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AvatarRoderich Tumulka

Here are some general remarks on the practical usefulness of Bohmian mechanics (in a rather random order). They don’t address Maximilian’s specific question but contribute to the overall theme of ways in which Bohmian mechanics is helpful.

* Consider two theories making the same predictions: one clear, precise, and simple, the other unclear, vague, and complicated. Which one is better? There is no doubt that Bohmian mechanics is clear, precise, and simple (a complete definition fits on a single slide). Copenhagen QM is not completely clear (Feynman: “Nobody understands quantum mechanics”), and it is vague and complicated because of the measurement axiom that refers to observers and measurements (vague concepts, as nobody has a precise definition of which physical objects count as “observers”).

* It is one of the goals of physics to find out how the world works; it would seem odd to say, now that Bohmian mechanics provides a coherent answer that standard QM could not provide, that we are no longer interested in this goal.

* In the same vein, it is one of the goals of physics to find explanations of the observed phenomena. Again, it would seem strange to say we are no longer interested in explanations of, say, the double slit experiment–even more so in view of the idea, introduced by Bohr and eloquently elucidated for the double slit in Feynman’s lectures, that such an explanation be impossible. Let me add, I agree that it is one of the goals of physics to make predictions (and another to develop better technology), but there are also the goals to find out how the world works, and to find the explanations of the phenomena that call for explanation.

* Bohmian mechanics has inspired some discoveries, e.g., Bell’s nonlocality theorem. (My own recent work on multi-time wave functions and interior-boundary conditions was inspired by Bohmian mechanics.) It may also be useful in the search for, e.g., a theory of quantum gravity.

* Bohmian mechanics is easier to learn for students than standard quantum mechanics.

* Bohmian mechanics has applications to numerical methods for solving the Schrodinger equation. (Experts find a higher efficiency if grid points are not evenly spaced but |psi|^2 distributed, and a higher efficiency if the grid points “move with the flow,” which is what Bohmian trajectories do.)

* Let me draw some parallels with the following questions: Do we need mathematicians? Should students learn proofs (say, of the Gauss integral theorem)? Well, for practical physics computations it is usually not relevant to know the proof of the Gauss theorem, while it is very relevant to know the theorem itself. That is, a limited level of rigor is often sufficient for getting the right answer and efficient for getting it quickly. Nevertheless, sometimes math can get very confusing, and then it is useful to know the details of math facts. (E.g., how exactly is the delta function defined? And what exactly does it mean to say that the Laplacian of 1/r is -4 pi delta?) So, it is good that there are mathematicians who are very careful when formulating statements and proofs. (And, after all, what they do is correct.) The situation is a bit similar with Bohmian mechanics: Even if it is usually not necessary for finding the correct predictions, it can be useful to have a precise version of QM, particularly when QM gets confusing.

* Bohmian mechanics provides some useful approximations for the computation of predictions, e.g., concerning the statistics of arrival times (when will the detector click?) and semi-classical approximations. While the predictions are the same in Bohmian mechanics and standard QM, and can be computed also without Bohmian mechanics, certain approximations are suggested by the Bohmian approach. To be sure, the approximation can equally be used if Bohmian mechanics is wrong (say, if collapse theories are right).

* Bohmian mechanics permits an analysis of quantum measurements, while they are taken as primitive and/or unanalyzable in standard QM. In Bohmian mechanics, one can prove theorems about measurement. E.g., positive-operator-valued measures (POVMs, also known as “generalized observables”) arise from Bohmian mechanics through an analysis but have to be postulated in QM as an extension of the theory. As another example, I once used Bohmian mechanics to give a simple, clean, and clear-cut proof of a superselection rule (i.e., that certain superpositions are indistinguishable from mixtures), while standard QM could offer only hand-waving talk in support of this rule.

* Bohmian mechanics also permits an analysis of issues with philosophical subtleties, e.g., (i) limitations to knowledge, (ii) quantum non-locality, (iii) tunneling times. (i) This is the phenomenon that some facts in the world cannot be completely revealed by any experiment; e.g., wave functions cannot be measured. (ii) Bell’s theorem says that entangled particles must undergo some action-at-a-distance (which cannot, however, be used for sending messages). (iii) Bohmian trajectories provide an obvious definition for how long a particle stayed inside the barrier during tunneling and a deconstruction of the allegation of faster-than-light motion inside the barrier. All of these examples have aspects that go beyond mere operational statements (“if we set up an experiment like this …, then the outcome will be x=… with probability p(x)=…”). And in all of these examples, the clear picture provided by Bohmian mechanics allows us to understand and deal with these aspects.

* Bohmian mechanics takes away the need for philosophical contortions when explaining QM or talking about what happens in certain experiments or what happens out there in the world. (As an example of such a contortion, Bell mentions: “Was the world wave function waiting to jump for thousands of millions of years until a single-celled living creature appeared?”)

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