The Huygens-Fresnel Principle: more on the External Reset

In a previous post we have described the rules that apply at each "External Reset", that is, when a particle encounters an external force-carrier, an external "boson". In particular, the span is reversed according to the 1D rule (1)

which is generalized to the 3D rule described in this post (2),

where the resulting spans are intended to be rounder at the nearest integer.

However, these rules alone are not sufficient to represent the emergence of quantum behavior such as self-interference and superposition. Thus, it is now time for some more details.

Let us come back to the double slit experiment. Consider such an apparatus as illustrated in the figure below, where O is the source of particles, S1 and S2 are the two slits, and P is the recording screen.

In standard QM the wave function at the screen is obtained as the superposition of two wave functions emitted at slits S1 and S2 and propagated to the screen. This is possible because of the superposition principle, of course. However, the fact that the actual source O can be ignored and S1 and S2 taken as the effective sources is a consequence of the Huygens-Fresnel principle.

In our local-realistic model, mechanisms (1) or (2) would indefinitely keep trace of the particle source, whatever interactions it experiences after its emission. That is clearly unrealistic, and moreover it would not comply with Huygens-Fresnel principle.

If these rules hold true exactly as they have been displayed so far in this blog, span at slit S1 would be equal to δ (see figure) and span at S2 would be equal to -δ. But between the slits and the screen, at a point, say, x, particles either coming from S1 or S2 would have the same span x as counted from the source O, that is the same for both rays. Thus there could be no Quantum Reset since span difference would always be equal to zero, and no boson creation. So, ultimately no interference.

Even considering interactions (i.e., External Resets) inside the slits with the slits' walls, the model would not be improved. Particles leaving S1 would have a span that is ±δ, depending on how many "hits" it has had with the slit walls. Similarly, span at the exit of S2 would be ±δ. The situation would be equivalent to three sources at 0, ±2δ, which clearly would not produce the QM behavior (that is, interference from two sources at ±δ).

The missing rule to complete the model is the following. At each External Reset, there is a small probability that span is not flipped according to (1) or (2), but it rather vanishes. So we have 1D rule (1a):

and analogously (2a) in the 3D case.

We shall denote as ER* these ER events that occur with probability ϵ. In addition, when an ER* occurs, the particle's "lifetime" (temporal distance form the source) also drops to zero together with its span,

In average, the expected span dynamics at each ER is thus

,                   

The result of this modified rule is that, after many ERs there is a sufficiently large number of ER* that occurred, so that both the span and the expected lifetime tend to vanish. The particle is now 'virgin' and it has no memory of its original source. Everything works as its source were at its current position and time.

Let us apply this improved model to the double slit scenario. Assuming that sufficiently many interactions occur inside the slits, particles exiting either of them will have zero span and lifetime, exactly as if they were actually emitted from them. Then, the mechanism described in this post creates the interference patterns that are experimentally observed.

In a continuous force field, span/lifetime reset plays no role as already discussed in this post, and thus (1) or (1a) are strictly equivalent.

In a discontinuous force field, such as a 'particle in a box' scenario, the particles that hit the boundaries and experienced an ER* (reset their span) cannot produce further QR with particles that, vice versa, hit the boundaries and experienced just an ER (flipped their span). This is because their lifetimes are not the same. Particles having had only ER are equivalent to particles emitted from one of the many virtual sources at either sides of the box (depending on how many hits). Particles having had an ER* are equivalent to particles emitted from the position and time of their last boundary hit. Thus, for the rules of QR, there is no interaction between ER*-particles and the trace left in the lattice nodes by ER-particles. In conclusion, only ER-particles build the interference patterns, which thus reflect the infinite virtual sources as detailed in the previous post on this subject.

By the way, we had seen there that momentum in a box tends to one of the allowed values as predicted by QM, which are all rational numbers (an integer divided by 2a, where a is the width of the box). We can now revisit the slits above and imagine them as 'boxes' where, after many hits (both ER and ER*) particle exit with

• zero span 
• zero lifetime (that is, 'virgin'), and
• a rational-valued momentum comprised between -1 and +1

This confirms our initial assumption (see here) that source momentum is a rational-valued quantity (and, consequently, momentum propensity, quantum momentum, etc. are also rational-numbers) and proves the correct representation of Huygens-Fresnel principel in our model:

a point in space (actually, a small region) where particles experience many interactions with the external world, becomes a secondary "source" of those particles

(Cover illustration by user Anne Nordmann (norro), Wikimedia Commos, https://commons.wikimedia.org/wiki/File:Refraction_on_an_aperture_-_Huygens-Fresnel_principle.svg, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license)