A general solution of the measurement problem

What is the measurement problem?

The measurement problem is that there is a conflict between the evolution following the Schrödinger equation and the description of a measurement.

The Copenhagen interpretation describes a measurement in the following way: We have a measurement device, which is located in the classical part, so that the pointer of the device has a classical position, which after the measurement points to the result of the measurement.

The measurement device as a whole is identified with some self-adjoint operator \(\hat{A}\). If this pointer, after the measurement, points to the result \(a\), then the wave function of the quantum system is in an eigenstate \(\psi_a\) of this operator \(\hat{A}\) with eigenvalue \(a\), thus, \[\hat{A} \psi_a(q) = a \psi_a(q).\] If the state was, before the measurement, in a state \(\psi = \sum_a c_a \psi_a\), the result \(a\) will be found with probability \(|c_a|^2\). The measurement problem is that this evolution is not a Schrödinger evolution. A Schrödinger evolution would define a single wave function with certainty, not a probability distribution of different wave functions.

If one, instead, describes the measurement instrument using quantum theory, one gets, as the result of the Schrödinger evolution, only a superposition of different states of the measurement instrument \[(1) \qquad |\psi_{0}\rangle = \psi^{dev}_0(q_{dev}) \left(\sum_a c_a \psi_a(q_{sys})\right) \to |\psi_1\rangle = \sum_a c_a \psi^{dev}_a(q_{dev}) \psi_a(q_{sys})\] where \(\psi^{dev}_a(q_{dev})\) would be a wave packet localized around the pointer pointing to a. This is what the Schrödinger equation gives, and there is no way to reach the mixed state which we observe (different possible outcomes with different probabilies) from that state of a superposition of macroscopic states.

To accept these states as states of reality is absurd, as shown by Schrödinger's cat. (That this absurdity is today embraced by the many worlds interpretation does, in my opinion, not show that it is not absurd, but only that the standards of science have degenerated in such a way that also absurdities have a chance to become accepted at least by parts of the mainstream.) So, the problem is how to get from the superposition (1) to the probability distribution \[ |\psi_1\rangle \langle\psi_1| \to \sum_a |c_a|^2 |\psi_a\rangle\langle \psi_a |\] which we observe in reality.

How assuming the existence of a trajectory solves the measurement problem

In fact, there is an easy way to solve this problem. All one needs is to accept that there is a split between a "classical" part and a "quantum" part, that in the "classical" part the state is described by a trajectory in the configuration space \(q(t)\in Q\), and the quantum part by a wave function \(\psi(q,t) \in \mathcal{L}^2(Q)\). Moreover, one has to accept that the split is arbitrary. This is natural if one assumes that it is a split in our description, not in reality.

In this case, it follows that there can be different splits describing the same physical situation. Once there is no difference in the physical situation, the two different descriptions should be compatible. Let's see how this looks like. Assume we have a classical part (the observer), an intermediate part (the cat), and a quantum part. It follows that for the intermediate part we have two valid physical descriptions, one with a trajectory, and the other with a wave function.

There is also a straightforward way to connect both descriptions. It connects the wave function describing the big wave function of the quantum part together with the intermediate part, the trajectory of the intermediate part, and the small wave function of the quantum part taken alone: \[ (2) \qquad \psi_{small}(q_{quant},t) = \psi_{big}(q_{quant},q_{int}(t),t).\]

Even if the big wave function follows the Schrödinger equation, the small function is not obliged to follow a Schrödinger equation. In general, it will follow a Schrödinger equation only if there is no interaction between the quantum part and the intermediate part. If there is such an interaction, then the evolution of the small wave function will not follow a Schrödinger equation. So, during a measurement, we have an evolution which does not follow the Schrödinger equation for the quantum part, but which is influenced by the trajectory of the intermediate part.

This solution requires an acceptance that, at least for such intermediate regions, such a trajectory \(q(t)\in Q\) exists, in addition to the wave function.

The realistic interpretations assume that it exists everythere, that means, in the quantum part too, and use the continuity equation which follows from the Schrödinger equation \[ \partial_t |\psi|^2 + \partial_i (|\psi|^2 v^i(q)) = 0. \] to obtain at least an average velocity. In dBB theory, one assumes that the velocity \(v^i(q)\) is even the deterministic velocity, but usually those interpretations are stochastic, and \(v^i(q)\) is interpreted only as average velocity.

This is also what the minimal realistic interpretation proposed here is doing.

But it is not even necessary to have an equation for the average velocity. The hypothesis that some continuous trajectory \(q(t)\in Q\) simply exists, even if the equations are unknown, is sufficient to write down equation (2) and to solve in this way the measurement.

That means the measurement problem is purely a problem of those interpretations which claim that quantum theory is complete.

About the incompleteness of quantum theory

What makes the classical-quantum split of the Copenhagen interpretation problematic is that the epistemic interpretation of the wave function is in conflict with some other ideas of the Copenhagen interpretation, namely that it is complete. This is what the Einstein-Bohr controversy was about: Einstein believed that quantum theory is incomplete, while Bohr defended its completeness.

Some defense of Bohr's position

I have read somewhere (forgotten where) the idea that Bohr may have been misinterpreted, not only by Einstein, but by essentially everybody else, in the following way: What he defended was not the idea that quantum theory is the ultimate, most fundamental theory, with no more fundamental theory being possible. It was, instead, simply the internal consistency of quantum theory. Such an internal consistency requires also, in some sense, a notion of completeness. All what can happen according to the theory - evolution in time, measurements, whatever else - can be described using the same formalism of quantum theory. In this, much weaker, sense, quantum theory is, indeed, complete. But this notion of completeness does not exclude at all the existence of some more fundamental theory.

Einstein was, instead, arguing for something different, namely that quantum theory is incomplete, in the sense that it does not describe reality, but only our incomplete knowledge of reality.

In this understanding, Bohr would have had no problem with the idea that quantum theory is only a theory about our incomplete knowledge of some yet unknown underlying reality. His point was only that quantum theory is an internally consistent, and in this sense complete, theory about our knowledge.

What makes the completeness problematic

The situation is different if we think about completeness in a wider sense, namely if one thinks that there is no underlying reality, no more complete more fundmental theory. Usually a theory of incomplete knowledge presupposes that there is something we do not know completely - an unknown reality. As de Broglie said, "It seems a little paradoxical to construct a configuration space with the coordinates of points which do not exist" (de Broglie 1928).

If there is no such underlying reality, the wave function itself maybe the complete description of reality. This makes the situation quite absurd - we have two different descriptions of reality, one for the classical domain, one for the quantum domain, and they have no clear connection.

Not that there would be a problem in general if different theories describe reality in a different way. But the two ways should be connected with each other, so that one should, in principle, be able to use the more fundamental description in the domain described by the less fundamental theory too.

But this fails, with Schrödinger's cat. It is a description of a quantum state, which can be easily created. If the wave function describes only knowledge, Schrödinger's cat is completely unproblematic. If the wave function describes reality, and, moreover, described it completely, Schrödinger's cat is absurd.

Not that absurdity is something unacceptable in modern physics - this particular absurdity is actually one of the most popular interpretations of quantum theory, the "many worlds interpretation". But, sorry, different from Bohr's position above MWI, is something which deserves only to be named "absurd" and otherwise ignored.