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3. The Instanton

3.1 Bosonic Unification S.EW.G --- S.E.W.G

The following derivations lead to a simplified string formalism as boundary- and initial conditions in a de Sitter cosmology encompassing the classical Minkowski-Friedmann spacetimes holographically and fractally in the Schwarzschild metrics.

The magnetic field intensity B is classically described in the Biot-Savart Law:
for a charge count q=Ne; angular velocity ω=v/r=2πf; current i=dq/dt and the current element i.dl=dq.(dl/dt)=vdq.

The Maxwell constant then can be written as an (approximating) finestructure:
μoεo =1/c2=(120π/c)(1/120πc) to crystallise the 'free space impedance' Zo=√(μoo)=120π~377 Ohm (Ω).

This vacuum resistance Zo so defines a 'Unified Action Law' in a coupling of the electric permittivity component (εo) of inertial mass and the magnetic permeability component (μo) of gravitational mass in the Equivalence Principle of General Relativity.
A unified selfstate of the preinertial (string- or brane) cosmology so is obtained from the finestructures for the electric- and gravitational interactions coupling a so defined electropolic mass to magnetopolic mass respectively.

The Planck-Mass is given from Unity 1=2πGmP2/hc and the Planck-Charge derives from Alpha=2πke2/hc and where k=1/4πεo in the electromagnetic finestructure describing the probability interaction between matter and light (as about 1/137).

The important aspect of alpha relates to the inertia coupling of Planck-Charge to Planck-Mass as all inertial masses are associated with Coulombic charges as inertial electropoles; whilst the stringed form of the Planck-Mass remains massless as gravitational mass. It is the acceleration of electropoles coupled to inertial mass, which produces electromagnetic radiation (EMR); whilst the analogy of accelerating magnetopoles coupled to gravitational mass and emitting electromagnetic monopolic radiation (EMMR) remains hitherto undefined in the standard models of both cosmology and particle physics.

But the coupling between electropoles and magnetopoles occurs as dimensional intersection, say between a flat Minkowskian spacetime in 4D and a curved de Sitter spacetime in 5D (and which becomes topologically extended in 6-dimensional Calabi-Yau tori and 7-dimensional Joyce manifolds in M-Theory).

The formal coupling results in the 'bounce' of the Planck-Length in the pre-Big Bang scenario, and which manifests in the de Broglie inflaton-instanton.

The Planck-Length LP=√(hG/2πc3) 'oscillates' in its Planck-Energy mP=h/λPc=h/2πcLP to give √Alpha).LP=e/c2 in the coupling of 'Stoney units' suppressing Planck's constant 'h' to the 'Planck units' suppressing charge quantum 'e'.

Subsequently, the Planck-Length is 'displaced' in a factor of about 11.7=1/√Alpha=√(h/60π)/e and using the Maxwellian finestructures and the unity condition kG=1 for a dimensionless string coupling Go=4πεo, describing the 'Action Law' for the Vacuum Impedance as Action=Charge2, say via dimensional analysis:

Zo=√([Js2/C2m]/[C2/Jm])=[Js]/[C2]=[Action/Charge2] in Ohms [Ω=V/I=Js/C2] and proportional to [h/e2] as the 'higher dimensional source' for the manifesting superconductivity of the lower dimensions in the Quantum Hall Effect (~e2/h), the conductance quantum (2e2/h) and the Josephson frequencies (~2e/h) in Ohms [Ω].

This derivation so indicates an electromagnetic cosmology based on string parameters as preceding the introduction of inertial mass (in the quantum Big Bang) and defines an intrinsic curvature within the higher dimensional (de Sitter) universe based on gravitational mass equivalents and their superconductive monopolic current flows.

A massless, but monopolically electromagnetic de Sitter universe would exhibit intrinsic curvature in gravitational mass equivalence in its property of closure under an encompassing static Schwarzschild metric and a Gravitational String-Constant Go=1/k=1/30c (as given in the Maxwellian finestructures in the string space).

In other words, the Big Bang manifested inertial parameters and the matter content for a subsequent cosmoevolution in the transformation of gravitational 'curvature energy', here called gravita as precursor for inertia into inertial mass seedlings; both however describable in Black Hole physics and the Schwarzschild metrics.

The Gravitational Finestructure so derives in replacing the Planck-Mass mP by a protonucleonic mass:

mc=√(hc/2πGo).f(alpha)= f(Alpha).mP and where f(Alpha)=Alpha9.
The Gravitational finestructure, here named Omega, is further described in a fivefolded supersymmetry of the string hierarchies, the latter as indicated in the following below in excerpt.

This pentagonal supersymmetry can be expressed in a number of ways, say in a one-to-one mapping of the Alpha finestructure constant as invariant X from the Euler Identity:

X+Y=XY= -1=i2=exp(iπ).

One can write a Unification Polynomial: (1-X)(X)(1+X)(2+X)=1 or X4+2X3-X2-2X+1=0
to find the coupling ratios: f(S)¦f(E)¦f(W)¦f(G)=#¦#3¦#18¦#54 from the proportionality

The Unification polynomial then sets the ratios in the inversion properties under modular duality:

(1)[Strong short]¦(X)[Electromagnetic long]¦(X2)[Weak short]¦(X3)[Gravitational long]
as 1¦X¦X2¦X3 = (1-X)¦(X)¦(1+X)¦(2+X).

Unity 1 maps as (1-X) transforming as f(S) in the equality (1-X)=X2; X maps as invariant of f(E) in the equality (X)=(X); X2 maps as (1+X) transforming as f(W) in the equality (1+X)=1/X; and X3 maps as (2+X) transforming as f(G) in the equality (2+X)=1/X2=1/(1-X).

The mathematical pentagonal supersymmetry from the above then indicates the physicalised T-duality of M-theory in the principle of mirror-symmetry and which manifests in the reflection properties of the heterotic string classes HO(32) and HE(64), described further in the following.

Defining f(S)=#=1/f(G) and f(E)=#2.f(S) then describes a symmetry breaking between the 'strong S' f(S) interaction and the 'electromagnetic E' f(E) interaction under the unification couplings.
This couples under modular duality to f(S).f(G)=1=#55 in a factor #-53=f(S)/f(G)={f(S)}2 of the 'broken' symmetry between the longrange- and the shortrange interactions.

SEWG=1=Strong-Electromagnetic-Weak-Gravitational as the unified supersymmetric identity then decouples in the manifestation of string-classes in the de Broglie 'matter wave' epoch termed inflation and preceding the Big Bang, the latter manifesting at Weyl-Time as a string-transformed Planck-Time as the heterotic HE(64) class.

As SEWG indicates the Planck-String (class I, which is both openended and closed), the first transformation becomes the suppression of the nuclear interactions sEwG and describing the selfdual monopole (stringclass IIB, which is loop-closed in Dirichlet brane attachement across dimensions say Kaluza-Klein R5 to Minkowski R4 or Membrane-Space R11 to String Space R10).

The monopole class so 'unifies' E with G via the gravitational finestructure assuming not a Weylian fermionic nucleon, but the bosonic monopole from the kGo=1 initial-boundary condition GmM2= ke2 for mM=ke=30[ec]=mP√Alpha.

The Planck-Monopole coupling so becomes mP/mM=mP/30[ec]=1/√Alpha
with f(S)=f(E)/#2 modulating f(G)=#2/f(E)=1/# ↔ f(G){f(S)/f(G)}=# in the symmetry breaking f(S)/f(G)=1/#53 between short (nuclear asymptotic) and long (inverse square).

The shortrange coupling becomes f(S)/f(W)=#/#18=1/#17=Cuberoot(Alpha)/Alpha6
and the longrange coupling is Alpha/Omega=1/Alpha17=#3/#54=1/#51=1/(#17)3.

The strong nuclear interaction coupling parameter so becomes about 0.2 as the cuberoot of alpha and as measured in the standard model of particle physics.

The monopole quasimass [ec] describes a monopolic sourcecurrent ef, manifesting for a displacement λ=c/f. This is of course the GUT unification energy of the Dirac Monopole at precisely [c3] eV or 2.7x1016 GeV and the upper limit for the Cosmic Ray spectra as the physical manifestation for the string classes: {I, IIB, HO(32), IIA and HE(64) in order of modular duality transmutation}.

The transformation of the Monopole string into the XL-Boson string decouples Gravity from sEwG in sEw.G in the heterotic superstring class HO(32). As this heterotic class is modular dual to the other heterotic class HE(64), it is here, that the protonucleon mass is defined in the modular duality of the heterosis in: Omega=Alpha18=2πGomc2/hc=(mc/mP)2.

The HO(32) string bifurcates into a quarkian X-part and a leptonic L-part, so rendering the bosonic scalar spin as fermionic halfspin in the continuation of the 'breaking' of the supersymmetry of the Planckian unification. Its heterosis with the Weyl-string then decouples the strong interaction at Weyl-Time for a Weyl-Mass mW, meaning at the timeinstanton of the end of inflation or the Big Bang in sEw.G becoming s.Ew.G.

The X-Boson then transforms into a fermionic protonucleon triquark-component (of energy ~ 10-27 kg or 560 MeV) and the L-Boson transforms into the protomuon (of energy about 111 MeV).

The last 'electroweak' decoupling then occurs at the Fermi-Expectation Energy about 1/365 seconds after the Big Bang at a temperature of about 3.4x1015 K and at a 'Higgs Boson' energy of about 298 GeV.

A Bosonic decoupling preceeded the electroweak decoupling about 2 nanoseconds into the cosmogenesis at the Weyl-temperature of so TWeyl=Tmax=EWeyl/k=1.4x1020 K as the maximum Black Hole temperature maximised in the Hawking MT modulus and the Hawking-Gibbons formulation: McriticalTmin=½MPlanckTPlanck=(hc/2πGo)(c2/2k)=hc3/4πkGo for Tmin=1.4x10-29 K and Boltzmann constant k.

The Hawking Radiation formula results in the scaling of the Hawking MT modulus by the factor of the 'Unified Field' spanning a displacement scale of 8p radians or 1440° in the displacement of 4lps.

The XL-Boson mass is given in the quark-component: mX=#3mW/[ec]=Alpha.mW/mP=#3{mW/mP}~1.9x1015 GeV; and the lepton-component: mL=Omega.[ec]/#2=#52[ec/mW] ~ 111 MeV.

A reformulation of the rotational dynamics associated with the monopolic naturally superconductive currentflow and the fractalisation of the static Schwarzschild solution follows. in a reinterpretation of the Biot-Savart Law.

All inertial objects are massless as 'Strominger branes' or extremal boundary Black Hole equivalents and as such obey the static and basic Schwarzschild metric as gravita template for inertia.

This also crystallises the Sarkar Black Hole boundary as the 100Mpc limit (RSarkar=(Mo/Mcritical.RHubble)=0.028.RHubble~237 Million lightyears) for the cosmological principle, describing large scale homogeneity and isotropy, in the supercluster scale as the direct 'descendants' of Daughter Black Holes from the Universal Mother Black Hole describing the Hubble Horizon as the de Sitter envelope for the Friedmann cosmology (see linked website references on de Sitter cosmology) for the oscillatory universe bounded in the Hubble nodes as a standing waveform.

The Biot-Savart Law: B=μoqv/4πr2oi/4πr=μoNef/2r=μoNeω/4πr for angular velocity ω=v/r transforms into B=constant(e/c3)gxω

in using acentripetal=v2/r=rω2 for g=GM/r2=(2GM/c2)(c2/2r2)=(RSc2/2R2) for a Schwarzschild solution RS=2GM/c2.

B=constant(eω/rc)(v/c)2oNeω/4πr yields constant=μoNc/4π=(120πN/4π)=30N with e=mM/30c for
30N(eω/c3)(GM/R2)=30N(mM/30c)ω(2GM/c2)/(2cR2)=NmM(ω/2c2R)(RS/R)= {M}ω/2c2R.

Subsequently, B=Mw/2c2R = NmM(RS/R){ω/2c2R} to give a manifesting mass M finestructured in
M=NmM(RS/R) for N=2n in the superconductive 'Cooper-Pairings' for a charge count q=Ne=2ne.

But any mass M has a Schwarzschild radius RS for N=(M/mM){R/RS}=(M/mM){Rc2/2GM}={Rc2/2GmM}={R/RM} for a monopolic Schwarzschild radius RM=2GmM/c2=2G(30ec)/c2=60ec/30c3=2e/c2=2LP√Alpha=2OLP.

Any mass M is quantised in the Monopole mass mM=mP√Alpha in its Schwarzschild metric and where the characterising monopolic Schwarzschild radius represents the minimum metric displacement scale as the Oscillation of the Planck-Length in the form 2LP√Alpha~LP/5.85.

This relates directly to the manifestation of the magnetopole in the lower dimensions, say in Minkowski spacetime in the coupling of inertia to Coulombic charges, that is the electropole and resulting in the creation of the mass-associated electromagnetic fields bounded in the c-invariance.
From the Planck-Length Oscillation or 'LP-bounce': OLP=LP√Alpha=e/c2 in the higher (collapsed or enfolded) string dimensions, the electropole e=OLP.c2 maps the magnetopole e*=2Re.c2 as 'inverse source energy' EWeyl=hfWeyl and as function of the classical electron radius Re =ke2/mec2=RCompton.Alpha= RBohr1.Alpha2=Alpha3/4πRRydberg= 1010{2πRW/360}={e*/2e}.OLP.

The resulting reflection-mirror space of the M-Membrane space (in 11D) so manifests the 'higher D' magnetocharge 'e*' AS INERTIAL MASS in the monopolic current [ec], that is the electropolic Coulomb charge 'e'.
This M-space becomes then mathematically formulated in the gauge symmetry of the algebraic Lie group E8 and which generates the inertial parameters of the classical Big Bang in the Weylian limits and as the final Planck-String transformation.

The string-parametric Biot-Savart law then relates the angular momentum of any inertial object of mass M with angular velocity ω in selfinducing a magnetic flux intensity given by B=Mω/2Rc2 and where the magnetic flux relates inversely to a displacement R from the center of rotation and as a leading term approximation for applicable perturbation series.

This descriptor of a string based cosmology so relates the inherent pentagonal supersymmetry in the cosmogenesis to the definition of the Euler identity in its finestructure X+Y=XY=-1, and a resulting quadratic with roots the Golden Mean and the Golden Ratio of the ancient omniscience of harmonics, inclusive of the five Platonic solids mapping the five superstring classes. Foundations and applications of superstring theory are also indicated and serve as reference for the above.