Phase Reality!

T-duality, a basic symmetry of superstrings, conceives reciprocally related variables through the aesthetics of mathematics. (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 quantum wavefunction, and secondly general relativity's metric tensor. This P-duality generalizes current physical models and introduces an "inside-out" world view as well. As envisioned by our common philosophy, the P-duality archetype manifests an omnipresent Machian center inhabiting 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.

String theory, which innovated T-duality, substitutes for "standard" quantum gravity parameters a wave compactification among hyperspatial dimensions. For more venturesome physicists the T-duality analog, P-duality, doubles the dimensionality available to the accustomed quantum wavefunction by establishing a spherical symmetry between phase space and its dynamic inverse. This inversion 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.

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 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.

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 as real quantum states are represented in traditional phase space through a conventional wavefunction.

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 or n_N^-, are simply action eigenvalues divided by h; thus, n_N=N/2, and n_N^-=2/N^-. 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. Because of this reciprocal symmetry, exclusively finite action entails there be no singular zero-valued spin.

(A quantum equation converts into its classical correspondent by replacing h, Planck's constant, with zero. There exists a continuum of intervening equations, wherein h eventually decreases to zero. This might suggest mathematical states of decoherence.)

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 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, here in P-duality that of least eigennumbers, localizes particles near S =h, downward from real, and upward from virtual actions, into the arena of mixed (shared) states. Real and virtual wavefunctions, correlated at and evolving from a primordial state, generate the family of subatomic particles from their interference,

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

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

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

with its inverted units.

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'

n ↔ 1/n'

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 surrounding conventional phase space. Entangled from a common cosmological original event, these phase spaces together enable single particles to "self interfere" coherently where N/2=2N^-. In other words, the particle mediates locally an inverse/conventional wavefunction interference, comparable to that accustomed of two conventional wavefunctions. 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 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. This phenomenon arises from the action shared between their virtual wavefunctions within the spacetime of measurement. So entanglement entails continuously evolving virtual phase surfaces coherently interfering with those actual. Radially nested, comparatively reciprocal phase spaces maintain probabilistic causality through spacetime 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 uncertainty spontaneous virtual particles can "bootstrap" potentially null regions to accommodate that constraint. 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). A statistically equivalent, quantum space would sustain its spin continuity through complementarity, associating observation with particles bootstrapped out of the vacuum. Otherwise, six compactified dimensions coupled reciprocally at Planck constant action to six outer dimensions could satisfy, within six apparent dimensions overall, 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 microscopically the entire range of macroscopic phase space.

Wavefunction "collapse" now manifests Born probabilities in an accessible, physical transformation, having lost its abstract, anomalous character to the inverse wavefunction φ^-[r], the local information equivalent to φ[r]. Measurement of observables coincides with an immediate event. 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.

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 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 overall unbounded in principle, overlies actions included in its new triad amongst real, virtual, and mixed states. Conventionally, a singular interval, as 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 the infinite self-energy of the singularity. For instance, summing every momentum state of inverse phase space gives an effectively infinite self-energy of compactification within the forementioned ellipsiod of uncertainty.

Virtual wavefunctions establish a paradigm with many potential predictions for virtual particles ("virtuosos") parallel to quantum field theory. In real phase space, quantum electrodynamics' archetypal Feynman diagrams depict virtual processes with half- and whole-integer spins (those of electrons and photons) as actual possibilities, namely, random walks. These diagrams comprise an ensemble, for which all interactions of equal order (the number of virtual photons portrayed) contribute equal probability. Its total statistical distribution relies on the relative count of possible interaction configurations, weighted among orders. Inverse phase space, conversely, embodies the composite path wherein reciprocal actions' eigennumbers n_N^-, upon normalization, themselves act as statistical weight. As with QED, higher order virtual photons result in greater configuration action, but now smaller n_N^- and less weighting. Reciprocal phase space transforms real Feynman series propagator actions into virtual parallel ones defining fractional crossections from initial to final state. For instance, the QED power expansion that estimates the anomalous magnetic moment of the muon or electron has converging terms of fine structure constant orders. Alternatively, virtual quantum numbers belonging to a given process describe propagator coefficients, enabling convergence of point-entity calculations. In general, this procedure shares the efficient aesthetic of the conventional Feynman representation, but may not require renormalization.

The real phase space of a double-slit experiment maps conformally through the aforenamed 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, 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.

Conjoined reciprocal phase spaces exhibit analogous structure and function to the Hartle-Hawking "no-boundary" (Lorentzian with Euclidean) spacetime. Chapter Three in Hawking's The Nature of Space and Time displays in Figure 3.12 the Hartle-Hawking trajectories reinterpreted here to the phase space quantum dichotomy. His connected Euclidean circular and Lorentzian hyperbolic worldlines suggest a comparable continuous conventional/inverse action path. Paths of real particles meet at h those of their virtual counterparts during interchange from inverse to conventional phase space. These nested, comparatively transposable phase spaces share symmetric trajectories across h in much the same way that Hawking radiation arises across the horizon between imaginary and real time.

Sharing mathematical properties initially attributed to strings by T-duality, P-duality derives phase-dependent action through a wavefunction symmetry. Additionally, it persuades our accustomed philosophy of modern physics more towards a new determinism. Before we enact the hypothetical, albeit beautiful, dimensionality of string or M-theory, compactification by inversion through action h may 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, as of the cosmological constant magnitude. Quantum interpretation governs two complementary, radially symmetric and reciprocally nested phase spaces. The philosophy of observer as participant must now realize the universal center and connective histories that reside within each of us.

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

δr=g^μν[r] x_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 observable universe radius R_H approximate the extrema of observable spacetime curvature. R_H/L*=(n_max)^1/2 represents the characteristic quantum number for spacetime. The absolute range of action eigennumbers is the square of that number - on the order of the cosmological constant, Λ. A real action S[r]=n_Nh≥h/2 infers 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 mutual spacetime event horizon.

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

δr'=g^μν[r'] x_μ (n_N^-)^1/2R_H/L*,

These geometrodynamic states of virtual action associate primarily with look-back, macrocosmic curvature - including an inverse image of the Big Bang. Such inverse spacetime approaching R_H 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_H. Ordinary spacetime reciprocated through L* effectively obeys the naked singularity "dress code" by shielding an otherwise discernable center beneath the inverted horizon.

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 singular 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).

Among actions S[r], there exist correction curvature radii r_c[S[r]], square-root integer factors of r_c[min]=L*/(2^1/2). In general, |δr|=r_c[S[r]]≤R_H. For a giant spiral galaxy, |δr|=10¹³cm, since (n_N)^1/2=(10⁷⁵erg-sec/10^-27erg-sec)^1/2=10⁵¹, x_μ=|r|=10²³cm and L*/R_H=10^-61. (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.)

Through the Higgs mechanism's spontaneous symmetry breaking, particles decay from the false vacuum into the true vacuum, with the acquisition of mass. A Higgs field configured as superimposed real (conventional) and virtual (inverse) spacetimes can justify the dilution of the vacuum energy ~10¹²² fold from the predicted Planck density to the empirical critical universal density. At the Higgs energy minimum (where r=L*) the scalar linear black hole mass density, c²/G=10²⁸g/cm, moderates the opposing spacetime geometries of the unstable Planck region with that of the condensing cosmos.

Under inversion, spherically symmetric spacetime bounded by L* and R_H correlates conformally its interstitial curvatures with those of inverse spacetime between L* and (L*)²/R_H. Therefore this transform alters the sign and magnitude of related geometrodynamic accelerations. Inverting the r<L* spatial and temporal radial spacetime components with Friedmann's ρ_M retains two compactified dimensions of 0 ≤M/r≤ 10²⁸g/cm, and obtains a brane analog ρ_Λ with two extradimensions apparent as 0≥M/r≥ -10²⁸g/cm. Macroscopically, these transposed compactifications have the effect of a locally inverted 2-D black hole geometry, evoking the vacuum (i. e., dark) energy and agreeing with the radially accelerating universal expansion that recent supernovae data indicate.

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 - otherwise arise from bivalent spacetimes, aboriginally evolving apart from a common, correspondent and collapsed state. The microscopic, reciprocal, and virtual spacetime represents interiorly the universal, traditional and real exterior geodesics. In other words, the cosmos projects inwardly - as quantum gravitational holographs - those phenomena which inflation would claim outwardly, but with even greater symmetry.

(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 acceleration for galaxies and larger bodies where the influence of so-called dark matter begins. 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_H=v²/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_H. If so, this would indicate a characteristic of baryonic matter's inertia to overcome an "ultraviolet catastrophe" (analogous to the blackbody's). Consequently, a_H tends to maintain the radial acceleration, and likewise the orbital velocity, of galaxies' outlying halos. The effects of "dark matter" on large-scale structures are predominantly due to compliance with discretized minimal acceleration, given by the ratio between speed of light squared and the cosmological horizon radius. The behavior of quantized acceleration might be measured in an atom trap.

Could the impetus for the ever-expanding universe lie in an exotic, long range repulsion between primordial matter and antimatter? If so, it might be possible to calculate the abundance of antimatter at a given distance and direction by variations in redshift from the Hubble law.

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 as 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.

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. This quasi-frictional effect would most likely manifest as a vacuum polarization, and may help explain the quantum nature of dark energy.

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, 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.

If gravity can radiate from complete gravitational collapse (a black hole singularity) while electromagnetism (light) cannot, then gravity does not frequency shift itself as it does light. Also, the interaction between curvature and mass (E/c²) indicates that spacetime geometry shifts light frequency, but not necessarily gravity's.

Grant that the free electric charges in the universe are comparatively sparse. Their overall randomness manifests a standard deviation N^1/2=10⁴⁰, where N=10⁸⁰ is the number of free, or ionized, charges (assumed an order less than the number of charges overall) in the visible universe. The deviation, or net free charge distribution, may represent a charge separation with dipole moment. Its field, on average, acts upon a test electron with force F. That electric force is approximately the gravitational force the electron would experience if acted upon by the mass of the universe. In other words,

(keq_dipole/Gm_eM_Universe)~1.

Bands. Musical in that their 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 yields "vibration" energy two Planck units, etc. A string does not differentiate between number of twists, and therefore represents a special case of bands. The width of the band is dualistic to its number of twists, as winding numbers are to vibration numbers in string theory. 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?

The thermal equilibrium between two relatively receding (c<v≤ 2c) cosmological horizons, justified by inflation's Higgs phase transition, could provide locally the randomness for correlations violating the Bell inequality through superluminal (c<v≤ 2c) "signaling."

Inflation reinforces a dimensional resonance at the geometric mean of cosmological horizons, (R_HL*)^1/2=10^-2.5 cm. If a characteristic separation for branes is on this order, its gravitational influence should be found by experiment in the next decade.

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

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

An inverted infinite quantum well would display spacetime properties of topological genus one.

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

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 is just as effective as compactified space in representing the resonances underlying physics.

Imagine quantum geometry as a projection of spacetime onto a discontinuous, exactly 3-dimensional (yet fractal) phase space - that is, one both probabilistic and embeddable.

One outcome of a black hole being a 2-D spatial "holographic" projection of 3-D space is that charge, mass and spin are properties unaffected by the transformation.

Primordial black holes are seeded at the inflationary era. Inflation, with its superluminal expansion, acted to reinforce shells of energy toward gravitational collapse in the early universe. A gravitational field with alternating relativistic regions shared velocity boundaries of integral c values (0≤v≤ c, c≤v≤ 2c, 2c≤v≤ 3c...). Regions of alternating expansion and compression manifested relative to each other. The potential and probability for forming microscopic black holes developed within those limits. These holes were characterized by the time period of inflation (~10^-35 seconds), whose growth factor perhaps exceeded 10²⁸. Inflation's superluminality conveyed initial Planck wavelengths on to further condensation of lower-energy resonances. The holes in turn seeded geons (stable, charged microminiature black holes) which likewise generated more familiar particles.

"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 10⁵⁶ cm. The universe that we experience outward is one of 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 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; 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 (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.

Inflation takes place during the early cosmos in a range of spacetime curvatures which can also be found presently near a black hole singularity, recapitulating the properties and processes of inflation. This region may provide a certain radiative signature (such as Higgs bosons) to us searching for evaporating black holes and for evidence of inflationary processes.

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. This hypothesis includes all entities in spacetime from individual vacuum quanta to the background radiation. The shell may interrelate observer or object, and eventually generalize to any 2-D manifold.

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 allowed variables we may know about a black hole (from the "No Hair" theorem): mass, spin, and charge (and derivations thereof). The extreme symmetry and Riemannian parallelism of the Schwarzschild black hole transfers coherently (much like an "isotropic laser") such information that is allowed about the singularity. This Bekensteinian data materializes at the black hole surface as those measurable quantities or their gradients, including that of temperature. Any black hole should exhibit at its horizon magnified characteristics of its interior, not only conventional "No-Hair" quantities, but also allowed gradients characteristic of interior structure.

Let two equivalent Schwarzschild black hole singularities separate by twice their conventional horizon radius. When so "contiguous," their gravity weakens by opposite attraction the gravitational field near their point of greatest symmetry (Lagrange point). Convergent black holes, then, create paradoxically a significant region of sublight escape velocity. The holes infalling together (from their imagined conventional horizon) may virtually lay bare their singularities. Many photons which impinge upon (the impact parameter of) such an individual black hole and become trapped would not when extrapolated to the two hole situation. The distortion of the mutual horizons' surface area indicates an overall increase in their entropy which is accounted for while evolving toward the final combined singular state.

A gravitational black hole abhors a "naked" mass singularity, but allows it the observable property of charge, with correspondent E-M field. Similarly, the horizon radius r for "electronic black holes" (where m_ec²=e²/4πε₀r) 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 energetically bound against electric repulsion at a given radius as to be reproduced by the energy of attempted measurement. As with strong force quark isolation, charge singularity (i. e., E. B. H.) measurement itself restricts 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 depend upon ordinary geodesic curvatures.

Mini black holes especially would alter the quantum field calculations depicted by Feynman diagrams. Not only might overall particle paths be finite on such a spherical spacetime, but vacuum polarizations could be skewed by non-Euclidean geometry to slightly lesser values.

Sequence of measurement, more appropriately than time, provides an objective and fundamental physical standard for assessing change in quantum cosmology. Quantum mechanics retains its space/time separation, and relativity its spacetime unity, with sequencing inherent to the dynamical changes in both. Between two recorded events, physics maintains as a transient, probabilistic dynamic. There are then the two states of change - observation, and statistical continuity between observations. Derived from the quantum wavefunction, action eigenvalues represent the possible sequential permutations (Planck steps) 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.

If we think of local time as absolute, not interval, we must consider global time bounded (with a referential beginning and/or end); or, if we think of local time as interval, not absolute, we must consider global time unbounded (without a referential beginning and end).

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.

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

2. Special relativity mediates observer-object action.

3. The wavefunction for macroscopic special relativity is assumed sinusoidal.

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

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

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

(John Archibald Wheeler, but not Stephen Hawking, argues 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 ε_F=0 for black hole physics.

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 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 H₀ (the Hubble constant) acts like Planck's constant.

Research the Sloan Digital Sky Survey for attenuation 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 5 x 10²⁷ beyond the universal event horizon 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 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 by a quasi-quantum effect. There may be extensive, radically peculiar regions of such space that from our perspective would create, e. g., fluctuations in the microwave background radiation.

Assuming inflation with its greatly superluminal expansion, there should be multiple spherical event horizons (cycles) alternating outward for each multiple of "c." Encountered first is our Hubble forward time universe, then a tachyonic reverse time spacetime, then forward, etc., reflected many powers of ten times relative to our observation. 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, now effecting forward time once more. This continues for n cycles, where n~10⁵⁵/10²⁸, the ratio between the ultimate inflationary and linearly expanding universe radii.

Inflation is also accessible here and now in the high energy (10^-24 cm), approaching Plank scale physics. That could indeed allow superluminal communication. It may be possible to tunnel into these higher "harmonics" subatomically to achieve faster-than-light by adjusting phase around the frequency 10³⁵ sec^-1 (that of GUT). Likewise, pre-inflationary photons in a sense govern our visible universe.

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

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

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

    =σh

σ=10¹⁰²=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 average Gaussian expected 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. With respect to internal angular momentum, this value for N signifies the superuniverse as maximal, statistically the most symmetric and random derivable (like an all-encompassing black hole?) from our known universe data.

Similar symmetry arguments apply to charge and matter/antimatter dichotomies.

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 derivation,

1/α=log₂(R_universe/r_proton)

or,

2^1/α=R_universe/r_proton

gives R_universe=1.3 ± 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 observable universe describes a cosmological black hole, while the proton is sometimes referred to as an "electronic black hole."

The fine-structure constant alpha is most fundamental. Base two is also fundamental*, perhaps representing a duality of spin inherent to a fractal alpha-dimensional space. Elsewise, inflation may account for the exponential relation between the fine-structure constant and the universal horizon radius/proton radius ratio, where α is a measure for the magnitude of inflation.

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

*Any (physical) system with a duality will depend on integral multiples of a base two. Take a system of otherwise identical N particles whose charge for each particle is randomly positive or negative. The possible permutations number 2^N.

See "Further Evidence for Cosmological Evolution of the Fine Structure Constant".

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_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 impulse 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 paraphrases 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 relatively real, accessible phase space within an entity of "observer-object." Here the observer provides herself as the locus for the missing complement to the quantum object. A compatible measurement retains the observer-object status quo, while a complementary measurement interchanges observables.

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 always occupied a participatory role in quantum mechanics and cosmology. The observer-object classicizes traditionally distinct quantal entities by treating them much like an observer and the cosmos she occupies. One might think that by including the observer, physics would become more subjective. However, by separating ourselves from our own observation we actually subjectify our measurement. The set of observers is itself the intersection of quantum and cosmos.

Conventional quantum mechanics is uncertain as to the simultaneous existence of complements. In the person of observer-object, however, complementary observables unify as both real and simultaneous. The domain for the formerly displaced complement exists here and now. As a totality, observers with their quantum objects actually constrain spacetime to uncertainty and nonsimultaneity. Simultaneity can hold in either observer-object phase space or relative spacetime, but not both for one spatial event. Observer-object states trade an accustomed uncertainty of action for that of interval. The Planck units embody this discreteness. So if the observer-object system is quantum mechanically objective and classical, it enjoys this for a subjective and uncertain relativity.

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 perceive, remember and predict patterns evolves from the anentropic components of observation.

The observer's constituent pattern memory is juxtaposed against the 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, it likewise generates locally ordered mnemonic networks. Again, a "random" process has as its complement anentropic memory.

3. What is the largest quantum number value between matching classical configurations in the observable universe? Do you find significance in the fact that beyond a certain complexity, statistics requires unique forms of physical entities? I. e., does the correspondence principle rely upon the (in)distinguishability of quantum configurations?

For instance, any two protons have a high probability of sharing near-identical spacetime sets of quantum numbers, which simple crystals are much less likely to share. Yet given two DNA molecules within the universe, one might in principle attain the most symmetric possible replication of quantum numbers.

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 this cosmic limit for identical sets of quantum numbers to enumerate the symmetry of the correspondence principle - to demarcate the quantum from the classical.

Reality, contrary to popular endeavor, is that which you cannot escape - 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 truth of measurement.

The quantum itself obeys many requirements of life: e. g., taxis (particle momentum), reproduction (cascading), metabolism (beta decay), or homeostasis (thermal equilibrium). Thermodynamically interpreted death, as absolute zero, is unattainable in the quantum realm. 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 universe, 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, a few molecules of those breathed by Christ, Hitler or others before us. Certainly we carry a heritage of stardust, of former predator and prey. Viruses are a living/nonliving interface, an uncertainty between complements. Quanta, inherent to all things, are thus universal to biological life.

In biology, the only certainties are death and taxis.

EPR measurements involve a process which satisfies quantum wavefunction ψ boundary conditions. The phase difference between measurement discontinuities is at first arbitrary. The initial measurements at x=a and at x=b, where ψ[b-a]=cos(b-a)≤ 1, can thence determine nonlocal spin correlation between a and b from virtual (b-a< 2π phase change) frequency shift, in lieu of wavefunction "collapse". This shift affects such measurements as spin correlation in Bell experiments, agreeing with the cos²θ probability dependence measured for the simple EPR setup, and simulating through quantum entanglement superluminal or otherwise anomalous processes previously associated with quantum mechanics.

The observer, while correlating EPR statistics between nonlocal termini a and b, has himself undergone a virtual frequency shift relative to the particle measurements, and challenged his objective interpretation. The observer's relation to the isolated experiment has previously been overlooked for the discontinuous mechanism of wavefunction collapse at those points. Instead, Feynman's path integral method may consider separately the paths of the original experiment, the observer's correlation process itself, or the complete circuit of both. The observer lies on a unique path of the infinity between a and b which give greatest probability, say, for a cosine squared probability dependence. Observers concur on correlation data because, by virtue of shared observation endpoints in phase space, they all occupy paths of similar virtual frequency shifted statistics. The EPR particle arrangement and its observational connection give equivalent overall statistics to both completing a circuit including same initial and final state. The distinct routes for shared observation and experiment conserve across all trials average probability, the square of correlation.

Evident in Bell-type experiments, the measuring apparatus contributes as much as the measured system to the statistical deviation from classical causality. By bounding the experimental phase 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 if not the perception of quantum reality itself.

Imagine a computer architecture which emulates the base "e," rather than the accustomed binary base 2, for rapid applications exponential in nature. (I believe it has been shown that the most fundamental base for an ideal computer is e). Base 2 is most practical for its natural "on-off" logic. Base 3 can be achieved in theory through a baryonic triad of identical quarks. Circuits which use base 2 logics and base 3 logics may be constructed with an overall characteristic base converging upon e.

Such a network of circuits would assign logic gates according to the power series coefficients of "e base 2" and "e base 3." For the former, a coefficient of 0 would require no branching and a coefficient of 1 would require a binary gate. For the latter, a coefficient of 0 would require no branching, a coefficient of 1 would require a binary gate, and a coefficient of 2 a tertiary gate. Some sort of binary-tertiary interaction could actually underlie physical structure in materializing the natural logarithm.

I once read that every electron is the same electron.

I would say, more accurately, that an observer cannot observe more than one quantum at a time. Even for classical observations, the number of photons N_p we collect over time T_p I believe to be "collapsed" individually such that N_p/T_p < 1/T*, where T* is the Planck time. This basically states that any measuring instrument cannot process more than one quantum simultaneously.

It may seem like a rehashing of the Copenhagen interpretation, but considers large scale interpretations and those involving multiple observers. Any autonomous sentient can at best share one wavefunction collapse simultaneously with you or me, according to the initial hypothesis. To confer with the sentient, then, requires a separate collapse. Thus, there are no contemporaneous events involving more than two observers, or beyond an observer and a quantum.

To say that we can appreciate only one quantum particle is, ironically, equivalent to saying that we as observers are one of the near infinity of quanta represented by its wave aspect. All intermediate wave packets are evolving towards particle or wave nature. One could say that the totality of cosmological quanta determines the discreteness of the individual quantum.

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=1.3 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. The first realistic chance for carbon-based life to form in our universe is demarcated at the cosmic temperature of approximately 100 ^oC (373K), the maximum temperature withstood by primitive DNA. Since the cosmic temperature today (13.7 billion years) is 2.7K, and since the radiative t vs T formula is

t^-2/3/(13.7 billion years)^-2/3=373K/2.7K,

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

4. 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?