I strongly disagree that quantizing a classically nonlinear field theory should be expected to lead to a nonlinear Schrodinger equation, either at the level of the global quantum state, or at the level of the little wavefunctions which appear in the course of a calculation.
Linearity at the quantum level means that the superposition principle is obeyed—that if X1(t) and X2(t) are solutions to the Schrodinger equation, then X1(t)+X2(t) is also a solution. Nonlinear Schrodinger equations screw that up. There are sharp bounds on the degree to which that can be happening, at least for specific proposals regarding the nonlinearity.
Also, I think the “argument from Haag’s theorem” is just confused. See the discussions here. The adiabatic hypothesis and the interaction picture are heuristics which motivate the construction of a perturbative formalism. Haag’s theorem shows that the intuition behind the heuristic cannot literally be correct. But then, except for UV-complete theories like QCD, these theories all run into problems at some high energy anyway, so they don’t exist in the classic sense e.g. of von Neumann, as a specific operator algebra on a specific Hilbert space.
Instead, they exist as effective field theories. OK, what does that mean? Let’s accept the “Haagian” definition of what a true QFT or prototypical QFT should be—a quantum theory of fields that can satisfy a Hilbert-space axiomatization like von Neumann’s. Then an “effective field theory” is a “QFT-like object which only requires a finite number of parameters for its Wilsonian renormalization group to be predictive”. Somehow, this has still failed to obtain a generally accepted, rigorous mathematical definition, despite the work of people like Borcherds, Kreimer, etc. Nonetheless, effective field theories work, and they work empirically without having to introduce quantum nonlinearity.
I should also say something about how classical nonlinearity does show up in QFT—namely, in the use of solitons. But a soliton is still a fundamentally classical object, a solution to the classical equations of motion, which you might then “dress” with quantum corrections somehow.
Even if there were formally a use for quantum nonlinearity at the level of the little wavefunctions appearing along the way in a calculation, that wouldn’t prove that “ontologically”, at the level of THE global wavefunction, that there was quantum nonlinearity. It could just be an artefact of an approximation scheme. But I don’t even see a use for it at that level.
The one version of this argument that I might almost have time for, is Penrose’s old argument that gravitational superpositions are a problem, because you don’t know how to sync up the proper time in the different components of the superposition. It’s said that string theory, especially AdS/CFT, is a counterexample to this, but because it’s holographic, you’re really looking at S-matrix-like probabilities for going from the past to the future, and it’s not clear to me that the individual histories linking asymptotic past with asymptotic future, that appear in the sum over histories, do have a natural way of being aligned throughout their middle periods. They only have to “link up” at the beginning and the end, so that the amplitudes can sum. But I may be underestimating the extent to which a notion of time evolution can be extended into the “middle period”, in such a framework.
So to sum up, I definitely don’t buy these particular arguments about Haag’s theorem and nonlinearity. As you have presented them, they have a qualitative character, and so does my rebuttal, and so it could be that I’m overlooking some technicality which gives the arguments more substance. I am quite prepared to be re-educated in that respect. But for now, I think it’s just fuzzy thinking and a mistake.
P.S. In the second edit you say that the modified commutation relations “require modifications to linear quantum mechanics”. Well, I definitely don’t believe that, for the reasons I’ve already given, but maybe this is a technicality which can help to resolve the discussion. If you can find someone making that argument in detail…
edit What I don’t believe, is that the modifications in question amount to nonlinearity in the sense of the nonlinear Schrodinger equation. Perhaps the thing to do here, is to take nonlinear Schrodinger dynamics, construct the corresponding Heisenberg picture, and then see whether the resulting commutation relations, resemble the commutator power series you describe.
I strongly disagree that quantizing a classically nonlinear field theory should be expected to lead to a nonlinear Schrodinger equation, either at the level of the global quantum state, or at the level of the little wavefunctions which appear in the course of a calculation.
Linearity at the quantum level means that the superposition principle is obeyed—that if X1(t) and X2(t) are solutions to the Schrodinger equation, then X1(t)+X2(t) is also a solution. Nonlinear Schrodinger equations screw that up. There are sharp bounds on the degree to which that can be happening, at least for specific proposals regarding the nonlinearity.
Also, I think the “argument from Haag’s theorem” is just confused. See the discussions here. The adiabatic hypothesis and the interaction picture are heuristics which motivate the construction of a perturbative formalism. Haag’s theorem shows that the intuition behind the heuristic cannot literally be correct. But then, except for UV-complete theories like QCD, these theories all run into problems at some high energy anyway, so they don’t exist in the classic sense e.g. of von Neumann, as a specific operator algebra on a specific Hilbert space.
Instead, they exist as effective field theories. OK, what does that mean? Let’s accept the “Haagian” definition of what a true QFT or prototypical QFT should be—a quantum theory of fields that can satisfy a Hilbert-space axiomatization like von Neumann’s. Then an “effective field theory” is a “QFT-like object which only requires a finite number of parameters for its Wilsonian renormalization group to be predictive”. Somehow, this has still failed to obtain a generally accepted, rigorous mathematical definition, despite the work of people like Borcherds, Kreimer, etc. Nonetheless, effective field theories work, and they work empirically without having to introduce quantum nonlinearity.
I should also say something about how classical nonlinearity does show up in QFT—namely, in the use of solitons. But a soliton is still a fundamentally classical object, a solution to the classical equations of motion, which you might then “dress” with quantum corrections somehow.
Even if there were formally a use for quantum nonlinearity at the level of the little wavefunctions appearing along the way in a calculation, that wouldn’t prove that “ontologically”, at the level of THE global wavefunction, that there was quantum nonlinearity. It could just be an artefact of an approximation scheme. But I don’t even see a use for it at that level.
The one version of this argument that I might almost have time for, is Penrose’s old argument that gravitational superpositions are a problem, because you don’t know how to sync up the proper time in the different components of the superposition. It’s said that string theory, especially AdS/CFT, is a counterexample to this, but because it’s holographic, you’re really looking at S-matrix-like probabilities for going from the past to the future, and it’s not clear to me that the individual histories linking asymptotic past with asymptotic future, that appear in the sum over histories, do have a natural way of being aligned throughout their middle periods. They only have to “link up” at the beginning and the end, so that the amplitudes can sum. But I may be underestimating the extent to which a notion of time evolution can be extended into the “middle period”, in such a framework.
So to sum up, I definitely don’t buy these particular arguments about Haag’s theorem and nonlinearity. As you have presented them, they have a qualitative character, and so does my rebuttal, and so it could be that I’m overlooking some technicality which gives the arguments more substance. I am quite prepared to be re-educated in that respect. But for now, I think it’s just fuzzy thinking and a mistake.
P.S. In the second edit you say that the modified commutation relations “require modifications to linear quantum mechanics”. Well, I definitely don’t believe that, for the reasons I’ve already given, but maybe this is a technicality which can help to resolve the discussion. If you can find someone making that argument in detail…
edit What I don’t believe, is that the modifications in question amount to nonlinearity in the sense of the nonlinear Schrodinger equation. Perhaps the thing to do here, is to take nonlinear Schrodinger dynamics, construct the corresponding Heisenberg picture, and then see whether the resulting commutation relations, resemble the commutator power series you describe.