When possibilities cancel. Add first, square last

The maths of chance began with dice games. In 1654 Blaise Pascal and Pierre de Fermat worked out a gambling puzzle in letters to each other. Since then one rule has held for every everyday kind of chance, from dice to weather forecasts: for two ways that cannot both happen, add the chances. Because no chance is ever below zero, adding a way in can never make an arrival rarer. Keep that in mind. The double slit breaks it.

For two ways that cannot both happen:

So P(A or B) ≥ P(A): a second way in can never lower the total.

Noise cancelling headphones carry a small microphone that hears the noise outside. A chip makes the same wave flipped upside down and plays it into your ear, and the two cancel. It works best on steady low hums, like an engine or a train. Sudden and high-pitched sounds are harder, which is why voices still get through. On a pond, the energy is not destroyed, only moved: the high places gain exactly what the flat places lose. Thin coatings on glasses and camera lenses cancel reflections the same way.

Two equal waves, the second one behind by a phase φ (a whole wave is 360°):

The combined height is 2A|cos(φ/2)|: 2A in step, zero at φ = 180°.

Max Born proposed in 1926 that the chance is the square of the height. In his first paper the square appears only in a footnote, added just before printing. He won a Nobel Prize in 1954, mainly for this idea. To keep the numbers simple, this page uses a scale where the brightest band scores 1, and shows only the middle of the screen, where each slit alone spreads its electrons evenly. The bright bands get twice what simple adding gives, and the dark bands get nothing.

Born's rule for two amplitudes a₁ and a₂ with a phase difference φ:

The last term is interference. Here |a₁| = |a₂| = ½, and the extra path d sin θ sets the phase:

cos φ averages to zero across the screen, so the average stays ½.

Explainers 3 and 4 showed this with a which-path switch. What removes the stripes is difference, not force: one possibility is now tagged left and the other right. The tag only has to exist, and nobody needs to read it. A faint tag, which tells the slits apart only some of the time, leaves fainter stripes, because the parts of the two possibilities that still match can cancel. Berthold-Georg Englert put this trade-off into a formula in 1996. The slider assumes a clean tag, with no extra noise.

Attach tag states t₁ and t₂ to the two paths (overlap taken real and positive):

A clear tag has ⟨t₁|t₂⟩ = 0, so P = |a₁|² + |a₂|². Stripe strength V and how well the tag tells the paths apart, D (a guess is right with chance ½(1 + D)), obey:

Feynman explained this in his 1985 book QED, based on lectures for a general audience. Each way something can happen gets an arrow, like a clock hand that turns as the possibility travels. Add all the arrows nose to tail, then square the length of the total. Because chances come from a square, possibilities interfere only in pairs, so three slits should show no extra three-way effect. In 2010 Urbasi Sinha and colleagues sent light through three slits and found none, to within about a hundredth.

Each arrow is a complex number: a length r and a direction θ.

Squaring leaves only pair terms. Sorkin's three-path test:

Sinha and colleagues bounded ε to less than about 10⁻² of the two-path interference.

Born's square rule has passed every test so far, yet nobody can say why nature squares. Some physicists try to derive it from deeper ideas, and others simply take it as a starting rule. The book questions the square itself on pages 27 and 73. What every experiment agrees on is the order: the possibilities add like waves first, and the square comes at the end.