Soft question: MWI and localization

Question

According to the many worlds interpretation when an electron passes through a diffraction slit we would approximately have a superposition of states.

For each place where the electron wave and the detector(the wall) got entangled(was detected) when an observer sees the results of the experiment they entangle with each state.

My question is, why are the results so localized?

In other words, if you tell me the detector has a good resolution I understand why you would only see a tiny blob where the electron it crashed, and thus our state would be a superposition of blobs.

Each blob is a place where a component of the electron wave got entangled with the wall in a detection event. The superposition is the superposition of the walls, each universe having a wall with an electron of in a different place.

The probability, ie the number of walls with an electron in position X, is proportional to the Born Rule of the original electron wave.

My question is, why when you do this experiment the resulting superposition has localized blobs? I mean photons don't leave such diffraction pattern.

Instead you get to see the whole diffraction pattern at the same time, a blur, while for electrons you see only a tiny blob.

I suspect this has to do with

1. The size of the photons, they are bigger cause their wavelength is bigger cause lower momentum,
2. Bosons and maybe pauli exclusion? Like you can have many photons colliding with the same positions and blur over while for electrons only one at a time?

So for photons you tipically get many photons and thus a blur; you would need very low intensity light to get the "only a tiny blob pattern, ie only one photon at a time", while for electrons this does not happen. So you need an statistical ensemble to discover the pattern.

I think I am probably wrong, what is the correct explanation?

Answer 1

In quantum theory a system $S_1$ can be described by a state $\psi(\mathbf{r},t)$ and it obeys some equation of motion such as the Schrodinger equation: $$ i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r},t)+V(\mathbf{r})\psi(\mathbf{r},t) $$

In practice no system $S_1$ is completely isolated from the outside world so it is actually part of a larger state $\Psi(\mathbf{r},t)=\sum_j\psi_j(\mathbf{r},t)\phi_j(\mathbf{r},t)$ where the $\phi_j(\mathbf{r},t)$ are states of the rest of the universe $S_R$ and this larger state obeys an equation of motion like the one above.

Interactions that copy information out of a system suppress interference between the different possible states of that system: this process is called decoherence:

https://arxiv.org/abs/1911.06282

For a large system like a screen, the system usually changes a lot more slowly than the processes that copy information out of it like air molecules or light interacting with the screen and interference is very heavily suppressed. This process picks out a set of states that don't interfere much with one another and those states are usually narrowly peaked in both position and momentum on the scale of everyday life because the equations of motion are differential equations with respect to position. Decoherence doesn't eliminate any of the states of the set it just suppresses interference between them so they evolve autonomously to a good approximation.

As a result if you take quantum equations of motion seriously as a description of how reality works, reality looks a bit like a collection of parallel universes in the circumstances of everyday life:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

The resulting states tend to be peaked in position and momentum because the relevant equations of motion distinguish between states with different positions and momenta and so copy information about those quantities into other systems and suppress interference between different values of those quantities.

Answer 2

I don't think the comments you've received have been very helpful so let me put it this way. We all know that classical models for point particles and classical models for fields (like the EM field) are very different. None of that changes after we "quantize". After accounting for quantum effects, it is still true that particles and fields do very different things. Saying that the distinction disappears because wavefunctions endow particles with "field-like" properties is a common elementary confusion.

1. The state of a quantum mechanical particle lives in a Hilbert space where the position operator is a good operator. Its eigenvalues, which are fixed $(x, y, z)$ co-ordinates, are the only outcomes for it which can be measured at a given time. The probability of each such measurement is described by the wavefunction.

2. The state of a quantum field can instead be acted upon by field operators. Their eigenvalues this time are delocalized functions and the propbability of each is described by a wavefunctional.

The statements above are part of quantum mechanics which means they are true in all interpretations including the most minimal one which is MWI. Knowing this, there are two questions I can distill from what you posted.

1. Why is the former case (single particle quantum mechanics) applicable to electrons in the double slit experiment? Because that theory was desigend in part to explain the double slit experiment. Many previous experiments in the 1800s were consistent with the electron being treated as a classical particle whereas the extreme conditions of colliders require a QFT treatment. The fact that the double slit experiment lies in an intermediate regime is a happenstance which was fortunate for science. In retrospect, we can say this is a consequence of QED being weakly coupled at the energies we can access.

2. Why does striking a wall approximate a measurement of the position operator? There is a long discussion of detectors that would be possible to have here but let's just say a "pixel" on the wall is a particle which has been prepared in such a way as to let us "know" when it interacts with the electron. It could be an atom of a semiconductor that allows electricity to flow when excited for instance.

So let us analyze a 3 particle system in the many worlds language. One incoming electron $x_0$ and two detector particles $x_1, x_2$. Since we are good at making rigid walls, the dynamics of the system are such that $x_1$ has to be close to $y_1$ and $x_2$ has to be close to $y_2$. In other words \begin{align} \int_{-\infty}^\infty \int_{-\infty}^\infty \psi(x_0, x_1, x_2, t) dx_1 dx_2 \approx \int_{y_2 - \epsilon}^{y_2 + \epsilon} \int_{y_1 - \epsilon}^{y_1 + \epsilon} \psi(x_0, x_1, x_2, t) dx_1 dx_2. \end{align} At any time $x_0$ is free to vary but the universes that have $x_1$ or $x_2$ different from what they were initially will come with tiny probabilities of us finding ourselves in them. We may therefore very well find ourselves in a universe where $x_0 \approx x_1$ (detection at $y_1$) or a universe where $x_0 \approx x_2$ (detection at $y_2$). But having $x_0$ be close to both of them is just not something that can happen in single particle quantum mechanics.

There can still be universes that have electricity flowing in both places but this would not follow inevitably from the setup like it would in an experiment with fields. Rather it would be a result of fine tuning. How can the wavefunction controlling the appearance of these universes attain a high value at some time? The main way would be for the evolution to be governed by a detector Hamiltonian which favours the "hopping" of a recently detected electron towards the next pixel of the detector.