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For the sake of simplicity, I’d like to believe that there is one master non-linear partial differential equation governing physics. In particular, consider a Klein-Gordon form:

$$ \frac{\partial^2 f}{\partial t^2}=v^2 \left (\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z ^2} \right ) - m^2f $$

The state function $f$ (with $\partial f/\partial t$) here is naturally a function of space $xyz$ and time $t$. But, let $v$ and $m$ depend on the local $f$ and $\partial f/\partial t$ (so that the equation is non-linear and probably impossible to solve analytically). It seems $f$ needs to be a two-number vector (to allow two light polarizations, or two electron spins), but I'd like to leave that unspecified for now.

Then, is there an argument why nature can't have this simple non-linear form (where quarks and other standard model elements appear as something like solitons)? I don't understand why most physicists are focusing on more-complicated 12D string theories.

(To simplify even further, it would be nice if $m=0$, but I believe Einstein and others sought such an "electromagnetic-only theory for the electron" without success, even after adding two more dimensions! If there's no general argument for my question above, I wonder if there is a specific argument for why nature can't have the $m$=0 form.)

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The idea that nature is described by a nonlinear system of equations was the idea that Einstein had in the 1920s, and motivated his search for a unified field theory. It doesn't work, and it's philosophically less worthwhile than current theories anyway, so even if it did work, it wouldn't be simpler than string theory, or as elegant.

The idea that you can describe what's going on with local equations is false, as is demonstrated conclusively by Bell's inequality violations. The Bell inequality tells you that you can send electrons to far-away locations with spins that can be measured in 3 directions, A,B,C. The spin of the two electrons in each direction are 100% correlated (it's actually anti-correlated, but same difference for the argument), so if you measure the spin in direction A, and one electron is up the other is 100% certain to be up. Same for direction B and C, the two electrons always report the same spin in any of the three directions.

The spin in directions A and B are 99% correlated, meaning if you measure A on one of the electrons is up, then B on the other electron is up 99% of the time, and B is down 1% of the time. The spin in directions B and C are 99% correlated, so if you measure B is down on one electron, C is up on the other electron 1% of the time.

From the 100% correlation of the electrons, you conclude that the nonlocal field state (hidden variable) on one electron has the property that

A and B are 99% the same, 1% different B and C are 99% the same, 1% different

From this you deduce that

A and C must be at least 98% the same

meaning that whatever field configuration is happening to make A, the field configuration for C can only give different results 2% of the time, the sum of those times when it gives different answers than B plus the times B gives different answers than A.

This bound is called Bell's inequality, and it is violated by quantum mechanics. A and C are different 96% of the time.

This means any type of local-in-space description, linear, nonlinear, complicated, simple, whatever, will never ever work to describe nature. Your description is either nonlocal in the sense of faster-than-light communication, or nonlocal in the sense of having a global notion of state which is entangled nonlocally by measurements. This is why nobody looks for nonlinear field equations to describe nature anymore. It can't possibly work.

But the main ideas of Einstein's nonlinear field theories have survived to inspire developments in later physics.

  • The pions are excitations of a sigma-model, which is a type of nonlinear field theory. They are small oscillations of the quark condensate in the vacuum.
  • The proton can be thought of as the topological soliton of the sigma model. In quantum mechanics, it can still be a fermion even though the sigma model has no fundamental fermionic variables.
  • The field equations of 11-dimensional supergravity, which are a central part of string theory, generalize General Relativity in pretty much the only nontrivial ways known--- they give the biggest extension of spacetime symmetry possible, and they include a new field, constrained by the supersymmetry.

So these ideas are not a dead end, but they cannot work without quantum mechanics by themselves. If you want to understand quantum behavior emerging from some sort of nonlinear dynamics underneath, this dynamics can't be local.

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Good point! But, let me be clear that I was envisioning a quantum interpretation of $f$ (please change $f$ to $\psi$ if it changes your interpretation). To handle Bell's 2-particle spin correlations, I think I could use a “four-number vector” for $f$ (instead of a “two-number vector”, as suggested in my original question but really “unspecified”, so I consider the question still open). –  bobuhito Jul 26 '12 at 6:11
    
In general, N-particle spin correlation seems to need a "$2^N$-number vector" which, I admit, does seem like a bad kluge...but, doesn’t string theory similarly blow up with multiple particles (e.g., one particle uses 12D, but two particles use 24D)? –  bobuhito Jul 26 '12 at 6:11
    
@bobuhito: Bell's inequality closes the debate on local realism, there is no local realism. All QM theories have this exponential blowup when you simulate them, in string theory it's a little different because the wavefunctions are defined asymptotically, so it's not 10d 20d 30d wavefunctions, but more like 9d 18d to describe the "in" state of scattering. Anyway, this is nature's blowup, and unless we find that highly entangled quantum systems don't exist in nature, so that quantum computers fail at a few hundred or few thousand qubits, it is not reasonable to give non-quantum descriptions. –  Ron Maimon Jul 27 '12 at 2:28
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The theory you've written down is linear in $f$. If $f$ and $g$ are solutions, then so is $f+g$. This means that your field theory has no interactions between the different modes of $f$.

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it's non-linear - please read everything –  bobuhito Jul 26 '12 at 3:59
    
My mistake. A new complaint, however: the equation fails to be Lorentz-invariant if $v$ depends on $f$. You're back to free theory if you want $m=0$ + Lorentz invariance. –  user1504 Jul 26 '12 at 4:11
    
It's possible that the speed of light is different in a high-field background...we've probably only tested the weak-field limit. –  bobuhito Jul 26 '12 at 5:35
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