Counterpart of Feynman integral in de Broglie-Bohm theory

In summary: Can you point me in the direction of the book?I cannot point you in the direction of the book, but I can give you a reference for the Regularization of Quantum Field Theory by taking a cutoff by Maximal Possibility of Particles.In summary, the author of the referenced article claims that the de Broglie path integral does not solve the mathematical difficulties of the Feynman integral, and that the two types of integrals are mathematically the same.
  • #1
ErikZorkin
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I am looking for good references / clarifications on the subject.
First of all, my question is concerned only with mathematical formulation of something that sort of plays the role of the Feynman path integral of the "standard" QFT. It is not concerned with the physical or philosophical difficulties of the de Broglie-Bohm theory (I guess, there was enough discussion here already).

There was a thread somewhere here mentioning the so called de Broglie path integral. I couldn't find any good references on that. What first appears in the search are there works: 1, http://www.academia.edu/download/32269761/Bohm_Feynman_LastV.pdf, 3.
But they are not published in reputable journals and contain mathematical inconsistencies (I think, Demystifier found some errors in the first work).

Nicolic addressed some QFT, but he seems to have used the usual path integral in eq. (28).
Tumulka even claimed (without details) that the de Broglie path integral didn't solve the notorious mathematical difficulties of the Feynman integral.

So what is this de Broglie path integral precisely? Aren't all trajectories deterministic and actual in this theory so the functional integration over all virtual paths is redundant?
 
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  • #2
ErikZorkin said:
Nicolic addressed some QFT, but he seems to have used the usual path integral in eq. (28).
There are no path integrals in this paper. I would know, I wrote it. :wink:
Eq. (28) is an ordinary integral, see Eq. (30).
 
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  • #3
Demystifier said:
There are no path integrals in this paper. I would know, I wrote it.

Glad that I came across an expert!

So, (30) is mathematically the same "measure" as the one used in the Feynman integral, I guess. "Measure" in quotes because there is no infinite-dimensional Lebesgue measure. The mathematical problem is thus the same as in the de Broglie theory and Feynman path integral formalism, correct?
 
  • #4
ErikZorkin said:
So, (30) is mathematically the same "measure" as the one used in the Feynman integral, I guess. "Measure" in quotes because there is no infinite-dimensional Lebesgue measure. The mathematical problem is thus the same as in the de Broglie theory and Feynman path integral formalism, correct?
I'm not sure I understand your question, but I will explain the difference between two types of integrals on an example. Suppose that one considers one particle moving in one spatial dimension. Classically, the particle has trajectory ##x(t)##. The analog of my Eq. (30) would then be
$$dxdt$$
By contrast, the path integral would have the measure
$${\rm lim}_{k\rightarrow\infty}dx(t_1) dx(t_2)\cdots dx(t_k)$$
 
  • #5
Demystifier said:
I'm not sure I understand your question, but I will explain the difference between two types of integrals on an example. Suppose that one considers one particle moving in one spatial dimension. Classically, the particle has trajectory ##x(t)##. The analog of my Eq. (30) would then be
$$dxdt$$
By contrast, the path integral would have the measure
$${\rm lim}_{k\rightarrow\infty}dx(t_1) dx(t_2)\cdots dx(t_k)$$

This is what I meant. It is not even a measure
 
  • #6
ErikZorkin said:
It is not even a measure
It is for a finite but arbitrarily large k.
 
  • #7
Demystifier said:
It is for a finite but arbitrarily large k.

Exactly. But the integral (28) used the pseudo-measure (30). How come (28) is an ordinary integral as you said in your 1st post?
 
  • #8
ErikZorkin said:
Exactly. But the integral (28) used the pseudo-measure (30). How come (28) is an ordinary integral as you said in your 1st post?
You are right that it is a pseudo-measure in the sense that ##n=\infty##. But it is an ordinary integral in the sense that it is not a path integral. In my (30) ##n=\infty## corresponds to the infinite number of particles, while in the path integral ##k=\infty## corresponds to the infinite number of the possible values of time even for a single particle.
 
  • #9
Demystifier said:
In my (30) n=∞n=\infty corresponds to the infinite number of particles, while in the path integral k=∞k=\infty corresponds to the infinite number of the possible values of time even for a single particle.

Right, this is what I thought of. I think, mathematically, those are the same things and, thus, problematic.
So, all in all, it seems you don't need the path integral, but an integral over all particles.

Have you, or anyone else, thought about lattice and/or finite-dimensional variants of the theory?
 
  • #10
ErikZorkin said:
Have you, or anyone else, thought about lattice and/or finite-dimensional variants of the theory?
If you mean a regularization of QFT by taking a large but finite cutoff in the form of a maximal possible number of particles, I have thought about it. In fact I have seen it in a book, so I can give you a reference if you are interested.
 
  • #11
Demystifier said:
If you mean a regularization of QFT by taking a large but finite cutoff in the form of a maximal possible number of particles, I have thought about it. In fact I have seen it in a book, so I can give you a reference if you are interested.

Yes, it sounds interesting.
 
  • #12
ErikZorkin said:
Yes, it sounds interesting.
Fujikava and Suzuki, Path Integrals and Quantum Anomalies, Sec. 3.4.

By the way, I have just discovered that this book contains also a derivation of fermion path integrals that @A. Neumaier revealed to me a half an hour ago from a different source. Amazing!
 
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  • #13
Demystifier said:
Fujikava and Suzuki, Path Integrals and Quantum Anomalies, Sec. 3.4.

By the way, I have just discovered that this book contains also a derivation of fermion path integrals that @A. Neumaier revealed to me a half an hour ago from a different source. Amazing!

Thank a lot! I've read this section and briefly browsed through the rest of the book. However, I didn't fully get how it addresses finite-dimensional QFT. I was still able to find this article and some other materials on QM on finite-dimensional Hilbert spaces for position and momentum.

Do you have an idea what happens in your derivation (in the article I cited in the OP) when you introduce a cutoff as a finite number of particles? Would it make sense at least as an approximation? (The background is, I am looking for (effective) theories without pseudomeasures ##\lim_{n \rightarrow \infty} \prod_i \mathrm d x_1 \dots \mathrm d x_n##)
 
  • #14
ErikZorkin said:
Do you have an idea what happens in your derivation (in the article I cited in the OP) when you introduce a cutoff as a finite number of particles? Would it make sense at least as an approximation? (The background is, I am looking for (effective) theories without pseudomeasures ##\lim_{n \rightarrow \infty} \prod_i \mathrm d x_1 \dots \mathrm d x_n##)
Yes, I think such an approximation would make perfect sense.
 
  • #15
Demystifier said:
Yes, I think such an approximation would make perfect sense.

Right. But in the case of the path integral, it doesn't always make that much sense -- the approximations don't converge (or misbehave).
I was thinking maybe in your application of Bohmian mechanics to QFT they would behave better.
 
  • #16
ErikZorkin said:
But in the case of the path integral, it doesn't always make that much sense -- the approximations don't converge (or misbehave).
Can you give an example when it misbehaves?
 
  • #17
Demystifier said:
Can you give an example when it misbehaves?

I think (not a specialist though) that this is quite often the case (the path integral cannot be well defined in the first place -- there is a theorem that forbids any complex-valued Lebesgue measure on an infinite-dimensional Hilbert space).
Furijawa wrote a book on time discretization approach to Feynman integral. There are certain limitations on the Lagrangian -- in particular, on the potential.
This article discusses discretization of the Lie formula. Already two-dimensional case seems not convergent.
I find also this answer quite nice and descriptive.

I suspect that if discretization had been always successful, we could have defined an appropriate Lebesgue measure. But this is not the case.
 
  • #18
Addendum: just came across this paper. See the example on the first page

Also this.
 
  • #19
But lattice QCD computations based on discretized path integrals get results which agree with experiments.
 
  • #20
Demystifier said:
But lattice QCD computations based on discretized path integrals get results which agree with experiments.

Yes. But it is mostly numerical studies an no convergence can be proven in the general case.

Actually, your integral might be something else because the particle quantity is at least countable. I am just wondering how convergence of your apparatus would look like with a finite particles cutoff.
 
  • #21
ErikZorkin said:
Actually, your integral might be something else because the particle quantity is at least countable.
Yes, that's an important difference.
 
  • #22
Demystifier said:
Yes, that's an important difference.

Perhaps, your theory has some mathematical equivalence to a lattice field theory with on an infinite space-time lattice?
 
  • #23
Found this interesting article which seems quite related to our discussion.

It is called

"Pilot-Wave Quantum Theory in Discrete Space and Time and the Principle of Least Action"
 

What is the counterpart of Feynman integral in de Broglie-Bohm theory?

The counterpart of Feynman integral in de Broglie-Bohm theory is known as the Bohmian trajectory integral. This is a mathematical tool used to calculate the probability of particle trajectories in de Broglie-Bohm theory, which is an interpretation of quantum mechanics that posits the existence of real, physical particles with definite positions and trajectories.

How does the Bohmian trajectory integral differ from the Feynman integral?

The Bohmian trajectory integral differs from the Feynman integral in that it does not incorporate the concept of quantum superposition, which is a central aspect of the Feynman path integral. Instead, the Bohmian trajectory integral calculates the probability of particle trajectories based on the positions and velocities of particles in de Broglie-Bohm theory.

Why is the counterpart of Feynman integral important in de Broglie-Bohm theory?

The counterpart of Feynman integral, or the Bohmian trajectory integral, is important in de Broglie-Bohm theory because it provides a way to calculate the probability of particle trajectories in a theory that posits the existence of real, physical particles with definite positions and trajectories. This allows for a more intuitive understanding of quantum mechanics and has implications for the interpretation of quantum phenomena.

Can the Bohmian trajectory integral be used to make predictions in de Broglie-Bohm theory?

Yes, the Bohmian trajectory integral can be used to make predictions in de Broglie-Bohm theory. It allows for the calculation of the probability of particle trajectories, which can then be compared to experimental results. However, it should be noted that de Broglie-Bohm theory is not the only interpretation of quantum mechanics and its predictions may differ from other interpretations.

Is there experimental evidence to support the use of the Bohmian trajectory integral in de Broglie-Bohm theory?

There is ongoing debate and research on the experimental evidence for de Broglie-Bohm theory and the use of the Bohmian trajectory integral. Some experiments have shown results that support the theory, while others have shown results that contradict it. The interpretation of these results is still a topic of discussion among scientists.

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