Shame on you all my physicist friends who read my last post and didn’t reply! I was expected to have at least a classic answer (which is really easy to provide one) and then go through it to generalise to quantum field theory and have a lot of more fun!  However, having no comments for that post after two weeks means the visitors are not interested in the Sina’s Award! Anyway, I’m not gonna give up! This post will, probably, be the most challenging one I’ve ever published on science. However, due to the insignificant number of visitors, I’m publishing this entry only for my own future reference. Therefore, if it is read by anyone except me myself, you might find it a little bit unclear, since I’m not planning to explain every detail for myself!

Roger Penrose, in his book, The Road to Reality, has a chapter under the title “Gravity’s role in quantum state reduction”. He argues why he believes that the quantum mechanics we already have is just an approximation of the real behaviour of the nature. He brings a lot of interesting arguments to show that there are some clues to think that a more precise quantum mechanics should be non-linear! I absolutely recommend anyone to go through this chapter.

On the other hand, relatively long time ago, I’d just told you about an awesome paper by Abrams and Lloyd which generally speaking says if quantum mechanics was non-linear then we could, in principle, have computers which are capable of solving NP-complete problems in polynomial time!!!

For the non-computer scientists who read this post, if there are any, I should emphasise that a world in which NP-complete problems are solvable polynomially would be a ridiculously boring place! In fact there wouldn’t be any room for human being! “Artificial intelligence software would be perfect since they could easily do exhaustive searches in a large tree of possibilities.” “Whenever a scientist has some experimental data, the computers would be able to automatically obtain the simplest theory (under any reasonable measure of simplicity we choose) that best explains these measurements. ” Therefore, all the scientists could be replaced by computers. “There would be no privacy in the digital domain. Any encryption scheme would have a trivial decoding algorithm. There would be no digital cash and no SSL, RSA, or PGP.”† I believe you can even find more ridiculous consequences of being in a world in which there exists a creature that we have absolutely no significance over it!

Although this is very sad, but it could be another clue to think quantum mechanics should be non-linear! Think of the problem of natural computation. I think all I want to say is in the title of my weblog! Just “let the nature calculate for you”! Imagine solving a dynamics problem in classical scale. Say calculating the maximum height of an apple thrown in (+ve) y direction!  One way is just writing down the equations and solve them to find the maximum height and the other is to actually throw an apply and see how high it goes. The second is an example of natural computation. You use the nature to find a value for you. Feynman in his famous paper (1982) questioned that we cannot (efficiently) simulate the quantum world on a classical computer. Because nature behaves quantumly in its roots. I believe the same argument (and its reversed version) is hold for the ultimate theory of everything! If the NP-complete problems are solvable polynomially using a ‘non-linear quantum mechanical computer’, maybe it is because nature behaves non-linearly in the quantum scales! As I mentioned before, there are even more non-computer science clues to think about a non-linear quantum mechanics seriously! With a non-linear quantum mechanics even general theory of relativity would be compatible with this new paradigm! Therefore, maybe this time we can really have a theory of everything! And as I mentioned before it would be the end of world for humans!

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† One of the best texts on complexity theory could be found here. I’ve copied a few sentences from that.