8.16 Wave-Particle Duality, Uncertainty Relations, Etc.

Wave-particle duality was an early but important concept in standard quantum mechanics, and turns out to be a core feature of our models, independent even of the details of particles. The key idea is to look at the correspondence between spacelike and branchlike projections of the multiway causal graph.

Let us consider some piece of “matter”, ultimately represented as features of our hypergraphs. A complete description of what the matter does must include what happens on every branch of the multiway graph. But we can get a picture of this by looking at the multiway causal graph—which in effect has the most complete representation of all meaningful spatial and branchial features of our models.

Fundamentally what we will see is a bundle of geodesics that represent the matter, propagating through the multiway causal graph. Looked at in terms of spacelike coordinates, the bundle will seem to be following a definite path—characteristic of particle-like behavior. But inevitably the bundle will also be extended in the branchlike direction—and this is what leads to wave-like behavior.

Recall that we identified energy in spacetime as corresponding to the flux of causal edges through spacelike hypersurfaces. But as mentioned above, whenever causal edges are present, they correspond to events, which are associated with branching in the multiway graph and the multiway causal graph. And so when we look at geodesics in the bundle, the rate at which they turn in multiway space will be proportional to the rate at which events happen, or in other words, to energy—yielding the standard E ∝ ω proportionality between particle energy and wave frequency.

Another fundamental phenomenon in quantum mechanics is the uncertainty principle. To understand this principle in our framework, we must think operationally about the process of, for example, first measuring position, then measuring momentum. It is best to think in terms of the multiway causal graph. If we want to measure position to a certain precision Δ x we effectively need to set up our detector (or arrange our quantum observation frame) so that there are O(1/Δ x) elements laid out in a spacelike array. But once we have made our position measurement, we must reconfigure our detector (or rearrange our quantum observation frame) to measure momentum instead.

But now recall that we identified momentum as corresponding to the flux of causal edges across timelike hypersurfaces. So to do our momentum measurement we effectively need to have the elements of our detector (or the pieces of our quantum observation frame) laid out on a timelike hypersurface. But inevitably it will take at least O(1/Δ x) updating events to rearrange the elements we need. But each of these updating events will typically generate a branch in the multiway system (and thus the multiway causal graph). And the result of this will be to produce an O(1/Δ x) spread in the multiway causal graph, which then leads to an O(1/Δ x) uncertainty in the measurement of momentum.

(Another ultimately equivalent approach is to consider different foliations, and to note for example that with a finer foliation in time, one is less able to determine the “true direction” of causal edges in the multiway graph, and thus to determine how many of them will cross a spacelike hypersurface.)

To make our discussion of the uncertainty principle more precise, we should consider operators—represented by sequences of updating events. In the space of the multiway causal graph, the operators corresponding to position and momentum must generate events that correspond to moving at different angles; as a result the operators do not commute.

And with this setup we can see why position and momentum, as well as energy and time, form canonically conjugate pairs for which uncertainty relations hold: it is because these quantities are associated with features of the multiway causal graph that probe distinct (and effectively orthogonal) directions in multiway causal space.