Hi @Wayfarer,
I try to answer in a non-relativistic point of view.
Short answer: the particle does not interfere with itself. In QM the interference pattern is predicted by the form of the wavefunction which can be inferred by previous observations or by theoretical assumptions.
But I then introduced the argument that this shows that the 'wave equation' is independent of time (and therefore space, as I had understood these two to be related as 'space-time' in relativistic physics). I said, in particular, that 'what is causing the interference pattern is outside, or not a function of, space-time'.
In my opinion here you are right. The wave equation is in fact independent of time in QM
in form. However the wavefunction is of course a function of the positions (which in QM are (hermitian) operators, i.e. observables) and time (in QM it is a parameter and
not an operator).However in non-relativistic QM the "squared modulus" of the wavefunction is proportional to the density probability to find a particle in a position at a given time.
However strictly speaking the cause of the interference pattern is not "outside".In this case, in fact, the wavefunction is a sum of two terms: there is a term that describes the path through the first slit and another relative to the other path. However the wavefunction itself is not an observable. What you can observe is the probability distribution. Forgive me if I use a simplified formula:
Let F be the wavefunction, f1 the part relative to the first path and f2 the path relative to the second.
F = f1 + f2. The wavefunction however is complex. So the probability density distribution is:
FF* = f1f1* + f2f2* + (f1f2* + f2f1*)
The first two terms (f1f1* and f2f2*) are always positive. The sum of the other two (the interference) however can be negative (hence the minima)! What you observe is that the particles follow that probability distribution (which of course is related to the interference pattern: the maxima of the interference pattern are the points where FF* has a maximum and the minima where FF* has a minimum). The interference therefore is given by the wavefunction itself: if our system is not time-invariant of course the probability distribution can change during time. The interesting feature of QM however is that you do not observe F (and so f1 and f2) itself but FF*. So it does not intefer with itself, simply the interference pattern depends on the (a-priori unknown) expression of the wavefunction.
The interaction is between the particle and the experimental apparatus. However the wavefunction has all the information about this interaction.
Remember that in a
single experiment what happens is that we observe only one "path". To observe all possible path and therefore to verify that there is a "probabilistic" law we have to do a lot of measuerements. Consider a simplified version ot the experiment. Suppose that the particle can be observed only in two points of space, P1 and P2. Prior to the experiment you cannot know where the particle will be observed. You perform the experiment and you find the particle at P1. You perform a LOT of experiments and you find that with a probability p1 you find the particle at P1. QM says that when you perform another observation you will find the particle at the position P1 with a probailty p1.
Of course you can also predict the form of the wavefunction theoretically (i.e. from theoretical assumptions) - and in fact it is what is almost always done in physics. In fact you can include in the wavefunction all the information about physical interaction (i.e. for example the influence of a electro-magnetic field).
Regarding the ontology there is a lot of views.
Some adherents to the Copenaghen interpretation think that the real is what is observed, therefore until observed we cannot say that the particle "exists". (the wavefunction is not real and until observed the particle does not "exist" - Recall Einstein objections)
Others adherents instead think that it is an epistemological issue. They think that we can only make meaningful statements on the observed. Therefore prior the observation the "ontic status" of the quantum system is unknowable. In both cases the wavefunction is a "predictive tool".
Then there is the MWI (many-worlds). They think instead that the only existing thing is the universal wavefunction which never "collpases". All possibilities are actualized (therefore the particle goes to both slits!). The fact that we observe only a determinate path is due to a sort of an illusion.
The original Bohmian-mechanics treated both the particles and (universal) wavefunction as real. The wavefunction guides all the particles of the universe. Again we observe the probability distribution because of our "ignorance" (we do not observe the whole evolution of the universe).
The "new" Bohmian mechanics instead treats the wavefunction as a "law". The only reality is given by the particles.
There is also the Relational Approach, Statistical interpretation and others.
I suggest this link:
https://plato.stanford.edu/entries/qm/
I hope to have been of help
:-#