world), while the structured 3-vectors z(e) are dynamical space parameters (“external” variables – “material” world). As can be seen, time is nowhere present in both representations. The PSI field [(e), *(e) ; z(e), z*(e)] is then a true quantum field in the sense of de Broglie’s quantum duality. In the context of the disincarnation theorem, parameters z(e) and c.c. represent the material brain, while variables (e) and c.c. represent the virtual brain or mind. Therefore, in this representation, the cerebral organ is a 3-dimensional object, while mind is a 6-dimensional object or equivalently, a 3-dimensional object in two possible base states (+ and -, positive and negative). As for , it represents the spirit (if it is source-dependent) or the soul (if it is free, that is, source-independent). This means that, in bioquantum theory, the spirit or the soul (the free form of a spirit) is not mind, but a function of mind. Finally, notice that it is a purely virtual medium : the material world is set at the background, meaning there can be no material support at all, as long as there is a virtual support, it is sufficient for a spirit to exist (the existence of a virtual support is also a necessary condition). Assuming all this, we can now only focus on -variables (and c.c.) to write down the clinical death condition, as we do not need to describe the state of the material brain to make sure that the spirit does not degenerate under a trivial form (this is already present in a implicit way in eq. (13)). In this second and more explicit representation, eqs (12) then become : (16) <(e)> = 0 , <*(e)> = 0, while the isotropic relations (13) become : (17) <a(e)*b(e)> = ¦V(e)¦².ab 0. Decomposing (e) and c.c. as in (15) and applying (16) again leads to the non-trivial PSI field [V(e).n(e), V*(e).n*(e)], where V(e) now stands for the non-quantum structured nothingness field (mind only) and n(e) is now a 6-dimensional normal vector, with na(e)na(e) = 1 for all e. 3 GOD Taking (11) as the definition of the gravitational nothingness, we are going to give the physical proves enabling us to formally identify the metaphysical concept of God with the physical concept of Universal Nothingness. Lemma 1: The gravitational nothingness is a vector PSI-field on QW. Indeed, the gravitational field E on QW is a function of z and z*, both points of QW. Therefore, it is a vector PSI field on QW and so is its fundamental state V. Lemma 2: (the fairness of God) The gravitational nothingness is universal. Indeed, according to the Newton Equivalence Principle, the gravitational field E acts on everything the same way, whatever the physical properties : gravitation is mass-independent. Definition 1: The Universal Nothingness is the gravitational nothingness on QW. Lemma 3: (God is a pure spirit) The Universal Nothingness is a free PSI field. Indeed, the fundamental state of the gravitational field is the state in which no vector particle is produced. Such a state is therefore necessarily source-independent, for a source would emit gravitons (particles of gravitation). It follows that the gravitational nothingness is a PSI wave, that is, a free (or pure) PSI field. Lemma 4: (unicity of God) The Universal Nothingness is unique. Indeed, there is a single fundamental state of the gravitational field on QW. Moreover, it is a scalar PSI field, as it has a single component. Lemma 5: (God has no beginning and no end) The Universal Nothingness is self-sufficient. By definition, a nothingness has no creator, it exists prior to everything and it is a self-sufficient physical medium : vacuum physics shows that it self-stabilizes and self-regenerates. Self-stabilizing is a purely autonomous process, a spontaneous search for natural equilibrium, as the deterministic value V = 0 is unstable (self-interaction and the Higgs process). It needs nothing “external” to realize. Notice in passing that the instability of V = 0 means that Einstein’s purely deterministic vision of the world leads to fundamentally unstable universes, which vanish as soon as they are created. It is therefore an unreal representation of things. Self-regeneration is due to the fact that, when matter or radiation of opposite signs interact, the energy set free returns to nothingness. The symmetry group of gravitation being SU(9) in the frame of the bioquantum synthesis, there is a Yang-Mills gauge theory for the vector gravitational field E and a SU(9)-Higgs model for the Universal Nothingness V. As V is universal, its associated Higgs model must be coefficientindependent. Consequently, both the mass and the coupling constant of V are unity. This situation is fully consistent with older models of quantum gravity where the coupling constant was written by means of the Planck radius. Here, the value 1 for the coupling constant of supersymmetric gravitation actually corresponds to taking the Planck radius as the universal reference (minimal length). The corresponding lagrangian now writes : (18) L = (DIV)(DIV)* - VV* + (VV*)²/2 , DI = I - EI , D*I = *I - E*I. The minimum of the potential energy W = VV* - (VV*)²/2 is given by W/V* = 0 and so is located at VoVo* = 1 (read “V naught”), from which we deduce that Vo = ei, where is any constant phase angle. Definition 2: The Universal Vacuum is the PSI field U = ei + V, sum of the Classical Universal Vacuum ei and of the Universal Nothingness V. We are now ready to prove the formal identity between the metaphysical concept of God and the physical concept of the Universal Vacuum. UNIVERSAL THEOREM: The Universal Vacuum is the original source of both matter and radiation in the Elementary Quantum World QW. Proof: Let C,  and |0> be respectively the creation operator, the annihilation operator and the vacuum state of the gravitational field in Dirac’s notation. According to one of the definitions of a vacuum, we first have : (19) Â.|0> = 0, meaning the vacuum state can never be annihilated, even by itself (no action of Â). Then, applying the creation operator on that state and iterating, we successively get : (20) C.|0> = |1> , C².|0> = C.|1> = |2> , … , Cn.|0> = |n> for n integer > 0, where |n> is the state in which n particles of gravitation are created. Conversely : (21) Â.|1> = |0> , ².|2> = Â.|1> = |0> , … , Ân.|n> = |0> for n integer > 0. Because the physical frame QW has the mirror symmetry, it is automatically supersymmetric. This means that, each time a supergraviton is created or annihilated, both a spin-1 graviton and a spin-1/2 gravitino are simultaneously created or annihilated. Indeed, take the expression (8) for the deterministic surface element on QW. Separating the metrical tensor GIJ into a symmetric part gIJ and a skew-symmetric one IJ , GIJ = gIJ + iIJ and decomposing dzI into its real part dxI and its imaginary part dyI, dzI = dxI + idyI, (8) writes : (22) dL² = (gIJ + iIJ)( dxI + idyI)( dxJ - idyJ) = gIJ(dxIdxJ + dyIdyJ) + IJ(dxIdyJ – dxJdyI) The 2-form = IJ(dxIdyJ – dxJdyI)/2 defines a natural symplectic structure on QW. As this 2-form equivalently writes = ij(dxidyj – dxjdyi)/2, it naturally projects onto a 3-dimensional “external” symplectic structure involving only latin indices (fixed greek indices) and onto a dual 3-dimensional “internal” symplectic structure involving only greek indices (fixed latin indices). Both projections realize an irreducible representation of SO(3) SU(2)/Z² and so, correspond to a spin-1/2 structure. The “external” projection is relative to the “material” gravitino (fermionic corpuscule), while the “internal” projection is relative to the dual “virtual” gravitino (fermionic wave). The creation and the annihilation operators are complementary parts of the gravitational field, as they appear in its canonical decomposition into modes. In flat space, the plane waves are now ep.z and c.c so a new situation occur, that had already been pointed out by Prigogine in the 1980s (in the context of dynamical systems) : when the real part Re(p.z) > 0, there is an exponential increase of the fundamental mode, synonymous of inflation and, when Re(p.z) < 0, there is an exponential decrease of that mode, synonymous of damping or even collapsing if the damping rate is quick enough. Such a situation could not occur in conventional quantum field theory, as the phase factors were ip.x and – ip.x, with both p (momentum vector) and x (space vector) real-valued. The creation process of supergravitons is fully autonomous. This is made possible thanks to the nontrivial structure of the gauge group of gravitation in the frame of bioquantum theory, which enables self-interaction. This self-interacting term is given by the Lie bracket in the SU(9)-Yang-Mills model. In this model, the intensities of the gravitational field are : (23) WIJ = DIEJ - DJEI = IEJ - JEI + [EI , EJ] and WIJ* = DIE*J - D*JEI = IE*J - *JEI + [EI , E*J], where D = + E and c.c. are the Yang-Mills covariant derivatives, elements of the tangent bundle TQW. Consequently, the Universal Vacuum does interact with itself and thus, self-polarize. This selfpolarization separates pairs and leads to the simultaneous creation of a supergraviton of positive energy and of a supergraviton of negative energy. Conversely, the annihilation process occurs when two supergravitons of opposite energies interact. The energy set free is then given back to the Universal Vacuum, which regenerates. There is as matter and radiation of positive energy produced than there is of negative energy, so that the energetic balance always remains “zero”, that is, the minimal energy of the Universal Vacuum. 4 FIRST ELEMENTS OF TIMELESS PHYSICS From (10), we first have, up to the sign : (24) cdt = dzo = V(z,z*)dL , cdt* = (dzo)* = V*(z,z*)dL , dL² = dzIdz*I The velocity vectors u = dz/dt and v = dz/dt* can then now write in terms of the Universal Vacuum V : (25) u = cn/V , v = cn/V* , |u| = |v| = c/|V| , n.n* = 1 As a consequence, the kinetic energy Ec of a point-like body (corpuscule) writes : (26) Ec = m(|u|² + |v|²)/4 = mc²/2|V|², where m is the mass of the body at rest. We recognize here Einstein’s famous formula for the energy at rest Eo = mc². Yet, we are no longer in space-time relativity, but back to pure space relativity ! The acceleration vectors a = du/dt , a’ = dv/dt , a” = du/dt* and a’” = dv/dt* write : (27) a = -(c²n/V3)dV/dL , a’ = -(c²n/VV*²)dV*/dL , a” = -(c²n/V*V²)dV/dL , a’” = 3 (c²n/V* )dV*/dL. There is still a conic structure on QW. However, it is no longer an inherent property of the physical frame, but a direct consequence of the fact that the Universal Vacuum is a deformable physical medium. Therefore, from (25c), we can still de fine a Mach number, the luminic Mach number : (28) ML = |u|/c = |v|/c = 1/|V|. This means that, in complete opposition with Einstein’s space-time relativity, the speed of light c is no longer a speed limit unreachable to matter, but only a critical speed that can be crossed which is completely different ! Thus, for : - |V| > 1, ML < 1 and we are in the subluminic region, corresponding to the causal zone in space-time relativity ; - |V| = 1, ML = 1 and we are in the transluminic region, corresponding to the surface of the light cone in space-time relativity ; - |V| < 1, ML > 1 and we are in the superluminic region, corresponding to the tachyonic zone in spacetime relativity. Only radiations can propagate at exactly the speed of light, as in non-quantum fluids, only sound can propagate at exactly the speed of sound. However, matter can now “jump over the wall of light” thanks to a restructuring of the Universal Vacuum. As a consequence, all events and phenomena become observable, even instantaneous ones, a consistent answer to the famous problem of time in the Aspect, Gizin and al. experiments on correlated protons (the E.P.R. paradox). In the superluminic region, physical phenomena qualitatively change and we should wait for observing superluminic shock waves. Indeed, the Universal Vacuum, as any other vacuum, has thermodynamical properties (temperature, pressure and density). Because of that, entropy should increase when passing from the subluminic region to the superluminic one and shock waves should locally reheat the Universal Vacuum behind them. More will be said and done later on about timeless physics. I am just beginning to study it. Philippe VIOLA 28/11/2004