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January 20, 2004

Bohmian Mechanics

More than anything else, I dislike philosophy in my physics. Mainly because physicists are not any good at philosophy. The two fields need to be kept separate. So I present Bohmian mechanics: Quantum mechanics without the philosophy.

And that is it, really. Those two equations Schrödinger's equation and an evolution equation predict all the results of orthodox quantum mechanics. No observers, no wave-particle duality, no wave-function collapse, no playing dice with the universe (although the non-stochastic nature of the theory is not its main motivation).

What I like best about Bohmian mechanics is its simple mathematical specification. Once you have the two basic equations, all the rest of quantum theory falls out in the math.

Here is a slightly more in-depth explanation by Sheldon Goldstein. I could put it more simply, but I do not love you all enough to create my own gifs for the equations.

Bohmian mechanics (or the de Broglie-Bohm theory) is the minimal completion of Schrödinger's equation, for a nonrelativistic system of particles, to a theory describing a genuine motion of particles. For Bohmian mechanics the state of the system is described by its wave function tex2html_wrap_inline1146 together with the configuration Q defined by the positions tex2html_wrap_inline1150 of its particles. The theory is then defined by two evolution equations: Schrödinger's equation for tex2html_wrap_inline1152 and a first-order evolution equation

  equation541

for Q(t), the simplest first-order evolution equation for the positions of the particles that is compatible with the Galilean (and time-reversal) covariance of the Schrödinger evolution [24, pages 852-854,]. Here tex2html_wrap_inline1158 is the mass of the k-th particle. (If tex2html_wrap_inline1050 is spinor-valued, the products in numerator and denominator should be understood as scalar products. If external magnetic fields are present, the gradient should be understood as the covariant derivative, involving the vector potential.) This deterministic theory of particles in motion completely accounts for all the phenomena of nonrelativistic quantum mechanics, from interference effects to spectral lines [23, pages 175-178,] to spin [25, page 10 of [1],], and it does so in a completely ordinary manner.

And here is an a very readable dialogue prepared by Roderich Tumulka which covers some obvious objections one might have to the theory.

This is Sheldon's Goldstein's homepage, where you can find a large number of his published papers on Bohmian mechanics as well as the latest pre-prints.

Like Quantum mechanics, Bohmian mechanics can be extended to include the creation and annihilation of particles. It is straightforward enough to create Bohmian Field Theories from Quantum Field Theories.

The most serious objection to Bohmian mechanics is that it is non-local. This creates serious problems for Lorentz invariance. Because Bohmian mechanics has exactly the same predictions as Quantum mechanics, it predicts the correct result for the Einstein-Poldalsky-Rosen Experiment. (Bohmian mechanics actually has some close historical ties with EPR. It is the same Bohm and Bell for both.) Bohmian mechanics cannot get around the implied non-locality in the way that QFT does because BM claims that the particles involved are all real, and that what you see is what you get.

You can find one paper on Goldstein's website, published in Classical and Quantum Gravity, that shows one method of dealing with this: Opposite Arrows of Time Can Reconcile Relativity and Nonlocality. However, I do not find the approach suggested to be at all fruitful. It is an interesting thought-experiment, but not much more.

I am not actually too bothered by the non-locality of Bohmian mechanics. To me, it is already a substantial departure from relativity to claim that Lorentz invariance says nothing about particles and such, but rather certain mathematical constructs which are what is really real. A modification of Lorentz invariance based on the idea that particles and all that are real seems an equivalent way of dealing with the consequences of EPR to me. In fact, the Bohmian way of dealing with this puts the problem in a much clearer light.

Either way, Bohmian mechanics is Quantum mechanics if Quantum mechanics were just another boring old theory with none of the philosophic romance. I happen to like it a lot more that way.

Posted by Thrasymachus at 11:12 AM