Taking back my words

In my previous post, I described the relational interpretation of quantum mechanics and made a criticism to it. I have realized now that my criticism rested on a (rather elementary) mistake. I said:

One criticism that comes to mind is that the relational interpretation is not very clear on which events count as "interactions" or "measurements". In the situation described above, in which system B "measures" system A and changes its state relative to it, at what time does this change occur exactly? From the external point of view no change has occurred: relative to an external observer the whole state has evolved continuously into a superposition. So there is a question that seems real enough, "Exactly when did B's state relative to A change?" but to which quantum mechanics provides no answer and no experiment will ever shed light on. This seems strange; if Rovelli has a good answer to this question, I am not aware of it.

Quantum mechanics does indeed provide no answer for "when did B's state change?" but that is not a problem for the relational interpretation: it's just an ordinary example of the indeterministic nature of QM. Let's spell out more clearly what is going on here. At the beginning system A is (relative to B and to an external observer O) in a superposed state K1 Ia1> + K2 Ia2>. System B is before interacting with A in a state Ib0>, which we can interpret as the default state of a measuring device before a measurement. Now A and B start to interact. Relative to O who does not interact with them, the state of B evolves deterministically into:

IS(t)> = C0(t) Ib0> + C1(t) Ib1> + C2(t) Ib2>

This means that at each time t there is an amplitude C0 for the measuring apparatus B to have failed to detect anything, and amplitudes C1 and C2 for detecting the states<>Ia1> and Ia2>. If B is a "good" measuring device we should expect C0(t) to approach 0 fastly after the beginning of the interaction, and C1(t) and C2(t) to have, at least for large t, the same relative weight that K1 and K2.

According to the relational interpretation, the state of A relative to B will change either to Ia1> or to Ia2> at some time after the interaction started, and the state of B itself relative to A will also change either to Ib1> or to Ib2> at the same time. But we should not expect the relational interpretation to tell us at what time will that happen. We know, from quantum mechanics, what is the probability of the transition having taken place at each time t, as encoded in IS(t)>. And that is all we can know, and being the world indeterministic it is all we can expect to know.

In summary, the mistake I made in the last post was to forget that not only the resulting states of transitions are indeterministic in quantum mechanics, but also the times at which the transitions take place.

There is a thread going on on the Smerlak-Rovelli paper at Physics Forums, where Marcus has some flattering words of praise for this blog.