# Nonlinear dynamics beneath quantum mechanics?

Yesterday I asked whether the Schroedinger Equation could possibly be nonlinear, after reviewing the answers and material given to me in that thread I feel like my question were adequately answered.

However could there still be nonlinear dynamics underneath quantum mechanics that could potentially explain the "weirdness" of quantum mechanics? I know of atleast one serious physicist who is developing an approach in this direction, Tim Palmer with his Invariant Set Postulate.

Just like the Liouville equation is linear, but there are "deeper" nonlinear dynamics beneath it, couldn't the Schroedinger Equation be the same? This is atleast one of Tim Palmers motivations for developing his approach and is also what Dirac said way back in the days.

I suspect this is what David Zaslavsky meant yesterday when he responded to my other thread (Could the schroedinger equation be nonlinear) when he said:

That being said, there's no reason that the "true" theory underlying QM would have to be linear. In fact, for exactly the reason you pointed out (i.e. that general relativity is nonlinear), it's commonly believed that we will need some kind of nonlinear theory to properly explain the universe at its most basic level.

Any views / opinions / thoughts / theories ?

-
It depends on if you think QM is a theory of real things, or just a theory of how we compute given constraints on what can be measured. I take the latter view, and think that QM is exact --- just like probability theory is exact. One can construct non-linear systems, and use QM to analyse it, but that's got nothing to do with QM being linear or not. For instance, the whole Standard Model is highly non-linear, and all theories of quantum gravity will be, but they all use QM to formulate themselves precisely. –  genneth Sep 8 '11 at 13:07
Thanks a lot. However if quantum gravity will be nonlinear, doesn't this disprove certain interpretations? –  SchroedingersGhost Sep 9 '11 at 11:46
You're still not understanding that QM can be linear, and a theory of quantum gravity can be not --- the quantum gravity is built in the language of QM --- states are still vectors/rays/density matrices, measurements are POVMs, etc. Nothing in QM specifies an actual physical theory --- the Hilbert space, what are the measurements, their commutation relations, etc. are the meat of the physical theory, and those are still to be specified. –  genneth Sep 9 '11 at 13:39
No I do understand, but wouldn't this affect the interpretational debate to a large degree? Im pretty sure MWI couldn't "survive" if it turns out that quantum gravity is nonlinear? –  SchroedingersGhost Sep 9 '11 at 20:44
Perhaps it is me who is not understanding. What does the system under study using QM formalism have to say about QM itself? –  genneth Sep 9 '11 at 20:47

There is exactly one phenomena in quantum mechanics that is inherently non-linear in the wave function: the measurement process, regardless of the interpretation that you choose to adhere, behaves as a (non-deterministic) map where a subset of eigenvalues get enormously amplified, while the rest become enormously attenuated. Although i am not sure how helpful would turn out to be to think of measurement in those terms.

-

There is an alternative way of introducing nonlinearity into QM, at the level of QFT, that your previous question did not admit. QFT is usually presented as a theory in terms of a quantum field such as $\hat\phi(x)$, but this is not an operator, it is an operator-valued distribution, which lets us construct operators by "smearing" with "test functions" (which in signal processing would be called "window functions", which I believe may be the best way for non-mathematicians to start to understand this formalism), $$\hat\phi:\mathcal{S}\rightarrow\mathcal{A};f\mapsto\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x.$$ $\hat\phi$ is a linear map from functions on space-time to operators, $\hat\phi_f$. We can construct self-adjoint observables using sums of products of operators $\hat\phi_{f_i}$, whereas, again, $\hat\phi(x)$ is not an operator.

To obtain a probability theory, we have to introduce a state over the $\star$-algebra of observables that is generated by the $\hat\phi_f$, for whatever test functions we choose to use. The state has to be a normalized, complex-linear, positive semi-definite map (which means that $\omega(1)=1$, $\omega(\lambda\hat A)=\lambda\omega(\hat A)$, $\omega(\hat A+\hat B)=\omega(\hat A)+\omega(\hat B)$, and $\omega(\hat A^\dagger\hat A)\ge 0$). That's enough for us to construct a Hilbert space (using what's called the GNS construction), and then we can introduce a Born probability interpretation. The elementary way to construct a state is to introduce a vacuum vector $\left|0\right>$ and the trivial action of annihilation operators on it, $a_f\left|0\right>=0$, and then to define $\hat\phi_f=a_{f^*}+a_f^\dagger$ and the vacuum state as $\omega(\hat A)=\left<0\right|\hat A\left|0\right>$, which, with the commutation relation between the creation and annihilation operators, gives you the vacuum sector of all the quantized free fields (but many other states can be constructed). Note, however, that although the state has to be linear over the algebra for us to be able to construct a probability interpretation, there is nothing that requires that the map from functions on space-time to the algebra of operators has to be linear. That is, we can construct a Hilbert space and a probability interpretation even if $\hat\phi:\mathcal{S}\rightarrow\mathcal{A};f\mapsto\hat\phi_f$ is nonlinear.

No reason is given, but the Wightman axioms insist that the quantum field must be a linear map from functions on space-time to the algebra of operators, and there is a more abstract insistence on something very similar in the Haag-Kastler axioms, so this option is essentially unconsidered in the literature. It opens up a different world if we consider them. Note that this is not so much a theory beneath QM, it is essentially about the relationship between whatever abstract operators we use in QM and whatever we take to correspond to them in space-time (that's sort of beneath, but is more an intrinsic question that has to be answered by any abstract approach to QM, such as the Aaronson paper that was mentioned in an Answer to your previous Question, which is entirely right as far as it goes).

AFAIK, the above has essentially no connection with quantum gravity, noncommutative space-times, or any of the more popular research directions to be found in the literature today.

-