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One of the fundamental postulates of Quantum Mechanics is that states of a system are linear combinations of ground/observable states. Could someone point out some of the experiments leading to such a model? Is there a more general theory that deals with states that are nonlinear functions of the observable states with Quantum Mechanics being its first order approximation?

Qmechanic
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darko
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2 Answers2

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First of all, let me point out this post: Is the universe linear? If so, why? where Ron Maimon explains why it is difficult to imagine quantum mechanics as a nonlinear theory.

Now, for experiments. First let me point out that this is difficult. How would you directly test whether quantum mechanics is linear? You can construct linear superpositions (which has been done), but that doesn't tell you that all states are linear superpositions. As long as your experiment confirms quantum mechanics, you can argue that it also confirms linearity, as it is so very fundamental.

The easiest way to test linearity is to have alternative theories to test against. Such theories are very difficult to construct as pointed out above (and because they often lead to inconsistencies).

Somewhat prominent theories include the one by Bialynicki-Birula and Mycielski and the one by Weinberg. Both theories can be attacked from theoretical grounds but have also been attacked on an experimental basis: The first one was tested (for instance) in the experiments reported here: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.44.765 The second one was tested (for instance) in the experiments here: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.63.1031 There are also more indirect tests. Polchinski pointed out that Weinberg's theory would lead to the possibility of superluminal communication. Every test trying to find superluminal communication could therefore (if testing the Polchinski scenario) test Weinberg's theory.

Finally, let me point out the article here: https://arxiv.org/pdf/1002.4673.pdf

Martin
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  • Just to add: not all superpositions seem to be possible, the superselection rules have to be observed. – lalala Apr 15 '17 at 15:33
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Not exactly a full answer, but some thoughts (I really want an answer to this question too...)

This paper https://arxiv.org/pdf/2302.13421.pdf makes the distinction between "kinematical linearity" about how states can "add" to each other, and can be broken down into "additive" linear combinations of basis states; and "dynamical linearity" which is about how states evolve over time seemingly by the action of a unitary linear operator. The ArXiv paper tries to argue that "if quantum states are epistemic...the dynamics must be linear". Perhaps see also "Epistemic and Ontic Quantum States " and " States vs. Changes of States: A Reformulation of the Ontic vs. Epistemic Distinction in Quantum Mechanics ".

Another justification of dynamical linearity appears in "Why quantum dynamics is linear" Thomas F Jordan 2009, which claims that

Can we prove that the change of density matrices is linear? Yes, if we assume that the system can coexist with another without interaction.

This Reddit thread https://www.reddit.com/r/math/comments/9n0dxc/why_is_linearity_so_important_in_quantum_mechanics/ discusses both linearities. Regarding dynamical linearity someone says

If there are non-linear operators in quantum mechanics (as suggested by Weinberg), then this allows for real faster-than-light communication with an EPR pair (as pointed out by e.g. Polchinski).

So dynamical linearity seems to have some physically/philosophically meaningful substance behind it.

Regarding kinematical linearity someone said

QM was designed to be linear to accommodate the superposition principle of quantum states, which was observed in the lab. For example diffraction patterns of electrons.

Indeed this "principle of superposition" (that we can "linearly add" physically meaningful things to get other physically meaningful things) is not just a QM "axiom", see: Is there any fundamental reason why acceleration is a linear function of external forces?, Why is the Principle of Superposition true in EM? Does it hold more generally?.

D.R
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