mercredi 22 février 2017

{Bohmian mechanics, is} [subtle, malicious] (?)

Here is my post consisting as usual in quotes from some scientific articles fully available online, underlining (or emphasizing with a bold font) selected parts in order to sketch a draft response to the question in its title. This time, I was mostly inspired by reading this post at another blog named Elliptic Composability.

Inconclusive Bohmian positions in the macroscopic way ...
Bohmian mechanics differs deeply from standard quantum mechanics. In particular, in Bohmian mechanics particles, here called Bohmian particles, follow continuous trajectories; hence in Bohmian mechanics there is a natural concept of time-correlation for particles’ positions. This led M. Correggi and G. Morchio [1] and more recently Kiukas and Werner [2] to conclude that Bohmian mechanics “can’t violate any Bell inequality”, hence is disproved by experiments. However, the Bohmian community maintains its claim that Bohmian mechanics makes the same predictions as standard quantum mechanics (at least as long as only position measurements are considered, arguing that, at the end of the day, all measurements result in position measurement, e.g. pointer’s positions).  
Here we clarify this debate. First, we recall why two-time position correlation is at a tension with Bell inequality violation. Next, we show that this is actually not at odd with standard quantum mechanics because of some subtleties. For this purpose we do not go for full generality, but illustrate our point on an explicit and rather simple example based on a two-particle interferometers, partly already experimentally demonstrated and certainly entirely experimentally feasible (with photons, but also feasible at the cost of additional technical complications with massive particles). The subtleties are illustrates by explicitly coupling the particles to macroscopic systems, called pointers, that measure the particles’ positions. Finally, we raise questions about Bohmian positions, about macroscopic systems and about the large difference in appreciation of Bohmian mechanics by the philosophers and physicists communities... 
Part of the attraction of Bohmian mechanics lies then in the assumption that • Assumption H : Position measurements merely reveal in which (spatially separated and non-overlapping) mode the Bohmian particle actually is.   
A Bohmian particle and its pilot wave arrive on a Beam-Splitter (BS) from the left in mode “in”. The pilot wave emerges both in modes 1 and 2, as the quantum state in standard quantum theory. However, the Bohmian particle emerges either in mode 1 or in mode 2, depending on its precise initial position. As Bohmian trajectories can’t cross each other, if the initial position is in the lower half of mode “in”, then the Bohmian particle exists the BS in mode 1, else in mode 2.

Two Bohmian particles spread over 4 modes. The quantum state is entangled... hence the two particle are either in modes 1 and 4, or in modes 2 and 3. Alice applies a phase x on mode 1 and Bob a phase y on mode 4. Accordingly, after the two beam-splitters the correlations between the detectors allow Alice and Bob to violate Bell inequality... Alice’s first “measurement”, with phase x, can be undone because in Bohmian mechanics there is no collapse of the wavefunction. Hence, after having applied the phase −x after her second beam-splitter, Alice can perform a second “measurement” with phase x′ .

... There is no doubt that according to Bohmian mechanics there is a well-defined joint probability distribution for Alice’s particle at two times and Bob’s particle: P(rA, r′A, rB|x, x′ , y), where rA denotes Alice’s particle after the first beam-splitter and r′A after the third beamsplitter of {the last figure above}... But here comes the puzzle. According to Assumption H, if rA∈′′1′′, then any position measurement performed by Alice in-between the first and second beam-splitter would necessarily result in a=1. Similarly rA ∈′′2′′ implies a=2. And so on, Alice’s position measurement after the third beam-splitter is determined by r ′ A and Bob’s measurement determined by rB. Hence, it seems that one obtains a joint probability distribution for both of Alice’s measurements results and for Bob’s: P(a, a′ , b|x, x′ , y). 
But such a joint probability distribution implies that Alice doesn’t have to make any choice (she merely makes both choices, one after the other), and in such a situation there can’t be any Bell inequality violation.
... Let’s have a closer look at the probability distribution that lies at the bottom of our puzzle: P(rA, r′ A, rB|x, x′ , y)... now comes the catch... as the Bohmian particles’s positions are assumed to be “hidden”... they have to be hidden in order to avoid signalling in Bohmian mechanics. ... it implies that Bohmian particles are postulated to exist “only” to immediately add that they are ultimately not fully accessible... Consequently, defining a joint probability for the measurement outcomes a, a ′ and b in the natural way: 
P (a, a′ , b|x, x′ , y) ≡ P (rA ∈ “a“, rA ∈ “a ′ “, rB ∈ “b“|x, x′ , y) (10) 
can be done mathematically, but can’t have a physical meaning, as P(a, a′, b|x, x′ , y) would be signaling.
In summary, it is the identification (10) that confused the authors of [1, 2] and led them to wrongly conclude that Bohmian mechanics can’t predict violations of Bell inequalities in experiments involving only position measurements. Note that the identification (10) follows from the assumption H, hence assumption H is wrong. Every introduction to Bohmian mechanics should emphasize this. Indeed, assumption H is very natural and appealing, but wrong and confusing.

To elaborate on this let’s add an explicit position measurement after the first beam-splitter on Alice side. The fact is that both according to standard quantum theory and according to Bohmian mechanics, this position measurement perturbs the quantum state (hence the pilot wave) in such a way that the second measurement, labelled x ′ on Fig. 4, no longer shares the correlation (9) with the first measurement, see [4, 5]...

From all we have seen so far, one should, first of all, recognize that Bohmian mechanics is deeply consistent and provides a nice and explicit existence proof of a deterministic nonlocal hidden variables model. Moreover, the ontology of Bohmian mechanics is pretty straightforward: the set of Bohmian positions is the real stuff. This is especially attractive to philosopher. Understandably so. But what about physicists mostly interested in research? What new physics did Bohmian mechanics teach us in the last 60 years? Here, I believe fair to answer: not enough! Understandably disappointing... 
This is unfortunate because it could inspire courageous ideas to test quantum physics. 

Probably surrealistic Bohm Trajectories in the microscopic world?

... we maintain that Bohmian Mechanics is not needed to have the Schrödinger equation "embedded into a physical theory". Standard quantum theory has already clarified the significance of Schrödinger's wave function as a tool used by theoreticians to arrive at probabilistic predictions. It is quite unnecessary, and indeed dangerous, to attribute any additional "real" meaning to the psi-function. The semantic difference between "inconsistent" and "surrealistic" is not the issue. It is the purpose of our paper to show clearly that the interpretation of the Bohm trajectory - as the real retrodicted history of the atom observed on the screen - is implausible, because this trajectory can be macroscopically at variance with the detected, actual way through the interferometer. And yes, we do have a framework to talk about path detection; it is based upon the local interaction of the atom with the photons inside a resonator, described by standard quantum theory with its short range interactions only. Perhaps it is true that it is "generally conceded that.. . [a measurement]... requires a ... device which is more or less macroscopic," but our paper disproves this notion, because it clearly shows that one degree of freedom per detector is quite sufficient. That is the progress represented by the quantum-optical whichway detectors. And certainly, it is irrelevant for all practical purposes whether "somebody looks" or not; what matters only is that the which-way information is stored somewhere so that the path through the interferometer can be known, in principle.

Nowhere did we claim that BM makes predictions that differ from those of standard quantum mechanics. The whole point of the experimentum crucis is to demonstrate that one cannot attribute reality to the Böhm trajectories, where reality is meant in the phenomenological sense. One must not forget that physics is an experimental science dealing with phenomena. If the trajectories of BM have no relation to the phenomena, in particular to the detected path of the particle, then their reality remains metaphysical, just like the reality of the ether of Maxwellian electrodynamics. Of course, the "very existence" of the Böhm trajectory is a mathematical statement to which nobody objects. We do not deny the possibility that some imaginary parameters possess a "hidden reality" endowed with the assumed power of exerting "gespenstische Fernwirkungen" (Einstein). But a physical theory should carefully avoid such concepts of no phenomenological consequence.  
B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther
received October 12, 1993  

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