Is there a realistic interpretation of Quantum Mechanics?

(a critical analysis of the Bohmian mechanics)

In the last two posts the merits of classical (realistic) description of Nature were discussed. The major disagreement was on the burden of proof. I contend it is the responsibility of any alternative explanation to prove it is better than quantum mechanics. Given the overwhelming and irrefutable evidence for the applicability of quantum mechanics in describing nature, this is simply impossible, but is there a realistic interpretation of quantum mechanics?

When discussing physics, there are three possible mathematical frameworks to choose from:

• Lagrangian formalism in the tangent bundle,
• Hamiltonian formalism in the cotangent bundle.
• Formalism in the configuration space.

For the Lagrangian formalism, I cannot do any better than Zee in Chapter I.2 of his "Quantum Field Theory in a nutshell":

"Long ago, in a quantum mechanics class, the professor droned on and on about the double-slit experiment, giving the standard treatment.[...] The amplitude for detection is given by a fundamental postulate of quantum mechanics, the superposition principle, as the sum of the amplitudes for the particle to propagate from the source S through the hole A1 and then onward to the point O and the amplitude for the particle to propagate from the source S through the hole A2 and then onward to the point O.

Suddenly, a very bright student, let us call him Feynman, asked, "Professor, what if we drill a third hole in the screen?" The professor replied, "Clearly, the amplitude for the particle to be detected at the point O is now given by the sum of three amplitudes [...]."

The professor was just about ready to continue when Feynman interjected again, "What if I drill a fourth and a fifth hole in the screen?" Now the professor is visibly losing his patience: "All right wise guy, I think it is obvious to the whole class that we just sum over all the holes."

But Feynman persisted, ""What if I now add another screen with some holes drilled into it? The professor was really losing his patience: "Look, can't you see that you just take the amplitude to go from the source S to the hole Ai in the first screen, then to the hole Bj in the second screen, then to the detector at O, and then sum over all i and j?"

Feynman continue to pester, "What if I put a third screen, a fourth screen, eh? What if I put in a screen and drill an infinite number of holes in it so that the screen in no longer there?" [...]

What Feynman showed is even when there is just empty space between the source and the detector, the amplitude for the particle to propagate from the source to the detector is the sum of the amplitudes for the particle to go through each of one of the holes in each one of the (nonexistent) screens. In other words, we have to sum over the amplitude for the particle to propagate from the source to the detector following all possible paths between the source and the detector."

So if we have to consider all possible paths, the notion of the classical trajectory is simply doomed and there is no realistic quantum interpretation in the Lagrangian formalism. One down, two to go.

We will make a quick pass onto the phase space formalism for quantum mechanics. This is very easy, Wigner functions contain negative probability parts, and hence there is no realistic quantum interpretation in the Hamiltonian formalism either. Two down, one to go.

However the configuration formalism is the tricky one. Here one encounters the Hamilton-Jacobi and the Schrodinger equation.

Let us start from classical physics. Consider 1-d motion in a potential V. The Hamilton-Jacobi equation reads:

\(\frac{\partial S}{\partial t} + \frac{1}{2m}{(\frac{\partial S}{\partial x})}^2 + V(x) = 0\)
and
\(p = \frac{\partial S}{\partial x}\)

if V=0, we consider \(S = W-Et\) from which one trivially obtains:

\(\frac{\partial p}{\partial x}=\sqrt{2mE}\). If the particle is at the moment \(t_0\) at \(x_0\) then the particle motion is unsurprisingly:
\(x-x_0 = \sqrt{\frac{2E}{m}}(t-t_0)\)

The key point is that we need the initial particle position to solve the equation of motion.

So what would happen when we replace the Hamilton-Jacobi equation with its quantum counterpart, the Schrodinger equation:

\(-i\hbar\frac{\partial \psi}{\partial t} - \frac{\hbar^2}{2m}{(\frac{\partial \psi}{\partial x})}^2 +V\psi(x) = 0\) ?
If
\(\psi=\sqrt{\rho} exp(i\frac{S}{\hbar})\)

then we get the Hamilton-Jacobi equation for S but with an additional term called the quantum potential, and a continuity equation for \(\rho\). Welcome to the Bohmian formulation of quantum mechanics!

To solve the equation of motion we again need an initial condition, just like in the classical case. But this should raise big red flags!!! In school we were taught that quantum mechanics is probabilistic, not deterministic, and also that the wavefunction is all there is needed to know to make predictions. How is this possible? Where is going on here? 

To make quantum mechanics predictions one needs an additional ingredient: the Born rule. It turns out that in Bohmian quantum mechanics the Born rule constraints the allowed distribution of initial conditions and this is the beginning of the end of the realism claim of this interpretation.

Max Born

But where is the Born rule coming from? With the advantage of about 100 years of quantum mechanics now we have two nice answers. The wavefunction lives in a Hilbert space, and in there we have Gleason's theorem which basically mandates the Born rule as the only logical possibility to make sense of the lattice of projection operators. But this is an abstract mathematical take on the problem. There is also an excellent physical explanation given by (surprise...) Born himself: http://www.ymambrini.com/My_World/History_files/Born_1.pdf I will not explain Born's paper because it is very well written and very easy to be understood today, but the main point is that Born's rule is incompatible with an arbitrary initial probability density. The supporters of the Bohmian mechanics are well aware of this and they call the consistent initial probability density "quantum equilibrium". Moreover they point out that after some relaxation time an arbitrary probability density "reaches quantum equilibrium" and so any discrepancy of predictions between Bohmian quantum mechanics and standard quantum mechanics could only occur a few seconds after the Big Bang, so problem solved, right? Wrong! It is rather ironic that the undoing of Bohmian's mechanics comes from a most unexpected direction: Bell's theorem!!! Ironic because Bell was inspired to discover his theorem by viewing Bohmian mechanics as a counter-example to von Neumann's no go theorem on hidden variables.

In quantum mechanics it is very easy to see that the position operator at different times does not commute. Why? because the position operator at time \(t-t_0\) involves the momentum operator: \((t-t_0)P\) and P and Q do not commute. However this means that there is no probability space for positions at different times. But in Bohmian mechanics, the particle always have a "real" position and as such in there such a probability space does exists. Hence we can detect in principle statistical violations in the predictions of standard quantum mechanics and that of Bohmian mechanics.

The proponents of Bohmian mechanics are very well aware of this problem and they have a solution: there are no comparison of measurements at different times! I have to agree this is a very clever, but the trouble is only swept under the rug and it does not go away.

The prediction discrepancy goes away in Bohmian mechanics only if the theory is "contextual". Because of this the way velocity is measured in Bohmian mechanics is not what we would normally expect: \(v = (x_1-x_0)/(t_1-t_0)\) and Bohmian mechanics is known for "surreal trajectories".

Surreal or not, violations of the speed of light or not, non-locality or not, the main trouble is in the sudden change of the probability density after measurement. In the Copenhagen formulation, the wavefunction collapses upon measurement and this is naturally explained as updated information. After all, the wavefunction does not carry any energy or momenta and is just a tool to compute the statistical outcomes of any possible experiment. But one of the advertised virtues of Bohmian mechanics is its observer independence: the measurement simply reveals the pre-existing position of the particle. But is this really the case?

The trouble for Bohmian mechanics is that its predictions for two consecutive measurements differs from that of the standard quantum formalism. Why? Because the "quantum equilibrium" for the first measurement is not a "quantum equilibrium" for the second measurement because wavefunction collapses during the first measurement. (by the way, this is the root cause of why no correct quantum field theory can ever be created for Bohmian mechanics.)

So how does the Bohmian supporters deal with this? The theory is simply declared "contextual" and valid only between preparation (with a pre-supposed quantum-equilibrium distribution) and measurement.

Without contextuality Bohmian mechanics is an inconsistent theory as it predicts violations of the uncertainty principle. With contextuality Bohmian mechanics becomes a time-bound equivalent formulation of quantum mechanics. Think of it as an R^n flat local map of a curved manifold. A field theory in the Bohmian interpretation is impossible the same way a 2-d map of Earth (which topologically is a sphere) cannot avoid distortions. (I see a cohomology no go result for Bohmian quantum field theory in my future ;)).

Because of contextuality the Bohmian interpretation cannot be called realistic.

Now we have exhausted all 3 quantum mechanics formulations.  Quantum mechanics is simply not a realist theory anyway we look at it. It would have been really strange if in one equivalent formalism of quantum mechanics there were a realistic interpretation, and not in the other two.

Basically it is either initial particle positions, or Born rule. But the Born rule is an inescapable self-consistency condition on the Hilbert space due to Gleason's theorem, and we must rule out unrestricted initial statistical distributions.

Quantum mechanics is complete and any addition of "hidden variables" spoils its consistency.