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Introduction to Quantum Relativity and the Quantum Geometry
by William MacArthur
Well let's take the first part - Imagine a line segment, and call it a
1-sphere. Now, it has no surface, because a surface is necessarily 2D.
Yet the surface has been defined by the length of the segment.. You'll
see why in a moment.
Introduce a second dimension and spin the segment around its centre, and
it traces out a circle. Now call the circle a 2-sphere. It has a
surface! Namely the perimeter, but it it has no volume, because a volume
is necessarily 3D. However, the circle defines the volume even so,
because if we introduce a third dimension and spin the circle around a
diameter, we get a ball!
Now the ball is a 3-sphere, right? It has a surface, and a volume, and
the volume defines - something else, but it isn't there yet because the
something else is 4D. Now spin the ball around - um - well, you have to
stop trying to visualise now because you actually spin the ball around
a planar section, but in such a way that it is orthogonal to all 3
visible dimensions, and you get a 4-sphere, only now there's a fourth
dimension to this thing, which I think is called a hypersphere (Tony?).
Now the fun thing is that you can actually start even lower than the
line segment, at a point, and call it a 0-sphere, and a "rotation" here
has to take the form of a double translation along a line to produce the
1-sphere, but we will call it an 0D rotation. Of course there is no
surface, no volume, nothing at all, except position. Its "rotation"
produces a 1-sphere (line segment ) that has position and magnitude.
Spin the 1-sphere around it 0D centre (the position) and you get the
2-sphere with position, magnitude and a surface. Now spin that around
its 1D axis and you get a 3-sphere with position, magnitude, a surface
and a volume. Now spin the 3-sphere around a suitable orthogonal 2D
section and it makes a 4-sphere, which has position, magnitude, surface,
volume and...... duration? See how we produce ever higher hyperspheres
we get more and more basic properties?
And so it goes on. As an aid to visualising, let's go the other way and
slice a 4-sphere with a 3D volume (don't try to visualise this or you'll
go mad) and you get a 3-sphere. Slice a 3-sphere with a 2D plane
surface and you get a 2-sphere or plane circle. Slice a 2-sphere with a
1D line and you get a 1-sphere, and lastly slice this with a point to
get the 0-sphere.
Tony is using brane terminology, but basically this is the operation
he's talking about I think The rest is a bit more abstract still and
rather mathematical. The speed of light is c and it is constant.
Furthermore, light as the one entity that actually moves at this rate is
a wave with frequency and wavelength, but also mass-energy (it's all
kinetic - there is no *rest* mass energy, so no inertia, and that's why
it can never accelerate, decelerate, or even exist unless it's buzzing
along at c.)
So - we have a constant manifest in real terms as a wave with a
frequency (inverse time) a wavelength (a space function), and a
mass-energy, and they are all inextricably entwined. The mass m is
E/c^ 2.
The energy is E = hν, and the frequency ν is 1/t, where t is a
time parameter yet to be determined.. The constant speed c is νλ where
λ is the wavelength, and you can use simple algebra
to get the other
dependencies.
So, this is a self consistent simultaneous set of equations in six
dimensions (but two are constant so you can set it up as 4D) that can be
represented using matrix algebra. You can describe such a system as a
matrix, and all matrices have certain powerful mathematical properties.
Things like discriminants, and very powerfully, an "eigenvalue" which is
unique to the matrix. If two matrices have the same eigenvalue, then
they are merely variants of the same system.
OK, now, setting this system up as a matrix, we can compute the
eigenvector and eigenvalue for the matrix, (don't ask me how - I'm a
conceptual physicist who visualises stuff, not a mathematician, so I
leave the donkey work to the mathos) and it turns out to be a function
of photon mass with the physical dimension of time.
That is, the eigenvalue which is a characteristic constant of the system
is fixed, and we can give it a name. Tony calls it an instanton, which
can also be regarded as a quantum of time for that system (obviously
though different photon energies will give different eigenvalues and so
the instanton is a variable when we consider all possible frequencies.)
Note though that the evolution of the system is such that you can take
time as the initial parameter on which everything else is going to
depend. From it we have a frequency, and from frequency we have
wavelength (i.e. space) and lastly we have energy (mass) from the Plank
energy equation expressed relativistically, E = h ν = mc^2.
Now - say we shrink our system to the smallest possible length and time
that makes any physical sense, namely the Planck scale. You now have an
instanton that is absolutely the very foundation of quantisable
spacetime. It is a true time quantum from which EVERYTHING else derives.
I think Jim would be very pleased with this outcome as he sees
absolutely EVERYTHING as being built from Planck actions, a notion with
which I personally agree wholeheartedly. And you should be pleased
because it strongly vindicates your concept of a primal clock underlying
all reality (incidentally, in loop quantum gravity, these "instantons"
are ticked out by spin vectors that serve to generate space, so again
your early intuitions seem to be vindicated).....
Even though Tony's terminology is a bit obscure - and I for one am not
into brane theory at all yet, so I have to do a double take with most of
his stuff - it's mathy and I'm no motho - when I said it makes sense,
it's because I can *see* it clearly in my mind's eye. It certainly does
seem to be the way things work..........
Postulates of Quantum Relativity:
1. Set of Postulates of SR (verbatim): Principle of Relativity
2. Set of Postulates of GR (verbatim): Principle of Equivalence
3. Set of Postulates of QR (A,B,C,...): Principle of Quantization
A: Modular Duality between vibratory and winded Energy Eigenstates or
EE's, allows the description of high frequency/small radius EE's
as physically equivalent to low frequency/big radius EE's
The EE's are themselves defined in the Planck-Scale of the various
mensuration parameters as given in postulates 1. and 2.
Description:
The Operator E=f/R+wR describes vibratory mode f and winding mode w as being
as being invariant under simultaneous exchange of f & w and R & 1/R.
f=3 and w=2 for R=10 -> E=3/10+2.10=20+3/10 is IDENTICAL to
f=2 and w=3 for R=1/10-> E=2.10+3/10=20+3/10
In Point-Particle Quantum Mechanics, displacement and momentum are related
by Heisenberg's Constant and Fourier Transform.
EigenState [x> of position on the circle radius R is defined in [x>= Σf .exp[ixp]]p>
where p=f/R and [p> is a momentum eigenstate as the uniform vibration state
of
the string with phase factor h/4 π<=Et for t~1/E~R.
x is periodic with period T=2 πR as position x on a circle radius R.
EigenState [X> of position on the circle radius 1/R is defined in [X>= Σw.exp[iXP]]P>
where P=wR and [P> is a momentum eigenstate as the uniform winding state of
the string with phase factor h/4 π<=Et for t~1/E~1/R.
X is periodic with period T=2 π/R as position X on a circle radius 1/R.
Two wavepackets x and X then evolve in time in proportionality of their respective
radii to the time required to complete a revolution of the circles and as given
in the
Heisenberg Relation.
This serves as a technical description for 'displacement' in modular duality.
B: Modular Duality is defined in the threshold of the GR postulates
as the 'quantumsmeared' singularity of the Planck-Scale (1. & 2.)
E=f/R+wR/L P^2, with LP=Sqrt(Gh/2πc^3)=Planck-Length remains invariant
under exchange of f & w, as well as R & L P^2/R.
C: The fundamental constants, which define the Planck-Scale are algorithmically
obtainable in considering a demetricated senario, in
which space and time are undefined and subject to definition by a
Machian Observer doing the defining.
The derivation of the Electromagnetic Electronmass below indicates this.
A detailed analysis is found in the Algorithmic Gravity Paper.
D: The engagement of the Machian Observer then results in the Planck
Scale and under postulates B and A reattains the sets of the postulates
as given in 1. and 2.
This is described in the 'Machian Momentum' for the Universe Paper.
E: The algorithm defining the fundamental constants is based on the
premise of a mathematical supersymmetry, thus becoming realised
as physical reality, defined in postulate A.
The Cosmic Wavefunction is the following Differential Equation:
dB/dT + αB(n) = 0; α being the Electromagnetic Finestructure as the
probability of light-matter interaction (~1/137).
This has a solution: B(n) = B o.exp[-α.T(n)];
B o=2e/hA from QR boundary conditions defining:
T(n)=n(n+1) as the Feynman Path-Summation of particular histories
under the pentagonal supersymmetry given in the identity:
XY=X+Y=-1=i^ 2=exp[iπ] and lim [n->X]{T(n)}=1
This allows the Normalisation of the [ Ψ]^2 wavefunction to sum
to unity in B(n)=(2e/hA).exp[- α.n(n+1)] with Functional Riemann Bound
FRB=-1/2, centred on the interval [Y,...-1,...-X,...-1/2,...(X-1),...0,...X].
Interval [Y,-1] sets F-Space; interval [-1,0] sets M-Space with uncertainty
interval [-X,(X-1)] and interval [0,n) sets the C-Space, encompassing OmniSpace.
n<0 is imaginary as real reflection of real n>0 of the C-Space, metrically
defined
at the coordinate n=0 mapping n=n ps, which is the instanton tps=fss=1/fps.
Cycletime n is defined in GR as dimensionless TauTime in curvature radius
Rc=c.dt/d τ for the pathlength of x=ct and become dn/dt=Ho, n=Hot in QR,
with H o the
nodal HubbleConstant defined in c=HoRmax=λps.fps.
The Feynman Path so sums both negative and positive integers
as: -n......-3...-2...-1...0...1...2...3......n =T(n) in absolute value to double
the
infinities as the entropy reversal of lightpath x=c.t=(-c)(-t) in the Moebius
Property of the 4 worlds as outlined in the Machian Momentum Paper.
Cantor Cardinality Aleph-Null is thus Unitised in Aleph-All, counting infinities
as if they were integers of the Feynman Path.
This allows the Feynman interpretation of Quantum Mechanics as alternative
to the formulations of Schroedinger (fermions 1/2spin) and Klein-Gordon (bosons)
as timeindependent and timedependent (free particle form inconsistent with SR
in Schroedinger in 1st order t & 2nd order x), formulations respectively.
The units of B(n) are 1/J, that is Inverse Energy, with A^ 2 an algorithmic constant
defining Current-Squared and 2e/h the Josephson Constant in Amperes/Joules.
B(n) as the universal cosmic wavefunction describes the universe as a potentially
infinite collection of 'frozen' wormhole- eigenstates at n=0.
The timeinstanton 'unfreezes' one such eigenstate and activates the protoverse
as
discussed in the Machian Momentum Paper.
This then allows the 'Mappings' of the C-Space 'real time n>0' from the F-Space
of the
'imaginary time n<-1' under utility of the M-Space interval as 'mirror-space'.
QR unifies electromagnetic and gravitational finestructures in F-Space using the
Planck-
Length-Oscillation L PSqrt(α)=e/c^2
from the subtimespace.
This yields the decisive mapping for the B(n):
Coulomb Charge e = L P.Sqrt(α).c^2 <-->2Re.c^2 = e* (StarCoulomb
Charge)
But the StarCoulomb is Inverse Energy by definition of the vibratory part of the
modular dual heterotic supermembrane HE(8x8)=E psEss.
........................,... Eps=hfps=hc/λps=(me/2e).Sqrt[2πGo/αhc]=me/{2emPSqrt(α)}=1/e*.
m P is
the Planck-Mass and Go is the initiatory Gravitational
Constant, defined in
the FineStructure-Relation, which defines the Planck-Length-Oscillation in the
unification of electromagnetic interactions with those of gravitational permittivity:
G o=4πεo=1/30c with dimensionless
c-ether constant [c]unified.
This defines the MacArthur-Gamma as: "LightMass-Constant" LMC=Mac γ=30[c]unified.
Thus Quantum Relativity is defined in the charge mappings between the OmniSpace
dimensions; F maps magnetocharges e* onto electrocharges e under agency of the
11D-Witten Mirror, which is a Onesided Surface moebian connecting 10D to 12D.
The all encompassing source energy quantum is the E ps-Gauge Boson, which manifests
as the Gauge mediator for the four elemental interactions, suppressing the weak
interaction
in a primary triplicity however to allow the Higgs Restmass Induction mechanism
to proceed in the defining qualities of the Unified Field of Quantum Relativity
(UFoQR).
The details are discussed in the aformentioned papers and the archives of the QR
website
and its links.
The paper below introduces some of those details.
CALCULATION OF THE ELECTRONMASS
The Main Idea is that the Electronmass is made up of two parts.
One part is the 'naked electronrestmass' and the other an additional part (a magnetocharged
part also called electromagnetic mass).
Together this gives a dynamic effective electronmass, which derives naturally from
the
other integerbased fundamental constants.
There are three such constants given by an algorithmic program, which relate directly
to
Energy concepts.
1. Einstein-Constant as: (9,10,16)=> 9x10^ 16=c^2.
2. Planck-Constant as: (15,10,32) => 1/(15x10^ 32)=h.
3. Boltzmann Constant as: (15,16,18) => 1/(15x16^ 18)=k.
They are all dimensionless in pre-quantum algorithm space (prePlanck-Phase).
In a transitionphase, relating to the 'Quantum Big Bang', a boundary between a
'quantum
realm' and classical physics became established in a quantum tunneling of a 'false
vacuum'.
This is described in de Broglie phase-inflation.
This phase inflation defines a closed curvature topologically equivalent to an
open
curvature and specifies modular duality between maximum/minimum conditions.
The dimensionless number constants became physically manifested and related to
many
other derived parameters relating to energy in the order of Temperature, Radiation
and
Mass; that is in the order of Boltzmann, Planck and Einstein.
One such relationship finestructures Maxwell's: 1/c^ 2=μoεo.
Another relationship defines the electronic radius as a quantum macroboundary in
terms of
the magnetic constant μo.
Another relationship sets the linearisation of a circular quantum continuum in
direct
relation to a prequantumscale, that is the supermembrane epoch from Planck-Time
to
Wormholetime.
The wormholetime derives from finestructuring the Action-Constant.
This formulation involves the electronic radius as a number constant.
From that electronic radius, the electronmass is implied as:
m eo.c^2=e^2/[4πεo.(electronic radius Re)].
Equality of dynamic electromagnetic mass in magnetic selfinduction to the relativistic
mass
increase, then leads to a finestructure for the effective electronmass (above)
as a distribution
and as set out below.
There are a number of parameters and fundamental constants which are given by the
pentagonal supersymmetry between four dimensionally nested worlds.
The lightconstant 'c' becomes dimensionless in: c^ 2=9x10^16 for example, giving a 'cosmic
standard' system where c=300,000 km*/sec* precisely.
Calibration to the SI-system differs often less than 1%.
The electronmass was not amongst the elementaries however, because all restmasses
are set
to 0 in the first world (here termed Omnispace of dimensions 10-11-12-(13=1)).
So we must derive the electronmass from the other given elementals.
It proceeds something like this.
One first finestructures the Maxwell-Constant into electric and magnetic parts
and then
applies the OmnispaceLaw {ACTION=CHARGE^ 2} to find dimensionless
(superconductive) expressions for electric permittivity εo and magnetic permeability μo.
This ACTION LAW maximises/minimises the Lagrangian in Omnispace, especially the
boundary condition of the Planck-Length-Oscillation, which maps the ForceCharges
between Omnispace and Linespace.
Linespace connects Omnispace in dimension 1=13, which then also becomes a mirrored
Timedimension in 1-2-3-4 D. Hyperspace is world #3 mapping 4-5-6-7 and Quantumspace
maps this as 7-8-9-10 into Omnispace as world #4).
There are two equal expressions for the Planck-Length Oscillation; allowing mappings
from
the creation world into the first manifestation.
Planck-Length-Oscillation: (e/c^ 2)=Planck-Length.SqrtAlpha.
This allows mapping of MagnetoCharges (e*) from 13D as Coulombic charges (e) as
1D
(say).
Since Magnetocharge is defined as Inverse SourceEnergy in the form:
e*=Classical Electronic Diameter.c^ 2 in units (m^3/s^2) or as [Volume.Angular
Acceleration] and also as:
E[ ps]=hf[ps]=h/f[ss]=1/e*,
we find modular duality in supersymmetric associations between magnetocharges and
electrocharges.
In particular, the vibrating Supermembrane (in 11D) E[ ps=primary sourcesink] is modular
dual to the E[ ss=secondary sinksource] and so relates the inversion properties of
Maximum Radius to Minimum Radius.
This is the backbone of M-Theory, which labels our supermembrane E psEss as the heterosis
between the HE-Superstring (8x8) in 10 dimensions with a Bosonic superstring in
26
dimensions.
In particular, you now find the mapping:
(e=Planck-Length-Oscillation.c^ 2) <=>(e*=Classical Electronic Diameter.c^2) for
electropole e and magnetopole e*.
The overall symmetry, crucial for deriving the electronmass is found in the Cosmic
Wavefunction:
B(n)=(2e/hA).exp(-Alpha.T(n)),
where A is an algorithmic constant relating to (Current^2); Alpha is the Electromagnetic
Finestructure Constant related to the Gravitational Finestructure Constant in direct
scaleproportion as:
ALPHA(GR)=ALPHA(EMR)^18;
and T(n)=n(n+1) is the Feynmann Summation, the Path-Integral over all possible
particular
histories, summing all absolute integers in a Cantor Cardinality of Aleph-Null
becoming
Aleph-All, the counting of infinities as integers differing in 1).
The crucial point here is, that this wavefunction defines an infinite number of
'frozen'
spacetimes, which upon a process of 'thawing' (we could call it a spiritual awakening)
can
ACTIVATE themselves in (coming alive); but redefining themselves as eventuating
'Baby
Universes'.
Renormalising those frozen wormholes/potential universes, then defines a Functional-
Riemann-Bound in the Complex Plane at imaginary cycletime n=H ot (Ho a nodal Hubble
Constant at 58 Hubble Units).
The statistical mean is at [n=-1/2] for the squared wavefunction as a density distribution.
This [FRB=-1/2] then relates to many fields in pure number theory, such as the
Riemann
Hypothesis of the Zeta Function and the Euler Harmonic.
B(n) is also a disguised form of Stirling's Formula as function of Alpha.
But we now have expressions for the magnetic constant, which relate to the propagation
of
the unified field in the form of 12 selfintersecting monopolic current-'knots'/junctions,
giving rise to all particle families,
bosonic and fermionic in the application of a colourtrinity, triplets of unification
(Red-
Green-Blue=White in Radiation) and (Yellow-Cyan-Magenta=Black in Paint/Mass).
Complementary triplets/doublets/singlets then define the unitary symmetries, such
as 8
gluonic intermediate mesonic states in {RC=MG=BY} and the permutation octet
(WWW,WWB,WBW,BWW,BBW,BWB,WBB,BBB).
In particular there are exactly 10^ 10/180 λ[ps] wavelengths in the Classical Electron-
Diameter, the latter
which is found modular dual in the expression: c= λ[ps]f[ps]=1/λ[ss].f[ss].
This links Gravitational Radiation to Electromagnetic Radiation as Lagrangian Boundary
Condition.
Because of this quantisation of the wormholeradius r[ ps]=λ[ps]/2π;
we can directly relate the
unified fieldpropagation (which is 4 λ[ps] or 8π radians) to the magnetic constant linking the
linearisation of the worm-holeperimeter to the fourfoldedness of the angular multivalued
expression.
Define the electronrestmass as equality between the relativistic massincrease and
its
dynamic magnetic selfinduction.
You have: c^ 2.(m-me)=μo.e^2.v^2/(8πRe), where me is the electronmass; e the
electropolecharge; v the electron velocity; μo the magnetic constant and
Re the Classical
Electronic Radius.
Now set: Constant A= μo.e^2/(8πRe.me) and let Constant B=(v/c).
This leads to the quadratic: B^ 2=([A-2]/2A)+-[A^2+4A]^1/2/2A.
This defines a distribution of B^ 2=(v/c)^2 relativistic velocity ratios in me.A=μo.e^2/8πRe.
Using our definitions, we find: m e.A=30e^2.c/e*=4.645263574x10^-31 kg*.
This implies, that for A=1, m e=me/2, where me=9.290527155x10^-31 kg* from the
prequantum algorithmic associations, based on the magnetic constant defining the
Classical
Electronic Radius.
Here me is the EFFECTIVE ELECTRONMASS and includes a MAGNETOCHARGED
PART; added to the 'Naked Electronmass m eo'.
As B^ 2>=0 for all velocities v, bounded as groupspeed (not de Broglie Phasespeed always >
c) in c for which B^ 2=1; a natural limit is found for the B^2 distribution
at A=1/2 and
A=Infinity.
For A=1/2: B^ 2=-3/2(+-)3/2 for roots x=0 and y=-3;
for A=3/4: B^ 2=-5/6(+-)(19/12)^1/2 for roots x=0.425
and y=-2.092;
for A=1: B^ 2=-1/2(+-) Sqrt(5)/2 for roots x=X and y=Y;
for A=Infinity: B^ 2=(1/2)[-](+-)(1/2)[+] for roots x=1[-] and y=0[-];
Letting B^ 2=n, we obtain the Feynman-Summation and the Binomial Identity gives the limit
of A=1/2 in:
A=1/2 - B^ 2{3/8 - 5B^2/16 + 35B^4/126 -...}.
But the FRB is defined in the renormalisation of B(n) exactly about the roots X,Y
and so the
model of the 'heavenly boundary conditions' is proved, (relative to the composer).
The unifying condition is: XY=X+Y=i^ 2=-1=exp(iπ).
One can now introduce the magnetocharge to derive the 'naked electronmass' by
perturbation and the
REDUCED CHARGE.
Let m e*=me-meo for me*=2me.e^2/3Re.e*=1.556642765x10^-32 kg*
{EQ.*}
as a first approximation and introduce the unitvolume V in: m e*=μo.e^2.V/(4πRe^2.c^2).
"Cubing the Sphere" in the ratio of an inscribed (or circumscribed) sphere of unitradius
and bounded by a cube of side 2-units then sets the unitvolume in an octagonised
coordinate
system.
V so becomes 4 π/3 in the ratio 4π/(3.8)=π/6 and we have {EQ.*}.
Using perturbation techniques, one then approximates the 'naked electronmass' from
its
magnetocharged and effective progenitor.
A complication arises because of the higher dimensional independence of the Alpha-
Finestructure Constant from electric permittivity.
The creation world form for Alpha=60 π.e^2/h as a direct reflection of the ACTION LAW
(h*=ee*).
This results in a measured Alpha-Dip for ce
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