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Questions that I hope are not completely devoid of physical meaning.

Firstly, about space. Let be a Hilbert space, in which we can by definition establish the existence of complete and orthonormal vector bases; and a Psi vector (state) that we write as a linear combination (a "superposition") of these base vectors. Here we can define a notion of distance, because this space is provided with a scalar product; for example, the distance between two vectors in the combination; but this 'distance', if I understand correctly, is only a measure of the similarity/dissimilarity between these two vectors (and cannot therefore be interpreted as a Euclidean distance). So my first question is: what role does this distance play in interference? Or, more specifically, is a large distance (and therefore a large dissimilarity between vectors/amplitudes) an obstruction to the fact that they interfere? Which also raises the question, I think, of whether the angle matters for interference, and not just the phase ratio.

Now to time. I'll put my question like this: is quantum interference a process that takes place over time (and is therefore a 'process' in the strict sense), or an instantaneous consequence of superposition? A physicist (I'm not a physicist by profession, as you'll have gathered) told me, without being more specific, that interference is not an 'interaction' in the strict sense of the word, in other words in the physical sense of the term. I can only guess at what is meant by this (but perhaps you'll disabuse me of the notion): an interaction would be a process during which two 'entities' (in this case two Psi amplitudes) act on each other and therefore exchange energy, impulse and, more generally, information. So interference would be nothing of the sort. But why? I can only make the following assumptions:

  1. Quantum interference does not involve the exchange of forces between probability amplitudes. The amplitudes are simply superimposed and interfere without exerting any force on each other.
  2. It does not require the transmission of information between them. The interference pattern is determined by the superposition of the amplitudes themselves, not by any communication between them.
  3. Finally, it can be considered an instantaneous process, occurring at the same time as the superposition of the states/amplitudes. There is no propagation of influence in time.

And it's this last point that interests me. Doesn't quantum interference emerge instantaneously (so to speak) from superposition? It is, so to speak, the logical consequence, without us being able to speak of a succession (the superposition's anteriority would just be logical, not chronological). I have in mind an analogy with quantum entanglement. In the same way that entanglement does not imply an interraction at a distance (no transmission of a signal or information) between two quantum systems, but translates the simple fact of their non-separability (the superposition of correlations between their respective states), in the same way, there would be no interaction and therefore no spatio-temporal process involved in interference; but because at least two states are superimposed, it 'follows' (logically) that they interfere, and do so differently according to their phase ratios (and their angles, then ?). Does this hypothesis, and the analogy with entanglement, simply make sense?

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    Interference does not appear at the level of vectors in Hilbert space, those vectors need to be represented as functions on 3D space, say, to talk about interference. And the Hilbert space distance has nothing to do with it. Interference is neither a process nor "instantaneous consequence" of superposition, it is a geometric pattern that some spatially represented states exhibit (sums of point sources of waves, for example). Entanglement involves two (or more) objects, interference involves states of a single object, so it is unclear what is even supposed to exchange information with what.
    – Conifold
    Commented Apr 24 at 6:26

2 Answers 2

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It was Schroedinger who considered the psi-function as the main mathematical tool to formalize quantum mechanics (QM).

  • Influenced by de Broglie’s concept of matter-waves Schroedinger searched for a suitable wave equation. The result is the Schroedinger equation for the time-development of the psi-function, a linear differential equation. The sum of two solutions of a wave equation is again a solution of the same wave equation: The interference of two psi-functions is again a psi-function

    psi = psi_1 + psi_2

    As you say, the relevant quantities for the sum are the amplitudes and the relation of the phases of psi_1 and psi_2.

    Interference of psi-functions is mathematical not different from interference of water waves on the surface of a pond. The diffence is the interpretation of the waves.

  • Entanglement is different from interference. Entanglement may happen when two or more particles are components of one common quantum system.

    The mathematics of entanglement is more complex than the mathematics of interference.

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  • Thank you for your reply! I agree about phase ratios and amplitudes. But regarding quantum entanglement,.it was only a question of an analogy relationship, on a precise point... First of all, do you agree with denying quantum interference the status of interaction? And for the three reasons I mentioned? Commented Apr 24 at 4:35
  • @Husserliana Interference of two psi-function of spacetime coordinates (x,t) results in psi(x,t) = psi_1(x,t) + psi_2(x,t), the same (x,t)-coordinates for all three functions. The psi-function does not carry energy, momentum or other physical quantities. Hence I agree with your “assumptions” 1) and 2). - Concerning assumption 3): Interference of psi-functions is(!) the addition of psi-functions. – In addition, one can further investigate your questions in the context of quantum field theory. And in the light of Rovelli’s relational QM.
    – Jo Wehler
    Commented Apr 24 at 6:20
  • Thanks again for these elements! And for the reading recommendations. I'd be inclined to say that between saying that "interference is superposition" and saying that "interference instantly results from superposition", the difference is only one of logical emphasis - in the sense that my second assertion clearly shows that the first results from the other, that it is the logical consequence (but perhaps I'm wrong? And that interference and superposition are perfectly synonymous). Commented Apr 27 at 11:25
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In what follows I am going to use quantum theory without modifying its equations of motion or adding a collapse postulate.

In quantum theory the evolution of a measurable quantity is described by an operator called an observable. The possible outcomes of measuring an observable are its eigenvalues. An observable in general is not a single measurable number, but is rather represented by a matrix that describes multiple instances of the system with different possible values. The state describes what observable you last measured and what its value was. There is a formula called the Born rule that gives the expectation value of the observable given the state.

An observable is said to be sharp if one of its possible values has a probability of 1. Interference involves taking a sharp observable, making it unsharp and then making it sharp again so that the final sharp value depends on the phases picked up by each of the intermediate unsharp values. See for example sections 2 and 3 of

https://arxiv.org/abs/math/9911150

There is only one known explanation of the result of a quantum interference experiment and it involves all of the possible values of the interfering observable. So all of those possible values exist because otherwise they couldn't interfere with one another.

If information is copied from a system undergoing interference this will suppress the interference, this process is called decoherence. As a result of decoherence on the scales of space and time we see in everyday life the world as described by quantum mechanics resembles a collection of parallel universes:

https://arxiv.org/abs/1111.2189

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

Interaction is usually reserved for situations in which two distinct systems are involved in an event so you might not want to call interference an interaction, but this is largely a matter of terminology. Interference is a process that takes place over space and time as with any other physical process and it does require that the different versions are made to interfere with one another. Your claim that

there would be no interaction and therefore no spatio-temporal process involved in interference

is false.

You also write:

entanglement does not imply an interraction at a distance (no transmission of a signal or information) between two quantum systems, but translates the simple fact of their non-separability (the superposition of correlations between their respective states), in the same way

Different quantum systems are separable even if they are entangled since you can separate them and interact with them independently of one another. When entanglement experiments are described in the Heisenberg picture of quantum theory it can be shown that there is an description of entanglement that only involves local interactions. The correlations only arise when measurement results on two entangled systems are compared. The information required to produce the correlations is carried in the measurement results in the form of locally inaccessible quantum information:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

So entanglement experiments are processes taking place over space and time and can be explained in terms of local interactions.

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  • Thank you very much! No doubt I'm a little too influenced by certain interpretations of my QM (rather that of relative states than multiple worlds), where both entanglement and decoherence require a multidimensional configuration space. It is therefore difficult to talk about an "interraction in space" (classical physics). This does not, of course, prevent special relativity from being respected. Commented Apr 27 at 11:31

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