That nonlinearity is in the classical equations of motion.
Sure, the equation of motion are non-linear in the FIELDS, which aren’t necessarily the wavefunctions. No one has solved Yang-Mills, which is almost certainly the easiest of the non-linear lagrangians, so I don’t think we actually know about whether the quantum equations would be linear.
The standard approach is to fracture gauge symmetry and use solutions to the linearized equations of motion as wavefunctions, and treat the non-linear part as an interaction you can ignore at large times. This is actually hugely problematic because Haag’s theorem calls into question the entire framework (you can’t define an interaction picture).
It seems unlikely that you can have the classical equations of motion be non-linear in the field without the wavefunction having non-linear evolution- after all the creation operator at leading order has to obey the classical equation of motion, and you can write a single particle state as (creation*vacuum). The higher order terms would have to come together rather miraculously.
And keep in mind, its not just Yang-Mills. If we think of the standard model as the power-series of an effective field theory, it seems likely all those linear, first-order equations governing the propagator are just the linearization of the full theory.
There are many arguments against many worlds, I was simply throwing out one that I used to hear bandied about at particle physics conferences that isn’t addressed in the sequences at all. And generally, quantum field theories are still on fairly weak mathematical underpinnings. We have a nice collection of recipes that get the job done, but the underlying mathematical structure is unknown. Maybe a miracle occurs. But its a huge unknown area that is worth pointing to as “what can this mean?” Unless someone has dealt with this in the last 5 or 6 years, its been awhile since I was a physicist.
Edit: Replying to your edit, I’m thinking of any higher order terms you can throw into your lagrangian, be they quadratic fermion terms, three or four-point terms in Yang-Mills,etc. These lead to non-linear equations for the fields, and (quite likely, but unproven), the full solution to the wavefunction will be similarly non-linear. This may have implications for whether or not particle-number is a good observable, but I’m tired and don’t want to try and work it out.
Edit Again: I re-read this and its pretty clear to me I’m not communicating well (since I’m having trouble understanding what I just wrote). So- I’ll try to rephrase. The same non-linearities that crop up in Yang-Mills that require you to pick a proper gauge before you try to use the canonical relationships to write the quantum equations of motion are likely more general. You can add all kinds of non-linear terms to the lagrangian and you can’t always gauge fix them away. Most of the time they are small and you can ignore them, but on a fundamental level they are there (and at higher energies then some effective scale they probably matter). This requires modifications to linear quantum mechanics (commutation relationships become power series of which the traditional commutator is just the lowest order term, etc).
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.
Sure, the equation of motion are non-linear in the FIELDS, which aren’t necessarily the wavefunctions. No one has solved Yang-Mills, which is almost certainly the easiest of the non-linear lagrangians, so I don’t think we actually know about whether the quantum equations would be linear.
The standard approach is to fracture gauge symmetry and use solutions to the linearized equations of motion as wavefunctions, and treat the non-linear part as an interaction you can ignore at large times. This is actually hugely problematic because Haag’s theorem calls into question the entire framework (you can’t define an interaction picture).
It seems unlikely that you can have the classical equations of motion be non-linear in the field without the wavefunction having non-linear evolution- after all the creation operator at leading order has to obey the classical equation of motion, and you can write a single particle state as (creation*vacuum). The higher order terms would have to come together rather miraculously.
And keep in mind, its not just Yang-Mills. If we think of the standard model as the power-series of an effective field theory, it seems likely all those linear, first-order equations governing the propagator are just the linearization of the full theory.
There are many arguments against many worlds, I was simply throwing out one that I used to hear bandied about at particle physics conferences that isn’t addressed in the sequences at all. And generally, quantum field theories are still on fairly weak mathematical underpinnings. We have a nice collection of recipes that get the job done, but the underlying mathematical structure is unknown. Maybe a miracle occurs. But its a huge unknown area that is worth pointing to as “what can this mean?” Unless someone has dealt with this in the last 5 or 6 years, its been awhile since I was a physicist.
Edit: Replying to your edit, I’m thinking of any higher order terms you can throw into your lagrangian, be they quadratic fermion terms, three or four-point terms in Yang-Mills,etc. These lead to non-linear equations for the fields, and (quite likely, but unproven), the full solution to the wavefunction will be similarly non-linear. This may have implications for whether or not particle-number is a good observable, but I’m tired and don’t want to try and work it out.
Edit Again: I re-read this and its pretty clear to me I’m not communicating well (since I’m having trouble understanding what I just wrote). So- I’ll try to rephrase. The same non-linearities that crop up in Yang-Mills that require you to pick a proper gauge before you try to use the canonical relationships to write the quantum equations of motion are likely more general. You can add all kinds of non-linear terms to the lagrangian and you can’t always gauge fix them away. Most of the time they are small and you can ignore them, but on a fundamental level they are there (and at higher energies then some effective scale they probably matter). This requires modifications to linear quantum mechanics (commutation relationships become power series of which the traditional commutator is just the lowest order term, etc).
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.