2. Ontological Basis of a Generalised Superspin Symmetry

i.) It is evident from 1. i.) and 1. ii.) above that an electron conceived as a local object is unable to support spin as an intrinsic period in any degrees of spacetime freedom available to it. Moreover the two-valued quantisation of this spin is classically unphysical as a gyroscopic rotation, being more like a polarisation, although that description is usually reserved for light-speed photons. So the orbital angular momentum correction demanded originally by the Paschen-Back effect in the Zeeman components of the yellow sodium line, by the Einstein-de Haas anomaly, and so on, cannot be accounted for by treating an electron as a particle with classical magnetic moment and angular momentum. Intrinsic spin is not even a properly localisable quantum variable either, since it becomes well-defined only in ensemble measurements. It is possible to conclude, then, that an electron may not 'have' an intrinsic property called spin, and this is a rational corollary of the fact that there is no evidence of local internal electron structure at each point where one half-unit of 'spin' is measured. But might there be a 'nonlocal internal structure' part-revealed at each of two points (let us call them nodes) where spin is measured? This would be equivalent to saying that a 'hidden variable' responsible for determining a binary quantity called spin is interior to 'an electron' whose interiority is redefined as a nonlocal object. (The discussion is of course abstract and this static 'object' gives little insight as yet into the form of any dynamical treatment.)

ii.) In this way we are attempting to build a nonlocal quantum object whose nature is always to interact at two points or nodes characterised by reciprocal spin vectors, which vectors enjoy a general transpositional symmetry. We will look more closely at this nonlocal object in a moment; but since we do not see its transpositional symmetry respected except in special cases (EPR-type correlations), the assumption would be that the metasymmetry is in general broken by interaction (meaning almost always, at the 'classical level') but that it persists in atomic Pauli spin pairs and may in special conditions be preserved over long distances in EPR pairs. A second assumption would have to be that although the metasymmetry is broken at arbitrary scales for the majority of these nonlocal objects (conventionally-speaking, these 'pairs of electrons') there will be something in the theory that mediates the hidden (broken) superspin symmetry. It is natural to suggest that the something which hides the broken spin or polarisation symmetry is the photon, which would make the carrier of superspin symmetry identical (or at least congruent) with that of charge. Briefly, the suggestion offered in Part 4 is that the superspin symmetry would be hidden in an imaginary 'super-rotation' of the plane of polarisation of a (linear polarised) photon. Although the photon would carry superspin it would not 'see' it, due precisely to its super-rotation. It would 'see' the restored superspin symmetry normalised (in null proper time) to that of a spin-zero scalar 'particle'; but an imaginary torsion in this particle (our nonlocal elementary object), which may a priori be any arbitrary fraction of phase, is preserved as a scalar potential in the limit of an electromagnetic vector field which is emergent in the breaking of superspin symmetry (See 4. iii. below). This could be thought of as analogous to a quantum 'Faraday effect', where instead of a magnetic field rotating the polarisation plane (continuously in Cartesian coordinates) of classical light waves traversing a dense medium, we have a 'superspin field' which is the zero sum of imaginary discontinuous polarisation-rotations of individual almost-monochromatic light quanta whose angles are normally regarded as constant under Lorentzian translations in vacuum (as found in Malus' experiment for classical waves, where the plane of linear polarisation is independent of the separation of the reflecting plates). One might then say that the fundamental quantum unit of boson spin, h/2p, has a function crudely analogous to Verdet's constant in the theory of magnetic rotation, so that

q/l = (h/2p)Hl

(2.1)

where wavelength l, pathlength l, and 'field strength' H are all normalised to unity and where q is like the specific rotation of the vacuum. (A connection to spacetime theory can be made by describing each quantised 'lurch' in the 3-space of paired spin orientations as a torsional stress in a properly-null geodesic interval whose projection on 4-space becomes a line element of the Einstein curvature tensor. Again, see 4.iii. et seq. for development of these issues.)

iii.) In these very general heuristic terms the picture that emerges makes sense of the spin-balanced, symmetric, two-electron atomic wavefunction, and of the related EPR-type pseudobosonic entangled pairs found in superconductors, where Pauli exclusion and Coulomb repulsion respectively are similarly suppressed. [Note 1] Indeed atomic electron pairs are more than casually similar to inertia-free BCS pairs in a cold superconductor. The electromagnetic gauge symmetry can be regarded as broken in a superconductor, electrons losing inertia and photons acquiring it. The implication is that the electromagnetic gauge symmetry is reasserted only where the general superspin symmetry is broken, and that it is in the emergence of the electromagnetic gauge symmetry that electron inertial mass also appears, in a way which can be regarded as an incorporation of the spin-zero scalar particle mentioned in 2.ii. above. (In this analogue of a Higgs mechanism a Goldstone boson in effect generates inertia through the radial form of the spin-zero scalar 'field' mentioned in 1.xvi., and so 'donates mass' directly to electron pairs through their photon coupling. Since this is an emergent property of the electrodynamical self-interaction of the 'field' there would be no intermediate massive Higgs boson in this theory. See 4.iii.)

Note 1. Kiesel, Renz and Hasselbach [37] have recently restaged the classic Hanbury Brown -Twiss photon correlation experiment using free electrons and find that anticorrelations are observed as predicted for antisymmetric fermion wavefunctions, i.e. these free electrons obey Pauli exclusion. Samuelsson and Buttiker [38] have recently proposed that superconducting electron pairs injected via a quantum dot into a pair of normal conductors would behave differently. Their calculations lead them to expect that as pairs split to enter the two conductors they will give rise to positive current correlations typical of Bose-Einstein behaviour.

iv.) Immediately the question occurs: 'Given that mass is held to be a fundamental property of a particle at rest and is a measure of its inertial coupling to some coordinate point of space, what would it mean to assert that our binary electron had a rest mass at two points of space?' It would appear to mean as much as to say that it had half a quantum unit of spin at two points of space, except that mass is conventionally a scalar lacking even the rudimentary 'up' or 'down' vector states of spin. But notice that there is a mass vector, whenever a system of particles is considered. In this respect mass, just like spin, acquires a well-defined vector during a participatory 'measurement'. Both are context-dependent. The mass vector is a directed 'force' which appears as an acceleration ('inertia', or 'weight') in the relativistic symmetry; but the local particle description reduces an energy-momentum four-vector to a scalar mass point for theoretical purposes, and thereafter spacetime geometry has to be substituted in order to recover mass as a directed quantity. Note that a pair of electron masses represents a constant action in any relativistic transformation, which can be represented perfectly consistently in terms of the locally-conservative Lagrangian of one homologous nonlocal object (see 2.v. below). In fact every un-ionised atom in the universe heavier than helium is a demonstration that the 'natural' minimum-energy condition for a pair of spin-balanced charges is a nonlocal state in which the function of a metric to define an ontological separation of points in space loses precise significance.

v.) But what are we to make of a fundamental nonlocal object as an element of a theory that has to give local answers? A local theory must be Lorentz invariant under transformation from frame to frame, but Lorentz invariance does not preclude 'absolutes' of which it can give no account. For example it does not preclude the existence of a preferred frame such as may be related to a hyperplane of constant time (or indeed an 'ether'). In fact the Lorentz group rotates around precisely such a frame defined at c, which is why special relativity demands that any elementary object must be nonlocal: Lorentz-invariant systems alone may be local, and a Lorentz-invariant system cannot be an elementary object. A classical particle or a quantum point particle are both nonlocal elementary objects, the latter being simply an extremal form of classical rigidity retained in the limit of SR. But although SR excludes extended classical rigidity it does not exclude extended nonlocal objects, and therefore does not limit an elementary object to a limit of scale, provided only that the elementary object is Lorentz-invariant insofar as it may be 'observed' with photons (or more generally, insofar as it is an interaction within the larger symmetry group of QED which includes spin). In other words SR allows nonlocal objects that mechanically transform with, or 'underneath', their local labels (i.e., an ether). Whether such a covariant nonlocal object has a useful theoretical function is a different issue; but in principle SR does not require its absence. Therefore the reason why point particles are acceptable (classically speaking) as a nonlocal element is not primarily that they don't transform, although they don't, but is for the subtler related reason that, being zero-dimensional, they will always interact zero-dimensionally. But one-dimensional objects may be constrained to do this too. This then is the general condition that nonlocal elements cannot violate consistently under SR, irrespective of scale: They must only interact with one another zero-dimensionally.

vi.) We can also show that applying this principle to one-dimensional elements irrespective of scale is the same as giving a generalisation of the Pauli exclusion principle, which, as expressed in standard QM, would now be retained only as an extremal case for short scales. These short scales assume a disproportionate significance in standard QM (we would now say) because of the focus on small local differences between terminal values of nonlocal objects which in fact have arbitrary scales.(As suggested in 1.ix. the same focus led to a failure to generalise the atomic Planck oscillator and thus eventually to the conceptual ambiguity of the quantum field.) Thus the spin-balanced symmetric wave function of a Pauli pair with the same orbital quantum number describes the unbroken spin-symmetry in an atom A between two nodes A' (up) and A" (down) which are also nodes ('ends') of n other nonlocal objects of arbitrary scale. Considering any two of these n other objects we can assign additional quantum numbers to them describing their scales and their relative polar coordinates as radius vectors. If these numbers are the same (within the brackets of quantum uncertainty) then we will find that their opposite terminal nodes will also form an equivalent Pauli pair B' and B" on some coordinate point we might designate atom B. Then [A'(up)«B'(down)] and [A"(down)«B"(up)] form two antiparallel complements of spin vectors, and the characteristic scale of Pauli exclusion which we take to define atomic scale is seen to be just the lower limit of a dipole (see also 3.vi below) which operates on any scale. Two binary objects like A'«x' or A'«y' may share one common node, but evidently they are unable to share two common nodes without being the same object, which is to state in more vivid terms the same exclusivity required by the condition that our two nonlocal objects may only interact (i.e. be in the same positional state) at their ends. Evidently it is this condition which imposes locality on a system of such objects, and it may be legitimate to go so far as to say that it is our binary construction which, by thus requiring a system of emergent angular relations, is the origin of the SR symmetry group as the invariance group for observables. (See Part 5.)

vii.) For our purposes, then, an elementary nonlocal object whose observable spin-dynamical states exist only at its ends does not conflict with SR because it behaves as a 1-space and interactions are linear (see 1.ix.). Similarly, some string theories with 'open' strings work by carrying their quantum labeling on the free ends, so that the strings only recognise one another end-on. Notice that this is an interesting scale-model of the nonlocal binary objects we are considering, which likewise would carry their quantum numbers on their ends. In general, string theories using extended elementary objects can be kept local because interactions only occur at a 'point', even for closed strings. Of course these theories are formulated down at the Planck scale, but the scale itself becomes significant in these theories for quite other reasons that have nothing to do with relativistic invariance. Within the limits of our heuristic model as it appears so far, we can make our nonlocal objects on any scale and because SR is only concerned with the Lorentz-invariance of these ends it is indifferent to nonlocal correlations of the states that are not Lorentz-invariant. It does not have anything to say about a space-like relation of these two states provided that they are consistently labelled. It simply doesn't explain this consistency. Superspin symmetry would be what absolves SR from having to explain such correlations, which it would have to do (self-destructively) if such correlations were the general rule. One could say that superspin protects SR symmetry from 'correlation pathology' by carrying the broken symmetry away invisibly (analogously to the way that the neutrino carries away the energy and angular momentum in electroweak interactions to protect conservation symmetries in beta decay.)

viii.) This is equivalent to a no-signalling condition for EPR pairs, but although spin entanglements are usually considered as the type of what Shimony [39] called 'passion at a distance' it is a case of a general principle that is robust for all quantum labels, without ever becoming pathological action-at-a-distance. Action-at-a-distance becomes a paradox from the point of view of SR if instantaneous actions are observable within SR (i.e. by local observers). Since the finite value of c in SR denies the possibility of instantaneous measurement for all observers ruled by SR no such paradox can arise within SR. This does not deny that in a preferred frame, such as that of c, all actions are in fact instantaneous. Again we can see that it is our 'end-on interactions only' locality condition, expressing a broken superspin symmetry as Lorentzian displacements, which protects SR from paradox. Or alternatively, c protects our binary locality condition from infinitesimal collapse; or h is the non-zero constant of superspin symmetry which prevents c being an infinite constant of Lorentzian symmetry. In general, what prevents the snake swallowing itself by its own tail is that any set of states found as measurements of end points by an observer moving < c will be by definition locally Lorentz invariant under SR without regard to their nonlocal object-connection, which is simply irrelevant as it does not contain a real time - just as a point particle does not contain a time, or as an ether does not contain a real time. However a point particle does not contain anything else, either, and is therefore so uninteresting as to be useless; whilst an ether contains altogether too much and destroys the clarity of our binary locality condition. In other words, as long as there exists a 'preferred frame' in which the null connection state of zero time appears, a linear object may be both sufficiently elementary and sufficiently interesting by virtue of carrying a pair of labels at A and B, which must be symmetrical under a nonlocal longitudinal rotation such that |AB| may be set effectively at zero within SR. This condition is represented by the photon speed c. It would be problematic if such rotation produced an asymmetry of mass or charge under vector transformation; but rotation within |AB| of electron mass and electron charge produce no local change of state as long as they are scalar magnitudes which for the purposes of SR are wholely specified locally/relativistically and the total energy of the Hamiltonian remains constant. Again SR does not explain mass and charge; but again it is not required to. The causal topology for a network of such pairs might in principle look very interesting in the 'preferred frame' in which the broken spin symmetry is repaired (which is the imaginary frame in which the photon super-polarisation would appear); but since the space-like null hyperplane of c can only be objectified by a wholly imaginary observer there is no deterministic local history involved and therefore no causal conflict in any particular case.

ix.) Problems with nonlocal objects only appear in local field theories which have to be defined at all points of a continuous space, when a number of those points are required to be occupied simultaneously by the nonlocal object. Our nonlocal objects are not going to be part of any such conventional field theory because they pre-empt the notion of a continuous background space. (One might say that they interpolate between those "points" rather than occupying them.) There are traditionally relativistic difficulties with extended fundamental objects in conventional Faraday-type field theories because events at different points on the object which are spacelike-situated might interfere with one another and violate causality. As mentioned, this difficulty is elided in superstring field theory by ensuring that when the string loops interact they only ever do so at a point, so that the spacelike-separated points do not disturb one another. (Having ensured that photon vibrations, for example, will not propagate down the string faster than light. This is from a string theory point of view just the same as decreeing a global locality condition for the spacetime in which the strings move, whereas in our proposal there will be no such classical manifold and the condition has to arise from primitive relations of observables.) The extended objects suggested here would differ in that they do not ever not interact, in fact they interact exhaustively; but they too only interact at a point (or two points, one at either 'end') and indeed they promise to define the meaning of a 'point' by their interactions. (Taken in the cosmic limit our nonlocal object plurally becomes space.) This may be another extravagant feature and problematic in terms of general relativity; but it may also be regarded as desirable that a continuous background space should be avoided if possible along with the potential problems of a nonlocal field theory.

x.) But, so far, so good (?): We have a fermionic one-space object with a two-valued nonlocal spin symmetry which appears mediated, when torsionally broken, by the gauge boson of a resulting local symmetry. The torsional breaking of EPR-type spin symmetry appears, arbitrarily with respect to the "up/down" spin vectors of the two fermionic "ends" of our object, as the imaginary rotation of the plane of polarization of an individual photon. We have yet to consider the case of an ensemble of these objects, but on general grounds we can see no reason not to expect that the manifest local symmetry group will remain the Lorentz group of electrodynamics, within which the broken generalised spin symmetry is carried as a hidden quantum variable, the imaginary rotation of the photon polarization. Insofar as polarisation defines space dimension, is it possible that introducing an imaginary angular momentum variable in some complex reformulation of SR might reproduce the tensor components of GR from super-rotations of the photon coordinate frame, as suggested in 2.ii.? This is of course no more than a pious hope; but that it is a hope at all can be taken to justify our decision to find a rationalisation of spin without resorting to the augmented coordinates of conventional supersymmetry. The reason for this is that the Lorentz transformations - both as formalised by Lorentz and as reinterpreted by Einstein - are historically and logically prior to their expression as rotations on the Minkowski [40] spacetime manifold. This geometrical representation was a rite of passage for a three-year-old SR and was subsequently reified by Einstein in the curvilinear space coordinates of the general theory. But this metrical continuity is in fact, like standard QM, a functional procedure, albeit more intuitive, derived from discrete observables which are primary. Hence the reason that attempts to quantise the manifold retrospectively have created legendary difficulty may be that they have attempted to quantise a procedure, not the primary actuality. As pointed out by Consoli and Siringo [41] GR can be considered an elegant device for calculating weak-field corrections to space and time measurements inside some some other theory to which the Equivalence Principle must be built in. GR expresses the experimentally-validated EP, but lacking an inevitable Machian form it does not produce it, so that tests of GR which only verify the EP and SR, neither of which depend directly on the GR tensor, are not necessarily evidence that the GR metric tensor formalism is fundamental. If so then an ontological interpretation of 'intrinsic spin' which eschews both the metric manifold and its still-further augmentation by abstract supersymmetric coordinates is at least not absurd on that account. An ontology of spin may not (yet) be a theory of gravity; but then GR is not a theory of quantum spin either, and there is no very obvious reason to expect that it ever would be. On the other hand a model of space distance which gives exactly the same metrical formulae as the non-Euclidean differential invariance of GR can be derived from an 'old fashioned' algebraic invariance suited to just such a construction of scale-free linear objects as I am suggesting here might rationalise 'intrinsic' spin (see section 6.ix. et seq.).