6. Reflections and connections
i.) The idea of a generalised nonlocal connectivity is not new, of course. It is as old as Newtonian gravity. It has been an issue in quantum theory almost since the theory was invented, and instead of being relegated to an nth-order problem nonlocality has become ever-more central to foundational questions during the last hundred years, despite the overwhelming dominance of the local field paradigm. In the 1940s Wheeler and Feynman [65] proposed to reinvent classical field electrodynamics as an action-at-a-distance theory, with some success, invoking a back reaction from a future event by mixing advanced and retarded wave solutions of the Maxwell equations; however problems with quantisation and an arbitrary boundary condition led to it being abandoned. The absorber theory demotes the field to a book-keeping device to avoid infinities but there still is classical Maxwellian radiation. The electromagnetic field is still there; the difference appears to be that the Feynman-Wheeler field is entirely determined by the particles - it has no independent degreees of freedom and thus the electron charge does not get acted on 'by itself' in the form of these independent degrees of freedom of the field. Hence the infinities would be elided. But because this classical field is not quantised into photons one then has to explain the photoelectric effect, spontaneous light radiation of atoms, and so forth, in addition to the cosmological requirement that all emitted radiation eventually gets reabsorbed in a closed universe. All of these problems are effectively addressed, in principle, within the new ontology outlined here, which removes the infinite degrees of independent freedom of the field but retains the function of photons inside a quantisation condition that automatically produces the Wheeler-Feynman boundary condition. Later when introducing his mature spacetime representation Feynman [66] pointed out that the fundamental first-order interaction equation strictly speaking only applies to virtual quanta, but that it was possible to generalise to 'real' photons because in any closed system all quanta can be considered as virtual - that is, every emission and absorption takes place internally and is an unobserved component of some exterior sum. He pointed out the complete equivalence of his description with the conventional quantum description in such a case. This was an heuristic argument, but again the same boundary condition can be seen to be the essence of a network system whose closure is analogous to that of a black body radiation cavity. It is suggested that the superspin network can be considered as having implications for QM that are dual with those of the transactional interpretation developed by Cramer [67] on the basis of the Wheeler-Feynman theory - i.e., almost complete practical triviality - but that whereas in Cramer's view the string-like picture of absorber theory interconnections remains a phenomenological device, the superspin network is a genuinely new ontology that makes distinctive predictions (although so far merely qualitative) beyond quantum electrodynamics.
ii.) Chiefly these arise from the non-field interpretation, and since the currently most successful theory of gravitation is conspicuously a field theory we can expect some differences. In particular the emergence of 'attractive' Newtonian gravitational interactions from the form of a primarily repulsive potential confined to the network lattice means that although the theory would predict gravitational radiation it would not predict gravitational waves. Gravitational wave observations therefore become a negative test of the theory.
iii.) To be specific about the distinction: Consider a spacetime interval from the points of view of GR and network theory. First, consider a single half-wave interval or string-segment. (This is an abstraction for the purposes of argument from an inherently pluralistic theory, of course, not a real possibility.) There is what we can call a 'space amplitude', which is a standing wave amplitude orthogonal to the time vector, but whereas the vector is a relativistically-real interval carrying a time-varying frequency the space amplitude is not time-varying: It is the fundamental mode of the oscillator and has no frequency. Obviously the dimension of this amplitude is what we recognise as the spin polarisation plane (see 4.i., note 4, p.28) and the rotation of this plane becomes equivalent to a curvature. Variations of this space amplitude (which has the same function as the phase speed c of a particle wave group with v < c) are imaginary. (They would correspond to an 'internal' space geometry of a quantum of curvature, which, if real, would assume local analysability of a quantum of gravitation into a continuously varying potential. This seems unintelligible.) This amplitude is a scalar in terms of any single interval and only becomes "time-varying" as a half-cycle of a train of a number of such phase waves, i.e. in a phonon excitation mode. Such a wave train, in turn, becomes like the gravitational wave-group dispersion (i.e., becomes like an 'object') of a wave of higher phase speed, and so on, until ultimately all gravitational wave-groups (all 'objects') are seen as dispersions of the standing-wave phase of all space. But this variation of the space amplitude is not a real time-variation leading to a real space frequency, i.e. to a measurable frequency, because although the space amplitude varies as a function of a global 'time', this global variable is in fact space-like and the time-variation is imaginary. This variation in a scalar 'time' is not an observable. The scalar limit of all real times for any given interval is of course c. Thus the change in distance will always be inversely proportional to a change in the speed of light by which it is cancelled out. (This is equivalent to saying that c is a constant of floating norm. See 4.vii, note 7, p.33.)
iv.) In GR and network theory the observables must of course be the same. But GR has a time-variation of imaginary space-amplitudes occurring within interval (that is, between observables), and a quantum theory of GR-gravity will naturally be a field theory of bosonic gravitons which then has to be coupled to arbitrary masses. That is, whereas the root principle of a quantum theory of radiation is that one boson goes wholly and uniquely to one fermion (i.e., in QED, one photon goes to one electron), the energy of gravitational wave radiation has to spread omnidirectionally and subsists (as problemmatical 'quanta of curvature') throughout space. According to GR an indefinitely large space field containing only two indefinitely small oscillating masses is filled with gravitational waves of indefinitely small amplitude at the frequency of oscillation, even though no test particle exists by which these waves might ever be detected or by which the relative motions of the masses might be otherwise gauged. The very meaning of 'mass' in such conditions is obscure. A real GR space wavefront is thus conceptually very difficult. But in a network theory the existence of a space wavefront which is an unobservable locus could have only an imaginary meaning. There is no real smooth substrate. Instead of a continuous space we have the line elements of a fractal boundary, and this 1080-dimensional boundary becomes the origin of the principle that only actions generated in the real relations of observables have meaning. Thus, instead of a time-variation of imaginary space-amplitudes occurring between observables as in GR, network theory has a space-variation of imaginary time-amplitudes occurring as observables, i.e., as pairs of states bounding intervallic string-segments; and the transformation algorithm for mapping an ensemble of intervals one onto another is special relativity, which gives us the electrodynamics of moving charges in the context of network quantisation.
v.) If LIGO and its sisters fail to observe gravitational waves this would be consistent with expectation, because the result of this space variation, with observables exclusively at all wave-nodes and intervals of quantised action carrying time-varying frequencies, whose transformations under the Lorentz group represent electric and magnetic potentials, is just what we already call quantum electrodynamics. One vivid (if ultimately very misleading) way of thinking about this is to say that gravitational wave radiation is ominpresent in matter but that its compression waves are always damped to extinction by induced opposite displacements due to the varying effective norms of electrodynamical constants. Evidently this would be equivalent to identifying photons as quanta of "curvature", and renormalisation in QED could then be seen as an accommodation of its invariance group to the effects of gravity. As mentioned, this is not in the end the most helpful point of view, but it expresses the idea that we ought not to expect measurable longitudinal space distortions over the optical path of even the longest laser interferometer.
vi.) Gravitational radiation, like any photon, will be invisible 'inside' any interval. Both become 'visible' only as some relation of charges. Network gravitational radiation will therefore be measurable as a displacement of mass. But mass radiation will not occur by a mass-charge coupling to a space field in a network theory, where there is neither a constant scalar charge nor a real space field. It will not propagate locally. It will dissipate nonlocally through the equilibrations of the whole network self-interaction, ultimately therefore being the origin of itself. Other than in this global metrical adjustment it appears only in a series or ensemble of intervals which is both the network analogue of a gravitational "wave train" and a series or ensemble of network "particles" - a supersymmetric phonon excitation, analysable into fermions and their exchange-force bosons according to how the odd/even spin rule applies over a given number of half-wavelengths. And because this radiation is not a local smooth field but a nonlocal fractal boundary we obviously don't expect space coherence in the form of a plane wave front: The cosmic gravitational radiation density is not a locus of disturbances in phase travelling in empty space, but rather tracks the cosmic mass-energy density, because it is simply the distribution of all "particles". In this sense one can say, picturesquely, that an inert gravitational wave detector is already gravitational radiation. Indeed the entire universe is the gravitational radiation from the Big Bang.
vii.) However the sensitivity of current-generation G wave experiments is extremely low. With expectation so close to zero for a positive outcome it may be difficult to demonstrate a negative. Is there any other possibility of an experimental check on G wave radiation? One controversial possibility is that of using a superconductor as a gravitational wave transducer. Network theory obviously predicts that this will not work. This may be a distinctive prediction, if - as is claimed - it is not clear that coonventional field theory contains a convincing reason why it should not work.
viii.) Chiao [72] proposes that superconductors will act as transducers of gravitational and electromagnetic waves and that the gravitomagnetic radiation field should be expelled from a superconductor just like the electromagnetic field. Essentially this says that the dynamical component of Einstein's spacetime - i.e., not the "static charge" (mass) but the relativistic effects due to moving electron masses - is driven out, in such a way that the ssuperconductor surface behaves like a plane magic mirror which reflects incident electromagnetic waves as gravitational waves with 25% efficiency. Sceptical physicists claim that a strong coupling between gravitational and electromagnetic waves in superconductors would already have been seen. [73] However Chiao proposes a simple experiment in which an electromagnetic wave incident on the surface of superconductor-1 should be transduced into a gravitational wave re-emitted towards superconductor-2 where it is transduced back into an electromagnetic wave and detected. If both "transducers" are isolated in Faraday cages then detection of an output from superconductor-2 signifies transmission of gravitational wave radiation.
ix.) There are no gravitational waves in our construction, and so the network prediction has to be that it will not work, because the "field coupling" is merely theoretical. The analogy between the electromagnetic field model and the gravitational field model appears pregnant because it discovers the approach to an identity between gravitation and electromagnetism in the low-temperature quantum limit. Expulsion of the dynamical gravitomagnetic 'field' means that the GR symmetry describing the relativistic relations of macroscopic mass charges is 'broken' in the superconductor and the bonding of 'repulsive' Cooper-pairs represents the identity of electrostatic charge and gravitostatic charge (mass) realised in the approach to thermal zero. This bond is equivalent to the vanishing (a false vacuum condition) of a static effective gravitoelectric field in the direction of the current as measured in the inertial frame of 'an electron'. The network analogue of 'gravitational radiation' will be simply the total fermionic emission of a system, which plays no part here, since an incident energy sufficient to stimulate significant emission would presumably turn superconductor-1 into a 'transducer' at the cost of destroying the superconductivity, and so would merely demonstrate the photoelectric effect.
x.) However this does not mean that there is no gravitomagnetic/electromagnetic coupling - far from it. It just means that its form is different. Chiao's direct coupling interaction Hamiltonian for superconducting Cooper pairs contains no length scale of course and is independent of the gravitational constant. It shows a coupling between electromagnetic and gravitational 'radiation' mediated by nonlocally-correlated pairs of electrons, with a coupling constant for the interaction equal to twice the electron charge. In the absence of a classical manifold our model does not support concentric gravitational waves, but it implies that precisely this kind of scale-free nonlocal 'transducer' - of which a Cooper condensate is one extremal case - is ubiquitous at all temperatures, but that it is precisely the ubiquity of gravitomagnetic 'transduction' which (if we wish to look at it like this) destroys the nonlocal correlation of generalised spin supersymmetry. If there were no such transduction all particles would be perfectly spin-correlated in a zero-energy false vacuum state incapable of thermodynamic work, all cosmic temperatures would be converted to gravitational heat, and (in classical-continuum terms) electromagnetic and gravitomagnetic vector forces would vanish to a scalar field. Thus, such transduction appears as the 'origin' of electromagnetic forces. As already pointed out (6.x above), we can express this in a picturesque way by saying that quantum electrodynamical phenomena suppress gravitational wave radiation. In fact we can say that quantum electrodynamical phenomena in effect contain gravitational radiation (which in a given case will be the rate of all fermionic losses from a system).
xi.) A further negative experimental prediction is possible in the area of sparticles. As re-emphasised in 6.ix a network theory would enjoy a radical dynamical supersymmetry. There is no need to rehearse here the rationale for this claim; suffice to say that one would not expect the discovery of a discrete class of supersymmetric partner particles. This is consistent with the failure so far of accelerator experiments to detect sparticles outwith a narrowing mass regime at the high end of theoretical expectations.
xii.) It has also been pointed out (see in particular 2.iii. 4.viii & 4.xi) that the theory would not indicate a need for a massive Higgs boson. Here again it is in conflict with predictions of the standard model that lie just outside the range of observations. This may not be undesirable given that the separation of mass and gravitation implied by the Higgs mechanism is at least inelegant and that some experimental doubts about the existence of the Higgs have been expressed [74]. Network modes can be identified analogous to the mass-donating function of Higgs particles, but these are not any truly distinct modes of the string, they are simply the fundamental modes of all spin pairs and as such are coterminal with the same nonlocal objects (string segments) otherwise identified as electrodynamical elements or gravitational elements. This exhaustive supersymmetry means that the mass-donating particle mode is identically the gravitational particle mode, a satisfying and rational economy lacking in the standard model.
xiii.) Consoli and Stevenson [75] and Consoli and Siringo [76] have explored a model in which an equivalent 'Higgs' particle occurs as a long-wavelength phonon excitation of a Bose-Einstein phion condensate. Consoli and Stevenson point out that in terms of such a mechanism it becomes possible to reconsider the Higgs field as candidate for the 'inflaton' scalar field of inflationary cosmology and argue that it improves earlier suggestions [77, 78, 79] that Einstein gravity could emerge from spontaneous symmetry-breaking. The gravitational constant would then be the vacuum expectation value of the scalar field. Consoli and Siringo further suggest how Newtonian gravitation may arise in a spontaneous symmetry-breaking phase transition from the symmetric phase of the phion condensate and remark that such a nonlocal theory, in which gravitational force has to be considered an effectively instantaneous long-wave excitation (1017c) is needed for a realisation of the equivalence principle (see 2.x. above). They suggest that such a description of gravity would turn out to be the same as a Feynman-Wheeler theory with nonlocal 'strings' and that Einstein gravity could be recovered as a weak-field effective theory generated by the underyling quantum theory. Further they point out that such a theory could address evidence of self-similar cosmic structure and might help resolve the dark matter problem. Certain resonances between the above point of view and the present thesis will be obvious to the reader. As mentioned in the Introduction, it is suggested that the present scale-free network concept represents a sketch of a non-perturbative non-field approach which may be dual with the underlying nonlocal field theory in their model.
xiv.) To recap this point: It has been suggested that the unbroken or 'unfolded' superspin phase of the network string is an inflationary mode. When folded on itself the scale-free repulsion gives rise to an 'attractive' short-range nonlinear inflaton self-coupling from the form of the folding. This form is emergent as follows: The string is the background-independent vacuum; its folding is cognate with a stepwise renormalisation of the vacuum gauge at each local node generated in the network of its self-interaction. The norm of c thus varies incrementally like the dispersion group velocity of the inflaton phase, which we can say is broken 'because' of a torsion in the string whose restoring potential becomes dual with a quantised 'curvature' in an effective field theory. We make superspin normative, which is analogous to setting as constant the 'specific rotation' of a gravitomagneto-optical 'Faraday effect' from which we derive a phenomenological factor h as the numerator of an imaginary angular momentum. The negative restoring potential of this super-rotation is a torsion in the photon linear polarisation plane which increases the space of quantum basis states for the dynamically-supersymmetric electron doublet. A nonrelativistic (i.e. nonlocal) superspin wave function would be degenerate in a pair of symmetric position states, a degeneracy preserved only in EPR-type correlations. In general the broken superspin symmetry expresses as a Lorentz-invariant transform, to a linear momentum, of an emergently real angular momentum, h/p, halved and quartered in the first and third partials of the inflaton. Thus a nonlinear self-coupling of scalar inflatons transforms spin-0 to an internal spin-1 coupling which can also be represented as dual with an induced attraction from a higher order phonon mode of spin-2.
xv.) A qualitative positive experimental prediction may be made with respect to gravitational anisotropy. In a network theory mass becomes a derivative product of the global distribution of momentum, dependent on the form of the distribution as measured in spherical coordinates locally at a point of measurement (and also on the scale of the interval between any two such points of measurement, which nonlocal dependency produces a scale-variant cosmic dipole responsible for small-scale mass attraction in the context of accelerating large-scale mass repulsion, a scenario with possible application to 'dark matter' and 'dark energy' models). It is possible that in supercooled experimental conditions certain materials may be reduced to a 'low'-energy dynamical equilibrium which is the network false vacuum state, in which they are rendered sensitive to an anisotropy in the global distribution of mass-energy. We can imagine that the local electrodynamical constraint reduces away at thermal zero, not to nothing but to a point of dual phase at which the underlying global constraint will begin to dominate. With a superconducting solid the natural dual point will not be reached; it may only be approximated by one dispersed phase of the material - conduction band electrons - which can be thought of as occupying a much higher false vacuum state due to the constraint of the crystalline atomic structure. But a gas of 'free particles' might exhibit instability at the point of dual phase due to a minute asymmetry in the global 'gravitational' constraint that is usually thermally masked. It is suggested that so-called 'bosenova' disruptions in Bose-Einstein condensates, which are not presently understood in terms of the well-developed standard theory of BECs and are believed by Weiman and some other workers in the field to require new physics [81, 82], could indicate such an effect due to the condensate being nudged off the false vacuum.
xvi.) If the particles in a nanokelvin Bose-Einstein condensate (BEC) are already thermalized as close to their lowest average energy per particle as the experimental regime allows, then the only way of shedding more thermal energy is by the BEC shedding some of its population. Where all particles were previously held in a false vacuum state by the local constraint at a temperature just a few billionths of a degree above the zero-point of thermal energy, now some will begin to experience the space-asymmetry of the global constraint dominating. In other words, particles in different regions of the BEC will begin to have 'weight' in various directions, arbitrarily with respect to the field-map of terrestrial gravity, and will 'fall' explosively out of the BEC with a range of kinetic energies far higher than any possible latent internal energy of the zero-point BEC. This is the non-local energy of the global constraint, the network 'vacuum potential'. Adjustment completed, a stabilised remnant of the BEC would then survive at the centre of an expanding cloud of debris. According to a network theory the gravitational radiation of a system is simply the sum of all fermionic particle emissions from the system. The energy of this radiated debris is neither thermal nor electromagnetic but represents the energy of the space configuration of the BEC at the zeroes of both. It represents the energy of a phase transition in the inflationary 'vacuum' constraint that donates the local mass energy of the BEC. To put it most dramatically, the 'bosenova' could be seen as pure gravitational radiation from a gravitational collapse in the laboratory. [85]
xvii.) Over the years a scatter of different values of G has been obtained in different laboratories around the world without a clear understanding of why the error-bars are so wide. Indeed Mbelek and Lachieze-Ray [86] have recently argued that although the errors are decreasing the values are not converging, so that the most accurate terrestrial measurements of G presently differ by ten times the intrinsic error. (Their suggestion is an unsuspected coupling between gravity and the earth's variable magnetic field; vide supra note 23, p.57.) And there is controversial direct experimental evidence of gravitational anisotropy. According to experiments carried out by Gershteyn et al. [87] the value of G varies with spatial orientation by at least 0.054%. If confirmed this would be extremely difficult to reconcile with existing gravitational theory. In a network theory as in conventional smooth-field cosmology the small deviations from uniformity in the gravitational energy density of the universe close to the 'last scattering surface' will be coupled to the mature cosmic matter architecture. But this coupling does not represent the time-evolution of an homogeneous average matter density from the 'early universe' to galaxies, and so is radically different from Machian interpretations of general relativity, where inertial mass is due to coupling with a local gravitational field. With 1/r2 field gravitation, distance variation would overwhelmingly dominate angular variation in any roughly isotropic cosmic mass distribution. But in a network theory the spacetime surface becomes a fractal boundary on which there is no average matter density and no real global distance scale; 1/r2 gravity is an emergent statistic; and local inertial mass becomes sensitive to angular variation in the cosmic particle distribution. The scale-free temperature fluctuations of the CMB as mapped by COBE/Boomerang/CBI etc. are then seen as shadowing just one component isolated in the limit of a time-free sky map (i.e., radically scale-free) which will be dominated by the density fluctuations due to the galaxy distribution. The fundamental network quantisation condition, prior to scale, is quantisation of direction and it is this complete nonlocal mapping from any given here-and-now which specifies - independently of radial distance scale - a local gravitational/inertial constraint whose departure from isotropy will be proportional to a varying angular node density on the celestial sphere.
xviii.) The COBE temperature fluctuation in the microwave background is 5.5 x 10-6, which insofar as it maps a nascent background fermion density would be a theoretical minimum degree of network gravitational anisotropy here-and-now. The dominant contribution would be from the much coarser fluctuations summed over in the galaxy distribution. So the observed anisotropy would be expected to reflect the total matter distribution on the sky (independently of radial distance scale) with 5.5 x 10-6 as the bottom of a range of variation equal to the range of values of the anisotropy of the cosmic fermion distribution at all epochs. If an increasing degree of 'roughness' due to gravitational accretion is assumed to be proportional to the 'age' of the epoch, then the dominant contribution to the characteristic gravitational/inertial anisotropy of a test system here-and-now might be expected to be of the order of 100 times that due to 'ancient' structures with the roughness of the CMB, or about 10-4. This expectation is not inconsistent with a claimed experimental minimum anisotropy of 5.4 x 10-4.