1.2 scope & motivation of the present work

To set out the motivation for considering a primitive trajectory-based approach we can begin by addressing the question: Why is such a description not a mere tautology?

An exhaustive register of all particle trajectories for all time would contain a full description of spacetime for any measurement purpose. But no useful theory can invoke such a register. A useful theory has to be an imperfect representation of the world, just because the complete description is not locally available to us. Obviously the meaning of the fact that for us, at any given here and now, a useful theory has to be predictive is that so much significant information is locked away in ‘future’ states. Further, it seems reasonable to state that the existence of a limit to predictability here and now is not of itself the cause of these future states being inaccessible: The quality of predictiveness belongs to the act of theorising; the inaccessibility is a given fact of the ontology.

The origin of the light cone runs ahead of past-timelike states to depict a local order, but is embedded in the complete causal structure. The requirement for self-consistency (expressed in the criterion of predictiveness) demands that somehow the apex of the here and now has to be representative of the information in future states as well as of information in past states. That future states are empty states, waiting to be filled by the outcomes generated by past states, is not at all clear. The past-timelike half of the causal structure is known to be insufficient on its own to determine quantum states; only a sum over all possible histories suffices to yield even a probability here and now. But presumably the sum of the probabilities of all cosmic events must in the end total to precisely unity. This suggests that the causal structure as a whole does not care about the philosophical distinction between determinism and teleology insofar as that distinction is concerned with the sign of the time variable. The universe in some sense just ‘is’, consistent with the idea that what is called the ‘microscopic reversibility’ of classical and quantum processes is an essential symmetry, which gets hidden in complex thermodynamic systems.

If the thermodynamical ‘breaking’ of time-symmetry is an emergent feature of large systems, whilst the underlying causal structure remains symmetrical, then it appears that a ‘complete’ theory must be able to show how a state at the here and now represents in some sense a resultant of contributions of all particle trajectories taken over all past and future times. Of course if this is conceptually correct then it becomes true by definition that a predictive formalism can never be more than an emulation of a complete theory, since the complete theory would be a botanical catalogue. But a theory need not be calculable to all orders for it to be physically non-trivial. The model of QED suggests that a promise of statistical predictability in some limited problems would suffice to identify it as a possibly useful theory (after all, GR cannot be solved realistically for the vast majority of physical interactions).

But surely a trajectory-based summation is merely tautological, even if incomplete and statistically predictive, unless it can show why the sum of all spacetime states of particles is not just the same as a sum over all other field quantities. For this, surely, is the function of the differentiable spacetime manifold whose infinite degrees of freedom we are proposing to replace, the mechanism that will make the theory a ‘theory of gravity’ in the sense of GR? The GR metric is exactly the sort of generalisation that one needs in order to get from a merely redundant description of trajectories to an interpretation of the correlation between mass and spacetime displacements which those trajectories embody.

This is all true, but problematical. A sum over the potentials of all other field quantities is required by the theory to be exactly proportional to the gravitational potential, since this sum is just the energy-momentum tensor which determines the metric tensor. But the fact that this proportionality is not a perfect isomorphism is a mystery. The metric tensor is something coupled with, but not found ‘inside of’, the particles themselves; it has autonomous components. That is, the components of the metric tensor do not all vanish even when the energy-momentum tensor is zero, i.e. when there are no mass-energy trajectories in it. In this respect the theory has an awkward unspecified degree of freedom even in its minimum condition. True, the energy of a quantum field is said always to be non-zero; but this is a poisoned chalice because quantum field theory predicts that the vacuum energy should destroy the universe, exactly on account of the fact that the spacetime manifold is a differentiable continuum.

It may be that GR is not the right generalisation. On the other hand, Einstein’s programme to make the gravitational/inertial ‘forces’ completely redundant inside a theory which identifies spacetime with the mass-energy distribution would be fulfilled by a non-trivial description of spacetime in terms of a geometry of non-differentiable trajectories. Eliminating undesirable components of the spacetime structure might also hold out the prospect of cancelling an unwanted vacuum energy. Is this possible in principle?

Since quantum theory requires all fields to be associated with particles there is an equivalent description in terms of trajectories, but not a reduction to them, since the trajectories are only particle-like in specific acts of measurement. In ‘between’ measurements the path of a particle cannot be determined instant by instant, so the trajectories are wave trajectories; momenta are carried by the field. This means that if trajectories are to be recovered from quantum field theory as primary entities, then plainly this cannot be done in terms of classical-deterministic particles because these are not adequate to support the whole causal structure.

This all leads, we will show, to the notion of a non-field trajectory, with no particle in it, which is an interesting object! In the semi-classical sense of a spacetime trajectory, eliminating the particle would leave an empty state. But ‘eliminating the particle’ from our explicitly non-classical trajectory cannot mean the same thing, since there is by definition no classical particle in it to start with. Instead we are removing the function of the particle. Since the ‘function’ of a particle is to represent the instantaneous space-time derivative of a total momentum along some semi-classical trajectory, we find that eliminating the particle means eliminating the instantaneous acceleration. In other words we are to work with the total momentum of the trajectory, which may change scale arbitrarily but is always a properly non-differentiable path. Further, if (for consistency) no sequence of such paths is to be considered differentiable, then it cannot be possible for the acceleration to go to zero on any path, meaning that no curve can be rectified into an infinite number of infinitesimal segments. Thus we lose the idea of continuity underlying the concepts of both field and point-particle, and arrive instead at the requirement for a non-zero minimum path associated with a non-zero minimum acceleration.

The rest of this article considers some of the detailed issues raised by this proposal, leading to some suggestions for the implementation of an unconventional mechanism for the gravitational action. So as to identify it, the framework is called parcellular mechanics or PM (defined in its place).

Meanwhile we can sum up some desiderata of the approach as follows: The world is to arise in the self-interaction of a causally time-symmetric system of non-differentiable trajectories. Each possible trajectory is to be somehow equivalent to a sum over all others in a path-integral approach to a finite perturbative theory of ‘quantum gravity’ where a) the trick would be to define ‘all possible’ so as to restrict the sum to a finite number of filled states and eliminate the degrees of freedom associated with empty states, and b) the register of all states needs to act nonlocally to determine each local trajectory.

Traditionally most effort focuses on trying to reproduce GR as a quantum field theory, but if GR is a classical approximation to some underlying quantum theory it is not necessarily the case that the underlying theory, even if correctly formulated, would ever reproduce the calculability of GR for astrophysical problems, any more than GR is a calculable theory for particle interactions. Neither is it at all to be expected, from this point of view, that the underlying theory need ‘operate on’ any of the elements that are part of the machinery of GR. In particular, an underlying quantum theory of gravity might be the begetter in common of the phenomenology modelled in GR and QM, yet be qualitatively different from either. The point I wish to emphasise is the corollary that quantising the gravitational ‘field’ (or the metric tensor) is not necessarily the route to this theory, meaning that the dimension of quantisation in the underlying theory, and the length scale(s) associated with quanta, are not necessarily those generated for the purpose by applying quantum theory to the spacetime of GR.

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