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About the equivalence of dBB theory in quantum equilibrium and Copenhagen - Printable Version

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About the equivalence of dBB theory in quantum equilibrium and Copenhagen - Schmelzer - 05-20-2016

One physical argument I have been able to identify in Bohmists' inequivalence & dishonesty was the following:
Quote:One omnipresent statement made by the Bohmists is that Bohmian mechanics is equivalent to quantum mechanics. That's obviously completely wrong. Equivalent theories must give the same predictions. That's pretty much possible only if they involve an "equally large set of degrees of freedom" that may be mapped to one another, that have the same physical meaning, and nothing is added or replaced on either side.
Let's compare what QM in the Copenhagen interpretation contains, with what dBB contains:
1.)  Above contain a classical trajectory.  The Copenhagen interpretation contains it only in the classical part.  Nonetheless, the Copenhagen trajectory of the classical part is simply the restriction of the Bohmian trajectory to this classical part.  So, we have a simple and natural map \(q^{dBB}(t) \in Q \to q^{Copenhagen}(t)\in Q_{classical}\). 
2.)  Above contain a wave function.  The difference is that dBB theory contains also a wave function of the whole universe, which includes the classical part.  But this does not prevent an identification with the wave function of the Copenhagen interpretation, which is restricted to the quantum part, because the two are related by a formula for an effective wave function of a subsystem:

\[ \psi^{eff}(q_{sub},t)  =  \psi^{full}(q_{env}(t), q_{sub},t).\]

This formula can be applied to define the effective wave function of the quantum part, using the wave function of the universe and the classical trajectory of the classical part:

\[ \psi^{Copenhagen}(q_{quantum},t)  =  \psi^{dBB}(q_{classical}(t), q_{quantum},t).\]

So, we have also a simple and natural map \((\psi^{dBB}, q^{dBB}(t)) \in\mathcal{L}^2(Q) \times Q \to  \psi^{Copenhagen} \in \mathcal{L}^2(Q_{quantum}) \). 

3.)  The physical meaning of the trajectory \(q(t)\) in the classical part of the Copenhagen interpretation is the same:  It describes what we see around us, as well as our own trajectory.  

4.) The physical equation for the wave function outside a measurement is the same Schrödinger equation for the quantum part.  

5.) The measurement process - an interaction between the classical and the quantum part in the Copenhagen interpretation - appears unproblematic too.  To see this, it is useful to remember one point of the Copenhagen interpretation, namely that the split between classical and quantum part is uncertain.  So, we can include the whole measurement device as into the classical part, as into the quantum part, and compare first of all the two resulting variants of the Copenhagen interpretation.  To connect the two Copenhagen variants, we have to use a variant  

\[ \psi^{sys}q_{sys},t)  =  \psi^{big}(q_{dev}(t), q_{sys},t).\]

This formula has to be unproblematic for the various variants of Copenhagen with different splits to be compatible with each other.  But it is also all we need to describe the collapse in the same way.  

6.) The equations for the trajectory in the classical part are identical too.  In dBB, these are the equations of the classical Hamilton-Jacobi formalism, which one obtains in the limit \(\hbar\to 0\) from
\[\frac{\partial}{\partial t}S+\frac{1}{2m_i}(\partial_i S)^2
- \frac{\hbar^2}{2m_i}\frac{\partial_i^2\sqrt{\rho}}{\sqrt{\rho}}
+ V(q) = 0. 
\]
and the guiding equation 
\[
m_i v^i(q,t) = \frac{\partial}{\partial q^i} S(q,t),
\]
in the Copenhagen interpretation this is simply some unspecified classical evolution.
7.)  The physical interpretation of the wave function \(\psi(q)\) as defining the probability of observing the configuration q via the Born rule \(\rho(q)=|\psi(q)|^2\) is postulated in the Copenhagen interpretation, and defines the quantum equilibrium in dBB theory.  The Born rule for all other measurements follows from the consideration of a general measurement process, see above, and, finally, the application of the Born rule for the configuration of the measurement device.

So, we have a simple mapping from dBB to Copenhagen, and have the same physical meaning of the parts identified by this mapping.  What about the existence of additional parts, and their role?  First, it is difficult to identify what are these "additional parts",  because these additional parts are nothing but the parts which are part of the Copenhagen picture too, if the split between classical and quantum part is made on the other side of the "additional part" in question.  Is there something special which, according to Copenhagen, cannot be in the quantum part?  The observer, may be?  But the observer consists of atoms, and these atoms may be and are regularly described inside the quantum part.  This works in the other direction too.  So, it is not even clear what these additional objects are supposed to be.  Of course, in comparison with every particular instance of a Copenhagen description of the world, dBB has additional parts:  the wave function of the classical part and the trajectory of the quantum part.  But in different Copenhagen splits these additional parts differ.  

Quote:But Bohmian mechanics also adds additional "beables", such as the classical particle positions, and uses these new classical degrees of freedom to decide what will happen in an experiment.  At this moment, the theory is already obviously inequivalent to quantum mechanics where all the predictions are made from the wave function – and never from some additional classical degrees of freedom.
Oh, the wave function before the measurement predicts the actual result of the measurement, and, additionally, the wave function after the collapse?   That's new for me.  What I have learned is that the wave function predicts only the probability of various outcomes.  And the real outcome is, then, described by the trajectory of the pointer of the measurement device, which, then, defines also the state of the wave function of the quantum system after the measurement. And this pointer of the measurement device is, at least in the Copenhagen version I have learned, part of the classical part and described by its classical trajectory.  

(One can imagine, of course, also Copenhagen variants which do not have classical trajectories in the classical part, but, instead, only sharply localized wave functions around some classical trajectory.  But this would presuppose some physical collapse mechanism, to avoid Schrödinger's cat, thus, would be some different, GRW-type, theory, so I would not think it would be accurate to classify such an interpretation as Copenhagen.)

Quote:It's remarkable how dense the Bohmists may be when it comes to the simple point that one either has an equivalent theory – so that he can't claim any improvement at all – or he changes something about the ways how things are predicted (or what can be predicted), and this general change means that the theory will make different or new predictions that are likely to disagree with the facts (given the fact that quantum mechanics agrees with all the facts).


I don't get the point.  There is an equivalence about the experimental predictions.  But the descriptions are quite different:  Copenhagen describes the world using an artificial, not well-defined split into a classical and a quantum part, describes above part in a quite different way, which seem to have nothing to do with each other, while dBB describes the whole world in the same way.  A unification of the Copenhagen classical and quantum part.  And Copenhagen has to postulate a lot of things which are derived in dBB theory, namely the whole measurement theory, as well as the quantum equilibrium (via Valentini's subquantum H-theorem).  These advantages have, obviously, nothing to do with predictions about observables.  They are theoretical simplifications and explanations, thus, conceptual advantages.

Quote:Bohmian mechanics incorporates both the wave function (rebranded as the pilot wave) and the actual positions. This immediately raises the problem that in general, we don't know which of these two "copies of the information" is observed, which of them affects other things. In almost every particular situation, you may see that every answer is a problem.
Motl may not know, but those who know dBB theory know this.  First, what we observe - the trajectories of measurement devices - are the configurations \(q(t)\).  Then, the wave function \(\psi(q)\) affects the trajectory \(q(t)\) via the guiding equation.  Then, the trajectory of one part of the world affects the trajectory of other parts of the world also via the guiding equation, but only if above systems are in a superpositional state, as defined by their wave function.  Then, the effective wave function of some subsystem - which is the wave function of the Copenhagen interpretation - is defined by the wave function of the universe and the configuration of environment.  

\[ \psi^{eff}(q_{sub},t)  =  \psi^{full}(q_{env}(t), q_{sub},t).\]

(Side remark: Wiki claims "The 1925's pilot-wave picture,[4] and the wave-like behaviour of particles discovered by de Broglie was used by Erwin Schrödinger in his formulation of wave mechanics.[5]", so the rebranding seems to have happened in the other direction.)

Quote:For example, take the simplest "relative success story" Bohmian theory for one electron and try to ask whether the electron emits electromagnetic radiation. The radiation is composed of photons which should also be associated with some "real trajectories" because the location of photons may be measured, too.
The idea that something should have a trajectory because it is can be measured is completely wrong.  In dBB theory, what can be "measured" is every operator which can be measured in the corresponding Copenhagen interpretation too, but only for the configuration there exists a trajectory \(q(t)\).  The results of all other "measurements" are not uniquely defined by the trajectory  \(q(t)\), but depend also on properties of the wave function \(\psi(q)\) as well as the initial value of the measurement device \(q_{dev}(t)\).  Sometimes (say, if the wave function is in an eigenstate of the operator measured) the result depends on the wave function only, so that one can say that a property of the wave function is measured.  In general, the result depends also on the initial value of the measurement device \(q_{dev}(t)\).  In this case, talking about a "measurement result" is misleading, it would be more appropriate to talk about the result of an interaction.  In particular, the rule "unperformed measurements have no results" holds in general in dBB theory: For an unperformed "measurement" we do not have a configuration \(q_{dev}(t)\) of the "measurement device", and, therefore, cannot compute any "measurement result".  

Quote:OK, these photons are emitted whenever a charge is accelerating. An obvious question arises: Should the "presence of acceleration" be decided according to the wave function (pilot wave), or according to the Bohmian trajectory? And analogously, should the Bohmian photons with trajectories be created only near the Bohmian electrons, or near all the points where the wave function (pilot wave) for the electrons is nonzero?
The question should, of course, be decided following the equations.  That means, first of all, one has to solve the Schrödinger equation.  Once the wave function has been defined, we can care (if we like) about the trajectories of the configuration variables of dBB theory.  

Quote:Just like it was impossible to meaningfully answer the question whether the synchrotron radiation is emitted according to the locus of the pilot wave or the Bohmian trajectories, it's impossible to define a meaningful formula for the energy. If the energy only depended on the pilot wave or only on the Bohmian trajectory, it could be easily proven that the conservation law is violated in some situations.
No problem, let's apply the rules above. The energy depends on the momentum variables, thus, is certainly not defined by the configuration alone, and therefore depends at least also from the wave function. If the wave function is in an energy eigenstate, it is the energy of this eigenstate which is measured. If not, the result depends also on the trajectory of the configuration itself as well as of the measurement device. But note that the resulting effective wave function of the system will be, after this, in an energy eigenstate, so that a repetition of the same measurement will give the same result again. What could lead here to a violation of the conservation law in dBB, but not in the Copenhagen interpretation, given that the predicted probability distributions of the measurement results are the same, is beyond me.

I hope this helps.