Is the wavefunction ontological or epistemological?
What is wrong with Bohmian interpretation?
Now we are almost ready to introduce a new quantum mechanics interpretation. There is one more
step to take: explain what is wrong with existing quantum mechanical
interpretations. We start with Bohmian mechanics.
Let me clearly state that each and every existing quantum
mechanics interpretation represent valuable contributions and provided new
insight in quantum mechanics. Bohmian mechanics was a core incentive for Bell
to develop his inequalities for example. But what is Bohmian mechanics? The
core idea is simple: the particle has a definite position at any single time
and it is guided by a “quantum potential”.
Since there are no local realistic models for quantum
mechanics, one can see right away that the quantum potential must violate
relativity. But can quantum mechanics and relativity coexists peacefully? What
if we pick Nonlocality as an axiom
for quantum mechanics? Popescu and Rohrlich considered such an idea http://arxiv.org/abs/quant-ph/9508009
. While this did not result in deriving quantum mechanics, it proved a fruitful
idea and it led to the introduction of a hypothetical device called a PR box
(PR from Popescu and Rohrlich). If classical resources can achieve the maximum
correlation given by the Bell
limit, quantum mechanics can achieve a higher correlation limit known as the
Tsirelson bound http://en.wikipedia.org/wiki/Tsirelson's_bound.
A PR box can go over the Tsirelson bound and achieve impressive feats http://arxiv.org/abs/quant-ph/0603017
.
Returning to Bohmian mechanics, there are internal
inconsistencies regarding reality. We already encountered in the prior posts
the problem of representing spin with a classical model. The biggest problem
however is the fact that Bohmian mechanics is contextual. What does
this mean? By Kolmogorov’s axioms, correlations cannot exceed Bell
limit. The only way out is to demand different statistical contexts. Suppose
one measures three variables A, B, C with A and B compatible measurements and A
and C compatible measurements, but not B and C. Contextuality means that the
random variable associated with A in the context of A-B experiment is different than the random
variable associated with A in the context of A-C experiment. So much for “objective
reality”.
Another problem I like to point out is the description of
the hydrogen atom in Bohmian mechanics. If the electron does have a definite
position, movement around the proton would radiate energy and the atom would be
unstable. The solution in Bohmian mechanics is that the electron and proton are
standing still at a fixed distance one from another. The problem is not that
the electron is stationary despite electrostatic attraction (after all Bohmian
mechanics is known for “surreal trajectories”) but the fact that this particular distance is distinguished from all other distances. This distance is
completely ad-hoc with no possible explanation except “God made it so”. Compare
this with the kinds of answers we get from quantum electrodynamics and it is
clear why no quantum field theory was developed for Bohmian mechanics.
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Bohm's interpretation of QM is "weak." For a nonrelatvistic system that has no spin, or where it is disregarded, and that is moving under the influence of some sort of potential it in some ways works. I think it is a fair to decent way to look at quantum chaos. Bohm just fails to account for the production of massive particles in relativistic interacting field theory and it is not able to work with quantum observables that have no classical analogue.
ReplyDeleteThe presumed contextuality of Bohmian QM is an “illusion” of sorts that some Bohmist fall into. They presume that the particle, called a beable or “be able,” is an objective thing. This is nothing more than a mathematical gadget. Much the same is the case with the quantum potential or the so called pilot wave. There is nothing about these things that have any more reality than what you have with waves in standard QM. One can perform canonical transformation on variables in Bohmian QM to illustrate some of this. In fact it is not hard to derive a form of the Feynman path integral from Bohmian QM this way.
Bohmian interpretation I think has some limited uses. In particular with chaos theory one has methods very amenable to particle perspectives and the Bohm QM has some advantages I think in the area. Bohm QM is though not worth much for most work. It is terrible for interacting fields in relativistic QM, it not convenient for most other areas of work from quantum optics to solid state physics.
Cheers LC
Bohmian interpretation is the easiest one to pick holes into (and that is why I started with it). But none of the other interpretations are free of blemishes either... I'll go through all the important ones and present their warts.
ReplyDeleteIt is not hard to show that the Hamilton-Jacobi equation with the quantum potential is
ReplyDelete-∂S/∂t = H – (ħ^2/2m)∇^2R/R, for ψ = Re^{-iS}
gives a Feynman type of path integral. If we have a canonical transformation p --- > P and q --- > Q, the Hamiltonian portion transforms as usual H(p, q) --- > H(P, Q). The quantum potential though transforms as well with
∇ = ∇_q --- > (∂Q/∂q)∇_Q + (∂P/∂q)∇_P
with summation of indices implied. The canonical transformation may be parameterized and it may be used in the product form
prod_i|q_i><q_i|,
for each position q_i related to q_{i-1} by an infinitesimal canonical transformation. We write this as q_i = q_i + ε(∂q_i/∂q_{i-1}). Keeping term to O(ε) means this product is a great sum over these generator elements, which ultimately Usp(n) group elements, and a Feynman path integral can be derived.
Of course all interpretations have weaknesses. I think what is needed is not an interpretation but an understanding of how QM is related to spacetime so the nonlocal aspects of QM can be understood according to the emergence of spacetime. The interpretations of QM we currently have are all lacking in many significant ways. This idea with the Grothendieck group, products and what I think are quantum homotopies seem like one possible approach to this.
Cheers LC
5 years ago I would have said I understood relativity but not QM. Today I'll say I understand QM intimately well but I have no (good) clue on the origin or meaning of space. Non-locality of QM is simply that fact that QM is completely blind to space. Or that configuration space and real space are completely independent things.
ReplyDeleteIn QFT forcing the space-time to become the de-facto configuration space results in Fock space and particle creation and annihilation. The Fock space and the spin-statistic theorem are effects of the "violence" committed against the Hilbert space when it is forced to be space-time. Mathematically this is the area of Hilbert modules and non-commutative geometry.
I agree that QM is blind to space, but only has a representation in space, or for that matter spacetime. There is in my opinion only a configuration space tht is a projection of fundamental quanta according to certain algebraic varieties. There really is no such thing as "real space."
ReplyDeleteCheers LC