Phase Reality!

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: The Second Superstring Revolution.) 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. P-duality extends these non-Newtonian theories and correlates nonlocal events with those local. As universally experienced, P-duality manifests a finite center to all observers, and connects our immediate to indirect experiences. P-duality simply recognizes phase in physics as having an analogous, reciprocal component. The essential challenge before us is to create a responsible and ethical philosophy of physics, testable yet not malicious.

String theory, which innovated T-duality, substitutes for "standard" quantum gravity parameters a wave compactification among hyperspatial dimensions. Consider that P-duality doubles the dimensionality available to the accustomed quantum wavefunction by establishing an elliptical symmetry between conventional phase space and its conformal dynamic inverse. ("No physical theory of local hidden variables [could] reproduce all of the predictions of quantum mechanics.") 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 spacetime by reciprocity within the local Planck radius.

Faster-than-light (non-local) actions a', of quantum mechanics, can be modeled simply with local actions a turned "inside-out" about the singular surface of radius h, the value of Planck's constant.

That is, h²=aa'.

Antigravitational (dark energy) spacetime intervals Δs', of general relativity, can be modeled simply with local spacetime intervals Δs turned "inside-out" about the singular surface of radius L*, the value of the Planck length.

That is, (L*)²=ΔsΔs'.

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^-. Such wavefunctions differ from their conventional counterparts, φ, by the interchange of their variable actions S with h (Planck's constant). I. e.,

φ[S, h] ↔ φ^-[h, S]. (Square brackets indicate a function in general unless otherwise noted.)

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: 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.

The inverse time independent free-particle wavefunction,

φ^-[r']=B'·exp(-2πi(h/r'p')),

generates action eigennumbers reciprocal to those of the conventional time independent free-particle wavefunction,

φ[r]=B·exp(-2πi(rp/h)),

and represents virtual, rather than real states. Action eigenvalues for the conventional wavefunction are

S_N[φ[r]]=r_Np_N=hN/2,

derived through Im[φ[r]]=0, from the condition of arbitrary phase. Likewise, action eigenvalues for the inverse wavefunction are

S_N^-[φ^-[r']]=(r'_N^-)p'_N^-=2h/N^-.

N and N^- are nonzero integers. Action eigennumbers, n_N and n_N^-, are simply action eigenvalues divided by h; thus, n_N=N/2, and n_N^-=2/N^-. Because of the n_N=n_N^- reciprocal symmetry, exclusively finite action entails there be no classically singular, zero-valued spin. Action eigenvalues of magnitude less than h/2 are defined as virtual, those between or including h/2 and 2h as mixed, and those greater than 2h as real.

Both wavefunctions φ[r] and φ^-[r'] share exclusively the action eigenvalues satisfying N/2=2/N^-; that is, corresponding to spins 1/2, 1 and 2 -- those of most familiar particles: fermions, photons, and gravitons, respectively. Photons, having spin one, manifest as maximally symmetric (i. e., of dualistic eigennumber ground state) between real and virtual phase space. The principle of least action localizes particles near S =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,

φ[r]·φ^-[r']=B·B'·exp(-2πi(rp/h+h/r'p'))=B·B'·exp(-2πi(n_N+1/n_N^-)),

The one-dimensional time dependent virtual Schrödinger equation,

(-(2πh)²/2m)(∂²[1/x']/(∂[φ^-[1/x',1/t']]²)) + V[1/x',1/t'](1/φ^-[1/x',1/t'])

=i2πh(∂[1/t']/(∂φ^-[1/x',1/t']))

describes the mechanics of reciprocal phase space through its corresponding time dependent virtual wavefunction solution

φ^-[1/x',1/t']=A'·exp(2πih(1/x'(p_x)'+1/E't'))

with its inverted units.

For a free particle in one dimension: conventional-real ↔ inverse-virtual

Wavefunctions (φ):

φ=C·exp(2πixp_x/h) ↔ φ'=C'·exp(2πih/x'(p_x')')

Operators (*):

(p_x)*=(-ih/2π)(∂/∂x) ↔ (1/(p_x')')*=(-i/2πh)(∂/∂[1/x'])

(x*)δ(x-`x)=(`x)δ(x-`x) ↔ ((1/x')*)δ(1/x'-1/`x')=(1/`x')δ(1/x'-1/`x')

where δ represents the Kronecker delta function

Commutators:

[x*, p*]=ih/2π ↔ [(1/x')*, (1/(p_x')')*]=i/2πh

Hamiltonians:

H*=((p_x)*)²/2m=-((h/2π)²/2m)∂²/∂x² ↔ (1/H')*=(1/((p_x')')*)²2m=-((2πh)^-22m)∂²/∂[1/x']²

Spectra:

E=nhν ↔ E'=hν'/n'

Photons passing "singly" through a double-slit apparatus develop an interference pattern at its screen similar to that projected there by "simultaneous" pairs of photons. Reciprocal phase space confers upon an individual photon the information needed to interfere with mutually nested conventional phase space. The projected wave pattern is essentially the same whether one photon at a time interacts with the slits, or an astronomical number are thus randomly effected. Entangled from a common cosmological original event, these phase spaces together enable single particles to "self interfere" coherently where N/2=2/N^-. In other words, quanta mediate 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.

An EPR experiment, starting with a singlet "zero" spin state, needs considered first the possible vacuum effects in that 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. Entanglement thus entails virtual wavefunctions interfering with actual ones: reciprocally nested phase spaces maintain probabilistic causality without necessitating superluminal signaling, "hidden" variables, or myriad obsolete universes.

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. So, 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 three-dimensional phase space mapped with associated spin vectors, requires topologically a four-dimensional zero-spin manifold within, whose boundary may partially define a spacetime horizon. Moreover, six compactified dimensions coupled reciprocally at Planck constant action to six outer dimensions could satisfy, within six apparent dimensions, the criteria for virtual phase space's nonzero spin.

Picture an ellipsoid in phase space given by

(rM*c)²+(pL*)²=h²

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

(rM*c)²+(pL*)²≥(h/2)²

without, and virtual/mixed phase space

(r'M*c)²+(p'L*)²≤(2h)²

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 locally all values: r≥L*/2 ↔ r'≤ 2L* and p≥M*c/2 ↔ p'≤ 2M*c. Their symmetry enables quantum measurement to include submicroscopically the entire range of macroscopic phase space.

Wavefunction "collapse" now manifests Born probabilities in an accessible, physical transformation, having lost its discontinuous, immaterial character to the inverse wavefunction φ^-[r], a local information equivalent to φ[r]. 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.

The Schrödinger cat thought experiment employs an enclosure surrounding a radioactive source, a Geiger counter, a vial of poison and a cat. The setup has the potential to measure the decay of the source, break the vial and thus kill the cat. To an observer outside, the cat would exist as a quantum superposition of dead cat/live cat -- whereupon the box is opened, revealing the near absolute mortality or vitality of the feline. The Copenhagen interpretation posits a wavefunction collapse when this box is opened, causing an immediate mathematical projection between complex states within the cavity and its wall lattice, realizing one macroscopic result. More materially, Bragg diffraction at the container demonstrates that the cat is never completely enclosed, thus allowing information (e.g. radioactive decay, signaling or brain waves) from within the closed box to be observed on its outside. Through the entire spectrum, any enclosure exhibiting Bragg diffraction would reveal its interior states. In general, radiation escaping the container interferes first with the quantum nature of its walls, determining that inner state(s) manifest either a live or a dead cat -- to an astronomical certainty.

Wavefunction reciprocal action eigennumbers may provide gravitational, electric or magnetic "singularities" continuity, since it can transform a real Dirac delta function into a virtual, smooth function. Upon inversion, Heisenberg's uncertainty principle -- as well as entropy -- reverses its inequality (with action eigenvalue limit of N=1 now that of N^-=1)

from ΔrΔp≥h/2 to Δr'Δp'≤ 2h.

Heisenberg uncertainty, now infinite upon inversion, overlies macroscopic actions included in the 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^- states to represent quasi-continuously their unbounded self-energy. Summing every momentum state of inverse phase space gives a practically infinite internal energy of compactification within the ellipsiod of uncertainty.

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.) outside 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-duality, sharing mathematical properties initially attributed to strings by T-duality, derives phase-dependent action through traditional and reciprocal wavefunction symmetries. It also guides 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 dual 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 (see the next article). Quantum interpretation oversees two complementary, aboriginally entangled and reciprocally nested phase spaces. Interpreting the philosophy of observer as participant indicates the quantum realm as dualistic yet unified.

Emeritus professor Frank Firk of Yale University, my alma mater, offers "Physics for Free", a challenging and disciplined tutorial for those who wish a glimpse of Ivy education.

The "semiclassical metric tensor," g^μν[r±δr], 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

δr=g^μν[r] Δx_μ (n_N)^1/2L*/R_H,

where n_N are real action eigennumbers (see the previous "P-Duality: Quantum Mechanics Inside-Out"), i. e. relative uncertainties, and δr is the action-equivalent radius of curvature. L*, the Planck length -- typical of Planck scale physics -- and the Hubble radius R_H approximate the extrema of measurable spacetime curvature. R_H/L*=(n_max)^1/2 represents the characteristic quantum number for spacetime. The absolute range of action eigennumbers is n_max -- on the order of the dimensionless cosmological constant, Λ. A real action S[r]=n_Nh≥h/2 implies a radial vector correction of conventional spacetime, and a virtual action S[r']=(n_N^-)h≤ 2h infers a radial vector correction of inverse spacetime; intermediate shared actions denote mixed states, of which S[r]=S[r']=h represents a shared, zero curvature event horizon.

For a giant spiral galaxy, |δr|=10¹³cm, since (n_N)^1/2=(10⁷⁵erg-sec/10^-27erg-sec)^1/2=10⁵¹, |Δx_μ|=10²³cm and L*/R_H=10^-61.

The n_N^- establish virtual states' "P-duality" formulation for the semiclassical metric tensor, where

δr'=g^μν[r'] Δx_μ (n_N^-)^1/2R_H/L*,

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_H manifests physics much as conventional spacetime approaching L*, only modified by curvatures proportional to their radii. Increasing action involves quantum resonances for both spacetime curvature outward from L* and inverse spacetime inward from R_H. Conventional spacetime reciprocated through L* effectively obeys the naked singularity "dress code" by shielding an otherwise discernable center beneath the Planck horizon. (A concerted effort, Cambridge Relativity, 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.)

The n_N and n_N^- comprise eigennumbers, solutions to the time independent free-particle wavefunction of arbitrary phase,

Im[φ[r]]=B·sin(-2πn[r])=B·sin(-2πrp/h)=B·sin(-2πS[r]/S*)=0,

Im[φ^-[r']]=B'·sin(-2πn^-[r'])=B'·sin(-2πh/r'p')=B'·sin(-2πS*/S[r'])=0,

and the n_N^- ↔ 1/n_N for nonzero n_N^- and n_N. g^μν[r±δr] fluctuates discretely in real space from g^μν[0] to g^μν[2r], whereas the vectors x_μ and x_ν vary continuously. The metric g^μν[0] refers to counteracting radial spacetime vectors (as on an asymptotically "naked" and extreme charged nonrotating black hole). That of g^μν[2r] refers to maximally reinforcing radial spacetime vectors (as toward the equator of an extreme Kerr geometry).

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 ~10¹²² fold from the predicted Planck density to the empirical critical universal density. The linear black hole mass density, c²/G=10²⁸g/cm, moderates spacetime geometry from the foaming Planck region to the outwardly accelerating dark energy.

Spacetime bounded by concentric event horizons can repeat reciprocally. Such transforms as [L*, R_H] ↔ [R_H, L*] alter sign and magnitude. There three compactified (embedded) dimensions of L* ≥ r ≥ (L*)²/R_H themselves directly embed two further compactified dimensions of (L*)²/R_H ≥ r ≥ (L*)³/(R_H)² which directly embed even further one dimension of (L*)³/(R_H)² ≥ r ≥ (L*)⁴/(R_H)³. Four+three+two+one=ten dimensions of holographic space(time) have greater symmetry than those of current string theory, with its four spacetime dimensions and six compactified dimensions.

Sequence of measurement, more appropriately than objectivity alone, provides a 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: measurement, and statistical continuity between measurements. Action eigenvalues represent the possible sequential permutations -- Planck steps, (n_N)^1/2L*/R_H -- 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 underdetermination within quantum mechanics.

The dual metrics (modified by the action-equivalent radii of curvature) symmetrize isotropic to the Planck and Hubble surfaces, correlating their conventional and inverted spacetimes. The solutions to cosmological constant, isotropy, flatness, magnetic monopole, horizon and dark flow problems -- tentatively addressed by the inflationary model -- may otherwise arise from the bivalent spacetimes, aboriginally radiating apart from a common, correspondent and singular geometrodynamic shell of radius L*. Submicroscopic, reciprocal and virtual geodesics represent interiorly those macroscopic, proportional and real exteriorly. Both inflation and inversion entangle the submicroscopic with the macroscopic, and which can be applicable to most inflationary solutions. In brief, the quantum cosmos could reflect inwardly those phenomena which dark mass-energy would claim outwardly.

(A beautiful account, The Cosmological Constant by Sean M. Carroll, explores thoroughly and generously the ramifications of Λ, Einstein's curious prescience.)

The "Hubble acceleration," a_H=c²/R_H=cH₀=6 x 10^-8 cm/s², demarcates a critical radial dynamic for galaxies' outlying halos and larger bodies, where the influence of so-called dark matter begins and spacetime expands (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 motion at approximately the rotational velocity v with radius r so that a_H=v²/r. This asserts that a radial universal expansion parameter -- i. e. the Hubble acceleration -- also affects rotational dynamics. The concept of "dark matter" may arise in large part from a quantizing of a_H. If so, this would indicate a characteristic of baryonic matter's inertia to overcome an "ultraviolet catastrophe" (analogous to the blackbody's). The effects of "dark matter" on large-scale structures may be predominantly due to compliance with quantized acceleration correspondence, given by the ratio between the speed of light squared and the cosmological horizon radius.

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, can correspond only to discrete points in spacetime, versus a continuum. Wavefunction collapse might occur, say, at such a singularity unique to a spacetime neighborhood, perhaps corresponding to the intersection of geodesics. Partial connectiveness between spacetime and quantum measurement could be fundamental to accommodating the unification of physics.

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 relative nature of dark energy.

M* is the characteristic mass of quantum gravity. This Planck mass demarcates exclusively quantum dimensions from those of black hole masses. 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, a black hole could be a real quantum "inside-out" - in terms of that scale - and vice versa:

(M_{black hole}·M_quantum)^1/2=M_Planck, where M is mass.

The averaged electric n-pole moment of the observable universe acts upon an electron with the same force as the averaged observable universe mass. In other words, where n=10⁸¹:

(keq_n-pole/Gm_eM_Universe)~1.

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.

Bell correlation between horizons: Do polarization cross-sections from opposite microwave background n-poles disobey the Bell inequality?

Quantum cryptography can, in principle, be broken by comparing a seemingly random string of qubits to its gravitational signature.

Is it possible for two photons to co-orbit stably, solely by attraction between their mutual energies (photonium)?

Is a universe, or black hole -- which can collapse to a singularity itself -- composed soley of singularities?

There would be singularities at the intersections of universes.

Was the big bang singularity "local" or "global"?

Does any spacetime region enclose a comparatively minimum energy density?

Consider, a problem in quantum geometrodynamics: the evolution of a zero curvature geodesic manifold in a massless, uniform vacuum potential.

Does general relativity, exclusively with photons in vacuo include a butterfly effect?

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.

Say the big bang event originated from the linear imaginary timeline interval (i∞, iT*], which was inverted within the interval [iT*, 0] and -- as continuous time through the spacetime singularity -- to real timeline inversion within the interval [0, T*] onto the linear timeline interval [T*, ∞).

"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 at least 10⁵⁶ cm. The universe that we experience outward is one of seeming finite 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, 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; these differ primarily in associated spacetime curvatures. 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 event 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.

Regarding the "black hole information paradox," a black hole's "singularity" is apparently 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 holographically into three dimensional information and vice versa.

Let two equivalent Schwarzschild black hole singularities separate by twice their conventional horizon radius. When so "contiguous," their gravitational field weakens by opposite attraction at the point of greatest symmetry (Lagrange point). Convergent black holes, then, create paradoxically a significant region of sublight escape velocity. Photons which impinge upon (the critical impact parameter of) such an individual black hole to become trapped would not when extrapolated to the two hole situation. Aside: how do black hole singularities relate to Lagrange points in spacetime?

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_ec²=e²/4πε₀r, r=2.81 x 10^-13 cm) limits what we may eventually know about the electromagnetic structure of a charged particle. An electronic black hole (E.B.H.), typical near 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. As with strong force quark color confinement, charge singularity (i. e., E.B.H.) observation itself denies direct ("naked") E-M information.

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.

A cosmic isotropy duality hypothesis states: construct a closed surface 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. This Gaussian law interrelates observer and object, applies to inverse-square radiation (that is, not finite-ranged) and generalizes to any spacetime 2-D manifold. The uncertainty of the radiation impinging upon the shell agrees with the quantal change in entropy across the varying manifold. This can correspond to a black hole surface area.

Comets occuring between some solar systems could have elliptical interstellar orbits, like a hypothetical trajectory with the Sun and Proxima Centauri as foci.

A photon is... ...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.

1. The Hilbert space observer state vector is orthogonal to that of the object.

2. Relativity mediates observer-object action.

3. The object state vector minus the observer state vector yields the complex relative state vector.

4. The complex relative state vector corresponds to a complex squared nonconjugated wavefunction, i. e., complex probability.

5. The complex squared conjugated wavefunction determines normalization of observer-object relativity.

(John Archibald Wheeler, but not Stephen Hawking, argued against using the imaginary number i as a relativistic reality.)

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:

n[B. H.]=n[F. D.]+n[B. E.]=csch(ε/κT)

where it is assumed that T is much greater than the T_F for this black hole.

The quantum within that normally designates Fermi-Dirac or Bose-Einstein statistics by its half- or whole-integer spin values has "lost its hair".

Note: Black hole equilibrium above requires the constraints put forth by Stephen Hawking in his seminal paper, Black Holes and Thermodynamics (Phys Rev D, 15 Jan 1976, p. 191-197).

Imagine the observable universe to be Minkowskian and modelled with a "spherical box" potential of event horizon radius R_H. 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_H. That solution predicts a minimum for quasar sightings near redshift z=.732, if 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.

Research the Sloan Digital Sky Survey for dilution of populations near redshift z=.732.

Alan Guth, in his original paper, Inflationary universe: A possible solution to the horizon and flatness problems (Phys Rev D, 15 Jan 1981, p. 347-355), discusses the magnitude of inflation to be at least 10⁸³ volumes greater than that of our observable universe. Thus we may physically realize continuous spacetime a linear factor more than 5 x 10²⁷ beyond the Hubble radius, R_H=1.4 x 10²⁸ cm. We can actually experience tunneling from this external spacetime where its velocity relative to us is less than the speed of light.

The universe's event horizon is defined by 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. There may be extensive, radically peculiar regions of such spacetime that from our perspective might create fluctuations in background radiation or gravitational lensing.

Consider Planck black holes to be at first seeded and later collapsed around the GUT era, inflation-bounded with its greatly superluminal expansion. There could be multiple concentric event horizons (cycles) alternating outward over many multiples of "c." The 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>10⁵⁵/10²⁸, the ratio between the ultimate inflationary and primarily localized universe radii. Regions of alternating expansion and compression would have 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^-35 seconds), whose radial growth factor (mentioned above) perhaps exceeded 10²⁸. Inflation's spacetime superluminality conveyed initial Planck wavelengths on to further attenuation as lower-energy resonances. These "geons" (Wheeler's stable, charged microminiature black holes) would in turn generate more familiar particles.

Inflation is also accessible here and now in the high energy (10^-24 cm) region, approximately the GUT scale. Consider communications guided through there, superluminal relative to broadcasters/receivers. By adjusting the signal phase, tunneling into this higher harmonic subatomically might achieve relatively "faster-than-light" communication around the associated frequency 10³⁵ sec^-1.

By considering the asymmetry of our observable universe, a maximally symmetric "superuniverse" may be realized.

J_max=the maximum known angular momentum of any structure in the observed universe.

    =10⁷⁵ erg-sec for a giant spiral galaxy (Borne, NASA).

    =σh

σ=10¹⁰²=the minimum number of angular momentum quanta equivalent to J_max.

Angular momentum can be right or left handed. Units of h combine constructively or destructively, so σ may be treated statistically as a deviation from a mean.

For large N, N=σ²=10²⁰⁴, where N is the Gaussian total units of h within a maximally symmetric superuniverse, and σ their standard deviation which assumes the maximum known cosmological angular momentum asymmetry.

J_super(avg)=Nh=10¹⁷⁷ erg-sec, the internal angular momentum of a superuniverse corresponding to N. This value for N signifies the superuniverse as maximal, spin-statistically the most symmetric and random derivable (like an all-encompassing Schwarzchild black hole) from our known universe data.

Similar symmetry arguments apply to +/- charge and matter/antimatter dualities.

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.

The third-degree coefficient can be considered arbitrary, since it determines solely the overall scale of the function.

p[x]=x³+bx²+cx+d

p'[x]=3x²+2bx+c

p''[x]=6x+2b

Inflection point: p''[x]=0

6x+2b=0

x=-b/3

Slope of inflection point: p'[-b/3]=3(-b/3)²+2b(-b/3)+c=b²/3-2b²/3+c=-b²/3+c

Relative maximum and minimum of p[x]: p'[x]=3x²+2bx+c=0

For cubic polynomials with a local maximum and minimum, b²>3c: x, x=(-b±(b²-3c)^1/2)/3 for those extrema.

The slope between them is: δy/δx,

and δy=x³+bx²+cx+d-(x³+bx²+cx+d)=x³-x³+b(x²-x²)+c(x-x)=(x-x)(x²+xx+x²+bx+bx+c) ,

δx=(x-x), so δy/δx=x²+xx+x²+bx+bx+c .

Substituting in the above quadratic solutions for x,x:

x²=b²/9-2b(b²-3c)^1/2/3+(b²-3c)/9 and x²=b²/9+2b(b²-3c)^1/2/3+(b²-3c)/9 ,

x²+x²=(4b²-6c)/9 and xx=(b²-(b²-3c))/9=3c/9 and b(x+x)=b(-2b/3)=-6b²/9 .

The slope between maximum and minimum is:

x²+x²+xx+bx+bx+c=(4b²-6c)/9+3c/9-6b²/9+9c/9=2/3(-b²/3+c)     Q.E.D.

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.

The reciprocal fine-structure constant, 1/α, is dimensionless and equal to hc/e²=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.

My conjecture,

1/α=log₂(R_H/r_proton)

or,

2^1/α=R_H/r_proton

gives R_H=1.4 ± 0.1 x 10²⁸ cm (the Hubble constant equivalent of 70 ± 1 km sec^-1 Mpc^-1, in very close agreement with supernovae data), for a commonly accepted value of r_proton=0.85 ± 0.05 x 10^-13 cm, the RMS proton radius. The Hubble radius yields a cosmological black hole radius minimum of R_H, while the proton is sometimes referred to as an "electronic black hole."

Of all the problems in "Big Numbers," few have a relation so simple yet exact.

The exponential relation between the fine-structure constant and the universal horizon radius/proton radius ratio might indicate a measure of inflation.

The classical electrical activity of a nerve cell suggests that nervous impulses (spikes) 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_ψ, arises from the basic parameters of nervous activity.

To find U_ψ, one calculates the minimum product of energy and time for the neuronal action potential across the synapse. This spike lasts 1 millisecond, with an average potential change of 40 millivolts. Assuming the minimum one electron involved, we have:

U_ψ=.001[s]· .04[V]· 1[e]· 1.6 x 10^-19[J/eV] = 6.4 x 10^-24[J-s] = 10¹⁰h.

The factor 10¹⁰ 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_ψ and object uncertainty h evaluate numerically a correspondence principle. The neurophysiological quantum number defines as N_ψ=U_ψ/h=10¹⁰.

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

ih(∂/∂t)ψ[x,t] = ((-h²/2m_e)∂²/∂x²+V[x,t])ψ[x,t]

to solve the probability wavefunction ψ[x,t]. ψ[x,t] describes the various states that an electron occupies in the synapse.

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."

Quantum measurement yields an object state whose complement traditionally remains in complex Hilbert space. Of the many quantum interpretations, none seems to suggest that the process of complementarity may unfold exclusively in the relatively real, accessible phase space of an "observer-object." Here the observer provides herself as the locus for the complement to the quantum object. A compatible measurement retains the observer-object status quo, while a complementary measurement causes a wavefunction phase shift.

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.

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, separating ourselves from our own observation actually subjectifies the measurement. The set of observer-objects is indeed the intermediary between quanta and cosmos. Large scale configurations complement a multitude of observers.

Simultaneity can hold in either observer-object quantized 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, if the observer-object system is quasi-classical, it interchanges a subjective and uncertain relativity with an objective and measured wavefunction.

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 states 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.

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.

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 locally unique forms of physical entities to exist? I.e., is the correspondence principle limited by the distinguishability of overall classical configurations?

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 to complexity (e.g. life itself) within our finite cosmos. There at most two macroscopic configurations of maximally identical quantum numbers likely 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.

A quantum equation converts into its classical correspondent by replacing h, Planck's constant, with zero. A continuum of intervening equations follows, wherein h gradually decreases to zero. These suggest mathematical states of decoherence.

Predictions interfere with outcomes - L. B.

In Many-Worlds, Schrödinger's cat would now represent parallel live and dead states. Life and death are incompletely described by means of quantum mechanics, though. Attempting to define them physically recalls koans like "What is below absolute zero?" "What is north of the North Pole?" or "What signals faster than the speed of light?" Even these may become physically realizable eventually, but the two metaphysical complements of mortality intersect only partially the truthvalue of measurement.

The quantum itself obeys many requirements of life: e. g., taxis (particle momentum), reproduction (cascading), metabolism (fission and fusion), or homeostasis (thermal equilibrium). Thermodynamically interpreted death, as absolute zero, is unattainable for a quantum body. Minimum nonzero spins, particles jumping in and out of the vacuum, entire universes bootstrapped into creation, or a "dead" state resurrected through observation's action bear witness to nonclassical animation. Spiraling photons of circularly polarized electromagnetic radiation self-propagate throughout the cosmos, not unlike the generating potential reflected in the winding pattern of DNA diaspora.

Microscopically, life can seem eternal indeed. Our bodies incorporate the legacies of lives past. Our lungs may contain, on average, numerous molecules of those breathed by Christ, Hitler or others before us. Certainly we carry a heritage of stardust, of former predator and prey. Viruses and prions are living/nonliving interfaces, uncertainties between complements. Quanta, inherent to all things, are thus universal to biological life. Life and its potential include the code which emulates them.

All life acts as observers, and in doing so, causes other life to react expansively. This may be the greatest test for life: neither Newton's third law, nor Einstein's principle of equivalence, nor Heisenberg's uncertainty principle, but a maximization of interactions rules the biosphere.

In biology, the only certainties are death and taxis.

The Bell inequality violates observer wavefunction ψ boundary conditions. The potential phase difference before measurement -- and between detectors at x₁ and x₂ -- is arbitrary. The initial measurements of θ_x1 and θ_x2, where ψ[θ_x2-θ_x1]=cos(θ_x2-θ_x1)≤ 1, can determine nonlocal spin correlation between θ_x1 and θ_x2 from virtual (0≤(θ_x2-θ_x1)< 2π) frequency shift, alternative to virtual phase shift. The former shift affects virtually such measurements as spin correlations in EPR experiments, agreeing with the cos²(θ_x2-θ_x1) Born probability dependence measured for the simple EPR setup. This simulates through quantum entanglement superluminal, strongly correlated or otherwise counterintuitive processes associated with nonlocal quantum mechanics.

The observer, while correlating Bell experiment statistics between nonlocal termini x₁ and x₂, has himself undergone a virtual frequency shift relative to those points. The observer's measurements of the isolated experiment have previously been overlooked for the discontinuous mechanism of wavefunction collapse at x₁ and x₂. Instead, Feynman's path integral method may consider separately the trajectories of the original experiment and the observer's direct correlation process. The observer lies upon a unique curve of the near infinity including x₁ and x₂ which give greatest probability, say, for a cosine squared probability dependence. Observers concur on correlation data because, by virtue of shared observation endpoints in experimental space, they all occupy paths of similar virtual frequency shifted statistics. The EPR particle trajectories and detector measurement connectivity give equivalent overall results. The distinct routes for observation and experiment conserve probability across each trial.

Evident in EPR-type experiments, the measuring apparatus contributes as much as the measured system to the deviation from classical causality. By bounding the experimental space, detectors induce wavefunction resonances unique to that observational setup. In defining these loci of correspondence, the apparatus selects continuity conditions, ascertaining Bell's probabilities.

Can Kirchhoff's laws simplify Feynman diagrams?

1. The set of exhaustive permutations on countable elements is minimally random.

2. Random numbers are discontinuous functions of the real number continuum.

3. Unbounded random numbers have uncountable differences.

4. Bounded random numbers have unequal differences.

Does the set of prime numbers, because of its lack of factorization, represent less information than other whole number infinite series?

Are primes p(n) determinedly nonrandom by the four rules of randomness above?

Is there any projection which transforms the uncountable into the countable?

Finite sets have fewer elements, countable sets the same, and uncountable sets more than the natural numbers. Consider their relative transitions to be fractal.

Do fractals represent countable or uncountable sets?

Are all possible combinations of countable points uncountable?

What is the cardinality of all sets of singularities in an uncountable dimensional universe?

The set of random numbers represents maximum entropy.

Null space, if it exists, would represent minimum entropy.

Can an infinite universe manifest uncountable physical entities?

What is the countability of real versus virtual infinite phase space?

Aside: Knot theory, not surprisingly, can optimize traffic patterns.

Do observers follow the same physical laws as objects?

Can one show that there are more than one observer? Is an observer conserved? Do observers evolve?

If an observer is an object, when is the converse true? Have all objects been observed?

What is the optimum dimensionality of an observer in spacetime? Are observers unitless?

Can an observer objectify any physical situation? Do observers require other observers for measurement?

Does any sentient being qualify as an observer? Does the act of observation animate its object?

How does an observer differ from a participant? Can a participant differentiate physics from mathematics?

Can an observational state be both true and false?

Is the Anthropic Principle "observational Darwinism"?

Can fractal information characterize an observer?

Can observers reconstruct decoherent classical images into coherent originals?

Can fractal (Hausdorff) dimensions be described by their relative entropies?

Does the number line form fractals with irrationals, rationals, whole numbers, algebraic numbers and/or transcendentals?

Can a true fractal duplicate a mapping upon any point? Can fractals of different Hausdorff dimensions map together as a unique fractal?

Do Hausdorff dimensions add as their fractal patterns? Do fractals evolve smoothly with continuously increasing/decreasing dimensions?

Aside: By itself, is a point nonmathematical?

The blackbody spectrum is accurate to the finite number of radiating quanta which compose it. Heat exchange around potential equilibrium moves the measured cosmic background radiation toward its perfect thermal curve, driving an increase in surrounding disorder. These fluctuations portray the "heat death" of the universe -- i.e. spacetime approaching a maximally symmetric, boundless and totally randomized state.

Besides the Schwarzchild black hole, are there any physical bodies whose entropy is directly calculable from their Bekensteinian quantum geometries? What if the Schwarzchild black hole were extremely charged, that is, with gravitational attraction exactly offset by surface electromagnetic repulsion?

The Bell experiment implies that there are three types of signaling -- one-way, probabilistic and informative. The first allows no interchange of radiation, the second allows only interchange of thermal energy and the third allows communication. Can entropy explain the non-local violation of the Bell inequality?

An "internal observer," relative to a black hole horizon, sees particles as we see the microwave backround radiation. A "surface observer" sees pair production from the Hawking radiation across the event horizon, some infalling and fewer infalling/outgoing. An "external observer" sees all events outside and relatively approaching a black hole horizon.

I believe that Hawking said the equivalent to a black hole is a white hole. The latter would radiate more mass-energy than it accepts. If they are equivalent, what is their inverse? A concave (look-back) universal perspective might be the relative inversion of all inner black/white holes' surface convexities.

There may be a point at which a supernova has an interior of coalescing black holes which Hawking-radiate free quarks.

Paradoxically, black holes have the upper universal limit for luminosity at their least mass-energy, but the lower universal limit for luminosity at their greatest mass-energy.

With help from an anonymous contributer.

1. The Anthropic Principle manifests itself most elegantly in the following way. The geometric average of the Hubble radius, R_H, and the Planck length, L*, is on the order of the size of a human zygote, the most primitive and symmetric human state:

(R_HL*)^.5=10^-2.3 cm=r_zygote

That is, we occupy a perspective central to the spatial extent of observation. This calculation is accurate on a log₁₀ scale to .2 out of 61, the error within the physical limits of observation, R_H and L*.

r_zygote=5 x 10^-3 cm       R_H=1.4 x 10²⁸ cm       L*=1.6 x 10^-33 cm

2. I encourage the SETI organizers to look out especially for well defined water (hydrogen bond) maser lines. Water is universal to life as we know it, and this particular amplified spectrum may well be used symbolically by intelligent life as a beacon to broadcast its presence on distant planets.

3. Those seeking to prove ESP tend to do so, and those seeking to disprove ESP tend to do so -- a statistical Cretan paradox. My personal belief is that ESP can neither be proved nor disproved -- it needs measurement for proof, but is confounded as a self-fulfilling prophesy. Maybe experiences which are unexplainable may be unscientific but not unreal. I speculate that quantum measurement comes as close as any physical perception to self-contradiction.

4. The maximum temperature currently withstood by protein-based life in Earth's atmosphere may correspond to that of emerging life in the early universe -- about 121 ^oC (394K). (Higher pressures at the time would allow cells to withstand greater temperatures before rupturing.) Since the cosmic temperature today (13.7 billion years) is 2.73K, and since the radiative t vs T formula is

t^-2/3/(13.7 billion years)^-2/3=394K/2.73K,

Aside: With a fantastically strong telescope, we could in principle see extrasolar beings living who had died many years ago.

5. There are many examples of convergent evolution here on Earth, such as the separate development of wings on birds, bats and insects. Could the midpoint of evolution be marked by an equivalence between convergent and divergent adaptations? For the universe as a whole, could isolated biological systems come to evolve convergently, such that the anentropic adaptation to their physical environments eventually recreates more similar than disparate characteristics?

As observers, we capture physical beauty with internal and external perceptions. We witness both microworld and macroworld, borne of quantum cosmology, not only to interact together but also to interchange with us. Our intimate universe, whose reality embodies both imagination and objectivity, projects itself upon our senses. The previous perspectives - most serious, and some idealistic - are all invented from such love for physics.