Super-Symmetric Direct Particle Interaction

Introduction

Below is a mostly non-mathematical outline of a version of Direct Particle Interaction with a slightly different starting point to that of Schwarzschild, Tetrode and Fokker. To distinguish it from STF DPI we will refer to this as a `super-symmetric` version of Direct Particle Interaction (SS DPI), though super-symmetry is an outcome, not an input to the theory. Several more detailed outlines of SS DPI can be found on the UT Mathematical Physics Archive, the most readable of these is probably this pdf.

Modification to the Theory of Schwarzschild, Tetrode and Fokker

The crucial modifications to the classical theory of Direct Particle Interaction by Schwarzschild, Tetrode and Fokker are

1. Remove intrinsic mass from the charges. The relativistic Newton-Lorentz equation of motion that would otherwise be part of the theory then goes away.
2. Constrain the charges to move at light speed.

The light-speed constraint is integrated into the action, and therefore the dynamics. It forces each charge to respond to the influence of all other charges (roughly: their EM fields) so as to maintain light speed motion, consistent with the force of those fields on the charge in question.

Before moving on let us address a possible objection that light speed motion is ruled out because it implies infinite energy. This objection is valid when the charge has an intrinsic rest mass rest \({m_0}\) say, because in that case the dynamic mass is \({m_0}/\sqrt {1 - {v^2}/{c^2}}\) and this becomes infinite as \(\left| {\bf{v}} \right| \to c\). The difference here is a starting point in which the charge has no intrinsic rest mass, so this objection does not apply.

Self-Consistency

Notice that there is no modification to the purely electromagnetic part of the STF DPI action, which is the only part of the action that distinguishes the STF theory as a whole from classical electrodynamics. The same modifications could be applied to CED therefore. However, unlike CED, the electromagnetic DPI action is intrinsically time-symmetric. Unlike CED it demands accommodation of the action of charges on the future light-cone of a nominally local charge, in addition to those on its past light cone. In other words, a charge in the present time can be respond to the influence of charges at both future and past times. Or again: the present is shaped by the past and the future.

This peculiar quality of DPI was always present in the STF presentation. The historical focus on the accommodation of purely retarded radiation in DPI can be understood as an attempt to eliminate - by contrived cancelation - any effect of the future on the present that is intrinsic to the theory at the microscopic level of individual pair-wise interactions between the charges.

It turns out self-consistency is quite easy to achieve with the modifications above. Whether or not self-consistency could have been achieved without the modifications above, i.e. with an intrinsic inertial mass satisfying a relativistic Newton-Lorentz Equation of motion, is unknown.

In any case the outcome with these modifications seems quite favorable. Two distinct modes emerge of the self-consistent sum of pair-wise interactions. If these modes are characterized as fields (necessarily approximately so in the case of DPI), then at a nominally local source they are either entirely time-symmetric or entirely time-anti-symmetric. The time anti-symmetric fields satisfy Maxwell-like equations though with a finite number of field degrees of freedom. And in the case of a single local charge a particular projection of the time-symmetric fields is the Dirac wavefunction, satisfying the Dirac Equation.

Super-symmetric Direct Particle Interaction

In summary, the new theory leads to something that is similar to Quantum Electrodynamics, even though the starting point is a classical implementation of Direct Particle Interaction. The diagram below summarizes the outcome.

Matter Modes

For isolated charges the matter modes can be `projected` onto a wavefunction that satisfies the Dirac Equation

\[\left( {{\gamma ^\mu }{\partial _\mu } + im} \right)\psi  = 0\label{Dirac}\,.\]

The \({{\gamma ^\mu }}\) are the usual \(4 \times 4\) Dirac matrices, in any representation. \(\psi\) is the Dirac bi-spinor as a column vector of 4 complex elements. A departure from standard quantum theory here is in the meaning of \(\psi\). In SS DPI \(\psi\) is electromagnetic. More specifically its elements can be written in terms of the 4-potential, the electric field, and the magnetic field of the modes that are time-symmetric at the local charge. In other words, \(\psi\) can be written in terms of self-consistent time-symmetric electromagnetic fields.

The interpretation of \(\psi\) engendered by SS DPI is a relativistic generalization of the Bohm model, wherein the single particle wavefunction is the generator of flow lines, only one of which is occupied by a real point charge. Unlike the Bohm model however, the projection of SS DPI onto a field theory gives these flow lines an electromagnetic origin, as described above.

SS DPI has yet to be developed to cover multi-particle fields. The early signs are that it is likely to converge with accepted theory (Quantum Field Theory) which here refers to second quantization of the Dirac field.

Radiation Modes

When approximated as fields in \({{\rm{R}}^4}\) the radiation modes of SS DPI can be expressed in terms of a 4-potential that satisfies

\[\begin{align}
\left( {{\partial ^2} - k_H^2} \right)A\left( x \right) &= 0 \label{HWE}\\
\partial  \circ A\left( x \right) &= 0\label{LGC}\,.
\end{align}\]

The Lorenz gauge condition is a consequence of the particular choice of interaction in the STF action. Gauge freedom can be restored by using a slightly different form.

The free potential of Maxwell theory (i.e. not explicitly coupled to a current) and that of the radiation potential projected out of SS DPI, \eqref{Dirac} and \eqref{HWE}, both satisfy a homogeneous differential equation. The latter is distinguished by an additional term involving \(k_H\). This term does not imply a photon mass, for which one would expect

\[\left( {{\partial ^2} + m_p^2} \right)A\left( x \right) =0\]

in units where \(\hbar  = c = 1\).

\(k_H\) is of order of the reciprocal length that is the radius of communication between the charges participating in the interactions. The latter is of order of the order of the Hubble radius, so \(k_H\) is  tiny therefore, making \eqref{HWE} and \eqref{LGC} mostly indistinguishable from Maxwell Theory.

Cosmology

The derivation of \eqref{Dirac} and \eqref{HWE} in (Ibison, 2020) \(k_H\) did not distinguish between past and future times, since by default both have the same standing in DPI. In the present cosmological era however the radius of communication to particles on the backwards light cone differs from that on the forwards light cone. Therefore those derivations need to be revised before they can be properly integrated into a cosmological scheme. In any case, the forward radius of communication - on the forward light cone to the future conformal singularity - is of the order of the Hubble radius, as is the radius on the backwards light cone. (The latter is also the `Particle Horizon`.) So it seems safe to presume that \(k_H\) will continue to be of the order of the Hubble radius when the calculation is performed in a cosmological background.

At present there has been no work on the integration of SS DPI with cosmology. The fate of the cosmological scale factor is decided by general relativity. It is a consequence of GR applied at the largest scale. The integration of SS DPI with cosmology is secondary to the integration of SS DPI with GR therefore, from which would follow some understanding of the relationship between SS DPI and gravity at all scales. There are already some indications that GR, or something similar, may have an electromagnetic origin within the Direct Particle Interaction framework (see below, and also jump to the page on gravity).

Arrow of Time

The presence of radiation modes is SS DPI overcomes historical objections to Direct Particle Interaction. SS DPI allows for matter to emit radiation without the requirement that radiation eventually be re-absorbed.

The mere presence of radiation modes does not guarantee an arrow of time, however. It remains to be explained why we see retarded radiation but not advanced radiation, though both are equally allowed by field theory and SS DPI. It is possible this electromagnetic arrow of time originates in the cosmological arrow of time. Though some contact between Cosmology and SS DPI has been made, Cosmology is still mostly outside of the scope of SS DPI as developed so far.

Possible Connections with General Relativity

Cosmological Red-Shift

Possibly the term involving \(k_H\) can be interpreted conferring an exponential decay (or rise) in time on the potential. If so, such behavior would roughly correspond to that expected of radiation in the Robinson-Walker coordinate system in our current cosmological era, which is approximately de Sitter spacetime.

MOND

Newton's Law of gravity is the non-relativistic limit of Einstein's theory of gravity (GR = general relativity). It remains a good approximation in some cosmological calculations, including for example the calculation of the motion of stars in galaxies, especially when far from the galaxy center.

In the outer arms of spiral galaxies it has been found the orbital velocity of stars is too high to be consistent with Newtonian attraction, given the observed mass in the galaxy inferred from observation of its constituent stars. The favored fix to restore compatibility with Newton's law - and therefore GR - is to presume the existence of dark matter in the galaxy. The matter is `dark` because, necessarily, it is invisible (non-interacting) except to gravitational forces.

The mass of a galaxy can also be inferred from how much it bends (lenses) light from more distant galaxies along the line of sight to the Earth. Dark matter is inferred to make up for the discrepancy between the observed mass of the galaxy and that necessary to account for the degree of lensing.

Modified Newtonian Dynamics (MOND) is as ad-hoc modification of Newton's Law of Gravity initially designed to explain / accommodate the observation of otherwise anomalous motion of stars in the outer arms of spiral galaxies, without appeal to Dark Matter. There has since been some progress in applying MOND to account for anomalous gravitation lensing, again without invoking Dark Matter.

Crucially, to account for observations, MOND was designed to cause a significant deviation from Newton's Law only when the acceleration of the affected body falls to around \(10^{-10} m/s^2\), which threshold we will refer to as \(a_0\). This acceleration can be associated with a length \(l_0 = c^2/a_0\ \sim 10^{27} m\). This length is a bit more than the Hubble Radius, which is around \(1.3 \times 10^{26} m\). A more appropriate comparison would be with the conformal radius (the conformal age times the speed of light) which is about  \(4.4 \times 10^{26} m\). Since the MOND threshold is only approximately \(10^{-10} m/s^2\) the two lengths are close enough to be suggestive of a connection.

Is Gravity conveyed by the Matter Modes of SS DPI?

The possible connection with electromagnetism, and in particular with an EM foundation of gravity is as follows. The conformal radius is the same order as the wavelength of EM radiation emitted by a charge moving at light speed with acceleration \(a_0\). Such radiation would be subject to \eqref{HWE} and therefore an otherwise anomalous dispersion at around this wavelength, approximating a cutoff. Distinct from conventional theories, SS DPI allows for electromagnetic radiation that is time-symmetric relative to the sources - the matter modes. It is likely, though not yet demonstrated, that these modes are subject to the same cutoff, suggestive perhaps of an electromagnetic foundation of gravity, conveyed by the time-symmetric matter modes of SS DPI.

It remains to be firmly demonstrated that time-symmetric `radiation` from a source in circular motion is not aberrated, which quality is required if such is to convey gravitational force. But this seems a likely outcome however, given the stability of the Bohr orbits say, which in SS DPI are maintained by flow lines generated from time-symmetric EM fields.

The mechanism by which time-symmetric modes could convey gravitational force is discussed here.

Correlation and Entanglement

The DPI picture differs from that of Quantum Theory as the latter is currently interpreted. The difference derives from the absence of true field degrees of freedom in DPI, and no \(x \in {{\rm{R}}^4}\). DPI is built instead on a theory that grants physical consequence only to points of `electromagnetic contact`. These points exist either end of light-like rays between particles. The rays themselves are not observable except at these contact points. If for a moment we forget that technicality, and grant physicality to the rays themselves, then the field of a charge is still not in \({{\rm{R}}^4}\). At a fixed time `fields` exist only on rays connecting the local charge to a finite number of other charges on the past and future light cones of that local charge. From the perspective of DPI field theory works because there are so many of these distant charges, and the rays so dense therefore, that in many circumstances one can approximate these rays with smooth continuous fields in \({{\rm{R}}^4}\) near the local charge.

An outcome is that DPI predicts that particle motions remain correlated across cosmological times and distances, both backwards and forwards in time, irrespective of the cosmological era. Viewed through the lens of quantum field theory it is conceivable that this correlation manifests as intrinsic entanglement, i.e. without special preparation by technical means. But this requires further investigation. The claim is likely to be in conflict the notion of a localized particle having its own isolated internal degrees of freedom. It is encouraging that the second quantized theory of Fermions is more naturally expressed in terms of Fourier modes (`field quanta`) that are intrinsically delocalized. But it might still be problematic that QFT supposes an infinitely dense Fourier space - the on-shell modes are in \({{\rm{R}}^3}\). From the perspective of SS DPI this must be an approximation, because in this theory there are a finite number of distinguishable modes. That number is expected to be of order of the number of visible particles, which is around \(10^{80}\).