My contention in this chapter is the following: Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call "probabilities" can be negative numbers. As such, the theory could have been invented by mathematicians in the nineteenth century without any input from experiment. It wasn't, but it could have been. (查看原文)
Let's say we want to invent a theory that's not based on the 1-norm-like classical probability theory, or on the 2-norm like quantum mechanics, but instead on the p-norm for some \( p \notin \{1,2\} \). Call \((v_1, ... v_N)\) a unit vector in the \(p\)-norm if \[|v_1|^p + ... + |v_N|^p = 1.\] Then, we'll need some "nice" set of linear transformations that map any unit vector in the \(p\)-norm to another unit vector in the \(p\)-norm.
It’s clear that for any \(p\) we choose, there will be some linear transformations that preserve the \(p\)-norm. Which ones? Well, we can permute the basis elements, shuffle them around. That’ll preserve the \(p\)-norm. And we stick in minus signs if we want. That’ll preserve the \(p\)-norm too. But here’s the little observation I made: if there are any line... (查看原文)
Now, my point of view is a bit different: computer science is what medites between the physical world and the Platonic world. With that in mind, "computer science" is a bit of a misnomer; maybe it should be called "quantittative epistemology." It's sort of the study of the capacity of finite beings such as us to learn mathematical truths. I hope I've been showing you some of that. (查看原文)
Student: What do you think the chances are that three Indian mathematicians will come up with an elementary proof?
Scott: I think it's gonna take at least four Indian mathematicians! We know today that if you prove good enough circuit lower bounds, then you can prove P=BPP... (查看原文)
