# The double-slit experiment

Based on the mathematical structure of quantum states and measurements and quantum correlations we introduced earlier, we can now explain how quantum theory provides the famous prediction of the double-slit experiment, which expresses the strangeness of quantum physics with its wave-particle duality.

Consider a photon going towards a plate with two slits.
It may be stopped by the plate, or go through the slits.
To simplify, let us assume that the case when the photon is stopped by the plate, is detected and eliminated from consideration.

The state of the electromagnetic field in each slit, is undetermined, in between the presence and the absence of the photon. The global system of both slits, is in a pure state of correlation of the slits.
And it is located "in the middle" between both states defined by (1 photon, no photon) and (no photon, 1 photon). Let us denote C the circle of points of the sphere "in the middle" (the equator) between them (taken as poles).

We have explained what are such "equatorial" points, elements of C: they are correlated states defined by an indirect isometry between both abstract spheres of presence/absence of the photon, exchanging the presence of the photon in the one, with the absence of the photon in the other.
Therefore, both equators are just following each other. Points of these equators correspond to different orientations of the electric field. Different elements of C give different correspondances between the electric fields of both slits: they say how much delay separates their oscillations (how late is one's oscillation with respect to the other). One whole way around C, makes this time difference change up to one whole period, coming back to the initial correspondence.

The photon has been sent to the slits from one specific direction, thus specifying the element of C involved (the phase difference between the slits).

After the slits, we have a screen with many points that are sensitive to the photon. So we have a measurement with many possible results (which point of the screen will detect the photon).
Each of these results consists in the pure measurement of a point of C. (The barycenter of these many points of C has to be the center of that sphere, for this to be a possible measurement.)

If we could detect which slit the photon was in (distinguish between (1 photon, no photon) and (no photon, 1 photon)), this would be a measurement along the axis of C (orthogonal in space), and would collapse the state of the system onto either (1 photon, no photon) or (no photon, 1 photon). It would no more be at its position on C. its probability of being detected by this measurement, thus as a pure measurement characterized as coming from some point of C (=whose probability cancels at the opposite point), is half the probability for this point by this measurement (=its maximum probability).

#### Further remarks on the double slit experiment

In principle, this experiment can be proceeded with any particle, however it becomes more and more difficult (sensitive to disturbance) as its mass increases. The biggest "particle" so far on which it has been made and interferences have been observed, is the fullerene (C60) molecule.

Instead of having a continuous range of possible positions of detections of the particle, corresponding to all points of C, it is possible to redesign the experience to have only two possible results, corresponding to diametrically opposite positions in C. This is an experience with photons going through mirrors and semi-reflecting mirrors (I forgot the reference).
So, a photon has a probability 1/2 of going through either path; if from only one path it would have a probability 1/2 of ending in either of possible final locations; but with this way of keeping both possibilities of paths, it can only reach one destination; but if a modification is done on one of these paths to inverse its phase, this operates a 180� turn of C so that the photon can only end up to the other destination. Strange conclusion: by only affecting the path of statistically half of the photons, the destination of all photons is changed !

### Some technical details about classical interference

For those who are not familiar (one might just ignore these details in the discussions, but I did once see someone playing on it to claim finding absurdities in the story of the double slit experiment and its interpretation just by complaining that these details failed to be specified and put at their right place in the story) :

When we measure by which slit particles go, then the result (identical with what we obtain by first letting them go in only one slit while hiding the other, then repeating with the other, and adding up the number of particles measured from both cases), does rather not consist in 2 spots (one spot from particles from one slit, the other from the other slit) but only one spot. At least we can make it just one spot, or if it is 2 spots (one from each slit) then they need anyway to overlap. Let us explain this in more details.

About the one slot from one individual slit: one might even argue that it is not one spot but an interference pattern with many spots already because of the phenomenon of diffraction. But this is only half-true, and must be corrected.
1. The result of diffraction is not many spots with similar brightness but one big central spot where most particles arrive (a fixed proportion like 70% or 80%, I did not to check which precise number, it must be known somewhere...), whose width (distance between the exact positions of pure darkness that separate spots) is twice those of all others.
2. The diffraction phenomenon is usually considered out of subject in discussions of interference because the width of spots due to diffraction is much larger that the width of spots from interference. Precisely because these widths are inversely proportional, respectively to the width of the slits and the distance between slits : by having slits with width much smaller than the distance between slits, we get spots (and thus diffraction patterns) much larger than the pattern of interference between slits.

So, interference can appear only where "both spots" from both slits overlap, and with visibility that is a function of the comparison of intensities coming from both slits : where without interference one intensity is twice the other, with interference the brightest places are 34 times brighter than the darkest ones. I just got this number from calculator : ((sqrt(2)+1)/(sqrt(2)-1))^2. From a ratio of intensities of 3, we get 14, that is ((sqrt(3)+1)/(sqrt(3)-1))^2. From 4 we get 9. From 100 we get 1.5

There is nothing interesting to say about the distance between the center of the main spot from one slit and the center of the main spot from the other slit because it is a completely independent variable, that we can reduce to 0 as well by means of proper optical devices (lenses).

### The Delayed Choice Quantum Eraser Experiment

As an introduction to the paradoxical concepts at play, 2 preliminary points of understanding are needed.

1. Consider the following simpler experiment.

Imagine an experiment where a particle has 1/2 probability to go in ways producing a given interference pattern, and the rest 1/2 probability to go in ways producing the exactly complementary interference, that gives luminosity exactly where the first interference gave darkness and vice versa. Imagine that the information of which way it went is recorded somewhere in a purely classical manner but you just don't look at it. As long as you ignore which of both possibilities it followed, you "observe an absence of interference". However, you discover retrospectively the interference that happened, just by informing yourself about that which-way information and reprocessing the picture by classifying the observed dots in the initial picture into 2 different pictures. There is no quantum mystery here, just elementary stuff of classical probabilities. In this case, what could be initially meant by "observing an absence of interference" ? Nothing.

2. Remember that the physics does not specify any "causality order" in the "spooky action at a distance" when entangled stuff is measured.

I described things here : http://settheory.net/epr pointing out that multiple interpretations for the physically same experiments (and effectively same predictions) are possible. And nature plays with us by its way of not having any naturally preferred interpretation. So when someone says that the future changes the past : well no, sorry, this might be said about past unmeasured stuff, but you go too far when extending this to the case of past measured stuff: it might be seen as one possible interpretation, but most probably a wrong one.

Let's start.

Anyway, the interference pattern does not destroy at all the "2 spots" pattern that may or may not appear in the absence of interference; instead, it comes as a supplementary effect (roughly as a multiplying this function by its own function).

What I mean is that when I see the 2 spots picture at 1:27 of the video, it gives the impression that the 2 spots are clearly separate from each other. But if they were indeed separate as they are shown there, no significant interference between them would be possible.

Be careful about the risk of over-interpretation of things in this video which seems to exaggerate the role of the observer. I guess that by only understanding what is said in this video as if things were a matter of conscious observation, without any deeper technical expertise on the subject, it would not seem clear what is wrong with the article http://arxiv.org/abs/1009.2404
(to which I replied there)

There is a wrong claim, as well as wrong pictures as I just explained, due to misunderstanding, in the phrase between 8:20 and 8:25. The fact is, there is no such a possible concept as an "observation of absence of interference" to be testified by the presence of 2 spots, because interference does NOT anyway remove the 2-spots pattern. Instead, it adds its own pattern on top of it.

at 6:20 it says "if it arrives at D1 or D2 they always display an interference pattern" yes but ONLY insofar as we make effective use of the information distinguishing between the D1 and the D2 cases to sort out the dots into 2 different pictures which are complementary interference patterns.
In these conditions, there is no more physical truth in saying that "the second photon arriving in D1 or D2 causes the interference pattern for the first photon" than in saying that "the precise position of the first photon on the screen changes the probabilities of destination of the second photon between D1 or D2". Both claims are equivalent.

The experiment presented around 9:20 comes with wrong interpretations. The fact is, it does not make direct sense to say what affects what. Instead, things should described as relative to the knowledge of a given observer. The video uses the words "before" and "after" (9:45 - 9:50) but you must understand that there is no physical role of time here, it is only an epistemological use of time, to describe the difference of probabilities of what you can guess about one thing "before" and "after" you inform yourself about something else. In what I just described before, can we say that the act of checking the which-way path so as to reprocess an non-interference picture into 2 complementary interference pictures, really acts on what happens (retrospectively creates an interference that was found to not exist) ? It doesn't. If you don't care about the fact that the same predictions or phenomena may be described in different ways which naively sound as if they come from distinct realities of "what happened", but in fact they are physically equivalent, so as to cast doubt on the question whether "what really happened" means anything at all as opposed with specific claims such as "it is already fixed" or "it is really changed retrospectively by observation", then you can end up in abusive conclusions.

Next: The EPR paradox - Quantum entropy - Quantum decoherence

Main site: Set Theory and foundations of mathematics - Foundations of physics (table of contents), with List of physics theories.