4. The Superspin Network. Symmetry-breaking and the Emergence of Local String Modes

i.) Superspin is supposed to be carried as a 'super-rotation' of the plane of polarisation of a (linear polarised) photon. The photon itself 'sees' the restored superspin symmetry normalised (in null proper time) to that of a spin-zero scalar 'particle' and is blind to an imaginary torsion which it carries over into the mass relations of a pair of spin-half leptons as a spin-one electrodynamical symmetry. Evidently 'spin' means different things to different particles. More revealingly, 'particle' means different things to different spins, which is a perspective requiring us to think in terms of polarisation. In this view the network is seen as a string supporting a complicated synthesis of vibration modes and their rotations. [Note 1] Thus if we see our elementary object as something like a stretched-out, open-ended superstring, then an ensemble of N two-valued 'particle' states becomes a network of such strings all interconnected at (2N)^1/2 vertices. This 3D lattice can then be unwrapped in the imagination to become a linear string of such elements all joined end to end. Thus each nonlocal elementary string can be regarded as one of N antinodes in this longer string containing N' local nodes, and in this way the whole string acquires a 'frequency' (though not as yet a real frequency, since we haven't specified any real time or space parameters - see 4.iii below). So we are imagining a complex network, or a complex standing wave, which has this inherent ambiguity: Either it is an excitation where the wave amplitude is the n^th partial harmonic of a zero-energy fundamental string mode of indefinite length; or it is a sequence of (n + 1) oscillators each supporting a fundamental amplitude driven by resonance. The essential point for the immediate argument is that a self-measurement can only ever be made on a node, because this is the locality condition. (This condition can be turned into a selection rule for real or virtual particle states; see 4.vi-vii.)

Note 1. The transverse dimensions of these modes are still schematic in this discussion. Briefly, the picture to be developed is that for any 'observer' an imaginary oscillation of a string segment only ever occurs locally as a real oscillation of string node A retarded from a real oscillation of node B. In photon time these connected states and all parallel connected states are contracted to a spacelike hyperplane containing projections of all orthogonal real oscillations. When this hyperplane is devolved back into a real time representation, the transverse imaginary vibrations living on the intersecting stringy line elements of its hypersurface are summed as photon amplitudes and projected over a dilated 4-space interval as components of a complex wave.

ii.) We can see by refolding the string node-on-node so as to reconstruct the 3D lattice that the lattice is an expression of this stringent self-consistency constraint on the string: That it can only self-interact at its nodes. This may sound 'obvious', inasmuch as wherever it 'happens' to self-interact there will by definition be a node, and there is no a priori reason why the folding should not generate an infinite number of nodes given an infinite local 'length' of string. But the useful result is that because the system only makes a 'measurement' on itself at an available node it becomes impossible for a new folding, a new act of measurement, to generate a new local node (i.e. by generating a new partial harmonic mode of an interval) unless the resultant frequency is implicit self-consistently in the the total conservative energy state of the whole string. In other words for a total number of N objects or string-segments, a new local node can only occur as an origin of a new set of (N+1)/(2[N+1])^1/2 pairs of observables commuting automatically with all other sets of observables, and an increased local vector potential due to a non-conservation of particle number implies that if the string's total energy is to be conserved then there must also be a compensating potential. Given that the local self-consistency conditions are all satisfied by those vector transformations in which higher spins appear, then the compensation will be such that the adjustment is fundamentally an increase in a global scalar potential due to a scalar 'particle' with no (locally conserved) angular or linear momentum. The compensation thus comes in as a negative scalar energy (i.e., negative 'mass'), meaning in turn that we can think of the fundamental mode of any (real) interval as an inflationary scalar particle or a spin-zero 'inflaton'.

iii. Cosmological inflation is normally thought of as a nonlocal function attached at an early 'historical' epoch of the evolution of a continuous local manifold. In a network this scalar inflaton particle represents the nonlocal interconnectivity provided by cosmological inflation but translated to a time-free function, which is arguably more intelligible. [Note 2] It would be a boson in terms of the usual spin statistics, and in a conventional field theory such an inflaton would also be said to give rise to an attractive coupling between 'inflaton charges' of either positive or negative sign. But as an element of a network the nested functions become more complicated, so that whilst the inflaton 'exchange' can be considered to be nonlinearly attractive between inflatons (and thus dual with a deflationary resultant of a global inflationary constraint which represents the network analogue of field gravitation [Note 3]) the coupling inside inflatons is a spin-one dipole between fermions, superimposed on the scalar inflationary function of the fundamental mode. There will be no pre-set inflaton charge or electron charge. The even-spin inter-inflaton coupling is plurally attractive, whilst the odd-spin intra-inflaton coupling is singly repulsive, and it is the resultant attractive or repulsive character of the action - an emergent function of a 'gravitational' constraint - which determines the local sign of the globally-neutral 'inflaton charge'; and this local action is evidently identical with the doublet electric charge, which is emergently either like/like or like/unlike. So although it is a boson, because of its supersymmetric expression as a mode of a real fermionic doublet the inflaton clearly also obeys Fermi-Dirac statistics. In fact one has to say that it obeys both sets of statistics 'simultaneously' (i.e. in photon proper time) in its complementary guises of charged electron(s) and uncharged photon (see, e.g., 1.ix. & 2.iii.) This ambiguity seems to be related to the Pauli-Weisskopf [46] interpretation of the Dirac negative-energy fermionic vacuum, which they showed has a duality with a scalar charged boson field described by the Klein-Gordon equation. (The Pauli-Weisskopf scalar field bosons, not being subject to antisymmetric exclusion, would be incompatible with the Dirac 'hole' theory as a model of antiparticles.) This neutral scalar string mode is then rather readily identified - at least functionally, if not formally - as a finite analogue of the zero-point oscillators of the Dirac vacuum.

Note 2. The local network potential is everywhere nonlocally donated, and the global 'epoch' is a spacelike projection of this function as an imaginary 'history' in which it becomes equivalent to primordial dissipative mixing . A flat overall gravitational potential would occur as the mean of a 'mass dipole ' locally coemergent with scale, where small-scale deflation arises in the context of large-scale inflation giving gravitational contraction inside an accelerating expansion. (See 5.ix) Because general relativity's field theory would be only an effective theory in a network cosmology there would be no point of singularity . A closer historical-geometrical analogy of the origin of these opposed curvatures would compare the flat scalar potential to a 'braneworld'. It should be noted ( although the point can't be developed here) that in the network the function of singularities is in general taken over by 4-surfaces, of which the old 'Schwarzchild singularity' of a black hole (presently regarded as only a light horizon embedded in a continuous manifold) would be the type.

Note 3. The factor N^1/2 for the ratio of global to local connectivity in a network of large N identifies the cosmically 'gravitational' nature of this function by producing the force constant as (10⁸⁰)^1/2 or 10⁴⁰ for a universe of 10⁸⁰ particles. See 4. vii., note, and Part 6.

iv.) In a network model of atomic electrons in equilibrium with a radiation field the origins within the network correspond to (2N)^1/2 fermion states, each of which subtends [(2N)^1/2-1] boson states, and each of these in turn represents the zero-point scalar fundamental of a potential photon vector, analogous to the zero-point vacuum oscillators which allow spontaneous emission of radiation in Dirac's theory. But the Dirac probability of spontaneous emission is proportional to a factor (N + 1)^1/2 for the condition of an infinite N. That is, Dirac assumed that there is no limit to the number of photons that may be promoted out of their zero states by a perturbation of the vacuum, and if that is true then there must be an infinite number of photons already in zero states. The model Dirac developed from this assumption was justified by its results, but it leads immediately to the interaction Hamiltonian becoming infinite and it is then rescued by setting some infinitesimal coefficients in order to keep the transition probability finite. Thus Dirac ushered in the picture which replaced classical empty space with a vacuum filled with zero-point oscillations of energy ½hv. But this vacuum is still a continuum, whose infinite degrees of freedom are evidently the origin of the infinity in the Hamiltonian. In a network model no continuum would be available to start with and Dirac's argument from an unlimited radiation density to an infinity of zero-point photon states would be inappropriate. The network would have photon zero states (the half-wave scalar particle modes) but there would be no need to avoid an infinite probability of emission because there is a limit to the possible number of real photons of a given state r_g coordinate at the node of 'emission', corresponding to [(2N)^1/2-1] fermion position states occupying equipotential levels with resonance r_g, and the underlying meaning of the coefficient of spontaneous photon radiation is that every fermion is a false vacuum of [(2N)^1/2-1] scalar particle states. [Note 4] Each of these scalar particles is an object or string-segment with no locally conserved spin, what we have identified (4.ii.) as the spin-zero inflaton mode of the string, partial harmonics of which will then correspond to further boson and fermion modes with locally conserved spin angular momenta. (See 4.iv. below. It will become important later that this model only allows one such scalar particle per interval as the real fundamental mode; all 'copies' in other modes are either spin-1/2 or virtual spin-2.) But note now that although the scalar inflaton may have no locally conserved spin (i.e., neither transpositional-fermionic nor rotational-bosonic) we are not saying that it has no spin. In fact we are suggesting that it has a superspin, the physical meaning of which will be that it is the negative or restoring potential of an imaginary torsion carried as a rotation of the plane of polarisation of a linear polarised photon. The photon mediates an electrodynamical coupling which, whilst a dipole, nevertheless always has a positive 'gravitational energy' which is attractive; and a network model suggests that this is because its spin-one vector potential always occurs as a cancellation of the negative inflationary potential of a scalar 'meson' which couples both to it and to an electron doublet, and so a cancelled negative potential appears as a positive energy of 'attraction'. Because this attraction always occurs in the third partial which includes both spin-1/2 electrons/positrons and their spin-1 vector boson its matter coupling can be said to be mediated by a spin-2 phonon excitation, which is the network 'graviton'.

Note 4. In a network ontology a vacuum state of an infinite number of photons of zero momentum has no meaning except for exchange probabilities involving an infinite number of states of electrons. Since an infinite number of such states (network vertices) cannot in principle be distinguished in any finite region, the probability at any point instant of finding a photon in any state, including zero, must be finite . The total possible number of photon states associated with each of (2N)^1/2 local electron position states becomes (2N)^1/2-1, and these are the zero states of both real and virtual photon modes. Thus any single real interaction path between any doublet of electrons A - B is a direct route which can be said to be equivalent to roughly this number of indirect routes each connecting A and B via one of the vertices C, D, E . . . N', like the set of all first-order perturbation amplitudes, and these are automatically summed over as the equivalent least action path A - B. In fact for finiteness and consistency all network diagrams to all orders must obviously sum over to the local action of any uninterrupted (least-action) network segment. What enables this condition is the fundamentally nonlocal renormalising of the vacuum gauge, segment by segment (i.e., superspin ), inside a constant scalar potential, which is equivalent to a vacuum gauge of constant norm inside a varying nonlocal scalar field potential. So one can see that this self-consistency condition expresses the emergent dipole potential of an underlying neutral scalar inflaton symmetry. The dipole balance of positive and negative 'corrections' due to the local form of the directionally-quantised inflaton 'field' represents the metrical accommodation of A - B to the sum of all possible perturbation orders . In this rather formal sense it is possible to say that a pathological divergence of virtual states is checked 'by gravitation'. See 4.vii, note 10, & Part 6.

v.) When the locality condition of self-interaction is applied to a set of such scalar elements the vector transformation requires Lorentz invariance to be emergent in the 'folding and refolding' of the string, node-on-node, under this multiplying scalar potential, and the real SR 'distances' which result must each express a changing real energy over a changing real time, preserving the action product as a constant of 4-space rotations. Now an intriguing surmise is that the untransformed (inflaton-mode) scalar potential of each interval is the origin of the Planck constant, recovered as the common extremum of all local action vector transformations. Beginning with the assumption that the spin-zero eigenvalue represents the QED basis state for a gauge particle of spin-one (and remember that physically this basis state enters as the fundamental mode, one half-wave-antinode long, of a string segment confined between self-interaction nodes) we introduce (see 1.ix., 4.iii.) the concept that the spin-zero function is actually degenerate in the two eigenstates of a super-rotation, q_s, which is unperturbed by the local magnetic field (being in fact its prior generator; see Part 5) but which may loosely be considered to be perturbed out of a false vacuum state by a superspin 'field' which does not couple with photons alone or with electrons alone but only with the dynamically-supersymmetric doublet state electron|­ñ + electron|¯ñ in the form of photon|­ñ + |¯ñ + |q_sñ. Because of the Pauli exclusion principle (appropriately generalised to a dipole - see 2.vi., 3.xi-xii.) this means that the degeneracy exists in a pair of 'electromagnetic field' coordinates and is lifted in the form of a displacement of a pair of local position observables linked by a photon (null signal line). By analogy with the classical magneto-optical rotation we suppose that this coupling effects an axial 'specific rotation' q_s = lQ_sH_sl, where Q_s is a constant, H_s is the (notional) superspin field strength and l is a path length. Where wavelength l, path length l, and 'field strength' H_s are all normalised to unity, the specific rotation of the vacuum state reduces to Q_s. Evidently therefore the scalar 'inflaton' mode sets a constant specific rotation Q_s equal to the common zero point of local spin angular momentum and magnetic moment. This means that whereas a classical analogy might suggest a 'gyromagnetic ratio', g_s, of our superspin doublet vanishing away in the scalar case, there is in fact still a degenerate eigenstate which is unperturbed but cannot vanish. Therefore we set this false-vacuum state of g_s as just the proportionality 1.0 so that all our other factors remain rational. Now the value of Q_s remains indeterminate, but since it is to be a rotation of phase we know it will be expressible as some multiple of 2p. The factor H_s will remain unity since we don't wish to import any continuous field potential as primary, and effectively it drops out. So, we can put

q_s = l(Q_s/2p)l

(4.1)

But with pathlength and wavelength both normalised to unity even the l and l drop out leaving simply Q_s/2p. To get from this imaginary scalar case to the first vectorial case of real interest we have to introduce a real time and a real energy. These will emerge along with a measurement of a wave number smaller than 1/1 made by a self-interaction of the folding string. But at present we have no idea what these values should be so we insert a phenomenological factor, h, an action containing an energy and a time, to be determined by experiment. Now, setting Q_s equal to one for that experiment and multiplying by h we thus get to h/2p which we can take to represent an extremum of every vectorial case. (We set Q_s = 1 because we know ex hypothesi that a photon in any local measurement does not 'see' a superspin Q_sh/2p and of course neither do we - directly. But we do see h.) This factor h is still arbitrary but it allows us to retain l also, and with l we can start to produce a series of quanta of length, as wave numbers, ratios of the half-wave spin-zero fundamental mode, giving

q_s = h/2p(l/2, l/3 . . . l/n)

(4.2.)

which relates discrete values of specific rotation to a series of increasing frequencies as a function of the quantum unit of boson spin.

vi.) Now at his point we should pause, because how we interpret higher partial modes depends upon how we characterise the process. According to the inherent ambiguity mentioned in 4.i. above we can say either that each new antinode is an excitation in the nth partial of a zero-energy fundamental string mode of indefinite length; or that it is one of a growing sequence of (n + 1) oscillators each supporting a fundamental amplitude driven by resonance (we will have to introduce and justify a selection rule in due course to resolve this ambiguity in realistic cases, but for now the underlying ambiguity is the important thing). For example, a whole photon wavelength contains a node at p, and terminates at a node after a further 180-degree rotation to phase at 2p, so that the spin-zero scalar fundamental now contains two antinodes, each fractally similar to itself, in the form of two photon half-wavelengths. Treated as belonging to a series of separate oscillators, each one of this pair of new antinodes represents only a scalar increase in a gauge potential in respect of local forces (see 3.x. above), so we have a degeneracy in the two states with the identical unperturbed energy eigenvalue. But this scalar potential also represents a quantum of superspin which is to acquire a value with the breaking of a generalised nonlocal symmetry to a restricted local spin symmetry so that its trace, h, appears as the energy/time dimensions of a spacetime action. In other words, although we say the superspin is carried as an imaginary rotation of the photon linear-polarisation plane (spin-zero transforming to spin-one as a local displacement but hiding an imaginary extra torsion that the photon does not 'see') it does have a real projection on the nonlocal axis of electron intrinsic spin as a function of space relations generated in relativistic electrodynamics.

vii.) In this view the new antinodes belong to an excitation at some increased standing-wave frequency of the entire string and are not simply two new scalar potentials but two components of a new partial mode with a relation to a frequency. Now an increase in the gauge potential becomes a contribution to a total real energy elicited according to the locality condition of the self-interaction in which SR time (interval) is emergent. Thus transformed with a time each re-gauging of the emergent vector potential at successive nodes is effectively a new zero-energy false vacuum state. Each successive zero-point raises the gauge by an imaginary advancement of phase, a superspin rotation (or its negative) which appears in the balance ledger of local forces as an effectively scalar lower bound to the production of energy with time or just the action constant h. [Note 5] But because time is emergent in relativistic measurement only as a function of photon wavelength (the momentum, and therefore the velocity, is completely indeterminate in less than a wavelength) there is a natural minimum of electromagnetic spin which we find by allowing the oscillators to be measured by the photon mode. For this measurement they become components of the whole photon wavelength, another excitation in another partial harmonic of the whole 'string', where they become twin antinodes in the third partial mode of the fundamental (we'll see what happened to the second partial in a moment) whose reflected phase angle rotates through a total, as measured in the effective unit time [Note 6], of h/4p, or spin 1/2. This is the electron-positron mode of the string, paired virtual amplitudes of the photon mode having equal and opposite spins that rotate into one another over 720 degrees of phase in unit photon time. So we have in the 'ground' states of each network element or string-segment: a spin-zero scalar boson, a photon, and an electron-positron pair, all scale-free, unit distance being normalised to one wavelength and velocity normalised to unit scalar speed (c). These will be the lowest modes of every segment.

Note 5. This gives rise to a range of equivalent arbitrary conventions for expressing fundamental 'constants'. If h is considered to be held constant under superspin transformation then it becomes possible to say that c is a constant of varying norm, or vice versa. Newtonian physics can be recovered from relativistic quantum physics by setting c at infinity and by setting h at zero. Thus, described as a cosmic 'history' in imaginary time, the incremental 'folding' of the string as it evolves through the series of modes can be thought of as a process analogous to a dispersion through a series of N different vacuum refractive indices, generating N emergent finite norms of c which sum over as the imaginary 'curvature' of GR spacetime with h notionally constant. Renormalizing c to a flat space would be equivalent to allowing a local variation in the norm of h for differently accelerated observers. Therefore since the total number of different real accelerations equals the number of 'folds', equals the number of network vertices, equals N^1/2, we can see that the negative or restoring potential of gravitation becomes a global renormalisation of the action gauge equivalent to a sum over the local restoring potentials of superspin. See 4.iv, note 9, & Part 7.

Note 6. Unit time can be any one of 10⁴⁰ projections of unit distance each specified at one of 10⁴⁰ network vertices, and unit distance is of course l/2. Unit time is therefore order of 10^-15 second for visible light . So the choice of unit time is arbitrary, but only to the extent of electing one of the finite number of self-observations allowed by the network. 'Observing' a node in a higher partial (at a higher frequency) represents a finer localisation of 'an electron' not because it illuminates some occult particle more precisely but just because 'an electron' is this act of self-observation and so 'occurs ' at any frequency at which an observation may occur. It is possible to think of an electron as just the sum of all its intersecting virtual photon modes.

viii.) In any possible local view, the inflaton mode would look like a spin-zero particle with a zero positive mass - a Goldstone boson, with a zero local potential - but its superspin is the origin of a nonlocal potential, a restoring potential which, as mentioned in 3.x., will be just the negative of the imaginary photon spin that carries the broken electron-spin correlation symmetry. A real photon spin being associated automatically with a positive-momentum real propagation vector it can be inferred that an imaginary photon spin is associated with an imaginary propagation vector and a negative momentum, a negative mass-energy, which is why it will be inflationary. (As mentioned in 2.iii. this mode does have a local function in this ontology analogous to that of a Higgs boson in the standard model, since it becomes the origin of inertial mass, but the mechanism is different in that it is identically the origin of gravitation as well. This satisfying ontological parity lacking is in the standard model. See Part 6.)

ix.) Consider the fundamental scalar mode of the string as 'seen' by a photon, whose local spin h/2p is the constant of a gauge symmetry which untwists local spin-zero (or a specific rotation q) to its normalised vector identity spin-one, as

q/l = (h/2p)

(4.3)

where wavelength, path length and magnetic field coupling strength are all set equal to one. That is to say, by analogy with the electron magnetic moment

m_eB = eh/4pm_e

(4.4)

we can suppose there would be a notional photon magnetic moment

m_gB = eh/2pm_g

(4.5)

which of course equals zero for a photon, even though it has spin-one, because the photon charge is not e but zero (i.e., the radiation statistics are linear, or two photon modes do not directly self-interact [Note 7]). From one point of view this is because its quantum spin state is degenerate in the two opposite electron-positron spin states, ±eh/4pm_±e. That is to say, a photon has these two spin eigenstates which are not (normally) both lifted out of degeneracy to acquire distinct eigenvalues. From another point of view the fact that the photon charge is zero means that its rest mass m_g in the denominator of (4.5) does not appear, which conventionally would be taken to mean that all of its mass is dynamical - it vanishes 'at rest'. But notice that m_g must be non-zero for the magnetic moment to be zero in (4.5). If m_g is not non-zero in the absence of charge then the magnetic moment becomes infinite. Conventionally this would mean that (4.5) could be dismissed as a meaningless over-complication of the fact that the photon carries no charge and thus cannot have a magnetic moment. But if one takes seriously the idea of increasing the number of basis states of the photon it becomes possible to suggest that there must be a degeneracy in the photon eigenstate for mass.

Note 7. Using intense lasers coherent photon modes can be made to couple inside certain materials.

x.) This degenerate photon mass can then be considered to be cancelled by the negative mass of a scalar boson, only appearing when the degeneracy is lifted and 'it' donates this mass to an electron in the emergent electrodynamical symmetry, as has already been suggested. This recalls the f-field theory of Pais [47,48] that would allow an electron to couple both to photons and to a compensating field of scalar mesons. The purpose of Pais' model was to rescue a finite electron self-energy, as was the equivalent scalar C-meson hypothesis of Sakata [49] and Tomonaga [50]. This 'cohesive meson' was introduced by Sakata to absorb ultraviolet divergences in the theoretical electron mass. Later, it was invoked by Tomonaga as a way of cancelling a divergence which appeared in calculating radiative corrections to the Rutherford scattering of electrons. Tomonaga pointed out that this divergence could be identified as a photon mass, but the problem with this interpretation was that there was no photon mass term in the Maxwell equations into which the divergence could be assimilated, so Tomonaga suggested that a hidden photon mass might be compensated by a Sakata-like C-field. But although these models were important in the development of renormalisation theory in the 1940s and gave clues to its eventual more 'technical' formalisation by Schwinger [51], the whole development presumed the underlying framework of a problem that can be split into an unperturbed radiation field and deviations from this unperturbed state due to interactions that are then superimposed upon it - the essentially classical metrical space field with quantum attributes imported. [Note 8] Consequently the relativistic gauge invariance of this ruling field-theoretic paradigm could not entertain a photon mass and Tomonaga's idea was discounted. Yet in the unified gauge theories of a later era the concept of spontaneous symmetry breaking shows that a photon can acquire and lose a mass. This is a phase change re-enacted in the laboratory during the warming transition of electrons out of the superconducting regime, when the broken electromagnetic gauge symmetry is recovered and photons which had been limited to short range by acquiring a mass now lose that mass so that the magnetic field penetrates back into the conductor and electrons reacquire inertia (see 2.iii.). In this sense an increase in the number of quantum basis states of the photon is an increase in the space of states for a supersymmetric doublet containing two (broken) superspin symmetry phases, and although one way of looking at electron inertia is to say that it represents a lifted degeneracy in the photon eigenstate for mass the proper perspective would be that of the superspin 'field' which couples with both electron and photon.

Note 8. In Schwinger's model for the interaction of a free matter field with a free radiation field the quantisation of the radiation field is effected by dissolving the metric manifold to an infinite number of geometrical points on an evolving spacelike surface , to each of which a time can be assigned - the 'super many-time formalism' based on the earlier many-time theory of Dirac and Fock [52]. This enabled the evolution of quantum states - hitherto represented by a common time variable for different spatial positions - to be made relativistically covariant for all observers, with the Schrödinger equation's common time now being associated to the infinitesimal displacement of a spacelike hyperplane whose point elements could be treated somewhat like particles. With this relativistic covariance in place the infinities can be handled as a constant background and subtracted from the experimental electron energies so as to preserve sensible answers. Feynman's [53] parallel development of the renormalisation procedure out of his action-at-a-distance-to-path-integrals programme involved summing discrete spacetime processes instead of differentiating over continua. It was essentially an S-matrix theory about collision processes , even though renormalisation is a perturbative tool of field theory. Looking at the function and the spirit of this whole enterprise, one could speculate that it is yearning towards the kind of radical deconstruction of the continuum offered by a nonperturbative scale-free network model.

xi.) The question of what possible meaning can be given to a state of a photon which is ontologically prior to the electrodynamical symmetry in which it emerges as the gauge boson is thus to be answered in terms of this superspin symmetry. The meaning of a photon rest mass which doesn't manifest will be basically the same thing as a zero-point photon state in the Dirac equation. The state in which a photon rest mass doesn't manifest can only be a state in which a photon doesn't exist at all! (In one sense, then, we are giving this rest mass back to the spin-zero scalar particle from which our photon derives, which in terms of this model becomes the equivalent of a 'Higgs particle'; but clearly this is not any distinct real mode of the string.) This state in which a photon doesn't exist is one of the complementary pair of 'stationary' states when a photon gains/loses a quantity of momentum p = e/c and appears/vanishes by 'emission'/'absorption' from/into an electron excitation. So the notional photon rest mass never appears although it is conserved within the doublet. In this sense the photon rest mass does exist as the energy of a stationary oscillation in the first partial mode of the spin-zero scalar Goldstone boson mentioned previously. From the equation E² = m²c⁴ + p²c² we can get the energy of a 'stationary photon' by taking the square root, and the momentum term drops out to leave just E/c² =± m_g. Proceeding as for Dirac hole theory we take both solutions; but instead of saying that one of a pair of alternative potential outcomes is almost always excluded by a highly asymmetrical context (like the 'sea' of occupied negative states in hole theory) we accept a general cancellation of two opposite simultaneous mass-energy eigenstates of this stationary wave (in the spirit of the Pauli-Weisskopf model; see 4.ii.) whose resultant is a massless spin-one photon. The lifting of the degeneracy in the photon mass is then due to a spontaneous symmetry-breaking generating a measurement context, such that whichever of the non-degenerate states appears in local 'measurement' will be by definition positive-real. And thus we arrive generally at a manifest positive mass-energy mc² being the rest energy of an electron - and bringing in the charge as in (4.4) - together with a negative mass-energy equal to -mc² remaining hidden, 'associated' both with the photon and with the electrons it connects - that is, with the null world-line of the whole doublet system - yet not appearing in any local audit of their positive energies. This system thus becomes a complex emergent state of a spin-zero inflaton whose imaginary amplitude varies like the phase speed of an emergently real photon wavelength.

xii.) A local measurement context in which this wavelength emerges elicits generally an electron (+m_e = -e), but the corresponding positron state (-m_e = +e) generally is not elicited locally because the superspin symmetry is broken locally. The resulting specific rotation q_s wrecks the correlated spin-one state in which virtual electron and positron are always paired, and it is this rupture which births the photon as carrier of both the electromagnetic gauge and the (hidden) torque of superspin. This role of the superspin can easily be seen from the fact that the rare events in which positrons are locally measured are always pair-production events in which a spin-one Bremsstrahlung or gamma g decays to an EPR-correlated electron-positron pair conserving the total entangled spin. Unlike the generality of spin states whose pairings are broken in the transition to local spacetime invariance, these entangled pairs should 'feel' no restoring torque from a hidden superspin, meaning (ex hypothesi) that they will not experience the normal electromagnetic gauge coupling of a pair of unlike charges. [Note 9]

Note 9. The implication being that recombination of a pair that remains purely entangled is only a virtual annihilation of virtually created particles. Interactions in which antiparticles are 'observed' - i.e., transfer real momenta - thus represent disentanglement, and restoration of the electromagnetic gauge within which locality is protected. When we use a neo-classical electromagnetic field as the 'environment ' for decoherence it brings in an infinite regress of virtual interactions. But we can see that 'virtual' states on the network are robustly virtual and that uncontained proliferation does not arise.

xiii.) Note that, as emphasised before, the coupling of matter to its scalar inflaton mode becomes the 'role' of the photon, in the sense that the real timelike displacement of charges in real < c observer frames, and the photon superspin, are dual representations of the gauge renormalisation between spin-0 and spin-1 embodied in the photon mode. Only when the nonlocal superspin symmetry breaks to a local electrodynamical symmetry of photon and electrons in this third partial mode of the string does a higher-order phonon spin emerge which represents a mode of induced attraction between inflatons whose nonlinear self-coupling is repulsive in the unbroken superspin phase. 'Normally' a spin-zero scalar field couples only to the trace of the stress-energy tensor and therefore would not couple to the electromagnetic field since the 4-space electromagnetic tensor is traceless. Thus it is conventionally held that because a scalar boson exchange cannot couple to a spin-1 photon then gravitation, which by definition does couple to the energy of photons, must be mediated by the next even-integer spin mode available - hence the spin-2 graviton. In our network model the 'graviton' becomes the spin-2 phonon mode of the induced inflaton attraction under spontaneous superspin symmetry breaking. Hence in entangled electron-positron pairs where that symmetry remains intact one expects the converse both of the normal electromagnetic and of the normal gravitational couplings to be observed - i.e., opposite charges whose electrical potential is generally considered attractive will energetically 'de-annihilate' and fly apart in a spin-singlet state with a total spin-angular momentum of zero, as observed, preserving the scalar mode of inflaton repulsion. Within strictly redefined limits, therefore, the network model implies that pair creation represents 'antigravity'.

xiv.) In Dirac's 'hole' theory positrons are negative energy states, and this holds in different representations. In Feynman's spacetime representation, for example, the antiparticle's action is a sum over negative energy states equivalent to inverting the time variable in the causal propagator, whereas in hole theory both positive and negative energy states evolve in a forward time direction. In any case the positron energy is -mc². Conventionally speaking the gravitational mass of -mc² is equivalent to that of +mc² because mass-energy is a scalar monopole charge and the gravitational potential is a global field. Experiment so far suggests that negative antiparticle masses 'fall' in this field just like positive particle masses [54] but this answer is not precisely pertinent to the question posed here a propos correlated singlets: The precisely pertinent prediction of Newtonian or GR-based field-gravitational theories is that the electron and positron exert a gravitational force on one another proportional to these two masses. In the network model gravitation is no longer a charge coupling with a field - it is one emergent pole of the scale-specific dipole and there is no direct dependency on an attractive global field potential. [Note 10] Rather the 'attractive' field potential is a construction put upon the mesoscale (classical) resultant of the generality of particle doublets where superspin symmetry is spontaneously broken in favour of +mc²; but this does not imply that 'particles' are sources of such central force potentials in miniature, and in our model the electron and positron do not exert a gravitational force on one another, although their action is of a kind which, in the large, is the generator of a local gravitational potential. Remember that a degenerate photon mass is considered to be cancelled by the negative mass of a scalar boson, only appearing when the degeneracy is lifted and 'it' donates +mc² to an electron and -mc² to a positron in the emergent electrodynamical symmetry (4.x.). Thus whilst the electron-positron pair preserves this cancellation this is because it fleetingly resists the breaking of its superspin symmetry, preserving an overall vacuum state of zero rest energy. This is equivalent to conservation of photon momentum in a virtual pair-creation, in the sense that all quanta in our closed network 'cavity' can be considered to be virtual; and the zero gravitational potential of such a pair 'explains' the zero cosmological constant of flat space, in the sense that the network contains only such virtual 'vacuum fluctuations' as may be considered to be contributions already factored into the 'experimental energy' of the whole.[Note 11] A total energy difference of 2mc² is not a gravitional potential because inertial mass will only become a locally emergent product of the global inflationary dipole in the same symmetry-breaking that maintains an asymmetry of positive real-time electron interactions via the annihilation of the positron. In other words, the gravitational potential is not generated by an internal attractive coupling due to mass - on the contrary, mass is an emergent local transform of the global inflationary potential, a renormative 'constant' varying reciprocally to c, and the pair's temporarily-preserved internal charge is the inflaton charge, expressed in the nonlocal entanglement of a correlated spin-singlet. The 'gravitional' coupling is a higher order phonon coupling across multiples of l/2 (see 4.xxi. below), and gravitational potentials will be emergent (locally) proportionally to a scale factor and (globally) proportionally to the asymmetry of 'matter' over 'antimatter'. (Thought of in terms of a network of Feynman-like diagrams this latter proportion would be the mean fraction of string segments with negative time variables. Cosmologically this implies an effective gravitational coupling varying proportionally to the mean free path of a positron, which can be used to index a varying cosmic mass-energy density. But in a network model there will be no globally-varying gravitational 'constant'. Instead of an objective history of a globally evolving space field we will have 10⁴⁰ observers each renormative in relation to floating 'constants'. There will be no true gravitostatic field coupled to a global mass-energy density. This cosmic field becomes an imaginary projection from local real gravitomagnetic displacements. We can think of the (electro)magnetic field as a relativistic deformation which is SR spacetime. By analogy we can characterise the gravitomagnetic field as a deformation which is GR spacetime, and extending the analogy we can suppose that where the electromagnetic field is a theoretical index of changes in relative +/- charge density for differently moving charges [see Part 5], so the gravitomagnetic 'field' is just a corresponding theoretical index of changes in relative +/- time density for differently moving masses.)

Note 10. Imminent (September 2002) experiments with the first substantial batch of cold antihydrogen at CERN are likely to assume such a condition and so test only for the 'conventional' equivalence of antiparticle masses under terrestrial gravity. A recent review for ESSA by Bertolami and Tajmar [55] identifies as worthwhile an International Space Station experiment to search for violations of the weak equivalence principle by antiparticles in microgravity, but again such a search would be insensitive to an inflaton coupling which is only manifest in the unbroken symmetry phase and is constructively the same as the total pair-momentum. There will be novel difficulties in understanding how to unify the spectrum of 'forces ' in a fundamentally monopolar network theory. These are clearly not only theoretical and practical but also semantic, owing to the ingrained assumption that GR must be a long-scale, low-energy limit of a short-scale, high-energy quantum gravity theory, such as Planck-scale loop quantum gravity. The network quantum theory imagined here would not be a short-scale theory but a scale-free theory, to which the connotations attached to the concept of a 'weak-field limit' would fail to apply.

Note 11. Inverting the usual theoretical philosophy:- The renormalisation of network constants node by node, which is the effect of superspin (see 4.vii., note 10), represents a physical identity between their effective values and their normative experimental values.

xv.) The energy of the vacuum and the energy of the network are obviously constrained to be the same quantity differently conceptualised, as we will now see. So far we have been considering the lowest modes of any given string segment, which are crudely-speaking equivalent to the first-order interaction terms of a field theory for a radiation field containing a pair of charges. Now it will be evident that as higher partial modes are considered a 'new' node may be either real or virtual depending on context. For example, it was said above that the first partial, with one node, gives a photon. We can say that considered as belonging to the first partial mode of the entire string this node cannot be real, because our locality condition forbids self-interaction without a self-consistently available node (there may be esoteric arguments around this, but cosmological implications can be discreetly left to one side for now). But once we have several objects or string-segments to work with we can start to have consistent 'observation' by self-interaction and very quickly large numbers of superposed modes become possible. Evidently any other fractally-identical spin-one pair of antinodes anywhere on the string can also be identified as a photon wavelength. In fact any sequence of such wavelengths can also be a considered a photon of arbitrary energy hn, since the quantisation condition contains an arbitrary time. Not all photons will be real (in the sense of being an observable), because the only meaning that SR and QM give to the question of whether a photon is or is not an observable is in the form: Is there, or is there not, an excitation of an electron to a set of higher energy eigenvalues? An observed photon therefore is a state of an observed electron, so that (one 'end' of) a real photon only occurs where there is (one 'end' of) a real electron doublet. This means that the mode of the string in which each is theoretically 'found' contains the same self-interaction node, which is a vertex of the network, a folding together of N^1/2 string segments, identified by changes of momenta. For electrons and photons their coupling is a transfer of momentum, which is possible because their modes share vertices. Coupling occurs at a resonant frequency because they live on odd-numbered partial modes. This is why the second partial mode between photon and electron modes is ignored in the analysis of 4.v. above: it is an even-numbered partial mode with three antinodes and is therefore a virtual mode corresponding to spin-3/2 (conventionally-speaking, a 'gravitino', the superpartner of the graviton) which does not have a direct electrodynamical coupling. [Note 12]

Note 12. This may be related to 'Furry's theorem'. As later rediscovered by Feynman and applied to Dyson's renormalised S-matrix, Furry's theorem states that all of the diagrams with odd numbers of loops automatically cancel out of the perturbation calculation and therefore do not contribute to the scattering potential. In terms of the network such 'loops' are automatically self-consistent routes within it, and it is an expression of this self-consistency that all contributing routes for electron/photon interactions will occur only in odd-numbered modes which have even numbers of antinodes, because only in these modes can the nodes of 'emission ' and 'absorption' coincide. Even-numbered modes which have odd numbers of antinodes (= loops) have no coupling and so are virtual in relation to a particular local scattering event, to whose amplitude they do not contribute. They do contribute globally, of course.

xvi.) There can be many higher-frequency non-resonating boson and fermion modes in any one string segment as well as many lower-frequency non-resonating phonon modes living on string lengths many segments long, none of which are observables at the natural frequency of the string segment in question. All of them may have real forms, either on a different group of string segments or at an altered natural frequency of the same string segment(s). Since that natural frequency is a function of the entire string network it may change, and a node may be generated, or the local Lorentzian geometry might change, bringing out a set of photon resonances over an entirely different range of frequencies in a new self-consistent 'observation'. The superposition of all possible modes will therefore be very complicated, with the following selection rule operating to distinguish real and virtual modes: In general, for any constant real length of confined string, the set of all odd-numbered string modes (1, 3, 5 . . . n, where the fundamental = zeroth mode) gives self-consistent interactions reinforced at one possible set of standing-wave nodes; whilst the set of even-numbered modes (2, 4, 6 . . . n) gives self-consistent interactions reinforced at another possible set of standing-wave nodes. In general these sets of modes will reinforce one another only occasionally, but because there is no such thing as a constant real length of confined string (the network is dynamical for any real observer) the distinction becomes academic, with odd and even string modes, and real and virtual frequencies with their different fermionic/bosonic spin modes transforming in and out of one another in an incalculable cosmic Fourier synthesis.

xvii.) When we focus on observables, transfers of momenta at vertices, then we normalise the network to a particular node where a self-interaction of the string equates (depending on context) either to the emission/absorption of a photon by an electron or to the annihilation of a photon and the creation of an electron-positron pair. If the node is self-consistently reinforced so that it approaches 'permanence' then repeated self-interactions of the string at that node measure 'an electron' and the interplay of wavelengths around it becomes analysable into real photons, virtual photons, and virtual electron-positron pairs, emitted and absorbed by it. If on the other hand the self-interaction is not a 'permanent' equilibrium condition but exists only as a fleeting resonance then a photon annihilates into a virtual electron/positron pair that each separate off into the network to spawn other photons, or vice versa. Overall this has obvious similarities to resonances of the Heisenberg scattering matrix and the Wheeler [56] S-matrix view, and pictures can be drawn that resemble interconversions in the Chew [57] 'democratic' hadron bootstrap. The elicitation of 'real' object states can be identified with (a) pairs of 'elastic scattering' events, pairs of nodes at which changes of frequency accompany changes of velocity, as opposed to (b) pairs of onwards inelastic scattering events without momentum transfer, which may be the same 'real' pair of nodes seen in a different scattering channel where the entire 'interaction' remains virtual. In the case of (b) there is (ideally) no momentum change along an uninterrupted straight channel, therefore no observable event, meaning that although each of the nodes can be said to correspond to some photon mode on that sequence of string segments they do not both correspond to the same partial mode. Each may be a terminus of a 'real particle trajectory' as paired with another node elsewhere, but they may not be paired as real with each other. In the former case (a) there is a dog-leg channel with two momentum changes, two real observables, and our local view may therefore be of two collisions on a single perturbed particle trajectory (for example), or of three different interconvertible particles, or of a vacuum particle promoted briefly to reality between two points of creation and annihilation by high-energy photons. These would then all be different context-dependent views of the same substructure - three complex linear oscillators or string segments - where the distinction between real and virtual particles, like that between bosons and fermions, disappears into a supersymmetric complex identity.

xviii.) Note the fact that where there is this dog-leg of momentum transfer between segments the Lorentz-invariant transformation of the underlying structure now looks a little like a problem concerning the rigidity or tension of three segments AB, BC, BD linked at B and C with the resulting force couple acting as a turning moment on a central element BC. But in this purely Lorentzian transformation it only looks like such a problem because of the locality. Because of the limit of the speed of light information about forces acting at the junction C is not available simultaneously anywhere else, say at the junction B; indeed if the messenger is characterised as a fermion the momentum information is carried by it even slower. Therefore a reaction at B should be causally independent of a reaction at C because, as we conceive it, B is 'in the past of' C and cannot be acted back upon by a force applied at C. Yet if B and C are local labels that live on a nonlocal object BC then there may be a nonlocal 'force couple' producing a turning moment on this object even though B is outside the light cone of C. [Note 13] This is a locally measurable effect because of a constraint which the nonlocal symmetry imposes on the local. We can expect this moment to be negligible (in relation to Lorentzian local forces) when the length scale of BC is large, but significant when the length scale is small, meaning that as the nonlocal object BC approaches characteristic 'inter-particle' dimensions rotation becomes general for pairs of local observables, and the nonlocal generator of this rotation will be found to be a constant, as we will now show.

Note 13. The Wheeler-Feynman action-at-a-distance theory of electrodynamics does include a back reaction from a future event through the radiation field, which represents a duality with the nonlocal network model. See Part 6.

xix.) Only the nonlocal 'preferred frame' of the scalar inflation is describable as a frame where, with perfect generality, the vector sum of all linear momenta is zero. So, when we now consider the underlying structure of three linked elements subject to a force couple acting as a turning moment on the central element we can see that this will in fact occur as the general case where there is a 'particle trajectory' between junctions of momentum channels (i.e., self-interaction nodes of the network) because the inflaton charge means that the arm of the 'force couple' must always be of non-zero length, even in the low energy limit, whilst the action of the couple will reduce to similar inflationary 'forces' at the points of application in the same limit. Therefore the context for the emergence of local forces includes, quite generally, a probability of a local turning moment due to a non-zero inter-vertical separation in the 'foldings' of the string, which itself expresses the nonlocal torsion of the superspin. Every element of our ontology will have different Lorentz-invariant projections for differently moving observers, which can be thought of as rotational foreshortenings of the linear momentum four-vector in 4-space where the total scalar energy transforms in every frame like a different time, and these transformations will converge to a common extremum setting a lower bound to the action of every interval which is the inflaton potential. But importantly this common extremum will have only real vectorial expressions for real observers. The constant inflaton potential acquires a constant metrical expression only in a theoretical scalar limit, and such a global/imaginary limit of the network is not approachable as a local/real state. In other words the 'Planck scale' is not a natural unit distance. The inflaton potential is a constant of a scale-free unit distance in this theory because unit distance is just l/2.

xx.) From one point of view this is because the superspin inflaton potential is an action. It is a vectorial constant of all real projections of unit distance, an expression of the specific rotation which we introduce again as a phenomenological factor set at 1.0. This will be equivalent to the relativistic 4-distance s² normalised for natural units of h, c and l. There is a real value to be discovered by experiment which will be different for every real <c observer, and this is the projective mapping of unit distance as a population of actions transforming onto one another by 4-rotations. Among this population of improper dilations of unit distance relative length scale emerges, expressing the proportionality of superspin to wavelength in terms of intervals of time. This 4-rotation contains the idea that there is a uniform translation of all objects through 4-space at c and that every real action containing a velocity < c is a transform of a 'hidden' angular 4-momentum. The domain of real dilated distances is therefore bounded by a limit at c in which no linear momentum has an observer transform as an angular momentum (i.e., an interval of particle translation is null, or s² goes to zero), and a limit at h in which no angular momentum has an observer transform as a linear momentum (i.e., particle 'intrinsic spin' is not a dynamical SR variable). The conservative domain between represents spacetime. But according to our conception 4-space becomes a device for modelling the intricate assembly of network spins in terms of averaged continuous functions. The underlying transformations of unit distance should be applied not globally but case by real case.

xxi.) Superspin therefore has a projection as a distance, in exactly the limit where angular momentum and linear momentum become identical; but this is not a limit of scale. That is, it is not itself a distance scale. It is a scale-free unit distance that occurs as the rotational 4-momentum of all linear 3-momenta. The limits of this projection occur always as the scalar half-wave fundamental mode of any string segment. All linear projections of all segments transform relativistically as 4-space rotations which are changes of momenta or foldings at the vertices of the self-interacting network. These local changes are by definition the real intersections or real nodes at which real spins also change. There are only three real fundamental spin modes disclosed in these interactions: zero, one half, and one. Other internal virtual spins and external phonon spins occur as multiples of these modes. The spin-2 phonon, for example, always includes one real change of momentum involving an electron-electron, electron-positron or electron-photon event at one network vertex, and the included SR angle is the generator of local spacetime action invariance, whilst the induced 'gravitational' coupling represented by spin-2 also appears locally but is generated globally as one pole of a 4-rotational dipole transform of the spin-0 inflaton. So gravitation, instead of being a tensor function of a scalar energy, becomes a function of vectorial changes of energy (changes of momenta) and so is radically generally relativistic in the relations of 'particles'. The tensor becomes a vector gradient of these discrete vertices, and so models an emergent statistical distribution. Each vertical change is associated with a renormalisation of h or c by an incremental rotation of the superspin gauge, allowing 'gravitation' to be expressed as a dispersion of linear momenta through a 1-dimensional vacuum of discontinuously-varying 'refractivity', generating an intricate quantised 'curvature' (folding) in a fractal global dimension.