# Relationship between Bohmian trajectories and Feynman paths ?

• bohm2
In summary, the relationship between "Bohmian trajectories" and "Feynman paths" is unclear. Some relations exist, but they are not very interesting.

#### bohm2

Relationship between "Bohmian trajectories" and "Feynman paths"?

Can someone summarize what (if any) relationship there is between these two? I've read that Bohmian trajectories are very different than Feynman paths but I've also come across papers suggesting that "Feynman method of summing over all paths can be constructed with the de Broglie-Bohm theory at its basis". It seems that these two views are incompatible.

I would like to see those papers suggesting that "Feynman method of summing over all paths can be constructed with the de Broglie-Bohm theory at its basis", even if they are not (yet) published in peer reviewed journals. I know moderators do not like links on unpublished papers, in which case you can send me a PM.

I've sent you the link (PM) as I doubt that the journal "The Waterloo Mathematics Review" would be deemed acceptable as per forum guidelines. Looking forward to your input (if any).

Thanks, bohm2. I have three negative comments on the paper.

First, the deBroglie-Bohm path integral (8.1) is not new.

Second, equation (8.1) is also quite useless. Namely, to calculate psi by this equation you must know L_q and v^psi, which is essentially the same as knowing psi. In other words, (8.1) can be used to calculate psi only if psi is already known.

Third, their derivation of Feynman path integral from deBroglie-Bohm path integral (8.1) is wrong. Namely, in 3th-4th line of page 29 they say that mv^psi=p is constant. But in fact it is not constant. It is constant only for a plane wave, but not for a superposition of plane waves in (8.2). Due to that error, their "proof" is incorrect.

To conclude, Feynman-path formulation of QM and deBroglie-Bohm path formulation of QM are quite independent. Some relations between them exist (after all, they are observationally equivalent), but these relations do not seem very interesting.

What I find more interesting is the fact that Feynman at one place uses Bohmian equations of motion in a totally different context:

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While I haven't read the piece or watched the video yet, I found a paper and this Perimeter video lecture suggesting some link between the two. I'm not sure if one would consider the relations between the two as interesting (assuming his arguments are meaningful):
Bohmian trajectories and Feynman paths are conceptually different objects from radically different views of quantum mechanics. Both offer different `particle pictures' in a subject that is based on wave mechanics. Some recent models of subquantum dynamical processes underlying the Dirac equation suggest that there may be an unexplored link between the two concepts via the quantum potential. We sketch the qualitative ideas involved and view some simple implementations that quantitatively illustrate the suggested link.
Bohm Trajectories, Feynman Paths and Subquantum Dynamical Processes
http://www.math.ryerson.ca/~gord/abstracts/PiQudos/PiQudoswebH/index.html

http://streamer.perimeterinstitute.ca/Flash/01a99ea2-1d23-4eb9-b7f0-b5dc3ae3cd13/viewer.html [Broken]

I'm not sure if this paper suggests a link also:

The path integral formalism is more intuitive and more powerful way of viewing quantum mechanics helping us to insight into subtle details of it. However, its "Achilles heel" is pathology of "infinite measure", "infinite sums of phases" with unit absolute values, etc. In particular, many of standard real-valued, time independent potentials which are used in modeling quantum systems are singular (for example, the attractive Coulomb potential) and do not fit with the theory. To avoid these obstacles, DeWitt, for example, determined a quantum corrector ΔVDeW=h2/6mR, R is scalar curvature. It is indispensable in order to derive the Schrodinger equation from the time evolution integral. By taking into account the Bohmian quantum potential we can find a corrector ~h2/2m for the path integral. It stems directly from the Schrodinger equation.

Bohmian trajectories and the Path Integral Paradigm. Complexified Lagrangian Mechanics.
http://arxiv.org/pdf/0808.1245.pdf

But these seem inconsistent with this argument by Tumulka or am I mistaken:
Note that the path integrals concern only the wave function, not the Bohmian paths...An obvious but basic fact that I want to emphasize is that the paths of the path integral have a very different status from the Bohmian paths: Feynman’s paths are mathematical tools for computing the evolution of ψ , while one among Bohm’s paths is the actual motion of the Bohmian particle, which exists in addition to ψ.
Feynman’s Path Integrals and Bohm’s Particle Paths
http://arxiv.org/pdf/quant-ph/0501167.pdf

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