The electron double-slit experiment, aka Young's slits (not to be confused with my new website), is a treasure-trove of nature. Some have said all of quantum mechanics can be considered from it.

A brief reminder: when a cathode fires electrons at a screen with two slits, beyond which is another screen, the pattern that builds up on the back screen is bands of light and dark, the dark bands being where few or no electrons strike, the light bands being where more strike. From this we deduce that the electron beam coming from the cathode is a wave.

If the voltage of the cathode is reduced such that only one electron fires out, say, per ten seconds, eventually the same pattern builds up. From this we deduce that each electron is a wave.



In non-relativistic quantum mechanics, this wave is described a the wave equation of the form:

$[E - V] u = \frac{1}{2m}[p - A]^2 u$

where u is the wavefunction, and all the other terms are operators: E is total energy (the Hamiltonian), A is magnetic potential, p is momentum, V is electric potential, m is mass, and our choice of units is such that all the decorative physical constants like h, c, e, etc. are 1. This is the quantum form of Newtonian mechanics $E=\frac{1}{2}mv^2$ extended to include electromagnetism.

This does not put time and space on equal footing. The solution requires knowledge of u at one time and two places. These are generally derived from physical considerations, e.g. where the electron wavefunction must vanish: two positions where the wavefunction is zero at one time: generally the start.

When u(r,t) is time-evolved according to the above, it spreads out from the cathode, goes through the slits, spreads out from the slits and interferes with itself. Consequently, the wavefunction is defined over many positions at the back screen. And yet when we measure, we see that the electron hit a precise point on the screen.

How the electron got from a field to a point is called the measurement problem, and different solutions to the measurement problem have yielded different interpretations of quantum mechanics. The oldest successful interpretation was the Copenhagen interpretation which states that, upon measurement, the electron wavefunction collapses probabilistically to a single position, the probability given by the absolute square of the wavefunction (the Born rule).

This idea of the absolute square is important. It is how we get from the non-physical wavefunction to a real thing, even as abstract as probability. Why is the wavefunction non-physical? Because it has real and imaginary components: u = Re{u} + i*Im{u}, and nothing observed in nature has this feature. The absolute square of the wavefunction is real, and is obtained by multiplying the wavefunction by its complex conjugate u* = Re{u} - i*Im{u} (note the minus sign). Remembering that i*i = -1, you can see for yourself this is real. We'll come back to this.

There are other probabilistic interpretations, and also some deterministic ones, such as Bohmian mechanics, wherein the electron always has a single-valued position and momentum (hidden variables), and Many-worlds interpretation in which the wavefunction does not collapse but, thanks to the mathematical rules of entanglement, you can never have a term in the wavefunction in which the electron hit the screen at position $y$ but you observed it at position $y' \ne y$.

I'd like to question here the physics of the Copenhagen interpretation -- a reference point for many non-determinists and God-botherers -- because it is incredibly simplistic and I have always thought so. The back screen is treated in an ideal way, which is something physicists often have to do to make problems tractable, but the artefacts of this idealisation are then taken as containing actual insight.

The back screen is a macroscopic object that cannot be treated precisely with quantum mechanics. However, we can still use QM to investigate the issue further. For instance: is the electron wavefunction the only quantity that affects where the electron can be found? Clearly it isn't. An electron cannot, for instance, occupy any part of the screen where an electron already is, unless that second electron can somehow vacate its position (Pauli exclusion principle). This will not reduce the potential sites the electron can occupy to 1, but it will, at any given time, for any given electron, stop the number of sites being a continuum. In the language of condensed matter physics (for the screen is condensed matter), an electron can only go where an electron hole exists. An electron hole in this context is any space where an electron could occupy but does not.

The back screen is a high-entropy object compared with the electron. That is, at any time, it may occupy one of hundreds of thousands or millions of microstates: particular configurations that are energetically equivalent to one another. At one instant t', a position on the screen r' may not admit an electron because it already has one there. At a subsequent instance t'', it might admit an electron at r'. The screen will explore these microstates in a thermodynamic way. i,e, in the same way that a box of gas will have different but energetically equivalent configurations of gas molecules one instant to the next.



[EDIT: To incorporate spin, just extend the concept of the coordinate r.]

So we've reduced the continuum of the screen down to some finite number of available electron holes at time t'. But the electron is again a physical thing: it has energy and momentum. It can't just stop when it fills a hole. It goes on to do other things and, again, this is not considered in the simplified version that sustains the Copenhagen interpretation. Because what it does next is also constrained by the precise microstate of the screen. If it scatters another electron such that it goes from state k --> k'', there has to be an existing hole with state k''. Further, whatever state the scattered electron ends up in, there has to be a hole for that too.



In this way we can reduce that finite number of acceptable positions on the screen ever more by continuing the process. Once it has scattered that first electron, it will scatter another and another, each scattering ever constrained by available states in the screen. We can go further. Eventually the electron, either alone or in an atom, will leave the screen entirely, go off on fun adventures in space and time, until eventually one day it finds a positron -- another kind of electron hole -- where it dies like the swine it always was.

Because the electron's birth and death are the true boundary conditions of its wavefunction! And here we turn to relativity. The relativistic wavefunction is of the form $[E - V]^2 = [p - A]^2 + m^2$. This puts time and space on equal footing (both energy and momentum are squared), requiring knowledge of the particle at two times, not just one. This is why in relativistic quantum theories, we do not proceed by specifying an initial state, time-evolving it forward, and asking the probability of spontaneously collapsing to a particular final state. Rather, we have to specify the initial and final states first, then ask what the probability is. This is constructed as a Green's function G(r, t; r', t'). You can see that we can swap the primed and unprimed coordinates around. If, for a given G(r, t; r', t') there exists a G(r', t'; r, t), the process X=(r,t) --> X'=(r',t') is reversible. (If the the magnitudes of the two are always equal, the system is in equilibrium.)

We are quite familiar in QM with specifying spatial boundary conditions. For instance, consider a tiny oven, a blackbody radiator. What photons can be emitted inside? We all know that only photons with wavelengths that are integer divisors of 2L, where L is the side of the box, can be sustained (B, C & D below). This is quantisation in a nutshell, and it pops out as solutions of the wave equation with the proper number of spatial boundary conditions.

But let's turn the question around. An atom in the oven emits a photon and later another atom absorbs it and re-emits it (more like A below). How does the first atom know to emit a photon such that an even number of its wavelengths will fit in the box? How does either the atom or the photon know how big the box is? Photon emission is just the de-excitation of an atom from one energy level to a lesser one, and the energy of the oven is given by its temperature, which could be anything.



The question becomes less mysterious when we treat time and space equally: the emitted photon doesn't just have a spatial endpoint, but a temporal one, i.e. the photon can only be emitted when it "knows" where and when it will be absorbed. But how could this be?

Photon emission/absorption is another example of a reversible process. Imagine atom 1, in an excited state, "spontaneously" de-excites, deterministically emitting a photon which then "spontaneously" collapses so as to be absorbed by atom 2 which deterministically excites. Run that movie backwards and you have exactly the same thing, only the thing that's spontaneous is now deterministic and vice versa. The Copenhagen description, though, is irreversible: wavefunction collapse is a loss of information that is not retrieved by simulating the reverse process.



The wave equation for the photon is just E = p. If we start with an initial wavefunction u = d(r), where r is its starting point, the wavefunction will expand and expand. To imagine the time-reversed proces, we begin with its final state. But if we just evolve the final state, that won't get us back to where we started. What can we evolve that will take us from time t' back to time t? To answer this, it's easiest to consider plane waves:

$u(r,t) = Ae^{ipr}e^{-iEt}$

What we want is the same thing but travelling in the opposite direction in space (p --> -p) and time (t --> -t):

$u_b(r,t) = Ae^{-ipr}e^{iEt} = u^*(r,t)$

This is just the complex conjugate of u, u*, the thing that, when we multiply it by u, gives us the most fundamental real thing QM has: the probability of finding the particle at position r at time t.

Is it a coincidence that the thing that makes a wavefunction real is also the thing that describes that wavefunction in reverse? Not according to the transactional interpretation of quantum mechanics, which holds that the actual trajectories a particle takes are not just determined by the retarded wavefunction going from time t to t', but also the advanced wavefunction going from time t' to t. (Advanced wavefunctions come up in standard QED as well, to yield the electron self-energy). In this interpretation, the complex conjugate is essentially a message from the future. The electron takes the real trajectories it does in part because it has information about where it's going or, from another viewpoint, where it's conjugate came from.



Both the retarded wavefunction going from t -> t' and the advanced wave coming from t' -> t may explore whatever places they want, but only in the trajectories where one is the conjugate of the other do their trajectories become real.

This is inconsistent with Copenhagen, but quite consistent with relativity. It treats space and time equally, requiring two temporal boundary conditions for the electron. Furthermore, it incorporates more physics. We can identify the complex conjugate of the wavefunction as the backwards-in-time emission of an electron hole in the back screen, which incorporates information about the microstate of the screen itself: for an electron hole to be advanced from (r', t'), the microstate of the screen must be such that there exists an electron hole at (r', t'). We would make such a demand of the cathode: for an electron to be emitted at point (r,t) there must be an electron at (r,t). It is only sensible that we do so for the hole the electron will occupy. (It's worth convincing yourself that this hole also has a history. For every electron that vacates position r to occupy position r', it leaves behind a hole and goes to where the hole previously was, describing an electron hole that vacates position r' to occupy position r. We do not need to consider it the same hole throughout, but there must be some conservation of hole-ness.) It conserves indirectly measurable quantum phenomena such as the interference effects in the double-slit experiment because the electron and its hole still go from r to r' or vice versa by every possible trajectory. And yet it eliminates the mysterious collapse mechanism which takes the wavefunction from a field to a singularity.

In a nutshell, my thesis is this... The true boundary conditions of any particle are its birth and death: where and when it was created, and where and when it will be destroyed. These are facts of each particle. This is the full time-dependent wavefunction of the electron which is, relativistically speaking, equivalent to a static 4D wave. Unlike in the QM of the Copenhagen interpretation, the conjugate solution (from death to birth) is also a solution when these boundary conditions are applied. This solution eliminates almost all of the trajectories possible (and expected) in Copenhagen QM. And this is just the single-particle picture.



As we expand the picture to include more bodies in the universe, especially the rest of the electronic field, more and more remaining trajectories are removed by things like scattering and Pauli's exclusion principle.



[EDIT: One can also see that single-particle trajectories that otherwise wouldn't be real could be made so by scattering. But Feynman showed that it is the closest paths to straight lines that contribute the most to the sum over histories.]

There is no guarantee here that this will eventually reduce the number of intersections with the screen to 1. But we are a long way from the original Copenhagen picture of an electron that might be found anywhere. I expect that, if we could solve the many-body Dirac equation for the universe (well, it would have to be some cosmologically-consistent generalisation of it), it probably would resolve to 1 intersection.

If not, we're left with a solution that looks like a hugely constrained version of the many-worlds interpretation. I name this the not-many-worlds interpretation of quantum mechanics.