This is not a religious tract, nor is it physics, nor is it philosophy. It is meant to be no less and no more than a rational discourse on some of conundrums at the outer limits of our known physical understanding today. Since its arguably B.S.; lets just call it metaphysics and be over with it. The intent is, however, serious. The exploration here is meant to be part of a puzzle piece, along with a parallel exploration of freewill and the existence of Platonic realms, and a critique of Heidegger. Some of this might be experimentally tested? And maybe we can even associate some mathematical equations with these ideas, giving them at least a little bit of traction.
Bunk. Of course this isn't right, and with the exception of a few brave individuals, (e.g. Julian Barbour) nobody but nobody actually believes that this is the case. We have a too-strong anthropocentric experience to accept this view: we all know what the past is, we all sense that we live in the present, and we all seem to agree that much of the future, or at least, the truly important parts, are unpredictable. Curiously though, in this popular view, it is never pointed out that in fact, the past fits the above description to a tee. The past is so perfectly "predictable" that, in fact, we don't even use that word: we "remember" the past, we don't "predict" it. The past is immutable and unchangeable: there is nothing that we can do to change the past. That is, the past is exactly like those immutable, unalterable equations of motion: predestined and inescapable. So it seems that the past is exactly like this Newtonian world-view: its just that something funny happens in the present, somehow making the future unknowable.
One might think of the past as a block of ice forming, as the liquid present freezes onto it. The present, the 'here and now', is like a wave of crystallization, like Kurt Vonegut's Ice-9, freezing, propagating through space-time, segregating what was from what might be. This might make for nice literary allusions, but doesn't fit with the Newtonian view. Differential equations know of no past, present, or future: they don't distinguish between these, and this is precisely where the metaphysical problem originates. Ordinary, 'classical' mechanics states that the future is just like the past, and this seems to be inescapable, even as we all know intuitively that this is not so.
Two modern developments in Physics seem to offer avenues of escape from the conundrum of pre-determination. The first, Quantum Mechanics, introduces a certain amount of randomness that provides the wiggle-room needed to make the future unpredictable, and offer at least a glimmer for free will. Unfortunately, Quantum is saddled with a number of messy interpretational problems that leave an unsatisfying taste in the mouths of anyone who cares to seriously try them. The second, Chaos Theory, doesn't offer immediate escape from classical dynamics, but does show us how a whole lot of unpredictable things can happen in a short amount of time.
The need for such an interpretation is due to some curious puzzles that arise in discussions of quantum measurement. We review some of these below, and after that, we dive in.
Lets review some popular quantum conundrums:
Lets review the popularly discussed quantum measurement proposals. All of these proposals seem to be lacking in that none provide any sort of detailed description of the mechanics of wave function collapse. They attempt to resolve the meta-physical paradoxes without proposing the physics.
There are various ways of dressing up this hypothesis in high-falutin' language. For example: during interactions with multiple bodies, certain paths that may have at one point contributed to the Feynmann path integral in fact bump into analytic 'cuts', and stop contributing in a phase-coherent fashion to the path integral, thereby causing a collapse of the wave function. These cuts in the analytic plane don't exist in two-body interactions, but absolutely litter the phase space when the N in N-body is large enough. In other words, for N sufficiently larger than two, most phases follow chaotic paths, and the phase relationships are no longer coherent, but become incoherent, thereby marking the wave function collapse. (For example, consider the chaotic regime of the forced harmonic oscillator). This view of wave function collapse seems at first to be very appealing, precisely because it can be dressed up with all sorts of flowery appeals to chaos and the like. But, as we mentioned, it founders because it is ultimately a local theory, and fails to explain EPR correlations.
We could attack the path integral from a second direction. For example, in thermodynamic problems, one averages over an avogadro's number of states, which, for all practical purposes, we can take as 'infinite'. This is because experimental evidence provides close support for the taking of thermodynamic averages. We also know that thermodynamics breaks down when dealing with dozens or hundreds of atoms, because by then, the averages no longer accurately approximate the situation. Unfortunately, we have few similar experimental explorations on the contributions of the path integral to second quantization. We know that it must be correct for the most part, because otherwise quantum mechanics in general wouldn't work. The Casimir effect, studies of dielectrics, and surface tension physics provide some experimental tests of second quantization, but the linkage is not precise. (In the Casimir effect, one sums over all standing waves trapped between a pair of metal plates. Or, at least, one sums up to a frequency at which the plates become transparent to EM waves. This summation is a kind of a path integral. The experiments work, and the summation is quite sensitive to the cutoff frequency/regulator. In certain geometries, the summations lead to infinities that need to be regulated or dealt with, much like the free-field QED theory. What's different is that in the Casimir effect or in surface-tension or dielectric calculations, the infinities are 'real' in the sense that they really do depend on the cutoff frequency, and different materials with different cutoffs can be measured in the lab. But there is still a big leap from these experiments to the path integral.
Abstract: By assuming that Planck-scale spacetime resembles a 'foam', it is deduced, by means of hand-waving, that most quantum phenomena are best understood as interactions classical geodesics on this foam. Furthermore, it is argued that the arrow of time, as something distinct from 3D space, and having a fixed past and unknowable future, is a side-effect of the reconciliation of anomalies in this foam.
Lets assume that spacetime, at Planck scales (10e-40), resembles a foam of worm-holes. Lets imagine geodesics on this foam. Now, lets imagine that these geodesics interacted like billiard balls, i.e. had point interactions with each other. This clearly leads to grandfather paradoxes throughout the foam: a future billiard ball could emerge from a wormhole in the past, and prevent itself from going into the very wormhole it emerged from.
Hypothesis: The space-time foam tries to organize or equilibrate itself so that such grandfather-paradoxes between billiard geodesics do not occur. Hypothesis: this act of organizing or equilibrating propagates through the foam as a 'wavefront', a 3D slab of a surface traveling along a fourth dimension. This thin 3D slab is what we call 'right now'; what lies behind this slab is 'the past', and what lies ahead is 'the future'.
Now let us pause to notice that this simple model does a decent job of 'explaining' second quantization. How does it do this?
First, note that the act of second quantization consists of writing a sum, a 'Feynmann Path Integral' over all possible paths, weighted by the exponent of the action. The action defines the classical path; as any given path deviates from the classical path, it is weighted away. With only the slightest of handwaving, it should be obvious that if we imagine geodesics on a space-time foam, that these could well be taken to be the 'paths' that contribute to the path integral. Now, a purist might start arguing about the Hausdorff Measure of geodesics on a foam, as compared to the 'uniformly distributed' Jacobian of a path integral, but I will only shoot back that any such argument is already on tenuous mathematical footing. The set of geodesics on a foam should probably form a complete enough set to work just fine as the domain over which a path integral is taken. (Although not usually studied by physicists, there are deep fundamental problems with integrals of stochastic differential equations, which are well know to 'quants' working in the financial industries. These difficulties have been overcome for simple arbitrage and options pricing formulas but continue to plague more complex financial models. With some handwaving, its clear that many of these problems apply to path integrals as well, especially when one considers the question of whether the 'paths' contributing to the integral are differentiable or even continuous.)
Next, note that the 'accidental' resemblance of various diffusion and stat-mech equations to various quantum equations is no longer so 'accidental': the statistical properties and distributions of geodesics on a foam should indeed have the same qualitative properties as Brownian motion in a hard-ball gas.
Thus, we've just hand-waved our way over a giant part of quantum mechanics. To be more precise, we will need to also cover why it is that Planck's constant is used in these path integrals, and how nothing in this handwaving contradicts the infinitely more rigorous theories of superstrings. Indeed, while we're handwaving, we should point out that talking of geodesics on a spacetime foam might be a lot like talking about the hydrogen atom is if it were a planetary system whose orbits are constrained to have integer-wavelength circumference. We know that Bohr Action-Angle formalisms gave over to solutions of the Schroedinger wave equation for the Hydrogen atom; in a similar way, we might think that this talk of geodesics is a mental place-holder for something that ultimately phrases itself quite differently.
Note that the probabilities of quantum wave functions are re-interpreted as the possible future geometrical arrangements of the Planck-scale foam. Some arrangements are very likely, some are very unlikely. There are no hidden variables here: all one can talk about is possible future arrangements based on a knowledge of the current/past history. (Question to be answered: if the past is not known, who does this affect the prediction? i.e. how would we envision a (non-relativistic) electron propagating in free space? What is happening to this 'wave' as this Planck-scale foam 'freezes' along? Generically, how does one derive the idea of a propagater from the foam? There are similarities to diffusion eqn, but how can one get more specific?)
Next, let us imagine what this world of geodesics implies for quantum measurement. There is a strong sense of non-locality that is induced by this foam. Imagine the classic spin-EPR experiment, where a singlet decays into a pair of spin-1/2 particles, whose spins are then measured, and found to be correlated. This is where, I think, my handwaving reaches a crux, and all the best stuff falls out. Here goes (this is a bit rocky):
Imagine that the spin-1/2 state is described by a collection of geodesics propagating on this foam. As this data propagates through the foam, a pair of measurements are made, and the data from these measurements are brought back together to roughly the same point in spacetime, where the experimenter can compare them and verify that e.g. Bell's theorem, has not been violated. Hypothesis: the geodesics in the space-time foam can only be made self-consistent in the backwards light cone. In particular, the geodesics from one measurement cannot be reconciled with those from the other until that have both entered into a common backwards light cone (i.e. the measurements have been brought together). When these paths are brought back together, the outcome 'freezes', i.e. become a part of the 'past'. This act of coming together defines not only the arrow of time, but defines time itself (or rather, distinguishes time from space).
The Past is Manifest Destiny. (Newtonian ...) The claim is that this is no accident. The creation of the past is what happens when geodesics are reconciled on a space-time foam.
The topological interpretation avoids the silly questions such as 'why do all electrons look alike'? They all look alike because they are all represented by the same set of self-consistent twists in the space-time foam. Asking 'why do all electrons look alike' is tantamount to asking 'why are all instances of the number two alike?' Or maybe better yet: 'why do all single-point dislocations of a lattice look alike?' Physical particles are configurations of an underlying relativistic 'ether'.
The above considerations would seem to dictate a program of topological research. However, this alone doesn't seem sufficient to resolve the issue of the arrow of time, or of the second law of thermodynamics. Topological arrangements would seem to be time-symmetric. Its too static in itself. If we are to view the past as 'fixed', and the future as 'indeterminate', then we need a language of topology where the past is fixed, the present is a mad scramble to arrange a topology so that it is consistent with everything that came in the past.
I know of no topology that acts like this. There is no study that asks and answers the question: 'how do I tie the knot right here, so that the strands 'back there' don't have to be rearranged'. Topological stability in crystal lattice dislocations is independent of time, and are arrangements in 3D space. Instead, we need to ponder the evolution of topological entities as they are brought 'close' to each other in space.
The other problem is that we still have no insight into what free-will might be all about. Clearly, free-will is a phenomenon of the present: I have no free-will in the past, nor in the future. At best, I can try to control what I am doing 'right now'. Topological arguments seem to be deterministic, however fanciful they may get. It seems to me that any theory of the flow of time is somehow incomplete until it also somehow addresses the question of free-will.
Another possibility is that upon wave function collapse in one location, an influence travels 'backwards in time', along one arm of the experiment, and thence forwards along the other, precipitating a change there. What is the form of this influence? Why, precisely that which can carry no information: a gauge-like rotation of the phase of the wave function. For space-like separated measurements, this communication via backwards-traveling geodesics helps keep each side in perfect correlation with the other, so that any interaction with a macroscopic, stochastic tank of particles that is the measuring instrument is 'simultaneously' echoed in the other arm.
Might it be possible to test this hypothesis experimentally, by placing two devices at space-like separations, blasting correlated particles into both of them, and then looking to see if they've achieved thermodynamic equilibrium? For, if the hypothesis is correct, then faster-than-light signalling is disallowed, but thermodynamic interaction is not. This might be experimentally testable.
However, this is tricky, and has a (fatal?) flaw: we can still do faster-than-light signalling. Lets say that experimenter at location A measures the temperature of a container that is getting getting correlated photons being shot into it. Experimenter at location B can put one of two containers in the path of the photons: a hot tank, and a cold tank. If quantum correlations could act to bring the two containers into thermodynamic equilibrium, then experimenter A would see the temperature rise, or drop, over time. With sufficient space-like separation, this could be used to perform faster-than-light signalling.
(Equally disturbing is to think of the situation in terms of entropy.)