Summary: Is Quantum Mechanics An Island In Theoryspace?
University of California, Berkeley CA USA
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.
"It is striking that it has so far not been possible to find a logically consistent theory that is close
to quantum mechanics, other than quantum mechanics itself." --Steven Weinberg, Dreams of a
Final Theory 13
The title of this paper should be self-explanatory, but if not: "theoryspace" is the space of logically
conceivable physical theories, with two theories close to each other if they differ in few respects. An
"island" in theoryspace is a natural and interesting theory, whose neighbors are all somehow perverse or