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Robert B. Laughlin, A Different Universe states the following concerning the relationship between superconductivity and quantum field theory. I do not understand why he says "the microscopic equations of quantum mechanics encrypted in the field theory are different from those of the real material and therefore wrong." Could someone explain?

Reductionist ideology has another fascinating manifestation in superconductivity theory that I call quantum field theory idolatry. Quantum field theory, a body of mathematics that grew out of studies of elementary particles, is commonly taught after conventional quantum mechanics as a special language for working in that subject—and also as kind of superior way of thinking. It is actually not a new way of thinking but simply a restatement of quantum mechanics in the context of the special limitations and conditions appropriate to the vacuum of space. These conditions make the formalism elegant and fun to learn—at least for mathematical types such as myself—but they also make it easy to hide the essence of a thing through manipulation. Sleights of hand can make some physical behavior appear to be caused by field theory when it is actually being caused by the manipulation itself. Shortly after the Bardeen-Cooper-Schrieffer theory was introduced, the language of quantum fields was discovered to be particularly well suited for describing important properties of superconductors—notably the supercurrents themselves, the Meissner effect, the above-threshold conductivity, and the collective sloshing motions of the electrons called plasma oscillations—precisely because it allowed one to postulate one’s way around messy, ultimately unimportant detail and get quickly to the meat of the matter. This eventually led to the practice of explaining all of superconductivity using field theory, and thus indirectly to the idea that quantum fields cause superconductivity. Even today one will find a great many people who secretly believe this. It is ridiculous—like believing that the weather is caused by the price of corn. In fact, quantum field theory works because the emergent universality of superconductivity makes it work, not the other way around. The microscopic equations of quantum mechanics encrypted in the field theory are different from those of the real material and therefore wrong. The only way one can start from wrong equations and get the right answer is if the property one is calculating is robustly insensitive to details, i.e., is emergent. Thus the lesson from superconductivity is actually not that quantum field theory is a superior computational technology but that quantum fields can themselves emerge.

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    $\begingroup$ I understand you're quoting here, but note how, on a website page, the text becomes a "wall of text". $\endgroup$ Commented Nov 12, 2023 at 14:33
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    $\begingroup$ @StephenG-HelpUkraine It was I who edited in the quote. It is a bit "wall of text" but that's how it is in the book and I felt obliged to quote the book exactly. $\endgroup$ Commented Nov 12, 2023 at 20:18
  • $\begingroup$ It was not intended as criticism, just to point out how internet writing seems to necessarily require a different approach than traditional writing (where that passage would not in any way be considered a wall of text). I'm sorry I wasn't clearer in my intent. $\endgroup$ Commented Nov 12, 2023 at 22:25
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    $\begingroup$ "one will find a great many people who secretly believe [that qft causes superfluidity]" How does one find them if they believe it secretly? It's like the description of these physicists is different from the real physicists and therefore wrong. $\endgroup$
    – tobi_s
    Commented Nov 13, 2023 at 6:59
  • $\begingroup$ @tobi_s I think what he meant is one can find such people by listening/reading them, even if they do not tell you explicitly their beliefs. It's similar to how when you listen to a person who makes odd political remarks/references, but never tells you their actual beliefs explicitly, perhaps because they do not have a good argument for them, or they are unacceptable in public discourse. After a while with such person, you may naturally formulate a hypothesis that they make these remarks because they believe various extremist/conspiracy theories, even if they prefer not to say out loud. $\endgroup$ Commented Nov 14, 2023 at 22:10

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It is very common in physics that the fine details of a system can be very complicated, but if we zoom out and look at the system from a larger scale it becomes much simpler.

The classical (in both senses) example of this is the Navier Stokes description of fluid flow. At the fine detail we have $10^{23}$ or so molecules in the fluid all interacting with each other, and no computer is powerful enough to calculate how the fluid will flow starting from individual molecules. However if we zoom out to a scale many times larger that the size of a molecule we can treat the fluid as a simple continuous medium, and then the equations that describe its flow are relatively simple (though notoriously hard to solve).

In the case of superconductivity the fine detail is even more complicated as we'd have to compute a many body wavefunction for all the particles in the solid. However as with fluid flow we can simplify the problem by ignoring the detail. Superconductivity happens because there is a (very small) attractive force between the conduction electrons. The origin of this force is a complicated interaction between the electrons and vibrations of the crystal lattice, but suppose we ignore the details and just assume some potential curve for this attraction. With this assumption we can write a quantum field theory, BCS Theory, that describes the behaviour of the conduction electrons in the presence of this attractive force, and it turns out this theory successfully describes the superconducting properties of the material.

This is where Laughlin's objection comes in. He is pointing out that the quantum field theory ignores the microscopic origin of the attractive force i.e. it is not describing the actual origin of superconductivity. It is an effective theory that works only because it ignores the details. I don't think he is objecting to the idea of an effective theory, because ... well ... most physical theories are effective to some degree. Rather he is incensed because he believes that physicists have been so seduced by the elegance of quantum field theory that they have forgotten that the theory of superconductivity is only an effective theory.

You didn't ask if Laughlin's objects are justified, but assuming the question is implied the answer is yes and no. Ultimately physicists are pragmatists. If a theory successfully predicts the results of experimental observations it's a good theory. If you start criticising the Navier-Stokes equations on the grounds they are an effective theory you will attract looks from aerospace engineers that are at best pitying. However Laughlin is right that we should always remember that our theories are just descriptions of our systems and we need to beware of taking them too literally.

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  • $\begingroup$ "You didn't ask if Laughlin's objects are justified, but assuming the question is implied the answer is yes and no." I just do not know what he means by "wrong", such is the reason for my question. By "right", does he mean to use the complete Hamiltonian or Lagrangian of the Standard Model? $\endgroup$
    – Hans
    Commented Nov 13, 2023 at 6:30
  • $\begingroup$ @Hans He means right would include the microscopic details of the interactions between the electrons and the lattice vibrations. I doubt he means you need to include the strong and weak forces in the analysis, though maybe he does. I would guess his point is that the interaction of the electrons with the lattice vibrations is not well described by a quantum field theory. $\endgroup$ Commented Nov 13, 2023 at 6:46
  • $\begingroup$ I use the Standard Model to stress my point of confusion as to which part Laughlin considers wrong, whether it is the fidelity to the actual interaction or something else. In case it was the fidelity --- he does say "[the microscopic equation of the quantum mechanics] of the real material" --- I want to leave no stone unturned. But if we are fine with the spirit, then the electrodynamic interaction amongst the atoms would suffice for being the right quantum mechanics of the real material. $\endgroup$
    – Hans
    Commented Nov 13, 2023 at 8:17
  • $\begingroup$ @Hans I think you're taking his comments more seriously than they deserve. $\endgroup$ Commented Nov 13, 2023 at 9:00
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    $\begingroup$ @Hans Physicists have been arguing about this since at least when I was a student forty years ago. Leave your friend with his opinion and get on with enjoying physics! $\endgroup$ Commented Nov 13, 2023 at 15:18
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I do not understand why he says "the microscopic equations of quantum mechanics encrypted in the field theory are different from those of the real material and therefore wrong." Could someone explain?

What he is trying to say is not super clear.

First, he seems to be arguing that quantum field theory is a sexy formalism and therefore is overused.

Next, he is saying, I think, that if you take some quantum fields and construct some hamiltonian like $$ \hat H = \int d^3x \psi^\dagger \left(-\frac{\nabla^2}{2m}\right)\psi + \lambda \psi^\dagger \psi^\dagger \psi\psi\;,\tag{1} $$ then you can derive some equations that describe a superconductor.

But the Hamiltonian in Eq. (1) is for damn sure not the hamiltonian of the real material. The hamiltonian of the real material has a Coulomb interaction between electrons, not some point interaction characterized by a $\lambda$, and the real hamiltonian has the interaction of the electrons with the lattice, and it is, generally, pretty complicated. The fact that we can get away with some relatively simplified field theory is, in his characterization, due to the emergent nature of the phenomena.

This is my take, as a mere mortal. Maybe better to just email him and ask directly.

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    $\begingroup$ Is there a theory that is not an effective one? After all, we cannot know everything, there is always a detail not yet discovered of which we are not aware. Will there ever be a real Hamiltonian of anything, or even if there be a real Hamiltonian will that be the end of all discoveries quantum or otherwise? $\endgroup$
    – hyportnex
    Commented Nov 12, 2023 at 15:06
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    $\begingroup$ It's all relative, the Hamiltonian in the above post is certainly less accurate than one including the Coulomb interactions. That Hamitlonian in turn is less accurate than one which includes the EM field as a dynamic entity...and on it goes. $\endgroup$
    – AfterShave
    Commented Nov 12, 2023 at 19:40
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    $\begingroup$ There are certainly ineffective theories... $\endgroup$
    – Jon Custer
    Commented Nov 13, 2023 at 19:41
  • $\begingroup$ @JonCuster: Good one! You just earned yourself a brownie point. $\endgroup$
    – Hans
    Commented Nov 14, 2023 at 1:23

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