Bohmian trajectories vs. Feynman paths, always continuous?

In summary, the conversation discusses the difference between Bohmian trajectories and Feynman path integral formulation. It is mentioned that Bohmian paths are assumed to be smooth and continuous, and that particle creation and annihilation is part of Quantum Field Theory where the fundamental thing is the field, not a particle. The idea of particle trajectories being stochastic is also brought up, and it is mentioned that the trajectories in a Wiener process are not differentiable but still continuous. Finally, a recent paper on the topic is referenced.
  • #1
asimov42
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After reading some of the other posts on the Forum, I'm clear on the fact that Bohmian trajectories (of the de Broglie Bohm formulation) and the paths of the Feynman path integral formulation are very different things.

I'm wondering (and it's a naive question, no doubt), when talking about Bohmian paths - if you have a particle at position A, and then later observe the particle at position B, are you assured that the path the particle took from A to B was continuous? That is, there is no uncertainty and the particle is assumed to 'exist' along the whole trajectory? (wondering in part how the de Broglie Bohm formulation deals with particle creation / annihilation along a trajectory)

Thanks all.
 
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  • #2
asimov42 said:
I'm wondering (and it's a naive question, no doubt), when talking about Bohmian paths - if you have a particle at position A, and then later observe the particle at position B, are you assured that the path the particle took from A to B was continuous? That is, there is no uncertainty and the particle is assumed to 'exist' along the whole trajectory? (wondering in part how the de Broglie Bohm formulation deals with particle creation / annihilation along a trajectory)

Yes that's true.

Particle creation and annihilation is part of Quantum Field Theory where the fundamental thing is the field - not a particle. Here is how it works. You take a field. Model it as a large number of blobs and apply quantum rules to those blobs. You then take the blob size to zero and you get a QFT. Particles then emerge from the field - but it's the field that is quantised. The field blobs are what you apply the BM interpretation to.

The particles arise in an abstract way similar to the quantum harmonic oscillator:
https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Thanks
Bill
 
  • #3
An argument for Bohmian trajectories to be smooth is given in http://arxiv.org/abs/1112.2034 "Furthermore, as (10) is a smooth function of its arguments, one expects that particle trajectories are smooth too."

I am not sure whether the "expects" is the physicists intuition or whether it can be stated as a mathematical requirement.
 
  • #4
atyy said:
An argument for Bohmian trajectories to be smooth is given in http://arxiv.org/abs/1112.2034 "Furthermore, as (10) is a smooth function of its arguments, one expects that particle trajectories are smooth too.".

It's a particle under the influence of the quantum potential. That its continuously differentiable is the usual assumption in mechanical systems where particles are subject to a potential.

Thanks
Bill
 
  • #5
Although the phase function is multivalued, its gradient is a single valued function and we have a velocity field picture where trajectories (integral curves) cannot cross or touch. The effective potential (Q + V) always acts so as to preserve this single-valuedness , which follows from the properties of the phase/Hamilton's principle function(action).
 
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  • #6
bhobba said:
It's a particle under the influence of the quantum potential. That its continuously differentiable is the usual assumption in mechanical systems where particles are subject to a potential.

But could a more generalized form admit non-differentiable trajectories? Before the comment on smooth trajectories, Nikolic says "In principle, the trajectories could be stochastic." So could one have something like a Wiener process in which the trajectories are not differentiable?

Edit: The OP asked about continuous, and the Wiener process trajectories are continuous even if they are not differentiable. So maybe asking whether the trajectory could be like white noise, in which the trajectories are not continuous, would be a better question.
 
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  • #7
atyy said:
But could a more generalized form admit non-differentiable trajectories?

It obeys a differential equation which means at least its first derivative must exist. Normally, in mechanics, it is assumed to be continuously differentiable.

See its guiding equation
http://plato.stanford.edu/entries/qm-bohm/#eqs

I suspect, similar to rigged Hilbert spaces, it would be fruitful to consider the physically realizable solutions as continuously differentiable and rapidly decreasing to all orders so as to apply Fourier transforms easily.

Thanks
Bill
 
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  • #8
atyy said:
Edit: The OP asked about continuous, and the Wiener process trajectories are continuous even if they are not differentiable. So maybe asking whether the trajectory could be like white noise, in which the trajectories are not continuous, would be a better question.
What's the actual upshot of a Wiener process being non-differentiable? Usually you have finite data and are computing finite differences to approximate derivatives, and the time series of a Wiener process is no exception, so it's not obvious to me there is any penalty for this fact.
 
  • #9
atyy said:
So could one have something like a Wiener process in which the trajectories are not differentiable?
Yes. See e.g. the recent paper
http://lanl.arxiv.org/abs/1510.06391
and references therein.
 
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1. What are the key differences between Bohmian trajectories and Feynman paths?

Bohmian trajectories and Feynman paths are two different interpretations of quantum mechanics. The main difference between them lies in their understanding of the nature of particles and their behavior. Bohmian trajectories view particles as having definite positions and trajectories, while Feynman paths view particles as existing in multiple states simultaneously.

2. How do Bohmian trajectories and Feynman paths explain the behavior of particles?

Bohmian trajectories explain the behavior of particles by positing that particles have definite positions and trajectories, and their movement is influenced by a quantum potential. On the other hand, Feynman paths explain particle behavior by considering all possible paths a particle can take, with the final path being a combination of these possibilities.

3. Are Bohmian trajectories and Feynman paths compatible with each other?

No, Bohmian trajectories and Feynman paths are not compatible with each other. They offer two different interpretations of quantum mechanics, and while they both can explain some phenomena, they have fundamental differences in their understanding of the nature of particles.

4. Can Bohmian trajectories and Feynman paths coexist in the same theory?

Yes, it is possible for Bohmian trajectories and Feynman paths to coexist in the same theory. This is known as the de Broglie–Bohm theory, which combines the deterministic nature of Bohmian trajectories with the probabilistic nature of Feynman paths.

5. Which interpretation of quantum mechanics is more widely accepted by scientists?

There is no clear consensus among scientists about which interpretation of quantum mechanics is more widely accepted. Some scientists prefer the deterministic approach of Bohmian trajectories, while others favor the probabilistic approach of Feynman paths. Ultimately, the choice of interpretation may depend on personal beliefs and the specific phenomenon being studied.

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