# QFT as pilot-wave theory of particle creation and destruction

There is this paper that someone posted in a thread:

http://arxiv.org/abs/0904.2287

Called "QFT as pilot-wave theory of particle creation and destruction"

Which talks mainly about Bohmian interpretations applied to QFT (the whole paper is in QFT scheme). However, in chapter 3, it talks about calculating probabilities of certain groups of particles of being in certain positions, while I read in several threads here that "particles have no positions" (this paper treats KG theory, phi4 theory, and some more in the end).

What do you think of this paper? Should I trust in it? (especially in chapter 3 and all the facts not directly related to Bohmian interpretations, which I dont care for now) Can, as this paper says, the probability of finding several particles in certain positions during certain intervals of time, be calculated? (I hope the answer is positive because I find hard to understand when someone states that "particles have no positions" when everything that I touch and see has a position!)

Thanks!!

## Answers and Replies

martinbn
In Bohmian mechanics particles do have coordinates.

Demystifier
Martinb, In chapter 3 things are independent of the interpretation (In fact, its title is "Interpretation-independent aspects of QFT"). The paper does not states that in Bohmian mechanics you can calculate the probability of certain particles of being in certain positions during certain intervals of time. What the paper states (If I didnt understand wrong) is that in QFT you can calculate the probability of certain particles of being in certain positions during certain intervals of time.

martinbn
Sorry, i haven't looked at the paper and was making an assumption, and also I misunderstood your question.

Demystifier
Gold Member
The_pulp, so far I think that your understanding of that paper is correct.

You may also be interested to see the attachment in
https://www.physicsforums.com/blog.php?b=2240 [Broken]
especially pages 10, 11, 20-24, 33-38.

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Ok, I find rewarding your answer and your pdf. But there is a lot of people around here claiming the opposite of what you are saying. A lot of people saying that particles "have no positions". In what sense is people saying that? I would say, particles "have no position" until we measure it and when we measure their position, we will find a value (distributed as the probability distribution that this paper described), and that is the position the particle had in one point of that interval of time (but now does not longer has it because of uncertainty in momentum). Is that what they are trying to say?

Ps: You are Nikolic???

Demystifier
Gold Member
Ok, I find rewarding your answer and your pdf. But there is a lot of people around here claiming the opposite of what you are saying. A lot of people saying that particles "have no positions". In what sense is people saying that? I would say, particles "have no position" until we measure it and when we measure their position, we will find a value (distributed as the probability distribution that this paper described), and that is the position the particle had in one point of that interval of time (but now does not longer has it because of uncertainty in momentum). Is that what they are trying to say?
What people are usually saying (when they know what they are talking about) in the context of relativistic quantum theory is not that particles do not have positions, but that that there is no position OPERATOR. More precisely, they say that the position operator is not Lorentz covariant, so that there is no LORENTZ COVARIANT POSITION OPERATOR (LCPO). And they are not completely wrong; there is no LCPO on the physical Hilbert space (i.e., space of dynamically physical quantum states). However, what I find out in those papers is the following: To have a well defined probability density of particle positions, it is not necessary to have a LCPO on the physical Hilbert space. Instead, it is enough to have LCPO on an extended kinematic Hilbert space, which contains the physical Hilbert space as a subspace. It turns out that LCPO on the extended kinematic Hilbert space exists, which allows to define probability density of particle positions in a Lorentz covariant manner.

If all that sounds too abstract to you, it can be simplified as follows. Other people say that there is no Lorentz invariant probability density on 3-dimensional space. I say: True, but there is Lorentz invariant probability density on 4-dimensional spacetime. Therefore, let us generalize the usual probabilistic interpretation in space to a new probabilistic interpretation in spacetime. In other words, I avoid the impossibility theorems other people use by looking things from a more general perspective where the theorems are no longer valid.

Very very loosely speaking, this can be compared with finding the solution of equation
x^2=-1
In elementary school you have seen the proof that a solution does not exist. But more precise statement is that the solution does not exist on the space of real numbers. If you "add one dimension more" called imaginary axis, then you can find two solutions
x=i and x=-i
The moral is: By extending the framework in which you look for a solution, you can solve problems which previously seemed impossible to solve.

Ps: You are Nikolic???
Yes.

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"...they say that the position operator is not Lorentz covariant, so that there is no LORENTZ COVARIANT POSITION OPERATOR (LCPO)..."

Do you know where I can find a demostration about it?

"... it is enough to have LCPO on an extended kinematic Hilbert space, which contains the physical Hilbert space as a subspace..."

"... Ps: You are Nikolic???

Yes. ..."

Wow, its really weird to be talking to the writer of the paper I was reading. Thanks for descending to the mud and clean some of our ignorance (thanks to you and to everyone who comes around here to teach us some physics for free!). Related to this, I find this approach to QFT (the interpretation independent part) much more clearer and direct than the traditional that could be read in, lets say Peskin & Schroeder (particle creation operators). I mean, its really similar to n dimentional QM. Why has not been developed until 2 years ago?

Instead, it is enough to have LCPO on an extended kinematic Hilbert space, which contains the physical Hilbert space as a subspace. It turns out that LCPO on the extended kinematic Hilbert space exists, which allows to define probability density of particle positions in a Lorentz covariant manner.
What exactly is this extended kinematic Hilbert space? Is it something like the rigged Hilbert space?

Demystifier
Gold Member
"...they say that the position operator is not Lorentz covariant, so that there is no LORENTZ COVARIANT POSITION OPERATOR (LCPO)..."

Do you know where I can find a demostration about it?
This is a very old result, which can be seen in the original paper
- T. D. Newton and E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949)
or in an old QFT textbook
- S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (1961)

"... it is enough to have LCPO on an extended kinematic Hilbert space, which contains the physical Hilbert space as a subspace..."

Well, I myself discuss it in the paper you mentioned
- http://xxx.lanl.gov/abs/0904.2287
in the paragraph around Eqs. (31)-(41), as well as in
- http://xxx.lanl.gov/abs/0811.1905
first paragraph of Sec. 2, and third paragraph of Sec. 3 (beginning with "Thus we see that eigenstates ...")

Related to this, I find this approach to QFT (the interpretation independent part) much more clearer and direct than the traditional that could be read in, lets say Peskin & Schroeder (particle creation operators). I mean, its really similar to n dimentional QM. Why has not been developed until 2 years ago?
To some extent it was developed before, but it is rarely mentioned in the literature because it is less practical in calculating the S-matrix elements. For example, a representation of QFT similar to n-dimensional QM can be found in the textbook by Schweber above. Another good textbook for this stuff is
- P. Teller, An Interpretive Introduction to Quantum Field Theory (1994).

Demystifier