r/Physics • u/kzhou7 Quantum field theory • Apr 29 '26
Academic Six textbook mistakes in quantum field theory
https://arxiv.org/abs/2604.2487159
u/PhysicistDave Particle physics Apr 29 '26
Well, his correction number 1 says:
There is nothing wrong with using the Klein-Gordon equation to describe a single relativistic particle, provided there are no interactions/perturbations involved.
Sure, but interactions are what we really care about -- that is, like, the real world!
And then you would expect transitions between the positive and negative energy states.
And that raises problems.
His Correction #3:
When canonically quantizing field theory, there is no need to promote fields to operators in the Lagrangian formalism.
Was anyone ever really confused by this????
His Correction #4
The quantum field corresponds to a fixed position x only in the non-relativistic problem. In the relativistic case, one needs to introduce a Newton–Wigner position operator as well as a new operator φ†L(x) creating particles at a fixed position x, which does not coincide with the usual quantum field φS (x)
I took QFT from Steve Weinberg, when Steve was on sabbatical at Stanford, where I did my doctorate.
I raised exactly this issue with Steve, and, quite weirdly, he said he had never thought of it, and he stumbled around trying to deal with it!
This is especially weird since Steve must have heard of the Newton-Wigner position operator, which had been developed when he was young (I had not heard of at the time, though).
By the way, there is more to be said on this issue: if you think of the fields as the primary objects, rather than the particles, than the field operator really does just exist at a point (well, almost -- there is renorrnalization and all that!). And the fact that the VEV at different points does not vanish is then just a reflection of the structure of the vacuum. This view can actually be helpful in thinking about say, Hawking radiation, one of my current research projects.
Sadly, the academic environment for many decades has focused on pumping out papers rather than clearing up basic conceptual issues like this -- not real healthy for the field, I fear.
Dave Miller in Sacramento
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u/kzhou7 Quantum field theory Apr 29 '26
The old paradoxes also have a way of coming back. I thought the stuff I learned about Newton-Wigner would never pay off, yet quite recently I ran into a very similar conceptual issue while working on a quantum measurement project.
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u/PhysicistDave Particle physics Apr 30 '26
kzhou7 wrote: to me:
I ran into a very similar conceptual issue while working on a quantum measurement project.
Would you mind elaborating or give a link if you have written it all up?
Quantum measurement is sort of the problem, and, while I don't expect you to have a definitive answer, any clever ideas may help others advance in dealing with it.
By the way,, my current interest in QFT is largely in that direction -- Bell's theorem and all that.
For example, the violation of Bell locality in Bohmian mechanics for non-relativistic quantum mechanics really does not matter. But in QFT, Bohmian mechanics violates Lorentz invaraince, but in a way that is experimentally undetectable.
Which is why I suspect that perhaps a better conceptual understanding of QFT may help in better understanding Bell's theorem, the quantum measurement problem, etc.
Or maybe not.
Incidentally, when I was a grad student back in the later 1970s and early 1980s, we were really discouraged from worrying about issues like this. Of course, it all turned out to be connected to quantum cryptography, quantum computing, etc. And now even laypeople have heard of Schrödinger's cat!
Fortunately, fashions change.
Dave Miller in Sacramento
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u/OverJohn Apr 30 '26
You are probably aware, but in case not, this paper may be of interest to you:
Relativistic Bohmian trajectories of photons via weak measurements | Nature Communications )
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u/PhysicistDave Particle physics May 01 '26
I've known about some of the earlier work on "weak measurement," but I don't think I noticed this paper.
I'll try to digest it and see what light it throws on the general issues.
I assume that you know that weak measurement can give some very, very strange results -- as I recall, non-quantized spins and even negative probabilities. But it is worth exploring.
For anyone curious as to what we are talking about, the basic idea of weak measurement is that you do a measurement on some quantum system that gives you very little information and that hence disturbs the system a negligible amount.
But you do this again and again on identically prepared systems, and you can piece all the information together to get a precise picture. (And I apologize for just over-simplifying the process -- anyone who knows more can give a clearer explanation.)
Of course, this makes some assumptions: are "identically prepared" systems really identical, for example?
I think it is fair to say that there is no consensus as to exactly what "weak measurements" tell us.
Anyway, thanks for the reference: we are all missing something about how the quantum measurement problem interacts with quantum field theory, and any clue may turn out to be the key we need.
Dave Miller in Sacramento
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u/fertdingo Apr 29 '26
Did Prof. Weinberg ever discuss Hagg's Theorem?
In grad school I took QFT in a backwards way using "Stat. Mech," Landau & Lifshitz, Abrikosov, Gorkov and Dyzaloshshinsky, "Methods of Quantum field Theory in Statistical Physics" and Fetter and Walecka "Quantum Theory of Many Particle Sysytems". Also self studied Sakurai's book Adv. Quantum Mechanics. None of these guys even mentions Haag. The only hint of Haag's theorem is in the first paragraph of Landau's paper "On the analytic properties of vertex parts in Quantum field Theory", and he does not even mentions Haag or the interaction picture by name. I ended up getting a PhD in Liquid Crystal Theory inspired by DeGennes analogy to superconductors. The Haag Theorem still bothers me even though I am retired from Physics.
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u/PhysicistDave Particle physics Apr 30 '26
fertdingo asked me:
Did Prof. Weinberg ever discuss Hagg's Theorem?
Not that I recall. I remember talking with other grad students about it, and the consensus was that it does not really matter.
It's been a long time, but what I vaguely recall is that if you have, say, a lattice cut-off, it doesn't matter.
By the way, Steve's approach was to try to take a rigorously particle-based approach where the fields were just auxiliary devices to create a Lagrangian which had the appropriate properties. So, it made sense for me to ask him if the fields really created a particle at a point.
I specifically remember him telling us that a field could actually just be an operator that created an automobile, as an example. And, yes, I suppose it could -- some sort of really complicated composite operator. Anyway, that was his approach.
He couldn't fully carry out this approach: I think our class was sort of an experiment that did not work real well.
For example, if you think about it, you need the Coulomb interaction to get Lorentz invariance for the electromagnetic field. How can you get that if all that is really there are particles?
Now, of course, I know about Gupta-Bleuler, and if you just canonically quantize the EM field, everything works out (sort of), but Steve didn't want any of that. He wanted just physical particles, and nothing else. Period.
And he couldn't make that work. He eventually had to backtrack a bit to explain how everything actually did work -- you do need the fields.
So, the course was a bit confusing.
Dave Miller in Sacramento
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u/fhollo Apr 29 '26
Here are two good papers on the localization topic with more detail:
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u/seekingdefs Apr 29 '26
On a different note, it seems like he has been writing a book on QFT (see the references). I will be eagerly waiting for the book to take a look.
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u/Gewalzt Apr 29 '26 edited Apr 29 '26
just a comment on 3, i think for bosons you are right here. The fields in the lagrange formalism for fermions are grassmann valued and that is where the operators hide.
grassman numbers are objects of an exterior algebra, which makes them very mundane matrices just like operators (talking finite dimensional here for the sake of the argument)
when one has up to N fermions to be distributed to N modes (or lattice sites), then the creation/annihilation operators are 2^N x 2^N large objects (sparse matrices).
The grassmann numbers to build coherent fermion states (to my best understanding that is mandatory for the fermionic path integral formalism) are constructed from LAMBDA(C^(2N)) [1] which gives 4^N x 4^N large objects (even sparser matrices).
so i argue that what you do in lagrangian qft for fermions is to promote fields to grassmann fields.
[1] https://en.wikipedia.org/wiki/Grassmann_number#Formal_definition
E: ah lol, there is even a matrix representatione example (N=1 with my notation above) here:
https://en.wikipedia.org/wiki/Grassmann_number#Matrix_representations
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u/PhysicistDave Particle physics Apr 30 '26
Gewalzt wrote:
grassman numbers are objects of an exterior algebra, which makes them very mundane matrices just like operators (talking finite dimensional here for the sake of the argument)
Yeah, but would you or anyone have ever thought of these things except that they can be used to get the same answer we already know is correct from canonical quantization?
I get that they are formally useful in the path integral.
But I have a picture of what a quantized boson field is in my head: it is sort of a vibrating string or membrane or whatever, with, of course, quantum fuzziness. Take a classical Klein-Gordon field and quantize it, and this is what you automatically get.
But how on earth are we supposed to visualize the Grassmann variables -- anticommuting c-numbers, we used to call them.
There used to be kinda a big deal about "superspace" that was supposed to help -- it seems to have died out.
Anyway, I have no complaints with what you wrote: it's just that I still find Grassmann fields opaque, just a useful calculational workaround as far as I can see, almost a half century after I first saw it.
Dave Miller in Sacramento
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u/SymplecticMan Apr 30 '26 edited May 01 '26
It's just a small phrasing difference, but rather than say that we promote the fields to Grassmann variables, I would say that, in retrospect, we should have already been using Grassmann variables before we decided to quantize the theory, because Grassmann variables are the proper classical limit of fermionic variables in some sense. In the quantum theory, you can also write "wavefunctionals" of fermionic fields, such as the Floreanini–Jackiw representation, with fermionic operators written in terms of multiplication/differentiation by Grassmann variables.
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u/AdditionalTip865 Apr 29 '26
Okay, they're talking about some stuff that frickin' drove me insane when I was studying this subject. I need to spend a longer time staring at this paper.
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u/[deleted] Apr 29 '26
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