Double slit

 The double slit experiment, electrons passing through interfere: electrons are wavelike.  Do it one at a time, still interference, how can that happen, the electron interferes with itself.  Detect which slit, the interference stops, and you get two bands corresponding to the two slits.

Exactly. That is the part of the double-slit experiment that makes it genuinely strange rather than merely a demonstration that electrons have some wave-like properties.

The crucial distinction is:

One electron at a time still produces an interference pattern.

You can send electrons through the apparatus so slowly that there is essentially no possibility of two electrons interacting. Each electron arrives as a single, localized detection—a little spot on the screen. Yet after many electrons, those individual spots build up the familiar interference fringes.

So, in the usual quantum-mechanical description, it isn't that electron A interferes with electron B. The electron's quantum state passes through both possible paths, and the two parts of its wavefunction interfere with each other.

Then comes the really interesting part: measure which slit it went through.

The interference disappears. The accumulated spots now form two broad distributions corresponding to the two slits.

And this is not simply because the measuring apparatus physically knocks the electron around. You can arrange the experiment so that the interaction is extremely gentle. What matters is that information about which path is available. Once the alternatives become distinguishable, the two components can no longer produce the same interference pattern.

There is a wonderfully compact way of expressing the difference:

    No which-path information: amplitudes from the two paths are added, then squared → interference.

    Which-path information available: probabilities for the two paths are effectively added → no interference.

And that "available" is important. Quantum mechanics doesn't require a conscious human being to look at the detector. If the physical world has recorded enough information to distinguish the paths, the interference is lost.

Which leads to the deliciously unsettling question: what, exactly, constitutes a measurement? That's where the double-slit experiment stops being a neat physics demonstration and starts getting into the foundations of quantum mechanics.

How does many worlds propose to resolve this?

Many-Worlds takes a rather radical position: nothing special happens at measurement. There is no collapse of the electron's wavefunction at all.

Suppose an electron encounters the two slits. Its quantum state becomes something like:

    electron went through slit A and electron went through slit B

—not in the sense that there are two classical electrons, but that the universal wavefunction contains both alternatives.

Now suppose a detector is placed at the slits. The electron becomes entangled with the detector:

Electron A + detector says A
Electron B + detector says B

And, crucially, the observer who looks at the detector becomes entangled too:

World/branch 1: observer sees "A"
World/branch 2: observer sees "B"

From the observer's perspective, there is now only one result. But the universal wavefunction contains both.
So why does interference disappear?

This is the clever part.

When nothing records which slit the electron used, the two components of its wavefunction remain coherent. They can subsequently recombine and interfere.

But when the electron becomes entangled with a detector, the two alternatives become correlated with different physical states of the detector:
∣A⟩∣DA​⟩+∣B⟩∣DB​⟩

The detector states DA​ and DB​ are effectively distinguishable. Through decoherence, the two components cease to behave as mutually interfering alternatives.

So Many-Worlds says you don't need a mysterious "conscious observation" to destroy the interference. The interaction with the environment does it. The universe effectively splits into noninterfering branches.
And here is the really interesting twist

In Many-Worlds, both outcomes actually occur.

You look at the detector and see:

    "The electron went through the left slit."

There is another branch of the universal wavefunction in which an observer indistinguishable from you sees:

    "The electron went through the right slit."

Neither observer experiences a splitting. Each experiences an ordinary, definite result.

And if you don't measure the slit, the two branches have not become irreversibly separated. They can interfere again. That's why the interference experiment works.

So Many-Worlds' answer to your original puzzle is essentially:

The electron doesn't choose a slit. Both possibilities continue in the universal wavefunction. Measurement entangles the alternatives with the rest of the universe, producing effectively separate branches.

The price of this explanation is, of course, rather substantial: you have to accept that those other branches are physically real.

And there's an even deeper problem: Many-Worlds explains rather naturally why there is no collapse, but explaining why we experience the particular probabilities given by the Born rule—why one branch has probability 60% and another 40%, say—is one of the major philosophical and technical issues surrounding the interpretation.

I have the feeling we are in Newton's shoes not conceiving of space-time as a flexible thing and therefore might do as he did and my no hypothises.

Yes—I think that's a very good analogy, with one important qualification.

Newton couldn't really conceive of space itself as something with dynamics. Absolute space was the stage on which matter moved. Then Einstein essentially said: what if the stage is part of the physics? Space and time aren't merely the background; they can curve, stretch, and participate in what happens.

And your "might do as he did and make no hypotheses" point is particularly apt. Newton famously resisted speculating about the mechanism behind gravity: "I frame no hypotheses" about its cause. He gave an extraordinarily successful mathematical description of what gravity does while declining to pretend that he knew what gravity was.

We may be in something rather similar territory with quantum mechanics.

We have an extraordinarily successful mathematical formalism. We know how to calculate the interference pattern. We know precisely what happens when which-path information becomes available. But we don't necessarily know what the underlying reality is that the mathematics is describing.

The various interpretations can then look rather like attempts to fill that gap:

    Copenhagen-ish approaches: don't ask what is "really" happening underneath; use the quantum formalism and its measurement rules.

    Many-Worlds: the wavefunction is the reality, and apparent collapse is an illusion produced by branching/decoherence.

    Pilot-wave theory: the particle really has a definite position, guided by a wave.

    Other approaches try to modify the quantum formalism itself.

And perhaps our descendants will look at all of these and say, rather sheepishly, "They were asking the wrong question."

There is a particularly striking parallel with your space-time example. We tend to assume certain things because they are built into the conceptual furniture of ordinary experience: objects have definite locations, events happen at definite times, an object takes one path rather than another. Quantum mechanics violates several of those intuitions.

It may be that "the electron went through one slit or the other" is itself a Newtonian-style assumption—an assumption imposed by our everyday picture of reality rather than something nature is obliged to provide.

And that's where I think your analogy gets especially interesting: perhaps the next conceptual revolution won't be a new equation at all. It may be a new answer to what the things in the equations actually mean.