But… Of course it's not the correct setting for QM. Before even talking about Turing computability and infinite set atheism (which rule out a continuous, infinite configuration space), configuration space is folded on itself around the identity axis.
Unless you think (a,b) is not the same configuration as (b,a), even though their amplitudes would add up before we have access to their square at the experimental level? Evidence towards "its the same configuration" looks quite overwhelming.
Or, could a "permutable" space, where (here with 2 dimensions) (x,y)=(y,x) for all x and y, be a Hilbert space as well?
Overall, I'm not sure what you're talking about. Can you be more explicit, or provide some links?
I don't know how you expect me to respond to this.
> Before even talking about Turing computability and infinite set atheism (which rule out a continuous, infinite configuration space), configuration space is folded on itself around the identity axis.
Infinite set atheism is basically Yudkowsky's reason for denying Hilbert space, so I don't know why we should talk before it.
Read the comments on "The Quantum Arena" -- Yudkowsky didn't even know whether the thing he was railing against as an uncountably infinite set was indeed infinite! (Presumably he has updated by now.)
> Unless you think (a,b) is not the same configuration as (b,a)
Well, it depends on the situation. I assume you're talking about the configuration space of the position of two indistinguishable particles, in which case of course I think they're the same configuration (that's what 'indistinguishable' means) and you're just beating down a straw man. If wavefunctions in general are members of a Hilbert space, then so are symmetric wavefunctions.
> Overall, I'm not sure what you're talking about. Can you be more explicit, or provide some links?
All of the wavefunctions for two particles described within are elements of L^2(R^2); the subset of physically realizable wavefunctions forms a subspace which is also a Hilbert space (answering your question about "permutable" spaces).
Well… I agree. But then again, I don't think he really was trying to teach it. The way I see it, he just lifted confusions you would have if you start to really learn QM.
Unless you think (a,b) is not the same configuration as (b,a), even though their amplitudes would add up before we have access to their square at the experimental level? Evidence towards "its the same configuration" looks quite overwhelming.
Or, could a "permutable" space, where (here with 2 dimensions) (x,y)=(y,x) for all x and y, be a Hilbert space as well?
Overall, I'm not sure what you're talking about. Can you be more explicit, or provide some links?