Issues on notation and concept of entanglement

[vanhees71]

Is there any relativistic causal non-local theory for $\lambda$? If not, you can of course speculate a lot without much scientific substance.

 

[Morbert]

Stepping away from the philosophy for a moment: Are there quantities Bohmian field theories can more readily compute? Or processes? I don't think a nonlocal skeleton $\lambda$ is automatically a bad idea, but does it sharpen any ambiguities or render some intuitions or computations more available?

 

[Demystifier]

Is there any relativistic causal non-local theory for $\lambda$?

No, but there is nonrelativistic causal nonlocal theory for $\lambda$ which makes the same measurable predictions as relativistic local QFT.

 

[vanhees71]

In nonrelativistic physics causality is not a big issue, because all you need is ordering in absolute time. How such a theory can make the same measurable predictions as relativistic local QFT is an enigma to me. In such a model I indeed could have instantaneous causal changes at far distant places, but according to relativistic local QFT I cannot.

 

[Demystifier]

Stepping away from the philosophy for a moment: Are there quantities Bohmian field theories can more readily compute? Or processes? I don't think a nonlocal skeleton $\lambda$ is automatically a bad idea, but does it sharpen any ambiguities or render some intuitions or computations more available?

So far nobody found computational advantages of Bohmian fields (BTW, there are examples of computational advantages of Bohmian particles), but it helps a lot to sharpen intuitions. In addition to offering a solution of the measurement problem, Bohmian fields offer a very simple solution of the problem of time in quantum gravity. (There is also a claim that it helps to solve the Boltzmann brain problem is cosmology, but I don't find it convincing.)

 

[Demystifier]

How such a theory can make the same measurable predictions as relativistic local QFT is an enigma to me.

I've tried to make it very simple in http://thphys.irb.hr/wiki/main/images/3/3d/QFound5.pdf

 

[martinbn]

By extending the set of interacting objects. Relativistic QFT excludes ftl interactions in a law for evolution of the state in the Hilbert space $\psi$. But minimal relativistic QFT says nothing about other possible interacting objects $\lambda$ that are not given by $\psi$. Bohmian interpretation is an extension of minimal relativistic QFT. It does not change the evolution of $\psi$, but it makes a concrete proposal for $\lambda$ and postulates a nonlocal law for evolution of $\lambda$. Since minimal and Bohmian QFT agree on equations for $\psi$, and since minimal QFT says nothing mathematical about the evolution of $\lambda$, there is no any mathematical contradiction between minimal and Bohmian QFT. The contradiction is only philosophical, because minimal QFT uses some philosophical arguments to argue that there is no $\lambda$ to begin with.

Or schematically:
Minimal QFT: $\psi$ local, period.
Bohmian QFT: $\psi$ local, $\lambda$ nonlocal.

What is the physical counterpart of $\lambda$?

 

[Demystifier]

What is the physical counterpart of $\lambda$?

In Bohmian mechanics it's a local beable such as particle positions or a field configuration. Note that local beables (ontic things with values at well defined points in space) have nonlocal interactions.

 

[martinbn]

In Bohmian mechanics it's a local beable such as particle positions or a field configuration. Note that local beables (ontic things with values at well defined points in space) have nonlocal interactions.

You need to write a dictionary. You said "By extending the set of interacting objects.", but particle positions are not physical object and cannot interact! Particles are objects that can interact. It seems that you uses phrases that should be used for the territory, but you use them for the map all the time. I know that you think I have a problem with nonlocality, but if everything was local I would have the same problem. It is just very hard to keep track of what is what.

 

[Demystifier]

You need to write a dictionary. You said "By extending the set of interacting objects.", but particle positions are not physical object and cannot interact! Particles are objects that can interact. It seems that you uses phrases that should be used for the territory, but you use them for the map all the time. I know that you think I have a problem with nonlocality, but if everything was local I would have the same problem. It is just very hard to keep track of what is what.

Did you try to read some other text (not written by me) on Bohmian mechanics? Did you have similar problems?

 

[martinbn]

Did you try to read some other text (not written by me) on Bohmian mechanics? Did you have similar problems?

It's worse. I can ask you, but I cannot ask a text.

 

[vanhees71]

I've tried to make it very simple in http://thphys.irb.hr/wiki/main/images/3/3d/QFound5.pdf

This is obviously a basic misinterpretation of the gauge-vector fields. These are of course not observables, because they don't fulfill the microcausality principle. Only gauge-independent quantities can be observables. To think there were an "ontology" of the vector potentials in gauge theories must be flawed and lead to wrong conclusions. Particularly it's clear that also the observable phase shifts in the Aharonov-Bohm effect are gauge invariant, because the magnetic flux entering them is gauge invariant and can be expressed with gauge-invariant fields, obeying the microcausality principle.

 

[Demystifier]

This is obviously a basic misinterpretation of the gauge-vector fields. These are of course not observables, because they don't fulfill the microcausality principle. Only gauge-independent quantities can be observables. To think there were an "ontology" of the vector potentials in gauge theories must be flawed and lead to wrong conclusions. Particularly it's clear that also the observable phase shifts in the Aharonov-Bohm effect are gauge invariant, because the magnetic flux entering them is gauge invariant and can be expressed with gauge-invariant fields, obeying the microcausality principle.

You missed the point. Suppose that some hypothetical civilization only discovered electrodynamics in the Coulomb gauge and never discovered the gauge invariance. I claim that they would never observe any contradiction between theory and experiment. If you agree with that statement (and I don't see any reason why shouldn't you), then it should be obvious they would have a Lorentz and gauge non-noninvariant theory that agrees with experiments.

 

[gentzen]

To think there were an "ontology" of the vector potentials in gauge theories must be flawed and lead to wrong conclusions.

An "ontology" is just a (mathematical) model in a certain sense, there is no need to draw premature conclusions. Using the vector potentials as model can indeed be dangerous, but the reasons are related to topology, not to the violation of Lorentz invariance. Gauge fixing always lead to valid local models, but topological obstructions often prevent getting a valid global model from such local models.

But I somehow have the impression that you had something completely different in mind when you wrote "... must be flawed and lead to wrong conclusions."

 

[Demystifier]

It's worse. I can ask you, but I cannot ask a text.

When you ask me, does it actually help?

 

[vanhees71]

You missed the point. Suppose that some hypothetical civilization only discovered electrodynamics in the Coulomb gauge and never discovered the gauge invariance. I claim that they would never observe any contradiction between theory and experiment. If you agree with that statement (and I don't see any reason why shouldn't you), then it should be obvious they would have a Lorentz and gauge non-noninvariant theory that agrees with experiments.

If they discovered (classical or quantum) electrodynamics they'd also have discovered gauge invariance, because electrodynamics makes only mathematical sense as a gauge theory, let alone its success in describing all electromagnetic phenomena.

 

[Demystifier]

If they discovered (classical or quantum) electrodynamics they'd also have discovered gauge invariance, because electrodynamics makes only mathematical sense as a gauge theory, let alone its success in describing all electromagnetic phenomena.

What exactly does not make sense if you compute everything in a fixed gauge?

 

[vanhees71]

Your (mis)interpretation of the Coulomb-gauge-fixed equations is a typical example. You claim there were actions at a distance, because you claim that the potentials were physical fields.

 

[Demystifier]

Your (mis)interpretation of the Coulomb-gauge-fixed equations is a typical example. You claim there were actions at a distance, because you claim that the potentials were physical fields.

How is that relevant from a scientific point of view if that makes correct measurable predictions?

 

[vanhees71]

It obviously doesn't, because what works in the "real world" is Maxwell's theory and not some distortion of it.

 

[Demystifier]

It obviously doesn't, because what works in the "real world" is Maxwell's theory and not some distortion of it.

But the Maxwell theory and its distortion differ only in philosophy. So what really disturbs you are some philosophical quibbles that have nothing to do with science.

 

[vanhees71]

How can this be: If you claim the potentials were physical fields you must necessarily have observables differing from Maxwell's observables, because in Maxwell's theory only the field $(\vec{E},\vec{B})$ is observable. If this is not the case, then your claim that the potentials were physical fields is unsubstantiated and you are back at Maxwell's theory which is a gauge theory.

 

[Demystifier]

because electrodynamics makes only mathematical sense as a gauge theory

Here is what one of your favored books (Schwartz, QFT and the Standard Model) says:

"8.6 Is gauge invariance real?

Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
Global symmetries are physical, since they have physical consequences, namely conservation of charge. That is, we measure the total charge in a region, and if nothing leaves that region, whenever we measure it again the total charge will be exactly the same. There is no such thing that you can actually measure associated with gauge invariance. We introduce gauge invariance to have a local description of massless spin-1 particles. The existence of these particles, with only two polarizations, is physical, but the gauge invariance is merely a redundancy of description we introduce to be able to describe the theory with a local Lagrangian. ...
In summary, although gauge invariance is merely a redundancy of description, it makes
it a lot easier to study field theory. The physical content is what we saw in the previous
section with the Lorentz transformation properties of spin-1 fields: massless spin-1 fields
have two polarizations. If there were a way to compute S-matrix elements without a local
Lagrangian (and to some extent there is, for example, using recursion relations, as we will
see in Chapter 27), we might be able to do without this redundancy altogether."

 

[Demystifier]

How can this be: If you claim the potentials were physical fields you must necessarily have observables differing from Maxwell's observables, because in Maxwell's theory only the field $(\vec{E},\vec{B})$ is observable. If this is not the case, then your claim that the potentials were physical fields is unsubstantiated and you are back at Maxwell's theory which is a gauge theory.

I did not say that potentials are physical. I said they could be ontic, but ontic is definitely not the same as observable. I can accept that you don't understand what ontic means, but then you don't understand even the Bohmian interpretation of nonrelativistic QM, in which case you have absolutely no chance to understand Bohmian interpretation of relativistic QFT.

 

[vanhees71]

Yes indeed, it underlines the importance of gauge invariance. Without gauge invariance you have unphysical degrees of freedom which make trouble (acausality and in the quantum case non-unitarity of the S-matrix).

 

[Demystifier]

Yes indeed, it underlines the importance of gauge invariance. Without gauge invariance you have unphysical degrees of freedom which make trouble (acausality and in the quantum case non-unitarity of the S-matrix).

That's not what Schwartz is saying in the quote above.

 

[vanhees71]

I did not say that potentials are physical. I said they could be ontic, but ontic is definitely not the same as observable. I can accept that you don't understand what ontic means, but then you don't understand even the Bohmian interpretation of nonrelativistic QM, in which case you have absolutely no chance to understand Bohmian interpretation of relativistic QFT.

Ok, if ontic doesn't mean physical aka observable, it doesn't make sense to me. Also Bohmian mechanics doesn't make much sense to me. I think it's just a curious attempt to reintroduce a kind of determinism through the backdoor but without any consequences for the physics.

 

[vanhees71]

That's not what Schwartz is saying in the quote above.

But that's what's behind what he is saying!

 

[Demystifier]

Also Bohmian mechanics doesn't make much sense to me.

I am fine with that, but then I can't understand why do you often praise the book on Bohmian mechanics by Durr.

 

[vanhees71]

It's the only book, where I thought to understand what's behind Bohmian mechanics. I can appreciate a good treatment of a theory even if I don't think that the theory is of much additional value in comparison to the standard theories.

 

[Demystifier]

But that's what's behind what he is saying!

The way I see it, you deeply disagree with Schwartz. He seems to be saying that gauge invariance is just a convenient mathematical trick which makes some calculations easier, while you seem to saying that gauge invariance is a deep truth without which it is absolutely impossible to get correct physical results.

 

[Demystifier]

Ok, if ontic doesn't mean physical aka observable, it doesn't make sense to me.

In another book by Durr, the title of the very first section in the very first chapter is "Ontology".
https://www.amazon.com/dp/3030400670/?tag=pfamazon01-20

 

[gentzen]

I am fine with that, but then I can't understand why do you often praise the book on Bohmian mechanics by Durr.

The book (Verständliche Quantenmechanik: Drei mögliche Weltbilder der Quantenphysik) is not just on Bohmian mechanics. But it is indeed well written, easy to read, and clears up quite some confusion.

But I also share your confusion, because that book also talks about relativistic Bohm-Dirac theory, but vanhees71 seemed pretty sure that there is no such thing.

 

[Demystifier]

But I also share your confusion, because that book also talks about relativistic Bohm-Dirac theory, but vanhees71 seemed pretty sure that there is no such thing.

Yes, but in this case I can understand @vanhees71 because this theory is a Bohmian interpretation of relativistic QM, not of relativistic QFT, while @vanhees71 thinks that relativistic QM is wrong even in an orthodox form.

 

[vanhees71]

The way I see it, you deeply disagree with Schwartz. He seems to be saying that gauge invariance is just a convenient mathematical trick which makes some calculations easier, while you seem to saying that gauge invariance is a deep truth without which it is absolutely impossible to get correct physical results.

Well, it may well depend on a reader's opinion which meaning he reads into the words of an author.

In my understanding what Schwartz does here is to underline the important difference between global symmetries, which indeed have observational consequences in terms of conservation laws a la Noether, while a local gauge symmetry indicates a redundancy in the description of the dynamics of a physical model.

Let's stick with electrodynamics and Abelian gauge symmetry. On the quite general level we are discussing here there's not so much difference for the more general non-Abelian case.

In classical electrodynamics the Maxwell equations summarize about 200 years of empirical knowledge about electromagnetic phenomena in terms of fields, at the time a brand-new concept discovered by Faraday to solve the old enigma of actions at a distance disturbing the physicists since Newton's gravitational froce law (including Newton himself). The Maxwell equations are written in terms of the observable fields $\vec{E}$ and $\vec{B}$ (the electric and magnetic field or taken together the electromagnetic field) and the charge-current densities $\rho$ and $\vec{j}$.

Using the homogeneous Maxwell equations you get the potentials, i.e., there exists a scalar field $\Phi$ and a vector field $\vec{A}$ such that (working with natural Heaviside-Lorentz units with $c=1$)
$$\vec{B}=\vec{\nabla} \times \vec{A}, \quad \vec{E}=-\vec{\nabla} \Phi - \partial_t \vec{A}.$$
Then it's immediately clear that the physical situation, given by $\vec{E}$ and $\vec{B}$, is not in one-to-one correspondence with the potentials, because for given $(\vec{E},\vec{B})$ any other potentials connected to one solution by a gauge transformation
$$\vec{A}'=\vec{A}-\vec{\nabla} \chi, \quad \Phi'=\Phi+\partial_t \chi$$
leads to the same fields, because
$$\vec{\nabla} \times \vec{A}'=\vec{\nabla} \times \vec{A}=\vec{B}, \quad -\vec{\nabla} \Phi'-\partial_t \vec{A}' = -\vec{\nabla} \Phi-\partial_t \vec{A} + \partial_t \vec{\nabla} \chi -\vec{\nabla} \partial_t \chi=-\vec{\nabla} \Phi-\partial_t \vec{A}=\vec{E}.$$
So the physical fields $\vec{E}$ and $\vec{B}$ determine the potentials only modulo a gauge transformation with an arbitrary scalar field $\chi$ and thus the physics is not in the potentials but only in the potentials modulo a gauge transformation. A gauge transformation is not a symmetry, because it just expresses the redundancy of the description of the theory in terms of the potentials, i.e., the physical situation is described not uniquely by the potentials but only modulo a gauge transformation. In other words there are more field degrees of freedom used than are physical dynamical fields.

This becomes clear using the action principle. It turns out that the canonically conjugate field momentum of $\Phi$ is identically 0, i.e., $\Phi$ cannot be a true dynamical degree of freedom. This leads to characteristic trouble when trying to use canonical quantization and you either have to fix a gauge and then quantize canonically or you use the general formalism for Hamiltonian systems with constraints a la Dirac.

The upshot is that indeed a local gauge symmetry is not a symmetry as a global one. While the global symmetry before introducing the em. field of the free matter fields (usually Klein-Gordon or Dirac), i.e., the symmetry under multiplication with spacetime independent phase factors, yields the conservation law of a charge-like quantity in the "gauged version", i.e., after introduction of the electromagnetic field charge conservation follows from the Bianchi identity and not as an independent conservation law.

Another important conclusion, not correctly discussed in almost all textbooks on QFT (the one exception coming to my mind is Duncan, The conceptual framework of QFT), including my favorite Weinberg, is that local gauge symmetries cannot be spontaneously broken. Indeed, if you try to spontaneously break a local gauge theory you do not end up with a degenerate vacuum/ground state but with an equivalence class of different representations of the ground state connected by gauge transformations, i.e., the ground state is not degenerate as you might expect from the analogous case of a spontaneously broken global symmetry. In the latter case you get massless Goldstone modes as physical degrees of freedom, while for a local gauge symmetry indeed there are no such massless Goldstone modes, because the "would-be Goldstone fields" are just absorbed via a gauge transformation into the gauge fields, providing for the additional 3rd degree of freedom of a massive vector field and so providing a mass to the gauge fields (or some of the gauge fields depending on the pattern of the would-be symmetry breaking of the gauge group, as in the electroweak part of the standard model where you have four gauge fields based on the gauge group $\mathrm{SU}(2) \times \mathrm{U}(1)$, which is "pseudo-broken" to $\mathrm{U}(1)$, making three of the four gauge fields massive by eating up 3 would-be Goldstone modes into three of the gauge fields, which then get the W and Z boson fields and keeping one massless, describing the photon).

Another argument comes from the representation theory of the Poincare group. If you want to construct a microcausal theory of massless spin-1 fields you realize that it must be a gauge theory if you don't want continuous polarization degrees of freedom, which indeed is not observed in Nature.

What I understand Schwartz means with a "mere convenience" is that in perturbation theory you like to work with manifestly Dyson renormalizable models to calculate (a priori not physical!) proper vertex functions as building blocks to get S-matrix elements (physical!). It depends on a gauge whether the theory on the level of the proper vertex functions is manifestly renormalizable or not. E.g., if you try to work with physical field degrees of freedom for a Higgsed gauge theory, i.e., with the would-be Goldstone's absorbed and only having the corresponding massive gauge bosons the usual power counting doesn't work out anymore, because the gauge-field propgator is not of counting order $-2$ but of order $0$ (because of the contribution $\propto k_{\mu} k_{\nu}/(m^2 k^2)$ for a massive spin-1 field). However, you can choose another gauge ('t Hoofts $R_{\xi}$ gauge, where the gauge-field propagator indeed has the "right" power counting)). Then you have the would-be Goldstone fields left in the formalism, but they together with the usual Faddeev-Popov ghosts conspire to eliminate order by order in the loop/$\hbar$ expansion of perturbation theory the also unphysical gauge-field degrees of freedom. The S-matrix is gauge invariant and the unitary gauge just is a limit of the $R_{\xi}$ gauges. This shows that the S-matrix depends only on physical degrees of freedom and is unitary and no causality/locality constraints are violated.