You misunderstood what I said. Bohmian QED consists of a two set of equations. One set is standard QED written down in a fixed Lorenz gauge, that's something what physicists like you do every day and there is nothing inconsistent with it, despite the mathematical acausality of the ##A^0## field. The second set of equations, which are the Bohmian field evolution equations, that's something that standard QED has nothing to say about.vanhees71 said:the Bohmian reinterpretation of QED is gauge dependent and thus acausal, i.e., it is intrinsically inconsistent.
You assume that only observables (self-adjoint operators) can be physical, which is true in standard quantum theory, but does not need to be true in any conceivable theory.vanhees71 said:If these Bohmian field evolution equations are acausal they cannot have any physical relevance, because they cannot refer to observables.
Newtonian gravity is a counterexample, which is acausal (in the sense you mean it) and consistent as a theory. If property A is fundamental, it does not imply that negation of A is inconsistent.vanhees71 said:... contradicts fundamental properties like causality and thus is inconsistent already as a theory.
Here you assume that relativity must be a fundamental principle obeyed by everything. But we don't know that with certainty. Maybe it's true, maybe it's not. So far experiments found no counterexample, but it doesn't imply that they never will. In principle it's possible that relativity is emergent rather than fundamental, but you don't accept that even as a possibility.vanhees71 said:The challenge for Bohmian physics or any other approach to nonlocal hidden variable theories within relativistic physics is to obey still causality. The solution in classical relativistic physics is the field picture to make the interactions local and fulfill causality. It still seems to be a problem to find non-local but causal dynamics within relativistic physics.
Yes, and that's exactly what in "BM for instrumentalists" I propose. More specifically, I suggest that future collider experiments (with energies higher than currently available) might see signs of Lorentz-invariance violation.vanhees71 said:Well yes, but then you Bohmian field theory should provide testable predictions, contradicting the principles of relativity, and then experiment has to decide. Then it would be a new theory making predictions contradicting standard local QFT, which obeys the principles of relativity.
Isn't it one reason why Einstein thought QM is incomplete? It is one reason Schrodinger had problems with QM as we know from his famous paper. We also know Schrodinger and Einstein corresponded and that Einstein used a similar argument (it isn't clear to me who used it first). Einstein spoke of "bombs" instead of atomic decay and dead cats if I recall. We also have the EPR paper and writings of his to understand other (related) reasons why Einstein thought QM to be incomplete. But there can be more than one reason Einstein thought QM to be incomplete.bhobba said:I was on you-tube and saw a video from Oxford on QM foundations. I didn't agree with it, but that is not an issue - I disagree with a lot of interpretational stuff. The video mentioned a paper they thought essential reading:
https://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/three_measurement_problems.pdf
Is it me, or has the author not shown the appropriate care? In particular, they claim theories that violate 1a are hidden variable theories. I thought - what - how does that follow. It may simply mean nature is fundamentally probabilistic. Or, in other words, the three assumptions are not inconsistent.
Specifically, I do not think the following is logically justified:
'And since we are interested in individual cats and detectors and electrons since it is a plain physical fact that some individual cats are alive and some dead, some individual detectors point to "UP" and some to "DOWN", a complete physics, which is able at least to describe and represent these physical facts, must have more to it than ensemble wave-functions.'
My response is - that might be your idea of what compete physics is, but it might be best not to assume that is everyone's idea of complete physics. Einstein, of course, thought QM incomplete, but I am not sure that is necessarily why.
Thanks
Bill
Minnesota Joe said:ETA: Better yet...is QM incomplete in Maudlin's sense? There are people who use the word 'complete' in the same way and definitions don't come from On High.
What acausality of the ##A^0## field in the Lorenz gauge? It gives ##\square A^\mu = j^\mu## for all components, thus, looks quite causal. ##A^0## is acausal in other popular gauges. But in the Lorenz gauge?Demystifier said:One set is standard QED written down in a fixed Lorenz gauge, that's something what physicists like you do every day and there is nothing inconsistent with it, despite the mathematical acausality of the ##A^0## field.
There is no acausality in the Lorenz gauge, but Lorenz gauge is not a completely fixed gauge. Coulomb gauge, for instance, is completely fixed and can be thought of as a special case of Lorenz gauge.Sunil said:What acausality of the ##A^0## field in the Lorenz gauge? It gives ##\square A^\mu = j^\mu## for all components, thus, looks quite causal. ##A^0## is acausal in other popular gauges. But in the Lorenz gauge?
Ok. But why would BM need a completely fixed gauge? It needs a well-defined evolution equation, anf it needs a potential because else the interaction cannot be defined. But it does not have the positivist prejudices against unobservables, thus, does not have to get rid of unobservables completely. As long as they follow a well-defined evolution equation (which they do in the Lorenz gauge) everything is fine.Demystifier said:There is no acausality in the Lorenz gauge, but Lorenz gauge is not a completely fixed gauge.
Ah, you missed the context. The context is my lecture in which I used Coulomb gauge in classical (not Bohmian!) electromagnetism to demonstrate that you may violate Lorenz invariance at the ontic level without violating Lorenz invariance at the measurable level.Sunil said:But why would BM need a completely fixed gauge?
Okay, thanks. This is an older paper of Maudlin's and I'd not read it before, so thank you for the link also.bhobba said:I think the discussion has answered my query. It has to do with what I was taking complete to mean and what Maudlin was. On that basis, I could close the thread but will leave it open if others have thoughts on the issue.
Thanks
Bill
There's no need for ##A^{\mu}## to be "causal", because it's not observable. What must be causal is the gauge-invariant electromagnetic field ##F_{\mu \nu}## which always is causal ("retarded solutions"), no matter in which gauge you calculate the potentials ##A^{\mu}##.Sunil said:What acausality of the ##A^0## field in the Lorenz gauge? It gives ##\square A^\mu = j^\mu## for all components, thus, looks quite causal. ##A^0## is acausal in other popular gauges. But in the Lorenz gauge?
Fine that you accept this.vanhees71 said:There's no need for ##A^{\mu}## to be "causal", because it's not observable. What must be causal is the gauge-invariant electromagnetic field ##F_{\mu \nu}## which always is causal ("retarded solutions"), no matter in which gauge you calculate the potentials ##A^{\mu}##.
Why shouldn't I accept the electromagnetic potentials?Sunil said:Fine that you accept this.
I've never denied this.Sunil said:First of all, it follows that you accept that such non-observable things like potentials have a place in physics, even if they are non-observable. Field theory with ##A^{\mu}## is much much simpler than field theory with ##F_{\mu \nu}## given that there is not even a reasonable Lagrangian for the interaction with a Dirac field using the ##F_{\mu \nu}##.
I don't use the word "real" anymore in discussions about Q(F)T, because I don't know its meaning anymore.Sunil said:The more problematic question is about their reality. The ##F_{\mu \nu}##, or whatever defines them, are real, given that they have observable consequences. But are the ##A^{\mu}## real? Positivists have problems with this. Realists not. They know and accept that realist theories are hypotheses and will remain hypotheses forever, that we cannot prove them, only falsify them. So, what is real is, as part of a realist theory, hypothetical too. Some criteria for preference are obvious: Simplicity, Ockham's razor. Giving reality to the ##A^{\mu}## gives much simpler realist theories than accepting only the ##F_{\mu \nu}## as real.
The equations of motion for the ##A^{\mu}## are of course gauge-dependent too. The ##F_{\mu \nu}## as gauge invariant fields are indeed the natural building blocks for local observables (fulfilling also the microcausality condition).Sunil said:What are then the equations for the ##A^{\mu}##? They have to explain the equations for the ##F_{\mu \nu}## which we can test in a much better way given observations. So we have simplicity and explanatory power as unproblematic guiding principles. If the ##F_{\mu \nu}## follow well-defined evolution equations with Lorentz symmetry, the simplest explanation would be that the ##A^{\mu}## follow such equations too. This would be ##\square A^{\mu}=0##. Why would a realist consider some more complicate realist theory? Don't complicate things without necessity.
You need the ##A^{\mu}## to formulate QED as a manifestly local QFT. Particularly you need a four-vector to have the usual transformation properties under Poincare transformations.Sunil said:What is, therefore, behind the "no need for ##A^{\mu}## to be "causal"? It is good old empiricism - the wish to derive the theories of physics from observations. We obviously cannot derive the equation ##\square A^{\mu}=0## from observation, given that there are a lot of other imaginable equations for the ##A^{\mu}## which give the same Maxwell equations for ##F_{\mu \nu}##. So, ##\square A^{\mu}=0## cannot be part of a theory based on empiricism.
Read your own quotations given below.vanhees71 said:Why shouldn't I accept the electromagnetic potentials?
It is quite easy: It is what a realist theory defines as the ontology. If it does not define an ontology, it is not a realist theory.vanhees71 said:I don't use the word "real" anymore in discussions about Q(F)T, because I don't know its meaning anymore.
Hm. They are not observables, about reality you refuse to speak, so what they are? Sort of something magical like "imaginary numbers" in the mind of Musil's Törless? Not real, not observable, only imaginable or so?vanhees71 said:For sure ##A^{\mu}(x)## do NOT represent observables. They don't obey the microcausality principle and are not gauge invariant.
Except that Bohm Aharonov shows that not all observables can be computed locally from the ##F_{\mu \nu}##.vanhees71 said:The ##F_{\mu \nu}## as gauge invariant fields are indeed the natural building blocks for local observables (fulfilling also the microcausality condition).
Except that the theory with the ##A^{\mu}## in the standard BRST approach is not even a quantum theory, but a physically meaningless mathematical exercise in an indefinite Hilbert space.vanhees71 said:You need the ##A^{\mu}## to formulate QED as a manifestly local QFT. Particularly you need a four-vector to have the usual transformation properties under Poincare transformations.
I know, but all these words have no clear meaning. I want to discuss physics not gibberish.Sunil said:Read your own quotations given below.
It is quite easy: It is what a realist theory defines as the ontology. If it does not define an ontology, it is not a realist theory.
The electromagnetic potentials are calculational tools. They don't represent observables directly but enable you to define operators that represent observables. That's standard textbook QFT.Sunil said:Hm. They are not observables, about reality you refuse to speak, so what they are? Sort of something magical like "imaginary numbers" in the mind of Musil's Törless? Not real, not observable, only imaginable or so?
In classical physics microcausality doesn't make sense. Of course in the Lorenz gauge you get the wave equation for the potentials and you can choose the retarded potentials, and you can choose the retarded solutions. This alone doesn't imply that they are directly observable. They are not but the electromagnetic field ##F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}## is.Sunil said:In the Lorenz gauge they would follow a local equation ##\square A^{\mu}=j^{\mu}## classically, so I would expect that microcausality would not be a problem too, not?
If you break gauge invariance, no matter how weakly, the entire theory becomes inconsistent. A gauge symmetry (in contradistinction to global symmetries) means that you use more degrees of freedom in your mathematical description than physical degrees of freedom. If you break gauge invariance the non-physical degrees of freedom become interacting, and fundamental features of QFT as a physical theory are violated, including the unitarity of the S-matrix, microcausality, Poincare invariance etc.Sunil said:The other problem is why are you so sure? Our QFTs are effective theories, and it follows essentially automatically that their symmetries may appear approximate symmetries. Approximate symmetries play an important role in QFT too, what is the problem of only approximately gauge invariant gauge fields?
I have no clue what you want to say with this statement.Sunil said:Except that Bohm Aharonov shows that not all observables can be computed locally from the ##F_{\mu \nu}##.Except that the theory with the ##A^{\mu}## in the standard BRST approach is not even a quantum theory, but a physically meaningless mathematical exercise in an indefinite Hilbert space.
I am not advocating Bohmian mechanicsm in any way, but I do agree with this point.Demystifier said:Here you assume that relativity must be a fundamental principle obeyed by everything. But we don't know that with certainty. Maybe it's true, maybe it's not. So far experiments found no counterexample, but it doesn't imply that they never will. In principle it's possible that relativity is emergent rather than fundamental, but you don't accept that even as a possibility.
No, the meaning is very clear. A realistic theory has to define an ontology. If it has a Lagrange formalism, the ontology of the system is defined by the trajectory in the configuration space ##q(t)\in Q## of the theory, in the Hamilton formalism by the trajectory in the phase space. And so on. These are quite well-defined things. If philosophers talk about realism and reality in a theory-independent sense, the result may be confusion, but the same holds for everything they talk about.vanhees71 said:I know, but all these words have no clear meaning. I want to discuss physics not gibberish.
In other words, they have no status at all. A "calculational tool" can be things with no relation to the theory at all.vanhees71 said:The electromagnetic potentials are calculational tools.
No. The Gupta-Bleuler formalism, resp. BRST quantization, which starts with a physically nonsensical indefinite Hilbert space for the ##A^\mu## fields, breaks down. One would have to start with real fields ##A^\mu##, and without exact gauge symmetry they would plausibly be observable. So, they should be handled in the mathematical formalism like usual observables. That means, they have to live in a well-defined (not indefinite) Hilbert space. This automatically clarifies problems with unitarity. You don't have to consider a factorization of some indefinite Hilbert space which may fail, but have to start from a definite Hilbert space. This gives some QFT. If gauge symmetry is approximate, then plausibly relativistic symmetry is approximate too, thus, causality will go back to classical causality in a preferred frame (which will be only approximately unobservable), and Poincare invariance will be only approximate too. No problem with QFT as a consistent physical theory.vanhees71 said:If you break gauge invariance, no matter how weakly, the entire theory becomes inconsistent. A gauge symmetry (in contradistinction to global symmetries) means that you use more degrees of freedom in your mathematical description than physical degrees of freedom. If you break gauge invariance the non-physical degrees of freedom become interacting, and fundamental features of QFT as a physical theory are violated, including the unitarity of the S-matrix, microcausality, Poincare invariance etc.
What I mean with the reference to the Bohm-Aharonov effect should be clear even from Wiki-Level information https://en.wikipedia.org/wiki/Aharonov–Bohm_effect which claims "The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, (Φ, A), must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, or using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded."vanhees71 said:I have no clue what you want to say with this statement.
To be described by a realistic theory which we can reasonably believe that it is true (that means, in particular, not falsified yet, but other criteria count here too) as part of the ontology of that theory.martinbn said:@Sunil What does it mean for ##A^\mu##(or anything else) to be real?
We have different perspectives, but IMO, Bells inequality is not the key anymore I think. Bells theorem is clear, but the ansatz is makes on equipartitioning has no place in how I understand things these days anway. It's good to kill of the original, old times objections to QM. But I see more modernt objections, where Bells theorem does not help.vanhees71 said:For me the issue is solved with the results of all the high-precision Bell tests.
Nobody has yet made clear to me, what a clear definition of "ontology" is. I don't care for philosophy. Physics is about well defined observable objective facts. There are no trajectories in phase space in quantum mechanics or quantum field theory.Sunil said:No, the meaning is very clear. A realistic theory has to define an ontology. If it has a Lagrange formalism, the ontology of the system is defined by the trajectory in the configuration space ##q(t)\in Q## of the theory, in the Hamilton formalism by the trajectory in the phase space. And so on. These are quite well-defined things. If philosophers talk about realism and reality in a theory-independent sense, the result may be confusion, but the same holds for everything they talk about.
These are all mathematical formulations of the theory, which also clearly defines what is representing observables, e.g., S-matrix elements in "vacuum QFT", thermodynamical quantities for different kinds of media with there different phases in many-body situations (quark-gluon plasma, neutron stars, usual matter around us), transport coefficients, etc.Sunil said:In other words, they have no status at all. A "calculational tool" can be things with no relation to the theory at all.
No. The Gupta-Bleuler formalism, resp. BRST quantization, which starts with a physically nonsensical indefinite Hilbert space for the ##A^\mu## fields, breaks down. One would have to start with real fields ##A^\mu##, and without exact gauge symmetry they would plausibly be observable. So, they should be handled in the mathematical formalism like usual observables. That means, they have to live in a well-defined (not indefinite) Hilbert space. This automatically clarifies problems with unitarity. You don't have to consider a factorization of some indefinite Hilbert space which may fail, but have to start from a definite Hilbert space. This gives some QFT. If gauge symmetry is approximate, then plausibly relativistic symmetry is approximate too, thus, causality will go back to classical causality in a preferred frame (which will be only approximately unobservable), and Poincare invariance will be only approximate too. No problem with QFT as a consistent physical theory.
That's wrong. What's observed in the AB effect is a non-integrable phase factor, which depends however on a gauge-invariant quantity only, i.e., the flux of the magnetic field through a surface intersecting the solenoid,Sunil said:What I mean with the reference to the Bohm-Aharonov effect should be clear even from Wiki-Level information https://en.wikipedia.org/wiki/Aharonov–Bohm_effect which claims "The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, (Φ, A), must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, or using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded."
The entire point of the BRST approach is to evaluate well-defined observables in a well-defined Hilbert space from a gauge theory, which by definition deals with a description with contains non-physical unobservable elements in the description. You introduce ghost fields. For an un-higgsed gauge symmetry these are the Faddeev-Popov ghost fields taylored such that in a given gauge-fixing the contributions of unphysical degrees of freedom of the gauge potentials ##A_{\mu}^a## to observables quantities are always precisely canceled by the contributions from the ghost fields. That makes that each "gluon" is described by 4 vector-field degrees of freedom, of which two are unphysical, and the contributions of the two unphysical fields to observable quantities is canceled by a corresponding contribution from the correponding two also unphysical degrees of freedom of the ghost fields.Sunil said:Where do you disagree with "Except that the theory with the ##A^\mu## in the standard BRST approach is not even a quantum theory, but a physically meaningless mathematical exercise in an indefinite Hilbert space."? In BRST, you start with an indefinite Hilbert space where the ##A^\mu## are defined, not? Together with the ##A^\mu##, which is IYO only a "calculational tool", this part of the BRST approach can be nothing more serious than a "calculational tool" too. In particular, it cannot be a quantum theory, because Hilbert spaces in QT are always Hilbert spaces and have a definite scalar product, and they have to have it because negative probabilities make no sense.
What is observer democracy? That each observer has his own reality, not equivalent to the reality of other observers?Fra said:I am not advocating Bohmian mechanicsm in any way, but I do agree with this point.
This is an important point and a special case of the question of observer equivalence vs observer democracy. What may be fundamental, could be democracy - not equivalence. The equivalence may well be emergent only, is the sense of an evolutionary steady state. Allowing oneself to relax fundamental things, and instead focus on observer democracy may add explanatory value to why the effective equivalence is observed, but also why they aren't perfect. It's like a "noise" at the level of physical law, but it need not be a bad thing.
I even think that anyone that thinks about the logical arguements of which relativity builds, may find that the observer democracy condition is most certainly more obvious, but a weaker conditions than equivalence. The equivalence condition is handy, but may simply be wrong. (I obviously think it is, but one does not have to buy into something I think to at least see that it's a logical possibility that is perfectly rational and sound)
/Fredrik
I think this is because we still do not understnad the theories deep enough. In a sense, we lack the physical understanding of the various calculations tools, because they tools are constructed in an extrinsic way, and they appear ad hoc; no matter how well corroborated. This gets more pressing one one tries to expand they theory an solve open questions. But I guess the ambition is that we could improve here.Sunil said:In other words, they have no status at all. A "calculational tool" can be things with no relation to the theory at all.
Being real is a primitive notion, it cannot be defined in terms of something else. Somewhat like the notion of set in set theory. Which gives me an idea! The notion of set is not defined, but there are some axioms (ZF) that sets obey. Analogously, perhaps it's possible to make some list of axioms that any real object must satisfy, given a collection of measurable predictions. And then we can have various models that satisfy those axioms, which are nothing but various interpretations of the collection of measurable predictions. In particular, we can have standard and nonstandard interpretation of QM, just like we have standard and nonstandard interpretation of axioms for real numbers.martinbn said:@Sunil What does it mean for ##A^\mu##(or anything else) to be real?
As I always seen it the general constructing principle between not only relativity but also any gauge interaction, is that what any observer actually observes is just as valid description of reality as that of another one. This also includes that any inference of the apparent laws made by an observer, must be as valid as that of another observer. This is what i call the observer democracy.Demystifier said:What is observer democracy? That each observer has his own reality, not equivalent to the reality of other observers?
Just to clarify. All observers are in the same world, there is only one universe in my view. (I'm not talking about MWI (which i never really got the real point of to be honest, it just seems weird and not solve any problems).Demystifier said:What is observer democracy? That each observer has his own reality, not equivalent to the reality of other observers?
What's the problem? Ontology - that's the definition what, according to thatvanhees71 said:Nobody has yet made clear to me, what a clear definition of "ontology" is.
Quantum theory is not a realist theory, but there exist realist interpretations, like Bohmian mechanics. These interpretations have trajectories in configuration space.vanhees71 said:Physics is about well defined observable objective facts. There are no trajectories in phase space in quantum mechanics or quantum field theory.
Do you read the text before you answer? Where I have claimed that there are gauge-dependent observables? The Wiki quote mentions the same thing as your formula.vanhees71 said:That's wrong. What's observed in the AB effect is a non-integrable phase factor, which depends however on a gauge-invariant quantity only, i.e., the flux of the magnetic field through a surface intersecting the solenoid,
$$\Phi_B=\int_{A} \mathrm{d}^2 \vec{f} \cdot \vec{B} = \int_{\partial A} \mathrm{d} \vec{x} \cdot \vec{A}.$$
It's an integral over the vector potential along a closed path, and this is gauge invariant. There are no gauge-dependent observables. This was a contractio in adjecto!
Again, do you read the text before you answer? Is it difficult to understand my "the Gupta-Bleuler formalism, resp. BRST quantization, which starts with a physically nonsensical indefinite Hilbert space for the fields, breaks down" which you have quoted.vanhees71 said:If gauge symmetry is not exact, the model doesn't make physical sense. There can be approximate global symmetries like chiral symmetry of QCD in the light-quark sector, which make physical sense, but if a local gauge symmetry is broken in any way, it doesn't make any sense anymore.
This has nothing to do with Poincare invariance. ... but as soon as the underlying local gauge symmetry is broken, it doesn't make any sense anymore. If you know the BRST formalism or Gupta Bleuler for QED, then this should be immediately clear!
Yes. Which obviously breaks down once the unobservable "calculational tools" become observable. Not because there is anything wrong with approximate gauge invariance, but because BRST uses an artificial indefinite metric for the ##A^\mu## which cannot make sense for observables.vanhees71 said:The entire point of the BRST approach is to evaluate well-defined observables in a well-defined Hilbert space from a gauge theory, which by definition deals with a description with contains non-physical unobservable elements in the description.
But the point was that this requires information about the ##\vec{B}## field inside the solenoid, where the electron is not. It is either on this trajectory or the other one - or nearby. Because of this point I have used the word "local".vanhees71 said:I meant what's wrong is the claim "The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field." Indeed in the AB experiment the particles don't have trajectories, and it's crucial that they don't have since the observable is a phase difference of the corresponding partial waves going the one or other way beyond the solenoid. The phase difference CAN be calculated with the observable field ##\vec{B}##, and that's the important point.
No necessity to cry given that I have not questioned this.vanhees71 said:The observable (the shift of interference fringes when observing an ensemble of many particles) IS a gauge-invariant quantity although the "local formulation" needs the use of the gauge-dependent potential.
Please read carefully what I write. Are you able to understand the difference between "physically nonsensical indefinite Hilbert space" and "ill-defined Hilbert space"? Of course, the indefinite Hilbert space is a well-defined mathematical object. It is physically nonsensical, because negative probabilities (however well-defined) are physically meaningless.vanhees71 said:The BRST formalism does NOT use an ill-defined Hilbert space but constructs an adequate well-defined Hilbert space to quantize a gauge theory.
Again: There is no doubt that the BRST construction breaks down, no need to repeat this.vanhees71 said:Again: A gauge theory, for which gauge invariance is only "approximate" (a contradiction in itself) is not a well-defined theory at all. The entire BRST construction of the adequate Hilbert space, leading to a unitary S-matrix breaks down then.
It is indeed an astonishing topological feature of fields that you have information about them when knowing them only along a much smaller subset of their domain. One of the most simple cases are holomorphic functions, whose values you know everywhere in an area given the values along its boundary (Cauchy's theorem), but this indeed saves both, "locality", i.e., a local description of the interaction between charged particles and the electromagnetic field in terms of the gauge potential (and this must be so, because the em. field as a massless vector field must necessarily be described as a gauge field, as can be inferred from the representation theory of the proper orthochronous Poincare group a la Wigner 1939).Sunil said:But the point was that this requires information about the ##\vec{B}## field inside the solenoid, where the electron is not. It is either on this trajectory or the other one - or nearby. Because of this point I have used the word "local".
I only wanted to emphasize that the BRST quantization formalism is necessary precisely because you need a well-defined Hilbert space and a unitary S-matrix to make physical sense of the entire construction.Sunil said:No necessity to cry given that I have not questioned this.
Please read carefully what I write. Are you able to understand the difference between "physically nonsensical indefinite Hilbert space" and "ill-defined Hilbert space"? Of course, the indefinite Hilbert space is a well-defined mathematical object. It is physically nonsensical, because negative probabilities (however well-defined) are physically meaningless.
Why are you then claiming again and again that a violation of gauge invariance. You are contradicting yourself with these two sentences, or can you quote a scientific paper where they successfully can interpret the gauge potentials as observable quantum fields?Sunil said:Again: There is no doubt that the BRST construction breaks down, no need to repeat this.
But this does not make a theory with approximate gauge invariance not well-defined. One has to use another approach to the quantization of such a theory, one which handles the ##A^\mu(x)## as observable quantum fields.
There are Hilbert spaces of holomorphic functions, but they have nothing to do with field theory.vanhees71 said:It is indeed an astonishing topological feature of fields that you have information about them when knowing them only along a much smaller subset of their domain. One of the most simple cases are holomorphic functions,
And that's wrong. You can start from a vector field ##A^\mu## too.vanhees71 said:I only wanted to emphasize that the BRST quantization formalism is necessary precisely because you need a well-defined Hilbert space and a unitary S-matrix to make physical sense of the entire construction.
Please quote me in a meaningful way. I do not claim "a violation of gauge invariance" but usually write complete sentences. Like the following: Theories with vector fields ##A^\mu## with some approximate gauge invariance are possible. They may be non-renormalizable, but this does not make them invalid as effective field theories. But these theories will certainly not use the Gupta-Bleuler resp. BRST approach, but start with a definite Hilbert space.vanhees71 said:Why are you then claiming again and again that a violation of gauge invariance. You are contradicting yourself with these two sentences, or can you quote a scientific paper where they successfully can interpret the gauge potentials as observable quantum fields?
And that's simply wrong. You can make physical sense of vector fields if you start with a definite Hilbert space. Of course, you need exact gauge invariance to be able to factorize, and without factorization you cannot make physical sense of the whole construction build on the indefinite Hilbert space. But you are not at all obliged to start with some indefinite Hilbert space.vanhees71 said:I only wanted to emphasize that the BRST quantization formalism is necessary precisely because you need a well-defined Hilbert space and a unitary S-matrix to make physical sense of the entire construction.