This recreational 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 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.Comment: 9 pages, minor correction
