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Say we have a state vector $|A\rangle$. Is it possible to have a theory where the evolution of $|A\rangle$ depends on the vector $|A\rangle$ itself? e.g.

$$ i\frac{\partial}{\partial t} \psi(t) = \hat{F}(\psi(t)) \psi(t) $$

On reason I was thinking about this was the idea that space-time is related to entanglement. But entanglement means knowledge of the state-vector. So if this were true, the state-vector would have to affect itself.

In other words is non-linear quantum mechanics possible?

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    $\begingroup$ Yes. See here en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation. $\endgroup$ – eranreches Nov 27 '18 at 14:31
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    $\begingroup$ "On reason I was thinking about this was the idea that space-time is related to entanglement. But entanglement means knowledge of the state-vector. So if this were true, the state-vector would have to affect itself." I'm not sure that this makes sense. Can you either elaborate on this reasoning or remove it from the question altogether? $\endgroup$ – probably_someone Nov 27 '18 at 14:34
  • $\begingroup$ @eranraches How do we reconcile that with the idea that quantum mechanics is based on linear operators? $\endgroup$ – zooby Nov 27 '18 at 14:38
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    $\begingroup$ Entanglement means that the state of a composite system cannot be meaningfully decomposed into the states of the fundamental systems that make it up - i.e. $\Psi(\vec r_1,\vec r_2)$ cannot be separated into $\psi_1(\vec r_1)\otimes \psi_2(\vec r_2)$. This has nothing to do with the linearity of operators. $\endgroup$ – J. Murray Nov 27 '18 at 14:43
  • $\begingroup$ @Murray That is true. But if space-time is determined by entanglement (as proposed by Susskind et al.). Then the movement of wave functions on space time is determined by the wave-functions themselves. Hence non-linear. $\endgroup$ – zooby Nov 27 '18 at 14:45
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The superposition principle, which is basically equivalent to a linear evolution equation of the quantum state, is very fundamental for quantum mechanics.

As eranreches mentioned in his comment, however, effective equations for many particles are often non-linear, e.g. the Gross-Pitaevskii equation or the Hartree-Fock equation. They feature an interaction term of the form $V * |\psi|^2$, where $*$ denotes the convolution. But there, the basic superposition principle is not changed and only the effective descriptions yields the non-linearity.

There is also a formulation of quantum mechanics in which the collapse is effectively modelled in the Schrödinger equation itself. This leads to a nonlinear, stochastic equation. These models are called collapse models. The first one was by GRW (Ghirardi, Rimini, Weber, see here) in and there are newer ones under the keyword CSL (continuous spontaneous localization). If such a theory is true - and they are not fully falsified yet - the superposition principle is violated a tiny little bit.

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  • $\begingroup$ Hi, I don't quite understand what you mean when you say the basic superposition principle remains unchanged? $\endgroup$ – zooby Nov 29 '18 at 19:56
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Is non-linear quantum mechanics possible?

No ─ quantum mechanics is at its very heart a linear theory; that's the core of what some like to call "the wave nature of matter" and what really distinguishes it from classical mechanics.

Of course, it's possible that there's a deeper theory that underlies QM which includes that type of nonlinearity and which only reduces to QM in some suitable limit (like relativity reduces to newtonian mechanics) and that we've yet to find the way out of that limit. We know that that could have very striking consequences and it's not impossible ─ but if it's true, we wouldn't call that theory QM.

All of that said, though:

Is a non-linear quantum-mechanics-like formalism possible?

Yes, absolutely. The simplest such version is the so-called non-linear Schrödinger equation, and its close cousin the Gross-Pitaevskii equation, $$ i\hbar \frac{\partial}{\partial t} \psi(\mathbf r,t) = \left[ -\frac12 \nabla^2 + V(\mathbf r) + g |\psi(\mathbf r,t)|^2 \right] \psi(\mathbf r,t). $$ This sees a lot of use in approximate methods in quantum mechanics, primarily in the study of Bose-Einstein condensates. However, it's also very useful in describing nonlinear phenomena in waves, such as e.g. intense laser beams in fibers, water waves in canals, and so on.

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