- #76

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I think changing a continuum field with a field on a lattice is a minor change of beables.Replacing Bohm's continuum model by a lattice model does not preserve Bohm's beables.

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- #76

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I think changing a continuum field with a field on a lattice is a minor change of beables.Replacing Bohm's continuum model by a lattice model does not preserve Bohm's beables.

- #77

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The change robs momentum modes from being beables, hence is drastic.I think changing a continuum field with a field on a lattice is a minor change of beables.

- #78

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I think those two field ontologies are equivalent. If one knows the actual value of ##\tilde{\phi}(k)## and if one definesThe change robs momentum modes from being beables, hence is drastic.

$$\phi(x)\equiv \int dk\, e^{ikx} \tilde{\phi}(k)$$

then one also knows the actual value of ##\phi(x)##.

It may be illuminating to see what is it analogous to in particle ontology. It is

Bohm in his paper is not explicit about that, but I think it's implicit in his Eq. (A1). Moreover, in the book by Bohm and Hiley (The Undivided Universe) it is quite clear that those two ontologies are considered equivalent.

- #79

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This works only in the continuum.I think those two field ontologies are equivalent. If one knows the actual value of ##\tilde{\phi}(k)## and if one defines

$$\phi(x)\equiv \int dk\, e^{ikx} \tilde{\phi}(k)$$

then one also knows the actual value of ##\phi(x)##.

From ##x## limited to a finite lattice (as in lattice models) one cannot go back to Bohm"s momenta ##p##. Thus lattice ontologies are essentially different from Bohm"s ontology.

- #80

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I think you are nitpicking. It's well known how to write the corresponding equation on the lattice. For a scalar field that's indeed very simple.This works only in the continuum.

From ##x## limited to a finite lattice (as in lattice models) one cannot go back to Bohm"s momenta ##p##. Thus lattice ontologies are essentially different from Bohm"s ontology.

- #81

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How do you get beables with arbitrary wave vector from the beables of a scalar field theory on a finite lattice??? It is well-known to be impossible by Nyquist's theorem.I think you are nitpicking. It's well known how to write the corresponding equation on the lattice. For a scalar field that's indeed very simple.

- #82

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The following is taken from the book "Quantum Fields on a Lattice" by Montvay and Munster. I think Eqs. (2.4) and (2.6) answer your question. Or do I miss something?How do you get beables with arbitrary wave vector from the beables of a scalar field theory on a finite lattice??? It is well-known to be impossible by Nyquist's theorem.

- #83

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What you miss is that ##p## is restricted to a finite number of values, unlike Bohm's beables, which are defined for all ##p##. Thus a lattice theory has too few beables compared to Bohm's.The following is taken from the book "Quantum Fields on a Lattice" by Montvay and Munster. I think Eqs. (2.4) and (2.6) answer your question. Or do I miss something?

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In addition, space is not periodic, and the argument you quoted only works in the periodic case.The following is taken from the book "Quantum Fields on a Lattice" by Montvay and Munster. I think Eqs. (2.4) and (2.6) answer your question. Or do I miss something?

- #85

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Bohm's beables on a continuum are defined for all ##p##, but Bohm's beables on a lattice are defined for some ##p## only. Bohm himself haven't study the lattice explicitly, but it's straightforward to modify the theory by replacing the continuum with the lattice, so it's justified to still refer to the resulting theory as "Bohm's".What you miss is that ##p## is restricted to a finite number of values, unlike Bohm's beables, which are defined for all ##p##. Thus a lattice theory has too few beables compared to Bohm's.

Perhaps the source of our disagreement is philosophical. While for you the continuum and the lattice are conceptually totally different things, for me the continuum is conceptually just a lattice in the limit ##a\to 0##.

So from my point of view, your objection is just irrelevant nitpicking.

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- #86

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In a field theory, the field beables are ##\phi(x)##. In the corresponding lattice model, the corresponding beables are ##\phi(n)## for ##n\in\mathbb{Z}^3_N\subset \mathbb{R}^3##. So it is simply a subset. There is also no problem with the momentum variables, which are in both cases simply ##\pi(x) = \dot{\phi}(x)## (and have nothing to do with the momentum of the particles, which are in a field ontology irrelevant pseudo-particles similar to phonons).Replacing Bohm's continuum model by a lattice model does not preserve Bohm's beables.

(Not sure what you name "Bohm's beables", but in

Bohm.D., Hiley, B.J., Kaloyerou, P.N. (1987). An ontological basis for the quantum theory, Phys. Reports 144(6), 321-375

field beables are used.)

- #87

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You'd read my posts within their context. See Post #59. Bohm's beables are the transverse momentum modes of the vector potential. These cannot be defined on a finite, nonperiodic lattice.Not sure what you name "Bohm's beables"

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Ok. That means, another proposal made by Bohm et al later is better, it does not have your problem:You'd read my posts within their context. See Post #59. Bohm's beables are the transverse momentum modes of the vector potential. These cannot be defined on a finite, nonperiodic lattice.

Bohm.D., Hiley, B.J., Kaloyerou, P.N. (1987). An ontological basis for the quantum theory, Phys. Reports 144(6), 321-375

Of course, for vector gauge fields the Wilsonian lattice variant is even better than the straightforward one.

- #89

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We were discussing an electron in an electromagnetic field, where @Demystifier referred in post #58 to this paper of Bohm. Instead, you propose references for scalar field theories that don't apply.Ok. That means, another proposal made by Bohm et al later is better, it does not have your problem:

Bohm.D., Hiley, B.J., Kaloyerou, P.N. (1987). An ontological basis for the quantum theory, Phys. Reports 144(6), 321-375

Where does this variant treat an electron in an electromagnetic field? Please refer to a paper that compared to #58 gives an improved answer.Of course, for vector gauge fields the Wilsonian lattice variant is even better than the straightforward one.

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The formulas have been given, for simplicity, for scalar field theory, but it is said that the field ontology works for other bosonic fields too. But, ok, here is a paper which describes explicitly the EM field:We were discussing an electron in an electromagnetic field, where @Demystifier referred in post #58 to this paper of Bohm. Instead, you propose references for scalar field theories that don't apply.

Kaloyerou, P.N. (1994). The causal interpretation of the electromagnetic field. Physics Reports 244, 287-358