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Physics Inside-Out: A Physics of Peace, by Loren Booda
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quantum interpretation, Planck scale physics,
quantum wavefunction, metric tensor, complementarity principle, quantum
cosmology, compactification, virtual states, T-duality, black hole physics,
quantum entanglement, quantum reality, philosophy of physi
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Quantum mechanics and relativity, upon
inversion of phase space through Planck
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<html>
<head>
<title>Physics Inside-Out: A Physics of Peace, by Loren Booda</title>
<meta name="description" content="Quantum mechanics and relativity, upon
inversion of phase space through Planck's constant, or spacetime through
the Planck length, reveal novel cosmological symmetries. Sensing beauty
in observation acknowledges physics' practical proof of reality.">
<meta name="keywords" content="quantum interpretation, Planck scale physics,
quantum wavefunction, metric tensor, complementarity principle, quantum
cosmology, compactification, virtual states, T-duality, black hole physics,
quantum entanglement, quantum reality, philosophy of physics">
</head>
<body bgcolor="FFBB22">
<P><align=top><font size="2">Decay of a mini black hole in ATLAS</font size></P>
<img src="Mini black hole.jpg" width=200 height=205 align=left align=top>
<img src="Ultradeep.jpg" width=200 height=205 align=right align=top>
<font color="0000FF"><P align=center><font size="7"><font face="Imprint MT Shadow">Physics Inside-Out</font face></font size></P>
<P align=center><font size="4"><font face="Imprint MT Shadow">A Physics of Peace</font face></font size></P>
<font size="1"><P align=center>© 2010</font size></P>
<P align=center><font size="4">by Loren Booda </font size></P></font color>
<br>
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<td><img src="EschSphII.jpg" width=291 height=189></td>
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<font size="3"><font color=Silver>
  <i>Our world reflects upon a sphere<br>
  Revealing silvered twin<br>
  From whose versed image shall appear<br>
  The universe within.</i><br>
</font color></font size></td></tr></table bgcolor></table>
<br>
<br>
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<td width="80%"><P><font color="0000FF"><b><u><font size="6">Phase Reality!</font size></u></b></font color></P></td></tr>
<br>
<br>
<P><font color="0000FF"><u><b><font size="5">Contents</font size></b></u></font color></P>
<ul>
<li><a href="#q1">1. P-Duality: Quantum Mechanics Inside-Out</a></li>
<li><a href="#q2">2. P-Duality: General Relativity Inside-Out</a></li>
<li><a href="#q3">3. Matters of Gravity</a></li>
<li><a href="#q4">4. Relativity's Complex Probability</a></li>
<li><a href="#q5">5. Black Hole Internal Supersymmetry</a></li>
<li><a href="#q6">6. Macromechanics</a></li>
<li><a href="#q7">7. Tunneling from beyond the Event Horizon</a></li>
<li><a href="#q8">8. Symmetry and the Superuniverse</a></li>
<li><a href="#q9">9. Creed</a></li>
<li><a href="#q10">10. The "Booda Theorem"</a></li>
<li><a href="#q11">11. Fine-Structure Constant Numerology</a></li>
<li><a href="#q12">12. Neurophysiological Uncertainty</a></li>
<li><a href="#q13">13. Configuration Complementarity</a></li>
<li><a href="#q14">14. Quantum Alive!</a></li>
<li><a href="#q15">15. Oceanus</a></li>
<li><a href="#q16">16. Quantum Phase Effect</a></li>
<li><a href="#q17">17. Logic <i>e</i> computation</a></li>
<li><a href="#q18">18. One Quantum at a Time</a></li>
<li><a href="#q19">19. The Cosmic Egg</a></li>
</ul>
<br>
<br>
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<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Quantum mechanics or relativity, upon inversion of
phase space through Planck's constant or spacetime through the Planck length, reveals
previously unexplored cosmological symmetries</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> T-duality, a basic symmetry of superstrings, conceives reciprocally related
variables through a Generalized Uncertainty Principle. (An informal introduction to
superstrings, and T-duality in particular, may be found at The Official String
Theory Web Site's String Theatre: <a href="http://www.superstringtheory.com/theatre/col14.html">The Second Superstring Revolution</a>.)
The two essays that follow below assert basic physics utilizing a T-duality analog, "phase-duality,"
to modify first the Schrödinger wavefunction, and secondly general relativity's metric
tensor. This P-duality generalizes current physical models and shifts the nonlocal
toward locality. As universally experienced, P-duality manifests a center omnipresent to
each observer, and connects our immediate to indirect experiences. The essential
challenge before us is to create a responsible and ethical philosophy of physics,
testable yet not malicious.</P>
<P> String theory, which innovated T-duality, substitutes for "standard"
quantum gravity parameters a wave compactification among hyperspatial dimensions.
For consideration by physicists, P-duality doubles the
dimensionality available to the accustomed quantum wavefunction by establishing an
elliptical symmetry between phase space and its dynamic inverse. This complementarity
enables many quantum interpretations - like a novel perspective on quantum
field theory - and explores semiclassical aspects of modern physics. Similarly,
P-duality modifies Einstein's spacetime metric tensor by means of a quantized
"action-equivalent radius of curvature," and compactifies four-dimensional spacetime
by reciprocity within the local Planck radius.</P>
<P> Faster-than-light (non-local) actions <i>a'</i>, of quantum mechanics, can be
modeled simply with local action <i>a</i> turned "inside-out" about the singular
surface of radius h, the value of Planck's constant.</P>
<P> That is, h<sup>2</sup>=<i>aa'</i>.</P>
<P> Antigravitational (dark energy) spacetime intervals Δ<i>s'</i>, of
general relativity, can be modeled simply with local spacetime intervals Δ<i>s</i>
turned "inside-out" about L*, the value of the Planck length.</P>
<P> That is, (L*)<sup>2</sup>=Δ<i>s</i>Δ<i>s'</i>.</P>
</td></tr>
</table>
<br>
<br>
<br>
<td width="80%"><font color="566D7E"><font size="5">These essays on duality were inspired
by Edward Witten, especially from his article "Reflections on the Fate of Spacetime," which
appeared in Physics Today, April 1996, on pages 24-30.</font size></font color></td>
<table>
<tr><td width="20%" rowspan=2></td>
<br>
<br>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q1>P-Duality: Quantum Mechanics Inside-Out</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i> Interchanging variable action and Planck's constant in a
real quantum wavefunction obtains a spectrum of virtual states
that invert standard eigennumber solutions</i></font color></font size></td><td width="80%" bgcolor="FFBB22">
<P>These essays on duality were inspired by Edward Witten, especially in his article "Reflections on the Fate of
Spacetime," which appeared in Physics Today, April 1996, on pages 24-30.</P>
<P> Inverted dimensions occur foremost in physical theory as part of the
crystallographic reciprocal lattice. Another application, inverse phase space,
contains all virtual states described by "virtual wavefunctions," φ<sup>-</sup>. Such
wavefunctions differ from their conventional counterparts, φ, by the interchange
of their variable actions <font face="Vladimir Script">S</font> with h
(Planck's constant). I. e.,
</P><br><P align=center><font color="660011">φ[<font face="Vladimir Script">S</font>, h] ↔
φ<sup>-</sup>[h, <font face="Vladimir Script">S</font>]</font color>.
(Square brackets indicate a function in general unless otherwise noted.)</P>
<br>
</P> "Conventional" (real) phase space transforms to "inverse" (virtual) phase
space, and vice versa, upon such a dynamic inversion through h. A virtual wavefunction generates action eigennumber solutions
reciprocal and symmetric to those of its conventional counterpart. Together
they obey both the de Broglie and Einstein postulates, and can be expressed in a
linear Schrödinger equation. The mathematical justifications for a
wavefunction entity are essentially identical for both "real" and "virtual"
wavefunctions.</P>
<P> The inverse (virtual) wavefunction introduces multifold applications to
the problems of virtual particles. Quantum field theory then considers
virtual states as arising from the virtual wavefunction. For some applications
in field theory, this wavefunction may be more efficient than its real counterpart.</P>
<P> The dual wavefunctions evolve particles from their mutual interference,
increasing geometrically the productivity of either wavefunction alone. The mirror
symmetry of P-duality is defined: <i>virtual quantum states are represented
in reciprocal phase space through an inverse wavefunction argument as real quantum states
are represented in traditional phase space through a conventional wavefunction argument.</i></P>
<P> The <u>inverse</u> time independent free-particle wavefunction,</P>
<br>
<P align=center><font color="660011">φ<sup>-</sup>[<b>r'</b>]=B'·exp(-2πi(h/<b>r'p'</b>))</font color>,</P>
<br>
<P>generates action eigennumbers reciprocal to those of the <u>conventional</u> time
independent free-particle wavefunction,<P>
<br>
<P align=center>φ[<b>r</b>]=B·exp(-2πi(<b>rp</b>/h)),</P>
<br>
<P>and represents virtual, rather than
real, states. Action eigenvalues for the conventional wavefunction are</P>
<br>
<P align=center><font face="Vladimir Script">S</font><sub>N</sub>[φ[<b>r</b>]]=<b>r</b><sub>N</sub><b>p</b><sub>N</sub>=hN/2,</P>
<br>
<P>derived through Im[φ[<b>r</b>]]=0, from the condition of arbitrary phase.
Likewise, action eigenvalues for the inverse wavefunction are</P>
<br>
<P align=center><font color="660011"><font face="Vladimir Script">S</font><sub>N<sup>-</sup></sub>[φ<sup>-</sup>[<b>r'</b>]]=(<b>r'</b><sub>N<sup>-</sup></sub>)<b>p'</b><sub>N<sup>-</sup></sub>=2h/N<sup>-</sup></font color>.</P>
<br>
<P> N and N<sup>-</sup> are nonzero integers. Action eigennumbers, n<sub>N</sub>
or n<sub>N<sup>-</sup></sub>, are simply action eigenvalues divided by h; thus,
n<sub>N</sub>=N/2, and n<sub>N<sup>-</sup></sub>=2/N<sup>-</sup>. Because of this
reciprocal symmetry, exclusively finite action entails there be no <i>singular</i> zero-valued
spin. Action eigenvalues of magnitude less than h/2 are defined as virtual, those
between and including h/2 and 2h as mixed, and those greater than 2h as real.</P>
<P> Both wavefunctions φ[<b>r</b>] and φ<sup>-</sup>[<b>r'</b>] share
exclusively the action eigenvalues satisfying
N/2=2/N<sup>-</sup>; that is, corresponding to spins 1/2, 1, and 2, those of
most fundamental particles: fermions, photons, and gravitons, respectively.
Photons, having spin one, manifest as maximally symmetric (i. e., of dualistic ground state)
between real and virtual phase space. The principle of least action localizes particles near
<font face="Vladimir Script">S</font> =h, downward from real, and upward from
virtual actions, into the arena of mixed (shared) states. Upon transforming
from a primordial singularity into a bivalent state, real and virtual phase
spaces maintain initially local entanglements. They generate the family of
subatomic particles from their own interference,</P>
<P align=center><font color="660011">φ[<b>r</b>]·φ<sup>-</sup>[<b>r'</b>]=B·B'·exp(-2πi(<b>rp</b>/h+h/<b>r'p'</b>)),</font color></P>
with action solutions based on the Fibonacci series. In this regard, n<sub>N</sub> and
n<sub>N<sup>-</sup></sub> compare respectively to vibration numbers and to winding numbers
of T-duality in string theory.</P>
<P>The one-dimensional time dependent virtual Schrödinger equation,
<P align=center><font color="660011">(-(2πh)<sup>2</sup>/2m)(∂<sup>2</sup>[1/x']/(∂[φ<sup>-</sup>[1/x',1/t']]<sup>2</sup>)) + V[1/x',1/t'](1/φ<sup>-</sup>[1/x',1/t'])</P>
<P align=center>=i2πh(∂[1/t']/(∂φ<sup>-</sup>[1/x',1/t']))</font color></P>
<P>describes the mechanics of reciprocal phase space through its corresponding time dependent virtual wavefunction</P>
<P align=center><font color="660011">φ<sup>-</sup>[1/x',1/t']=A'·exp(2πih(1/x'(p<sub>x</sub>)'+1/E't'))</font color></P>
<P>with its inverted units.</P>
<br>
<br>
<P><b>Free particle in one dimension: conventional-real ↔ inverse-virtual</b></P>
<P>Wavefunctions (φ):</P>
<P>φ=C·exp(2πixp<sub>x</sub>/h)
↔ φ'=C'·exp(2πih/x'(p<sub>x'</sub>)')</P>
<P>Operators (*):</P>
<P> (p<sub>x</sub>)*=(-ih/2π)(∂/∂x) ↔
(1/(p<sub>x'</sub>)')*=(-i/2πh)(∂/∂[1/x'])</P>
<P>(x*)δ(x-`x)=(`x)δ(x-`x) ↔
((1/x')*)δ(1/x'-1/`x')=(1/`x')δ(1/x'-1/`x')</P>
<P>where δ represents the Kronecker delta function</P>
<P>Commutators:</P>
<P>[x*, p*]=ih/2π ↔ [(1/x')*, (1/(p<sub>x'</sub>)')*]=i/2πh</P>
<P>Hamiltonians:</P>
<P>H*=((p<sub>x</sub>)*)<sup>2</sup>/2m=-((h/2π)<sup>2</sup>/2m)∂<sup>2</sup>/∂x<sup>2</sup> ↔
(1/H')*=(1/((p<sub>x'</sub>)')*)<sup>2</sup>2m=-((2πh)<sup>-2</sup>2m)∂<sup>2</sup>/∂[1/x']<sup>2</sup></P>
<P>Spectra:</P>
<P>E=nhν ↔ E'=hν'/n'</P>
<br>
<br>
<P> Photons passing "singly" through a double-slit apparatus develop an
interference pattern at its screen similar to that exhibited there by
"simultaneous" pairs of photons. Reciprocal phase space confers upon the
individual photon the information needed for interference with neighboring
conventional phase space. Entangled from a common cosmological original
event, these phase spaces together enable single particles to "self interfere"
coherently where N/2=2/N<sup>-</sup>. In other words, the particle mediates
locally an inverse/conventional wavefunction interference, comparable to that
accustomed of two conventional wavefunctions. For instance, a radioactive atom's
decay is determined through interference associating its underlying inverted
eigenstates with their correlates, the real phase space states with which we
are familiar.</P>
<P> An EPR experiment, starting with a singlet "zero" spin state, needs
considered first the possible vacuum effects in that initial neighborhood.
These primarily determine, before decay, the half-life of the parent
particle through the real/virtual phase space interference unique to it.
This experiment progresses much like the double-slit situation. The paired
photons emitted sustain quantum entanglement, a statistical correlation
inherent to the intersecting phase histories of each. So entanglement can entail
virtual wavefunctions interfering with actual ones. Elliptically nested,
reciprocal phase spaces maintain probabilistic causality without
necessitating superluminal signaling, "hidden" variables, or myriad obsolete
universes.</P>
<P> Any zero spin field quantization would seem to violate the finitude
of corresponding inverse phase space actions. Classically speaking, spin zero
is not allowed in P-duality, but through quantum uncertainty,
spontaneous particles can "bootstrap" potentially null regions to
overcome that constraint. Thus, a statistically equivalent
quantum space could sustain its spin continuity through complementarity,
with particles bootstrapped out of the vacuum. Otherwise, a six-dimensional hyperellipse, that of classical
phase space mapped with associated spin vectors, requires topologically a
four-dimensional zero-valued spin manifold within (which boundary may partially
define a spacetime, below). Moreover, six compactified
dimensions coupled <i>reciprocally</i> at Planck constant action to six outer
dimensions could satisfy, within six apparent dimensions, the criteria
for virtual phase space's nonzero spin.</P>
<P> Picture an ellipsoid in phase space given by</P>
<P align=center><font color="660011">(rM*c)<sup>2</sup>+(pL*)<sup>2</sup>=h<sup>2</sup></font color></P>
<P>where M* and L* are mass and length Planck units, r is radial displacement,
p is its corresponding momentum, c the speed of light and h, Planck's constant.
Let that ellipsoid represent a photonic harmonic oscillator
mediating real/mixed phase space</P>
<P align=center><font color="660011">(rM*c)<sup>2</sup>+(pL*)<sup>2</sup>≥(h/2)<sup>2</sup></font color></P>
<P>without, and virtual/mixed phase space</P>
<P align=center><font color="660011">(r'M*c)<sup>2</sup>+(p'L*)<sup>2</sup>≤(2h)<sup>2</sup></font color></P>
<P>within. Both quantum locality and uncertainty are depicted by this Planck
scale physics. This pairing of the inner, inverted phase space with the outer,
conventional phase space recovers <u>locally</u> all values: r≥L*/2 ↔ r'≤ 2L*
and p≥M*c/2 ↔ p'≤ 2M*c. <i>Their symmetry enables quantum measurement to include
microscopically the entire range of macroscopic phase space.</i></P>
<P> Wavefunction "collapse" now manifests Born probabilities in an accessible,
physical transformation, having lost its discontinuous, intangible character to the
inverse wavefunction φ<sup>-</sup>[<b>r</b>], a local information equivalent
to φ[<b>r</b>]. Measurement duality coincides symmetrically around h.
Correlated from the original Planck vacuum, mutually reciprocal phase spaces
enable seemingly simultaneous action-at-a-distance in spacetime.
A system such as Wheeler's "galactic interferometer" translates as compactified
within a virtual ellipsoidal phase space proximal to the observer, describing
why the observed interference effects appear instantaneous.</P>
<P> Inverting wavefunction action eigennumbers may provide gravitational,
electric or magnetic "singularities" continuity, since it can transform a real,
discrete displacement into a virtual, smooth one. Upon inversion, Heisenberg's
uncertainty principle <i>reverses</i> its inequality (with action eigenvalue limit of
N=1 now that of N<sup>-</sup>=1)</P>
<br>
<P align=center> from <font color="660011">ΔrΔp≥h/2</font color>
to <font color="660011">Δr'Δp'≤ 2h</font color>.</P>
<br>
<P> Heisenberg uncertainty, now overall unbounded in principle, overlies actions
included in its new triad amongst real, virtual, and mixed states. Conventionally,
a singular interval, where Δr→ 0, complements a momentum uncertainty
Δp→∞. Alternatively, inverse phase space allows 0≤Δp'→∞ for
Δr'→ 0. This enables a local spread of virtual momentum N<sup>-</sup> states
to represent quasi-continuously their unbounded self-energy. Summing every momentum state of
inverse phase space gives an effectively infinite internal kinetic energy of compactification
within the ellipsiod of uncertainty.</P>
<P> The real phase space of a double-slit experiment maps conformally through the
ellipsoid of uncertainty, revealing a "photon's eye view" which transposes
actions such as those at the slits, screen or light source. The photon's ellipsoidal
perspective interrelates the subluminal aspects of that experimental phase space with the
superluminal virtual states, in effect turning the universe inside-out. Now the Casimir experiment
incorporates cancellation or reinforcement of virtual waves with the environment (lab walls,
particles in space, observer, Earth etc.) <u>outside</u> its plates. The virtual wavefunction
of an infinite well potential has energy states with values reciprocal to those of the conventional
infinite well wavefunction. Solutions of virtual wavefunctions with their inverted
geometry thus juxtapose standard quantum field theory calculations.</P>
<P> P-duality, sharing mathematical properties initially attributed to strings by T-duality,
derives phase-dependent action through a reciprocal wavefunction symmetry.
Additionally, it persuades our accustomed philosophy of modern physics towards a new
realism. Before we enact the hypothetical, albeit beautiful, dimensionality of
string or M-theory, compactification by inversion
through action h can obtain a new universe of accessible phase space states. Not
only do many principles of quantum mechanics fit this "inside-out" nature in accord with
P-duality, further study reveals parallels to possible spacetime reciprocity.
Quantum interpretation oversees two complementary, aboriginally entangled and
reciprocally nested phase spaces. The philosophy of observer as participant
includes a real/virtual center and connective events that represent the universality
of any measurement.</P>
<br>
<P>Emeritus professor Frank Firk of Yale University, my alma mater, offers
<a href="http://www.physicsforfree.com/">"Physics for Free"</a>,
a challenging and disciplined tutorial for those who wish a glimpse of Ivy education.</P>
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q2>P-Duality: General Relativity Inside-Out</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>This approach to general relativity calculates a first order "action-equivalent
radius of curvature," whose quantum correction (10<sup>13</sup> cm for a giant
spiral galaxy) alters the radial vector of the accustomed metric </i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> The "semiclassical metric tensor," <font color="660011">g<sup>μν</sup>[<b>r</b>±δ<b>r</b>]</font color>,
generates intervals of quantum spacetime divergent from the Δs
of conventional general relativity. This modified tensor includes the first-order correction, an
action-equivalent radius of curvature</P>
<br>
<P align=center><font color="660011">δ<b>r</b>=g<sup>μν</sup>[<b>r</b>] Δx<sub>μ</sub> (n<sub>N</sub>)<sup>1/2</sup>L*/R<sub>H</sub></font color>,</P>
<br>
<P> where n<sub>N</sub> are real action eigennumbers (see the previous "P-Duality: Quantum Mechanics Inside-Out"), i. e. relative uncertainties, and δ<b>r</b>
is the action-equivalent radius of curvature. L*,
the Planck length - typical of Planck scale physics -
and the observable universe radius R<sub>H</sub> approximate the extrema of observable spacetime
curvature. R<sub>H</sub>/L*=(n<sub>max</sub>)<sup>1/2</sup> represents the characteristic quantum number for spacetime.
The absolute range of action eigennumbers is n<sub>max</sub> -- on the order of
the dimensionless cosmological constant, Λ. A real action
<font face="Vladimir Script">S</font>[<b>r</b>]=n<sub>N</sub>h≥h/2 infers a
radial vector correction of conventional spacetime, and a virtual action
<font face="Vladimir Script">S</font>[<b>r'</b>]=(n<sub>N<sup>-</sup></sub>)h≤ 2h infers a
radial vector correction of inverse spacetime; intermediate shared actions denote mixed
states, of which <font face="Vladimir Script">S</font>[<b>r</b>]=<font face="Vladimir Script">S</font>[<b>r'</b>]=h
represents a mutual spacetime event horizon.</P>
<P> The n<sub>N<sup>-</sup></sub> establish <i>virtual</i> states' "P-duality"
formulation for the semiclassical metric tensor, where</P>
<br>
<P align=center><font color="660011">δ<b>r'</b>=g<sup>μν</sup>[<b>r'</b>] Δx<sub>μ</sub> (n<sub>N<sup>-</sup></sub>)<sup>1/2</sup>R<sub>H</sub>/L*</font color>,</P>
<br>
<P> These geometrodynamic states of virtual action associate primarily with look-back,
macrocosmic curvature - including an inverted image nearing the Big Bang.
Such inverse spacetime approaching R<sub>H</sub> manifests physics much as conventional
spacetime approaching L*. Increasing action involves quantum resonances for both spacetime
curvature outward from L* and inverse spacetime inward from R<sub>H</sub>. Ordinary
spacetime reciprocated through L* effectively obeys the naked singularity "dress code" by shielding
an otherwise discernable center beneath the inverted horizon. (A concerted effort,
<a href="http://www.amtp.cam.ac.uk/user/gr/public/">Cambridge Relativity</a>,
from a world-class university, describes the ins and outs of curved and singular spacetime
with a plethora of informative visuals and basic concepts detailed.)</P>
<P> The n<sub>N</sub> and n<sub>N<sup>-</sup></sub> comprise eigennumbers, solutions to the
time independent free-particle wavefunction of arbitrary phase,<P>
<br>
<P align=center><font color="660011">Im[φ[<b>r</b>]]=B·sin(-2πn[<b>r</b>])=B·sin(-2π<b>rp</b>/h)=B·sin(-2π<font face="Vladimir Script">S</font>[<b>r</b>]/<font face="Vladimir Script">S</font>*)=0</font color>,</P>
<br>
<P align=center><font color="660011">Im[φ<sup>-</sup>[<b>r'</b>]]=B'·sin(-2πn<sup>-</sup>[<b>r'</b>])=B'·sin(-2πh/<b>r'p'</b>)=B'·sin(-2π<font face="Vladimir Script">S</font>*/<font face="Vladimir Script">S</font>[<b>r'</b>])=0</font color>,</P>
<br>
<P> and the n<sub>N<sup>-</sup></sub> ↔ 1/n<sub>N</sub> for nonzero n<sub>N<sup>-</sup></sub> and n<sub>N</sub>.
g<sup>μν</sup>[<b>r</b>±δ<b>r</b>] fluctuates discretely in real space from
g<sup>μν</sup>[0] to g<sup>μν</sup>[2<b>r</b>],
whereas the vectors x<sub>μ</sub> and
x<sub>ν</sub> vary continuously. The metric g<sup>μν</sup>[0]
refers to counteracting radial spacetime vectors (as on an asymptotically "naked" and singular extreme charged
nonrotating black hole). That of g<sup>μν</sup>[2<b>r</b>] refers to maximally
reinforcing radial spacetime vectors (as toward the equator of an extreme Kerr geometry).</P>
<P> Among actions <font face="Vladimir Script">S</font>[<b>r</b>], there exist
correction curvature radii r<sub>c</sub>[<font face="Vladimir Script">S</font>[<b>r</b>]],
square-root integer factors of r<sub>c[min]</sub>=L*/(2<sup>1/2</sup>).
In general, |δ<b>r</b>|=r<sub>c</sub>[<font face="Vladimir Script">S</font>[<b>r</b>]]≤R<sub>H</sub>.
For a giant spiral galaxy, |δ<b>r</b>|=10<sup>13</sup>cm, since
(n<sub>N</sub>)<sup>1/2</sup>=(10<sup>75</sup>erg-sec/10<sup>-27</sup>erg-sec)<sup>1/2</sup>=10<sup>51</sup>,
|Δx<sub>μ</sub>|=10<sup>23</sup>cm and L*/R<sub>H</sub>=10<sup>-61</sup>.</P>
<P> Through the Higgs mechanism's spontaneous symmetry breaking,
particles decay from the false vacuum into the true vacuum with the acquisition of mass. Real (true) and virtual (false)
spacetimes can justify the dilution of the vacuum energy ~10<sup>122</sup> fold from the
predicted Planck density to the empirical critical universal density. The linear black hole mass density, c<sup>2</sup>/G=10<sup>28</sup>g/cm,
moderates the spacetime geometry of the unstable Planck region with that of the expanding cosmos.</P>
<P> Under inversion, spherically symmetric spacetime bounded by horizons at L* and R<sub>H</sub>
correlates conformally its interstitial curvatures with those of inverse
spacetime horizons between L* and (L*)<sup>2</sup>/R<sub>H</sub>. Such a transform alters the sign and
magnitude of correspondent geometrodynamic accelerations. Inverting the (L*)<sup>2</sup>/R<sub>H</sub>≥r≥L*
radial spacetime components within Friedmann's ρ<sub>M</sub> retains two compactified
dimensions of 0 ≤M/r≤ 10<sup>28</sup>g/cm, and obtains a brane analog ρ<sub>Λ</sub> with two
extradimensions apparent as 0≥M/r≥ 10<sup>28</sup>g/cm. <i>Macroscopically, these
transposed compactifications have the effect of a locally inverted 2-D black hole geometry, evoking
the vacuum (i. e., dark energy) with the radially accelerating universal expansion that recent
supernovae data indicate.</i></P>
<P> Sequence of measurement, more appropriately than time alone, provides an objective and fundamental physical
standard for assessing change in quantum cosmology. Quantum mechanics retains its space/time dichotomy
and relativity its spacetime unity, with sequencing inherent to the dynamical changes in both. There are
two states of change: observation, and statistical continuity between observations. Action
eigenvalues represent the possible sequential permutations -- Planck steps, (n<sub>N</sub>)<sup>1/2</sup>L*/R<sub>H</sub>
-- that observation's action follows. Successive compatible [complementary] measurements define null (0) [unit
(±h)] eigenvalues. These quantum numbers order time-independently overall, obeying a random
walk statistic. It is indeed possible for retrograde, sequential action. A partial history of
Planck steps lost to statistics represents the overdetermination within quantum mechanics.</P>
<P> The dual metrics (modified by the action-equivalent radii of curvature) symmetrize spherically
across the Planck surface, correlating their compactified inverse and macroscopic conventional
spacetimes. The solutions to cosmological constant, isotropy, flatness, magnetic monopole and horizon
problems -- tentatively addressed by the inflationary model -- may otherwise arise from bivalent spacetimes,
aboriginally evolving apart from a common, correspondent and singular state. The microscopic, reciprocal,
and virtual geodesics represent interiorly those macroscopic, proportional and real exteriorly. In
other words, the quantum cosmos projects inwardly those phenomena which inflation would claim outwardly.</P>
<P> (A beautiful account,
<a href="http://nedwww.ipac.caltech.edu/level5/Carroll2/Carroll_contents.html">The Cosmological Constant</a>
by Sean M. Carroll, explores thoroughly and generously the ramifications of Λ, Einstein's curious prescience.)</P>
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q3>Matters of Gravity</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Speculations on curved spacetime</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> The "Hubble acceleration," a<sub>H</sub>=c<sup>2</sup>/R<sub>H</sub>=cH<sub>0</sub>=6 x 10<sup>-8</sup>
cm/s<sup>2</sup>, demarcates a critical radial acceleration for galaxies and larger bodies
where the influence of so-called dark matter begins and the dimensionality of spacetime diminishes downward
(the lower limit for acceleration due to inertial quantum mechanics is considerably less). Following the
rotation curve for a given galaxy, one notices the departure from conventional luminous matter dynamics at
approximately the rotational velocity v with radius r so that a<sub>H</sub>=v<sup>2</sup>/r.
This asserts that the radial universal expansion parameters, i. e., the Hubble acceleration,
also affect rotational dynamics. The concept of "dark matter" may arise in large part to a
quantizing of a<sub>H</sub>. If so, this would indicate a characteristic of baryonic matter's
inertia to overcome an "ultraviolet catastrophe" (analogous to the blackbody's). Consequently,
a<sub>H</sub> tends to maintain the radial acceleration, and likewise the orbital velocity,
of galaxies' outlying halos. <i>The effects of "dark matter" on large-scale structures are
predominantly due to compliance with quantized acceleration, given by the ratio
between speed of light squared and the cosmological horizon radius.</i> </P>
<P> Does quantum mechanics hold only at discrete points in spacetime? A unified theory
of physics has been evasive because physicists have considered only a continuum of unification.
In other words, we have attempted to relate all spacetime to all quantum dynamics - that they
are inclusively connected. Quantum measurement, however, may correspond only to discrete points
in spacetime, not a continuum. Wavefunction collapse might occur, say,
at such a singularity unique to a spacetime neighborhood, perhaps corresponding to the intersection
of quantized geodesics. Partial connectiveness between spacetime and quantum measurement could be fundamental to
accommodating the unification of physics.</P>
<P> At extragalactic distances, matter flows (co-moves) with the expanding spacetime,
thus experiencing little disparity in relative motion between the two. At
distances less than the radius of one's neighborhood galaxy, however, spacetime flows past
significantly coherent (by definition, unaffected by the Hubble expansion) masses. If so,
there should be a measurable difference between far field, static vacuum-matter interactions
and near field, kinetic vacuum-matter interactions. Such a quasi-frictional acceleration could
manifest as a vacuum polarization, and may help explain the quantum nature of dark matter.</P>
<P> M* is the characteristic mass of quantum gravity. This Planck mass demarcates
exclusively black hole masses above from those of quanta below. Symmetry between these regions
implies a duality for the two classes of entities. The Planck black hole, with its mass M*,
itself shares and interrelates properties of black holes and quanta. Since inverting the mass scale
around M* compares black holes and quanta one-to-one, <i>a black hole could be a real quantum
"inside-out"</i> - in terms of that scale - <i>and vice versa</i>:</P>
<P align=center>(M<sub>black hole</sub>·M<sub>quantum</sub>)<sup>1/2</sup>=M<sub>Planck</sub>, where M is mass.</P>
<br>
<P> The averaged effective charge of the observable universe acts upon an electron with the
same force as the averaged observable universe mass. In other words, where n=10<sup>81</sup>:</P>
<P align=center>(keq<sub>n-pole</sub>/Gm<sub>e</sub>M<sub>Universe</sub>)~1.</P>
<br>
<P> Bands' vibrations can be counted by their number of twists in
spacetime. For instance, an untwisted band has "vibration" energy zero Planck units.
A typical one-twist Moebius band has "vibration" energy one Planck unit. Two twists
yield "vibration" energy two Planck units, etc. The width of the band
is dualistic to its number of twists, as winding numbers are to
vibration numbers in string theory. A string does not differentiate between
number of twists, and therefore represents a special case of bands. Zero Planck length width, characteristic of strings
and classically forbidden due to its divergent energy, yields its virtual self for less than
a corresponding Planck time. A one Planck length width sustains a "winding" number of energy one Planck
unit. A two Planck length width sustains a "winding" number of energy one-half Planck unit, etc.</P>
<P> Bell correlation between horizons: Do polarization cross-sections from opposite microwave background n-poles
disobey the Bell inequality?</P>
<P> Is it possible for two photons to co-orbit stably, solely by attraction between their
mutual energies (photonium)?</P>
<P> Consider, a problem in quantum geometrodynamics: the evolution of a zero curvature
geodesic manifold in a massless, uniform vacuum potential.</P>
<P> Quantum cryptography can, in principle, be broken by comparing a seemingly random
string of qubits to its gravitational signature.</P>
<P> Rather than be constructed with the familiar "compactified" dimensions on the Planck scale,
superstring extradimensions of fractal values could exist interstitially to those integral-valued.
Fractal space, like compactified space, can represent the resonances underlying physics.</P>
<P> "Looking" back in time to the edge of our universe, one effectively sees a world turned
inside-out. Relative to our position, the "singular" space that made up the first moments of
the cosmos is now inflated across the sky at a distance of around 10<sup>56</sup> cm. The universe
that we experience outward is one of seeming <u>finite</u> density towards the big bang. Physicists learn that to look
out is to look back, but not the obvious, that the retrospective big bang is there magnified overall. The
ultimate spacetime horizon delineates a perspective on the eventual inversion of the universe,
and reveals itself as the initial singularity surrounding us. While looking outward involves looking
back in time to the initial singularity, looking inward similarly recalls energies whose resonances
reach back to the cosmological origin. The quantum mechanical interactions of today emulate the
synthesis of elementary particles in that early universe. Observers relate universal dynamics soon
after the big bang with those in their own subatomic neighborhood; <u>these differ primarily in associated
spacetime curvatures</u>. As particle accelerators explore shorter and shorter wavelengths, they not only
reveal processes of QCD, GUT and Planck regions evolving locally, but affirm likewise their correlation
to the mechanics nearer the big bang. Microverse approaching local singularity maps conformally onto macroverse
reaching toward its (singular) horizon. The boundary of our cosmos was, at least once, zero-dimensional according
to general relativity. The vacuum of present spacetime embodies a dynamic archetype of primordial expansion.
Immediate to us, a virtual big bang is born anew at every moment.</P>
<P> A cosmic isotropy duality hypothesis states: Construct a spherical shell in spacetime. Mass-energy
without and mass-energy within move overall to preserve their common isotropic projection upon the shell,
including position and apparent spin. The Gaussian shell may interrelate observer and object, and
eventually generalize to any 2-D manifold.</P>
<P> Regarding the "black hole information paradox," a black hole's "singularity" is apparently not
point-like, and may be a composite of Planck structures. Information about the "singularity" would manifest
at the black hole horizon as the only variables we may know about a black hole (from the "No Hair" theorem):
mass, spin, and charge (and derivations thereof). The extreme symmetry of the Schwarzschild black hole
transfers coherently (much like an "isotropic laser") such information that is allowed about the singularity.
This Bekensteinian data manifests interior structure at the black hole surface as No Hair quantities or their
gradients, including those of temperature. This suggests that two dimensional mass, spin and charge structure
translates into three dimensional information and vice versa.</P>
<P> Let two equivalent Schwarzschild black hole singularities separate by twice their conventional
horizon radius. When so "contiguous," their gravitational field weakens by opposite attraction the
near its point of greatest symmetry (Lagrange point). Convergent
black holes, then, create paradoxically a significant region of sublight escape velocity.
Many photons which impinge upon (the impact parameter of) such an
<i>individual</i> black hole and become trapped would not when extrapolated to the two hole
situation. The distortion of the mutual horizons' surface area (with its compromised sphericity) indicates an overall increase
in their entropy before evolving toward the final combined singular state.</P>
<P> A gravitational black hole abhors a "naked" mass singularity, but allows it the observable
property of charge, with correspondent electromagnetic field. Similarly, the horizon radius r for
"electronic black holes" (where m<sub>e</sub>c<sup>2</sup>=e<sup>2</sup>/4πε<sub>0</sub>r,
r=2.81 x 10<sup>-13</sup> cm) limits what we may eventually know about the electromagnetic structure of a charged particle.
An electronic black hole (E.B.H.), typical below the scale of a proton, has a particular
charge whose electrical potential magnitude equals its associated rest mass-energy. E.B.H.'s are entities
so gravitationally bound against electric repulsion at a given radius as to be reproduced by the energy
of attempted E-M measurement. <i>As with strong force quark isolation, charge singularity (i. e.,
E.B.H.) observation itself denies direct ("naked") E-M information.</i></P>
<P> Hawking radiation propagates from a black hole's event horizon. Lightlike geodesics
in ordinary spacetime might also demarcate Hawking pair production, with the density of real
particle production greater within their curvature, and lesser without. An arbitrary region
of spacetime actually creates such a density gradient of particles whose Hawking probabilities
would arise from ordinary geodesic curvatures.</P>
<P> A photon is...<br>
...the unit of information
...the standard of measurement
...the equivalent of mass
...the definition of spacetime
...the quantum of uncertainty
...the mediator of charge
...the carrier of light.</P>
<spacer type=vertical size=40>
<P> *A state-of-the-art review in PDF, <a href="http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:astro-ph/9906463">The Cosmic Triangle: Revealing the State of the Universe</a>,
by likewise distinguished authors, deserves exceptional note here for those wondering how
mass, expansion, and curvature combine critically to determine the fate of our cosmos.*</P>
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q4>Relativity's Complex Probability</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Complex spacetime relates to complex probability, the nonconjugated wavefunction squared
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P>1. The Hilbert space observer state vector is orthogonal to that of the object.</P>
<P>2. Relativity mediates observer-object action.</P>
<P>3. The object state vector minus the observer state vector yields the complex relative state vector.</P>
<P>4. The complex relative state vector corresponds to a complex squared <i>nonconjugated</i> wavefunction,
i. e., complex probability.</P>
<P>5. The complex squared <i>conjugated</i> wavefunction determines normalization of observer-object relativity.</P>
<P> (John Archibald Wheeler, but not Stephen Hawking, argued against using the imaginary number "i" as a relativistic reality.)</P>
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q5>Black Hole Internal Supersymmetry</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Statistics of quanta in black holes relies on a supersymmetry there between fermions and bosons
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> Conventional black hole physics has sole extensive measurable quantities charge, mass, and angular momentum
(the "No Hair" theorem). From these, the Hawking temperature, T, can be found. The statistical
distribution n[B. H.] is a function of T, and predicts the occupation of the
hole's internal quantum states with unobservable quanta:</P>
<P align=center>n[B. H.]=n[F. D.]+n[B. E.]=csch(ε/κT)</P>
<P align=center>where it is assumed that T is much greater than the T<sub>F</sub> for this black hole.</P>
<P> The quantum within that normally designates Fermi-Dirac or Bose-Einstein statistics by its
half- or whole-integer spin values has "lost its hair".
<P> Note: Black hole equilibrium above requires the constraints put forth by Stephen Hawking in his seminal paper,
<u>Black Holes and Thermodynamics</u> (Phys Rev D, 15 Jan 1976, p. 191-197).</P>
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q6>Macromechanics</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Cosmological redshift data may indicate discrete resonances of a "spherical box" universe Bessel function
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> Imagine the observable universe to be Minkowskian and modelled with a "spherical box" potential
of event horizon radius R<sub>H</sub>. The solution for the special relativistic
wavefunction in this case is from a spherical Bessel function of order zero. The significant,
second harmonic of this function inscribes a "null" shell of radius .5R<sub>H</sub>. That solution
predicts a minimum for quasar sightings near redshift z=.732, <u>if</u> the universe were indeed
Minkowskian. Chaotic motion would rarefy classical objects at this radius, and the wavefunction
there would equal zero, attenuating quanta. Such equilibrium could have established and maintained
large scale structures since the nebulous early universe. Compare these conclusions with those
of Prof. William Tifft of Steward Observatory in Arizona. His interpreted "resonances" of observed
redshift data indicate quantum-like multiplicities of matter distribution in the universe, where
H<sub>0</sub> (the Hubble constant) acts like Planck's constant.</P>
<P> Research the <a href="http://www.sdss.org/dr5/products/spectra/getspectra.html">Sloan Digital Sky Survey</a>
for dilution of populations near redshift z=.732.
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q7>Tunneling from beyond the Event Horizon</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Blueshifted sectors outside the universal event horizon may tunnel information across classically forbidden spacetime
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> Alan Guth, in his original paper, <u>Inflationary universe: A possible solution to the
horizon and flatness problems</u> (Phys Rev D, 15 Jan 1981, p. 347-355), discusses the magnitude of
inflation to be at least 10<sup>83</sup> volumes greater than that of our observable universe.
Thus we may physically realize continuous spacetime a linear factor 5 x 10<sup>27</sup> beyond the universal
event horizon radius, R<sub>H</sub>=1.4 x 10<sup>28</sup> cm. We can actually experience tunneling from this
external spacetime where its velocity relative to us is less than the speed of light.</P>
<P> The universe's event horizon is defined at the radius where spacetime's recessional velocity from
the observer equals light speed. What if, beyond this horizon, there exists a region of space whose
peculiar velocity (that deviating from the global velocity-distance relationship) toward us causes a
blueshift relative to the horizon? Its velocity relative to us might be subluminal, tunneling
information to us from outside our classically observable universe. By virtue of its
relativistic potential, this tunneling allows us to see through a classically forbidden zone through a
wormhole. There may be extensive, radically peculiar regions of such space that from our
perspective would create, for instance, fluctuations in the microwave background radiation or gravitational lensing.</P>
<P> Consider Planck black holes seeded before inflation with its greatly superluminal expansion,
then reinforced toward gravitational collapse in the GUT era. There could be multiple
concentric event horizons (cycles) alternating outward over many multiples of "c." Our initial horizon
is defined by where the conventional-time cosmos first expands away from us at the speed
of light. Beyond that a mirror tachyonic spacetime (c<v<2c) reverses the direction of time,
as does the third region of spacetime (2c<v<3c) over that of the second, establishing forward
time once more. This continues for n cycles, where n~10<sup>55</sup>/10<sup>28</sup>, the ratio between
the ultimate inflationary and linearly expanding universe radii. Regions of alternating expansion and
compression manifested relative to each other. The potential for forming microscopic black holes developed within those
limits. These holes were characterized by the time period of inflation (~10<sup>-35</sup> seconds), whose
growth factor perhaps exceeded 10<sup>28</sup>. Inflation's superluminality conveyed initial Planck wavelengths
on to further condensation as lower-energy resonances. The holes in turn seeded geons (stable, charged
microminiature black holes) which likewise generated more familiar particles.</P>
<P> Inflation is also accessible here and now in the high energy (10<sup>-24</sup> cm) region,
approximately the GUT scale. Consider communications detouring through there superluminally relative
to broadcasters/receivers. By adjusting signal phase, tunneling into this higher "harmonic"
subatomically might achieve faster-than-light communication around the associated frequency
10<sup>35</sup> sec<sup>-1</sup>.</P>
<P></P>
</td>
</tr>
</table>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q8>Symmetry and the Superuniverse</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Cosmological parametric asymmetry suggests statistically the existence of a maximally symmetric superuniverse
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> By considering the asymmetry of our observable universe, a maximally symmetric "superuniverse" may be realized.</P>
<P> J<sub>max</sub>=maximum known angular momentum of any structure in the observed universe.</P>
<P>     =10<sup>75</sup> erg-sec for a giant spiral galaxy (Borne, NASA).</P>
<P>     =σ<s>h</s></P>
<P>σ=10<sup>102</sup>=number of angular momentum quanta equivalent to J<sub>max</sub>.</P>
<P> Angular momentum can be right or left handed. Units of <s>h</s> combine constructively
or destructively, so σ may be treated statistically as a deviation from a mean.</P>
<P> For large N, N=σ<sup>2</sup>=10<sup>204</sup>, where N is the Gaussian
total units of <s>h</s> within a maximally symmetric superuniverse,
and σ their standard deviation which assumes the maximum known cosmological angular momentum
asymmetry.</P>
<P> J<sub>super(avg)</sub>=N<s>h</s>=10<sup>177</sup> erg-sec, the internal
angular momentum of a superuniverse corresponding to N. This value for N signifies the superuniverse as maximal, statistically the
most symmetric and random derivable (like an all-encompassing Schwarzchild black hole) from our known universe data.</P>
<P> Similar symmetry arguments apply to charge and matter/antimatter dichotomies.</P>
</td>
</tr>
</table>
<br>
<br>
<br>
<P><font color="0000FF"><font size="6"><P align=center><a name=q9>Creed</a></font size></P>
<P> <P align=center><font size="4">by Loren Booda</font size></font color></P>
<table align=center>
<tr>
<td>
<i>
If man can prove that science is knowledge,<br>
That wisdom reigns absolute in his college,<br>
The professor, who utters a vague "Cogito..."<br>
Is unsure of himself but desires to know.<br>
Know of the limit of science he loves,<br>
Know the unbounded, which his heart proves,<br>
Know of uncertainty, shadows in fear,<br>
To embrace his beliefs with the truth he holds dear.</i>
</td></tr></table>
<br>
<br>
<br>
<br>
<table>
<tr><td width="20%" rowspan=2></td>
<td width="80%"><font color="0000FF"><b><font size="6"><a name=q10>The "Booda Theorem"</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
<br>
<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Cubic polynomials relate to their derivatives' solutions
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> Prove: For a cubic polynomial with local maximum and minimum, the slope of the line connecting them is 2/3 of the slope at the inflection point.</P>
<P> The third-degree coefficient can be considered arbitrary, since it determines solely the overall scale of the function.</P>
<P> p[x]=x<sup>3</sup>+bx<sup>2</sup>+cx+d</P>
<P> p'[x]=3x<sup>2</sup>+2bx+c</P>
<P> p''[x]=6x+2b</P>
<P> Inflection point: p''[x]=0</P>
<P> 6x+2b=0</P>
<P> x=-b/3</P>
<P> Slope of inflection point: p'[-b/3]=3(-b/3)<sup>2</sup>+2b(-b/3)+c=b<sup>2</sup>/3-2b<sup>2</sup>/3+c=<font color="660011">-b<sup>2</sup>/3+c</font color></P>
<P> Relative maximum and minimum of <spacer type=horizontal size=5>p[x]: p'[x]=3x<sup>2</sup>+2bx+c=0</P>
<P> For cubic polynomials with a local maximum and minimum, b<sup>2</sup>>3c: x, <u>x</u>=(-b±(b<sup>2</sup>-3c)<sup>1/2</sup>)/3 for those extrema.</P>
<P> The slope between them is: δy/δx,</P>
<P> and δy=x<sup>3</sup>+bx<sup>2</sup>+cx+d-(<u>x</u><sup>3</sup>+b<u>x</u><sup>2</sup>+c<u>x</u>+d)=x<sup>3</sup>-<u>x</u><sup>3</sup>+b(x<sup>2</sup>-<u>x</u><sup>2</sup>)+c(x-<u>x</u>)=(x-<u>x</u>)(x<sup>2</sup>+x<u>x</u>+<u>x</u><sup>2</sup>+bx+b<u>x</u>+c) ,</P>
<P> δx=(x-<u>x</u>), so δy/δx=x<sup>2</sup>+x<u>x</u>+<u>x</u><sup>2</sup>+bx+b<u>x</u>+c .</P>
<P> Substituting in the above quadratic solutions for x,<u>x</u>:</P>
<P> x<sup>2</sup>=b<sup>2</sup>/9-2b(b<sup>2</sup>-3c)<sup>1/2</sup>/3+(b<sup>2</sup>-3c)/9 and <u>x</u><sup>2</sup>=b<sup>2</sup>/9+2b(b<sup>2</sup>-3c)<sup>1/2</sup>/3+(b<sup>2</sup>-3c)/9 ,</P>
<P> x<sup>2</sup>+<u>x</u><sup>2</sup>=(4b<sup>2</sup>-6c)/9 and x<u>x</u>=(b<sup>2</sup>-(b<sup>2</sup>-3c))/9=3c/9 and b(x+<u>x</u>)=b(-2b/3)=-6b<sup>2</sup>/9 .</P>
<P> The slope between maximum and minimum is:</P>
<P> x<sup>2</sup>+<u>x</u><sup>2</sup>+x<u>x</u>+bx+b<u>x</u>+c=(4b<sup>2</sup>-6c)/9+3c/9-6b<sup>2</sup>/9+9c/9=<font color="660011">2/3(-b<sup>2</sup>/3+c)</font color>     Q.E.D.</P>
<P> <font color="0000FF">This theorem (suggested by Dewey Allen, then of Arlington, Virginia, under the tutelage of Wilbur Mountain) was first proved by the author in 1976, when he was 16.</font color></P>
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<td width="80%"><font color="0000FF"><b><font size="6"><a name=q11>Fine-Structure Constant Numerology</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
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<br>
<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>The fine-structure constant reveals itself numerically as a measure of inflation between quantum and cosmos
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> The reciprocal fine-structure constant, 1/α, is dimensionless and equal to
<s>h</s>c/e<sup>2</sup>=137.0360... . α represents the relative strength of the
electromagnetic vs the strong force, and is one of the most accurately measured parameters in physics.</P>
<P> My conjecture,</P>
<P align=center>1/α=log<sub>2</sub>(R<sub>universe</sub>/r<sub>proton</sub>)</P>
<P align=center>or,</P>
<P align=center>2<sup>1/α</sup>=R<sub>universe</sub>/r<sub>proton</sub></P>
<P> gives R<sub>universe</sub>=1.3 ± 0.1 x 10<sup>28</sup> cm (the Hubble constant
equivalent of 70 ± 1 km sec<sup>-1</sup> Mpc<sup>-1</sup>, in very close agreement
with supernovae data), for a commonly accepted value of r<sub>proton</sub>=0.85 ± 0.05 x
10<sup>-13</sup> cm, the RMS proton radius. The observable universe describes a cosmological
black hole, while the proton is sometimes referred to as an "electronic black hole."</P>
<P> Of all the problems in "Big Numbers" (universal numerology), few have a relation so
simple yet exact.</P>
<P> <i>Inflation may account for the exponential relation between the fine-structure constant and
the universal horizon radius/proton radius ratio, where </i>α<i> is a measure for the
magnitude of inflation</i>.</P>
<P> See <a href="http://physicsweb.org/article/news/5/8/11">"Further Evidence for Cosmological
Evolution of the Fine Structure Constant"</a>.</P>
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<td width="80%"><font color="0000FF"><b><font size="6"><a name=q12>Neurophysiological Uncertainty</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
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<i>A classical analog to quantum uncertainty develops for nervous electrical discharges
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P> The classical electrical activity of a nerve cell suggests that nervous impulses are to the nervous system
what electrons are to a physical system: "quanta" ruled by a principle of uncertainty. Analogous to Planck's constant,
h, a "neurophysiological uncertainty," U<sub>ψ</sub>, arises from the basic parameters of
nervous activity.</P>
<P> To find U<sub>ψ</sub>, one calculates the minimum product of energy and time for the
neuronal action potential across the synapse. This impulse lasts 1 millisecond, with an average potential change of
40 millivolts. Assuming the minimum one electron involved, we have:</P>
<P> <P align=center><font color="660011">U<sub>ψ</sub>=.001[s]· .04[V]· 1[e]· 1.6 x
10<sup>-19</sup>[J/eV] = 6.4 x 10<sup>-24</sup>[J-s] = 10<sup>10</sup>h.</font color></P>
<P> The factor 10<sup>10</sup> approximately equals the number of active neurons in the human brain. The neuron
appears to be an electron integrator and a brain differentiator; namely, the action of the neuron is to that of the
brain as the action of the electron is to that of the neuron. Together, observer uncertainty
U<sub>ψ</sub>, and object uncertainty h, evaluate numerically a correspondence principle.
The neurophysiological quantum number defines as
N<sub>ψ</sub>=U<sub>ψ</sub>/h=10<sup>10</sup>.</P>
<br>
<P> The action potential is described in the solution to the differential equation derived by the neurophysiologists A. L. Hodgkin and A. F. Huxley. It may replace the potential V[x,t] in the quantum mechanical Schrödinger equation</P>
<P> <P align=center><font color="660011">ih(∂/∂t)ψ[x,t] = ((-<s>h</s><sup>2</sup>/2m<sub>e</sub>)∂<sup>2</sup>/∂x<sup>2</sup>+V[x,t])ψ[x,t]</font color></P>
<P> to solve the probability wavefunction ψ[x,t]. ψ[x,t] describes the various states that an electron occupies in the synapse.</P>
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<td width="80%"><font color="0000FF"><b><font size="6"><a name=q13>Configuration Complementarity</a></font size></b></font color></td></tr>
<tr><td width="80%"><font color="0000FF"><b><font size="4">by Loren Booda</font size></b></font color><br><br></td></tr>
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<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Observer and object as quantum complements affect physical objectivity and uncertainty
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P>1. Complementarity is the compare and contrast of the physical world. In his book "At Home in the
Universe," John Archibald Wheeler paraphrased Bohr's definition of complementarity (page 18): "The use of certain
concepts in the description of nature automatically excludes the use of other concepts, which however,
in another connection are equally necessary for the description of this phenomenon."</P>
<P>Quantum measurement yields an object state whose complement traditionally
remains in complex Hilbert space.
<i>Of the many quantum interpretations, none seems to suggest that the process of complementarity
may unfold exclusively in relatively real, accessible phase space of an "observer-object."
Here the observer provides herself as the locus for the complement to the quantum
object.</i> A compatible measurement retains the observer-object status quo, while a
complementary measurement reverses its phase.
<P>We measure directly the momentum of the quantum object, only to react with an
uncertainty of displacement upon our immediate personal perception. As momentum complements position, an
observer's state complements that of her quantum object. Upon measuring directly a displacement of Planck
length L*, the measurer would receive a momentum reaction
equal to h/L*, or 4,000,000 gm-cm/sec, beyond the kick of a mule. Normally, though, the observer does not
appreciate the physical sensation of complementarity.</P>
<P>The observer has occupied a participatory role with quantum mechanics and cosmology. The observer-object
classicizes traditionally distinct quantum entities by treating them much like an observer within the living cosmos
she occupies. One might think that by including the observer, quantum reckoning would become
subjective. However, <i><u>separating</u> ourselves from our own observation actually <u>subjectifies</u>
the measurement.</i> The set of observer-objects is indeed the intermediary between quanta and cosmos.</P>
<P>Simultaneity can hold in either observer-object phase space or relative spacetime, but not both
for any given event. Observer-object states trade an accustomed uncertainty of action for that of
interval. So, <i>if the observer-object system is quantum mechanically objective and quasi-classical,
it enjoys this for a subjective and uncertain relativity</i>.</P>
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<tr><td width="20%" bgcolor="0000FF" valign=top><font size="3"><font color=Silver>
<i>Observer memory complexity complements observed macroscopic structure, and thus retains a "reverse
multiverse"
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P>2. "We exist through observation." An observer is one who finds
patterns from seeming disorder - even entropy describes a definable
process. Entropy depends not only on the states of a configuration,
but also on the network of interconnections (entanglement) between
states. Anentropic by nature of retrospection, this latter "pattern
memory" potentially surpasses entropy's information exponentially
in magnitude. Configurations correlate the entropic effect of
measurement upon the environment with the incorporation of information
in the observer's pattern memory. Our ability to predict, perceive, and remember
patterns evolves from the anentropic components of observation.</P>
<P> The observer's constituent pattern memory is juxtaposed against her
sequence of observations. Our physical system is one that maximizes the
number of interconnections overall. The classical measurer's ignorance of
physical future contrasts her own lookback, interconnective history - a
"reverse multiverse" pattern memory. Just as a well-ordered closed system
of states is bound statistically to convert to disorder overall, it likewise generates
locally ordered mnemonic networks. Again, a "random" process has as its
complement anentropic memory.</P>
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<i>Quantum mechanics corresponds to classical indistinguishability
</i></font color></font size></td>
<td width="80%" bgcolor="FFBB22">
<P>3. What is the largest set of quantum numbers in common between different classical
configurations in the observable universe? Beyond a certain complexity, do statistics
require cosmologically only unique forms of physical entities to exist? I.e.,
is the correspondence principle limited by the distinguishability of classical configurations?</P>
<P> For instance, any two hydrogen atoms have a relatively high probability of
sharing most quantum numbers, while organic molecules are less likely to. It follows
that there exists an upper bound of complexity within our finite
universe where at most two macroscopic configurations of maximally identical
quantum numbers occur. Consider sets of quantum numbers approaching this cosmic
limit as enumerating the symmetry of the correspondence principle - to demarcate
the quantum from the classical.</P>
<P> <i>A quantum equation converts into