5. Superspin Interpretation of the Field
i.) When we try to understand spin in conventional QM the spin eigenstates are in a first approximation degenerate and commute with the energy eigenstate, the Coulomb force being spin independent. This degeneracy is removed when the electron is considered to be 'in a magnetic field' and the available states acquire different eigenvalues corresponding to opposite orientations 'in the magnetic field'. In our terms the two-valuedness of our objects is the doubling of the space of spin eigenstates, but the reason for the non-degeneracy in eigenvalues is less clear, as is the meaning of this condition called 'being in a magnetic field' itself, since the heuristic model we are exploring not only contains no classical current loops but also deprecates the reality of any continuuum of point potentials.
ii.) One is encouraged here by the fact that physics does not contain any really well-defined idea of what magnetic field is other than some phenomenological description of relative speeds and distances of 'moving charges'. If the field is just a convenient device to generalise calculations of the 'force' on an arbitrarily moving test charge, then it will obviously be possible to discuss magnetic forces solely in terms of the relative motions of charges. In Maxwellian theory this is difficult. But if classical electrodynamics is adjusted so that this is possible, what we get is exactly the special theory of relativity! Given Coulomb's law just about the whole of electromagnetic theory can be derived from SR. (Einstein [58] himself stated that SR grew directly from intuiting that a magnetic field was only an electric field in a moving frame.) The magnetic 'field' then is just a Lorentz-invariant transform of the Coulomb field for moving charges ('observers'), which has a special interest because, even though it is local, the force depends everywhere on the velocities of all charges. So what is the meaning of an atomic orbital's orientation, or an alignment of electron spins, 'in a magnetic field'?
iii.) Evidently its only practical meaning is a direction in relation to some specified drift of charges. In an example given by French [59] the drift velocity of charges in a typical wire carrying a 10 amp current would be barely perceptible to the eye (a couple of metres per hour!), yet for a test charge moving alongside the wire at the same drift velocity an unbelievably tiny relativistic Lorentz contraction of the inter-atomic distance of about 1 part in 1023 alters the positive/negative charge density in the wire sufficiently to produce a significant 'force' of magnetic 'attraction'. The magnetic field is entirely relativistic. From an SR point of view it would be more empirically transparent to say that a motion of some test charges, or a certain alignment of electron spin vectors, is some region of a magnetic 'field' rather than being 'in' a magnetic field. Thus an intimate association of spin-orbital magnetic moment with the structure of SR spacetime that I want to bring out is very natural. The extension to an association between superspin and (in effect) GR spacetime may not at first seem so natural!
iv.) A 'fundamental' value of this magnetic fraction v/c occurs as the spectroscopic fine structure constant (alpha ~ v/c @ 1/137), which can also be expressed in terms of several other combinations of 'fundamental constants'. One definition is that it is the electromagnetic coupling constant, whose smallness makes atoms 'large' and their electrons loosely bound, and makes it possible to calculate their behaviour in a non-relativistic approximation. In this treatment, alpha determines the magnitudes of the available orbital angular momenta and hence the Zeeman absorption lines. But it can be considered established that this ratio v/c is not a classical gyroscopic velocity, although v/c does represent a magnetic moment. In an illustration given by Eisenbud [60] the first of the two D0.01 Ängstrom position measurements required for even a rough specification of electron orbital velocity would lead to a momentum uncertainty equivalent to perhaps 1000 times the ionisation energy. Going on from what should be called the 'gross structure constant' to the line bifurcations of 'intrinsic spin' eigenstates leads to greater difficulty. What kind of a 'rotation' now remains? But from a network point of view the question is not: 'How far does an intuitive model of classical moments carry over into weird quantum particles?' but rather the opposite: 'How does this weird magnetic moment arise in an intuitive theory of linear objects?' Or, what turns the effective scalar field of equilibrium charge into the non-equilibrium vector field of electromagnetism? Or in still other words, how does Lorentz-invariance emerge out of a broken superspin symmetry?
v.) Under local measurements, each of our nonlocal objects has a basic two-valuedness of position which also entails a basic two-valuedness of spin, expressed at opposite nodes reciprocally. The simple symmetry group of rotation for this 'intrinsic spin' is in general broken for these local measurements in such a way that 'up/down' acquires an indefinite number of possible local orientations defined with respect to the local magnetic field. When the local magnetic field is weak its direction for electron spin generally is the direction defined by the total angular momentum vector which is the invariable axis of the atom, so intrinsic spin can be approached crudely as if it were a correction to the allowed orbital angular momentum eigenvalues, a small quantitative increment of the same kind. But this is not right. The orbital angular momentum eigenvalues, though integer quantised, are local dynamical variables that can have infinitely many measurement outcomes; they are in some sense energy states of local space rotations, even though the spectrum of possible states is not continuous. Even though a meaningful electron orbital velocity is unmeasurable in principle the relativistic ratio v/c does determine the orbital magnetic moment as though for a ballistic particle. But the intrinsic spin eigenvalue is qualitatively quite different in that it is fixed, and there is no way at all of representing it as a Lorentz-invariant rotation. The correct way of looking at it, therefore, is to say that this nonlocally intrinsic property called electron spin is carried over as a component into a more complicated group of rotations which emerges in the locally broken symmetry, but it is not itself a local dynamical variable.
vi.) To labour the point, the unanswered question is this: If intrinsic spin is not a local dynamical variable, that is to say if it does not transform with the symmetry group of SR; if it doesn't have a spectrum of eigenvalues in a magnetic field; if it is an imported constant action whose direction alone is set by the magnetic field; if all these are true, intrinsic spin cannot be an electrodynamical phenomenon except as it supervenes on electrodynamics in some limit. Given this, what then is it that removes the degeneracy of the doubled eigenstates in the first place? How can a fixed quantity be said to couple dynamically to a local field without coming unfixed and yet avoid violating energy conservation? The approach taken here is that the problem seems to be clarified if we propose that this is all back to front: Direction is fundamentally quantised in the structure of the network. Direction is not 'set by' the field; direction is rather the real essence of the quantisation condition. The imaginary 'field' is a projective medium useful in the mathematical treatment of discrete directions.
vii.) Intrinsic spin is therefore the more fundamental property, and does not belong inside SR. We propose that intrinsic spin is a glimpse of what generates SR. Contrary to what the relative scales of nested spectroscopic multiplets might superficially suggest, intrinsic spin is not a 'fine correction' to the relativistic orbital energy, any more than GR is a 'fine correction' to Newtonian gravity, or quantum theory is a 'fine correction' to classical mechanics. It opens a different window on nature. This perspective would explain the strangeness of how a vectorial component of a nonlocal statical constant can appear to be 'set' by a local magnetic field. The answer is that its direction isn't 'set' by the magnetic field; rather the magnetic field emerges as a local map of spin alignments and, through the spin-orbit coupling, atomic orbit alignments. In general 'the magnetic field' occurs as the breaking of the superspin symmetry, which remains imprinted in its local transform as a common limit of action underlying all projections of unit distance (i.e. all observer-specified relativistic times). The spin-orbit coupling does not arise in the first ('strong field') case which is equivalent to the angular momentum of our nonlocal torsion remaining hidden in the 'superspin field', i.e. as electron intrinsic spin. The projections of the superspin and the orbital spin vectors on the field are quite independent. Although this is called the strong field case what we are really seeing is not the application of a strong ordering to some weak, arbitrary states, but on the contrary the damping or reducing away of the Lorentz-invariant local spacetime order to reveal a strong spin-correlation order in its limit. The underlying symmetry breaks first to a spatially rudimentary, strong spin-correlation case, preserved in special domains such as EPR pairs. The emergent local symmetry is a spatially complicated, weak-spin case, the symmetry of electromagnetism which holds between the Ö(2N) nodes/origins which represent 'measurements' of electron states for N objects. Among sets of these electron states, considered as moving charges, there emerges a relativistic 'drift' of current whose 'magnetic' torque defines a collective rotational direction. From this point of view we can see that the 'orbital' angular momentum begins as a transform of a set of linear momenta, and its quantisation can be seen as a consequence of the fact that angular relations are not secondary selections made from some pre-existing field of all possible space-rotations; rather, angular relations are assembled out of linear elements (or in the folding and refolding of the string under the locality condition of its self-interaction, if we wish to put it this way). Intrinsic spin thus enters as a relativistic term, but it is not a special relativistic variable; it is a limit on locality but is not itself local; it is a determinant of magnetic field orientation rather than a response to it.
viii.) Now it is possible to say that insofar as spacetime is defined as the Lorentzian manifold of special relativity then it is dynamically constituted entirely by the 'magnetic field', because the electric field of a system of static like charges in equilibrium is effectively a scalar. 'Effectively' because a field in which a test charge placed at some position feels a directed force is formally a vector field; but this test imports tacitly the non-equilibrium dynamical transformation which is assumed for any real case, and the vectorial character of the field is emergent only in those 'placements of test charges' etc. which are actually electromagnetic. In an imaginary Coulomb field of undisturbed like charges in equilibrium there is by definition no preferred direction, no preferred observer frame and no concept of velocity, just a field of equidistant charges whose number on the surface of a spherical shell of radius r centred on any charge O goes up as r2 whilst the strength of the force centred on O goes down like 1/r2, and it is only the relativistic dynamical transformations of the Coulomb force law associated with non-equilibrium motions that produce the magnetic vector field as the fraction v/c of the electric field, due directly to the constant finite speed of light. This gives us the flat Minkowski manifold of SR along with an electric field which is actually a dipole, so now it is a vector field with emergent relative scale. Relativistic electrodynamics is coemergent with the dipole character of the 'field'. The implication of our new view would be that this happens because the dipole is itself generated in the transformation of monopolar supercharges (inflatons) to a magnetic vector field. This would allow us to explain 'positive' and 'negative' charges as phenomenological labels attached to local mappings of one nonlocal action which is itself a neutral scalar. The justification for this procedure is that it conforms to the emergence in parallel of a 'gravitational' dipole with positive and negative signs, placing the mass field in the same symmetry with the charge field.
ix.) It is not difficult to defend in casual terms the implication that spin is more general and more fundamental than SR spacetime. SR preserves the angular momentum of all rotations for all local observers as it preserves the action; but it does not itself give any account of the origin or significance of angular momentum. It is a description of electromagnetic rotational moments that do arise, but it does not say why there is such an interesting dynamical equilibrium in the first place. It is not an explanation of why the universe isn't describable as an homogeneous and isotropic scalar field of overall zero potential - or in other words, it is not a cosmological theory. Obviously it doesn't attempt to be such an explanation; it is only required to describe magnetic transformations of an electric dipole field of varying potentials without explaining where this dipole comes from. This limitation of its account of angular momentum in terms of the magnetic field is in fact explicit: It doesn't 'include gravity' - i.e., the electromagnetic field is spacetime with gravitational mass and inertia reduced away. However, a network model suggests that this dipole would be coemergent with SR space scale just as a gravitational/inflational dipole is coemergent with space scale. One certainly expects that positive and negative charges should unify cosmologically since they are relativistically transformable in the CPT mirror, and do indeed physically unify (as neutral 'charge', say as a photon, and in an extremal mass/charge limit) and it would be satisfactory that they should unify inside a theory which is also a theory of gravity and reconciles the dipole/monopole disparity of charge and mass. Our network model implies an electromagnetic charge dipole which is a relativistic transform emergent with the magnetic field and rooted in a neutral monopole inflationary 'field' which also gives rise to a mass-charge dipole. In both cases the local sign of the charge (a phenomenological label) would be nonlocally context-dependent and correlated with the emergence of metrical scale. This is why we do not in general encounter 'negative mass' (with the theoretical exception of particle-antiparticle pairs as indicated in 4.xiv): Only an 'experiment' on a cosmically significant scale would need a theory in which m changes sign for macroscopic bodies.
x.) A negative scalar inflationary field of electrons would be indistinguishable from a positive scalar inflationary field of positrons. Unperturbed, this indiscernability would be an identity. The meaning of displaced equal and opposite charges is a relational meaning bound up with the meaning of scale. The relational meaning of scale is by definition not an absolute and must be an emergent meaning, and the selection rule must by definition be connected with the emergent spacial ordering. It is very natural therefore to suppose that a selection rule governing the emergent electrodynamical dipole is connected with the coemergent dipole of inflation/gravitation, which is considered fundamental to the spacial ordering of inertial mass-energy. In itself this 'unification' of electrodynamics with gravitation is just an obvious desideratum. But normally when considering the origin of inertia one looks to some variant of GR cosmology to supply the necessary negentropic ordering, and the result is a somewhat complicated redundancy of neo-classical fields, further textured by superposed quantum fluctuations, and then tweaked with global inflation, dark matter, dark energy and so on, not to mention unknown physics required to account for the cosmological constant problem and evidence of fractal galaxy distribution. By applying our nonlocal-object network model, however, we get the idea that an inflationary superspin which is prior to quantum electrodynamics offers the prospect of a tranformation of SR which captures the essential function of the conventional transformation SR ® GR but which might give answers of a novel character to such questions as why so much of the mass of the universe is spinning, why gravitomagnetic forces arise, why the mass distribution looks self-similar, contradicting GR-based cosmological models [61], why these models do not elegantly represent the gravitational dynamics on galactic and cluster scales etc. (see Parts 6 & 7). The essence of the proposed duality of such a model with an effective field theory version of GR would be that the projection in 3-space of an orthogonal 4th-dimensional curvature between two points is cognate with a torsion in a null geodesic connecting them. Thus the Einstein stress tensor would be identified with the restoring potential of a broken superspin, and the string-segment would be represented as a geodesic line element of the projective 4-space surface.
xi.) As to what prospect there may be of turning such an idea into a useful formal theory, it is worth pointing out that since about 1860 there has existed the basis of a complete algebraic alternative to the now-familiar holomorphic spacetime of GR in the form of the Cayley-Klein construction of 'distance'. Cayley [62] showed that cross-ratios of linear ranges of points have an invariance under projective transformation which is analogous to that of Euclidean angles and distances. The cross-ratio of four points is crucially dependent on the order of the points, which defines a sequence of operations determining either a negative or a positive result, and the same applies to a pencil of rays projected from these points, which are said to be harmonically related when their cross-ratio is -1. Cayley defined distance between points on a line in terms of an equation of angular measure whose odd property is that for two points P and Q it yields a natural unit distance d(P,Q) corresponding to p/2 when the coordinates corresponding to P and Q, which may be p and q respectively, necessarily satisfy the equation pq + 1 = 0. (Notice that in the special case where P and Q coincide and the distance d(P,Q) is zero, p = q, and therefore since pq = -1 both p and q become imaginary.) Since Cayley's equation contains the term pq + 1 (equal to zero) as a numerator, the distance d(P,Q) becomes proportional to an angle whose cosine is constrained to be zero, and so the very strange situation emerges that whatever the coordinate separation of P and Q they are always at unit distance p/2. Cayley's equation may be proved to satisfy the expression d(P,Q) + d(Q,R) = d(P,R), which is the common-sense conception of additive distance where (say) two half-metre intervals make one metre interval; but in Cayley's conception of "distance" all intervals become expressions of the same projective invariant.
xii.) What does Cayley’s invariance theorem have to do with a model of gravitation based on the invariance of action among a network of linear nodal relations? The trigonometrical and geometrical formulae obtained for surfaces and higher figures on the basis of the Cayley-Klein definition of distance are exactly those obtained in the non-Euclidean geometries of so-called "curved space". It was Klein who followed this emerging trend, a trend which accelerated especially in Germany at this time owing to the work of Riemann. Cayley himself was not very inclined to take seriously the idea of the violation of Euclid's "parellel postulate" in curved space. And he may have been right. The most telling argument is, of course, that in the light of quantum theory the Riemann-Einstein programme is known to be fundamentally flawed in its assumption of a smooth spacetime. It is pertinent to observe that the European pursuit of a geometrical theory of invariance in a sense set the fashion for what finally emerged as Riemann's general analytic transformation. Without this legacy it is possible that the differential invariance that underlies the continuous transformations of fields, and which fell fully-formed into the lap of a relieved Einstein, might not so completely have overshadowed the algebraic invariance of Cayley. It is possible to speculate that a generalisation of the theory of relativity might then have taken a very different form.
xiii.) The implicit functional identity of the Cayley-Klein theorems and the new characterisation of space was explicitly set out by Felix Klein [63], but Cayley was well aware of this identity. Cayley had shown how to extend his definition from the line to the plane, or from the plane to the solid, by postulating arbitrary sections in the higher spaces of which his "coordinates" can be taken as projections in the lower. Interestingly, it happens that the whole of what we might call the space of the absolute or, with justice, the "Cayley space" of plane Euclidean geometry is a complex non-Euclidean space with real and imaginary forms: When the absolute is real the metrical formulae are exactly those obtained for Lobachevskian or hyperbolic geometry (open, negative-curvature, saddle-surface spaces), and when the absolute is imaginary the formulae are the same as those of elliptical geometry (closed, positive-curvature spaces, of which the spherical space of Einstein is a special case). And it is very interesting to see how the Euclidean case, in which space is open and flat, reappears from this complex superposition and cancellation of real-negative and imaginary-positive curvatures. The point is that whilst these abstractions may be technically useful they may be inessential. And if a network theory is true then all continuous manifolds become abstractions, not just those non-Euclidean forms that happen to offend the parallel postulate.
xiv.) This becomes clearer if we say that where Cayley held his absolute surfaces to be abstract when conceived in more than three dimensions, we hold them to be abstract when conceived in less than 1080 dimensions. That is to say, in geometrical terms "the line" may be taken as the Cayley absolute of "the point" (i.e., of the null signal line) and a figure in "the plane" is the absolute of "the line", just as the absolute of the solid is in turn a quadric surface, and so on. But none of these neo-Cartesian abstractions are discovered elements of the world at all; in geometry they are imposed structures, derived ultimately from analogy with structures in nature which are emergent structures. The actual underlying relations out of which they emerge (the quantum condition underlying an effective field theory of gravity) appear to respect a quite different definition of structure, a non-local projective structure of linear elements each of which is in effect the Cayley absolute of a null interval, and whose further projections define emergent spaces of 2, 3, 4 . . . n dimensions. We should regard these generalised geometrical structures as abstractions from merely topological relations and not as prefabricated matrices of coordinates. On the cosmological scale notions such as the "celestial sphere" and the "light horizon" convey rather vividly the impression of a bounded 3- or 4-space volume centred on the observer; but when we look closely at what such abstractions are built up on we see that they are used to generalise the operation of making projections of the null geodesics of specific photons according to certain rules. The portmanteau of these theoretically and observationally interdependent rules applied to the generality of all possible photon projections (or light rays) is spacetime. A topological procedure is generalised as a geometrical principle. This may be very convenient, but it may also be very misleading when it tempts us to construct a cosmology like a global history by joining an infinite number of possible points of measurement smoothly together by means of continuous non-Euclidean functions.