Particle vs Wave Interpretations of QM

[jeffn1]

Sure, classical physics also contains different interacting entities.

My point was not that interactions are mysterious or impossible. Rather, the history of physics contains several examples where entities initially regarded as distinct turned out to be different aspects of a more unified structure.

For example, electric and magnetic fields were once viewed as separate entities and are now understood as components of the electromagnetic tensor.

So when I see several fundamental fields interacting with each other, I naturally wonder whether they are truly fundamental and distinct, or whether a more economical underlying description might exist.

For what its worth, Art Hobson states, in Fields and Their Quanta, "The universe is a set of at least 25 fields, each of which fills the entire universe. These fields are probably facets of a single universal field." (P. 183)

 

[Matterwave]

For what its worth, Art Hobson states, in Fields and Their Quanta, "The universe is a set of at least 25 fields, each of which fills the entire universe. These fields are probably facets of a single universal field." (P. 183)

I wonder how he counts 25.

 

[PeterDonis]

I wonder how he counts 25.

6 leptons, 6 quarks, 12 gauge bosons, and the Higgs in the Standard Model. Of course this counting leaves out a lot of things. Hobson does say "at least", which might be a gesture in the direction of those things.

 

[Matterwave]

6 leptons, 6 quarks, 12 gauge bosons, and the Higgs in the Standard Model. Of course this counting leaves out a lot of things. Hobson does say "at least", which might be a gesture in the direction of those things.

What is the reason to count only 6 quarks? We count only the doublet from the weak SU(2) interaction and not the triplet from the strong SU(3) interaction? Naively I would have thought 18 -- 3 generations by SU(2) doublet by SU(3) triplet. And this is not even getting into chirality and spin.

Or do you think something like this is what Hobson was trying to account for with "at least"?

 

[PeterDonis]

What is the reason to count only 6 quarks?

Up, down, strange, charm, top, bottom.

Yes, that counting leaves out the quarks' color charge. But note that, for example, when we say the proton and neutron are each made of 3 quarks, that's how we're counting. We're not taking into account the color degrees of freedom, because hadrons are colorless, and the valence quarks inside them are not in any particular color eigenstate. Nor are there multiple different kinds of protons and neutrons with different color mixtures among the quarks.

We count only the doublet from the weak SU(2) interaction

No, we're not really counting that way. The leptons are the electron, muon, tauon and their neutrinos. But the electron, muon, and tauon aren't just their parts of the left-handed SU(2) weak doublet; they're also the right-handed SU(2) singlet. Similar remarks apply to the quarks. That's one of the reasons I said the counting I gave leaves out a lot of things.

And this is not even getting into chirality and spin.

Yes, that's not really taken into account, just as color charge on the quarks is not. Again, I said the counting I gave leaves a lot out. But it is one that appears in a lot of pop science sources, and even some not so pop science sources.

 

[Matterwave]

I'm going to have to think on this more but it seems highly non-trivial to me to even count the number of fields.

 

[PeterDonis]

I'm going to have to think on this more but it seems highly non-trivial to me to even count the number of fields.

Yes, it is. Hobson did say "at least" 25 fields.

 

[Matterwave]

Yes, it is. Hobson did say "at least" 25 fields.

Semantic question -- when you say "yes it is", you are saying it's trivial to count or it's not trivial to count?

I might have phrased my point better as it's not trivial what you count as distinct fields.

 

[PeterDonis]

Semantic question -- when you say "yes it is", you are saying it's trivial to count or it's not trivial to count?

Yes, it is not trivial. As in, yes, we have no bananas:

https://en.wikipedia.org/wiki/Yes!_We_Have_No_Bananas

 

[Roberto Pavani]

Isn't the counting somewhat convention-dependent? Maxwell's electromagnetic field already incorporates electric charge via its sources, and GR incorporates mass-energy via the stress-energy tensor. What is the criterion for counting these as separate fields rather than aspects of a more unified structure?

 

[Matterwave]

Isn't the counting somewhat convention-dependent?

"The counting" that I was referring to was the one in Art Hobson's quote mentioned in post #241 which I assumed was meant for fields in QFT.

Indeed, even Hobson himself said "at least 25", suggesting ambiguity.

What is the criterion for counting these as separate fields rather than aspects of a more unified structure?

In relativity, the electric and magnetic fields of Maxwell were unified into the electromagnetic field via the Faraday tensor $F_{ab}$ which is an antisymmetric (0,2) tensor. This tensor has 6 independent components which account for the 3 components of the electric field and 3 components of the magnetic field.

What is called "the electric field" in one reference frame may show up in another as part of "the magnetic field" and vice versa.

 

[Roberto Pavani]

$E$ and $B$ mix under Lorentz transformations.
However, for a Coulomb field generated by an electric charge, there is always a rest frame in which $B=0$, whereas $E$ cannot be transformed away.
For a free electromagnetic wave, neither field can be eliminated by any physical Lorentz transformation; doing so would require moving at $c$.

 

[PeterDonis]

Maxwell's electromagnetic field already incorporates electric charge via its sources

No, it doesn't. The fact that the field is generated by the sources does not mean it incorporates the sources. The field and the sources are separate degrees of freedom. Similar remarks apply to spacetime curvature vs. stress-energy in GR.

What is the criterion for counting these as separate fields rather than aspects of a more unified structure?

Um, because in the theories we have, they are separate things?

If someday we discover a Super Grand Unified Theory in which everything is just aspects of one thing, then you at least might have an argument for saying that "number of fields equals 1". But we have no such theory now.

 

[PeterDonis]

for a Coulomb field generated by an electric charge, there is always a rest frame in which $B=0$, whereas $E$ cannot be transformed away.

But this is a special case, and you can't base a general count of "number of fields" on special cases. The $B$ field doesn't cease to be a field just because it happens to satisfy $B = 0$ in a particular special case. In QFT terms, those modes of the field are simply in their vacuum state; they haven't disappeared.

 

[Roberto Pavani]

I meant that charge distributions and EM waves are already described within Maxwell's theory of the electromagnetic field. If there were some additional field beyond the electromagnetic one, then Maxwell's equations in their present form should break down wherever that extra field contributes additional $E$ ,$B$, or source terms.

 

[PeterDonis]

I meant that charge distributions and EM waves are already described within Maxwell's theory of the electromagnetic field.

No, charge distributions and EM waves are already described within Maxwell's theory of the electromagnetic field and its sources. I doubt you will find a single physicist who will claim that charge-current distributions are "the same thing" as EM fields or are "already incorporated" in them. If you can find a valid reference that makes such a claim, please post it. Otherwise, please do not clutter the thread with statements that you cannot support.

 

[Roberto Pavani]

I've only said that EM global field is only one and does obey Maxwell equations.
But I can't assert that EM is for sure fundamental.

 

[PeterDonis]

I've only said that EM global field is only one and does obey Maxwell equations.

"Only one" in what sense? It's an antisymmetric tensor field with six independent components, as @Matterwave said in post #251.

Also, "the EM field obeys Maxwell's Equations" is by no means all you've said. You said it as part of a more comprehensive claim, parts of which I objected to. If you're now backing off from those parts, that's fine.

But I can't assert that EM is for sure fundamental.

EM by itself is already known to not be fundamental; it's just one piece of the electroweak interaction, which happens to separate out nicely at low energies.

 

[Roberto Pavani]

Perhaps I am looking at Maxwell's equations in a less usual direction. Normally one starts from the sources and computes the field. My point was only that, formally, the equations also allow one to reconstruct the corresponding four-current from a given global field configuration:

$J^\mu = \partial_\nu F^{\nu\mu}.$

So I was not claiming that fields and sources are the same degree of freedom, only that the latter can be inferred from the former, at least formally.

 

[PeterDonis]

My point was only that, formally, the equations also allow one to reconstruct the corresponding four-current from a given global field configuration

Yes, if you already know the field globally, that determines what sources are necessary to produce that field. But of course that's true in the other direction as well: if you know the sources globally, you know the field they produce. So I don't see that this viewpoint makes the field any more fundamental than the sources are. Maxwell's Equations just assert a definite relationship between those two things; they don't say anything about which side of the equations is "more fundamental" than the other.

 

[Roberto Pavani]

if you know the sources globally, you know the field they produce.

I understand your point. My only reservation is that the converse does not seem completely symmetric: $J^\mu = 0$ still admits nontrivial solutions with $F_{\mu\nu} \neq 0$, such as free electromagnetic waves.

 

[Matterwave]

I understand your point. My only reservation is that the converse does not seem completely symmetric: $J^\mu = 0$ still admits nontrivial solutions with $F_{\mu\nu} \neq 0$, such as free electromagnetic waves.

Differential equations require initial value conditions or boundary value conditions to be uniquely specified.

 

[Roberto Pavani]

I understand your point. My only reservation is that the two directions do not seem completely symmetric. Maxwell's equations determine $J^\mu$ locally from $F_{\mu\nu}$ through

$J^\mu = \partial_\nu F^{\nu\mu},$

whereas recovering $F_{\mu\nu}$ from a given $J^\mu$ requires additional initial or boundary conditions and, in general, global information.

So perhaps the issue is not whether $F_{\mu\nu}$ determines $J^\mu$, but whether the notion of a globally well-defined classical $F_{\mu\nu}$ should be expected to exist once one moves beyond classical Maxwell theory.

 

[Matterwave]

My only reservation is that the two directions do not seem completely symmetric.

Do you have a problem with differential equations being used to describe physics in general? I'm having a hard time figuring out why you seem so hung up on "not completely symmetric" (whatever that means to you).

So perhaps the issue is not whether $F_{\mu\nu}$ determines $J^\mu$,

You literally wrote this equation out.. $J^\mu = \partial_\nu F^{\nu\mu}$

but whether the notion of a globally well-defined classical $F_{\mu\nu}$ should be expected to exist once one moves beyond classical Maxwell theory.

What?

 

[Roberto Pavani]

I am not objecting to Maxwell's equations at all. On the contrary, I am perfectly happy to take Maxwell's equations as valid at all scales, even if the classical description in terms of a definite classical $F_{\mu\nu}$ becomes fuzzy.

The point about the two directions is a separate, rather trivial observation about differential equations: given $F_{\mu\nu}$, one can obtain $J^\mu$ locally and directly. Going in the other direction, from $J^\mu$ to $F_{\mu\nu}$, requires solving the differential equations and therefore specifying the appropriate initial/boundary data (in addition to the sources within the region being considered).

My other point was precisely about the premise that classical Maxwell theory might cease to be applicable at some scale. If that is the position being taken, then I am asking what happens to the classical $F_{\mu\nu}$ at and beyond that scale. I am not proposing that Maxwell “breaks”; I am asking whether it still makes sense to regard $F_{\mu\nu}$ as a globally well-defined classical field if Maxwell theory itself is assumed to no longer be the appropriate description in that regime.

 

[Matterwave]

I am asking what happens to the classical $F_{\mu\nu}$ at and beyond that scale.

It was not at all clear this was your question. It probably deserves its own thread if you would like to discuss this particular issue. I can make a couple of very short comments here.

I am asking whether it still makes sense to regard $F_{\mu\nu}$ as a globally well-defined classical field if Maxwell theory itself is assumed to no longer be the appropriate description in that regime.

From a geometric viewpoint, AFAIK the field tensor $F$ is globally well defined. Here I'm taking "globally well defined" to mean across the entire spacetime manifold. At least I don't know of what kind of topological defects there might be to make $F$ not globally well defined. (Someone can correct me here if I'm wrong)

But it seems to me that what you mean by "globally well defined" is more like "can we use it in QM". In the quantum regime we do broadly 2 kinds of things:

1. In "standard" QM (non-QFT), we use the classical descriptions of EM to add terms to our Hamiltonian in the Schroedinger equation and we solve from there. This is what we do to solve the equations for the Hydrogen atom for example. Here we use the EM potentials $(\phi, \vec{A})$ rather than the fields themselves. The Hamiltonian is an energy-based picture (where potentials are natural) and also the Aharanov-Bohm effect shows that, in non-simple geometries, using just the $\vec{E},\vec{B}$ fields leads to problems. Specifically, the wave function picks up a phase shift as it moves around a solenoid even if it stays in regions where $\vec{E}$ and $\vec{B}$ are 0. See https://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485 for a (much) more complete discussion.
2. In QFT we view $A$ as the connection of the principle $U(1)$ fiber bundle relating to an internal symmetry over the spacetime. The field strength $F=dA$ is then the curvature of this fiber bundle. What we quantize is the connection $A$ which is also called the gauge field. More complexity arises as this picture is the low energy limit of a more broader theory (the electroweak theory) in which, at ultra high temperatures, the gauge field $A$ does not exist independently. In standard QED, the Lagrangian includes (contraction of) the field strength $-\frac{1}{4} F_{\mu\nu} F^{\mu\nu}$. In electroweak theory, it gets more complicated. This is a really deep rabbit hole.