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- Scott Aaronson, Is Quantum Mechanics an Island in Theoryspace?This paper investigates what happens if we change quantum mechanics in several ways. The main results are as follows. First, if we replace the 2-norm by some other p-norm, then there are no nontrivial norm-preserving linear maps. Second, if we relax the demand that norm be preserved, we end up with a theory that allows rapid solution of hard computational problems known as PP-complete problems (as well as superluminal signalling). And third, if we restrict amplitudes to be real, we run into a difficulty much simpler than the usual one based on parameter-counting of mixed states.
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The "usual story" regarding molecular chemistry is that it is roughly an application of quantum mechanics. That is to say, quantum mechanics supplies everything necessary and sufficient, both ontologically and epistemologically, to reduce molecular chemistry to quantum mechanics. This is a reductive story, to be sure, but a key explanatory element of molecular chemistry, namely molecular structure, is absent from the quantum realm. On the other hand, typical characterizations of emergence, such as the unpredictability or inexplicability of molecular structure based on quantum mechanics, do not characterize the relationship between molecular chemistry and quantum mechanics well either. A different scheme for characterizing reduction and emergence is proposed that accommodates the relationship between quantum mechanics and molecular chemistry and some initial objections to the scheme are considered.
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