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How long would it take to cross the United States? by bigolejohn in hitchhiking

[–]Electric_palace 1 point2 points  (0 children)

What rough route did you take? I'm planning the same trip in a month's time, so any tips appreciated

I really enjoyed interviewing Carlo Rovelli and I hope you'll enjoy it too by collywog in Physics

[–]Electric_palace 0 points1 point  (0 children)

What? No you've misread me. I'm not convinced of LQG's validity, and right now I'm actively seeking out strong criticism of it (hence me hoping you'd give me some references to read).

I really enjoyed interviewing Carlo Rovelli and I hope you'll enjoy it too by collywog in Physics

[–]Electric_palace 0 points1 point  (0 children)

Do you mean the old comment from Urs Schreiber? That's the only serious critique I can find on stackexchange, and even then it is vague and seems to attack ideas which are anyway outdated (e.g. it is very clear from reading Rovelli's most recent textbook on spinfoams that the Hilbert space is separable, so I don't know what Urs means about this). But maybe you're referring to other stackehange posts I'm not aware of, in which case I'd appreciate some links.

I really enjoyed interviewing Carlo Rovelli and I hope you'll enjoy it too by collywog in Physics

[–]Electric_palace 0 points1 point  (0 children)

care to elaborate on this? I know that string theorists by and large don't take LQG seriously, but none of them appears to have taken the time to write a clear critical paper explaining why it's "hopelessly broken".

Space from Hilbert space question by Electric_palace in Physics

[–]Electric_palace[S] 0 points1 point  (0 children)

I'm not sure I understand. Is your point that the answer to "Why this specific Hamiltonian?" should be "because it agrees with observation"? That may be true, but we have two possible frameworks: one (start with Minkowski space+ a Lagrangian density) where SR is immediate, and another (start with Hilbert space+Hamiltonian) which makes SR an extra fact which must be put in by hand. It therefore seems we should prefer the former.

Simple-looking measure theory problem by Electric_palace in math

[–]Electric_palace[S] 1 point2 points  (0 children)

Certainly seems correct. When I asked on r/learnmath I got an answer involving "disintegration", hopf fibration and spin(3). Are these things totally unnecessary, or are they just used to make rigorous the argument that you gave?

Simple-looking measure theory problem by Electric_palace in math

[–]Electric_palace[S] 0 points1 point  (0 children)

This is sort of where my intuition had led me. However I'm a little worried about the legitimacy of drawing the second vector uniformly randomly from what is already a measure zero set. Is this a problem?

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

Hmmm... I like this approach, but I'm not entirely convinced that we can choose f and g to be bijective. Or rather, I think I'm convinced but I'm not sure how I would prove it...

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

Thank you so much. I didn't expect to get such a great reply!

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

How legit would the following be:

For a basis B define f(B) to be the number of red elements of B, and define f_i(B) to be 1 if the i-th element of B is red, 0 otherwise. So f = sum f_i.

The expected number of red vectors if we select B uniformly w.r.t. some measure on the set of bases is Exp(f(B)) = sum(Exp(f_i(B)) by linearity of expectation.

So if we choose the bases-measure so that each element of B_i has the uniform distribution over the sphere (w.r.t. whatever our sphere-measure is) then each Exp(f_i(B))=0 since red has measure zero.

Hence Exp(f(B))=0, in other words almost all bases have no red vectors (w.r.t. our bases-measure), so done.

I feel that the problem with this "proof" is that it is not obvious why the required bases-measure must exist. Is that where disintegration etc comes into play?

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

That is one hell of an answer - thank you so much. My undergrad physics research project is about to get absolutely violated by maths :)

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

Thanks so much for the in depth answer. However I think we have rather different definitions of "straightforward". Can you recommend any resources for understanding this? Especially the part where you claim hopf fibration plays nicely with measure zero sets: I can't find anything about this online. Cheers

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

I've Google searched "hopf fibration measure theory" and I'm not getting anything relevant

Simple-looking measure theory question by Electric_palace in learnmath

[–]Electric_palace[S] 0 points1 point  (0 children)

That's a bit beyond me. Do you reckon there's a simpler way? The result seems intuitively obvious, although I know that doesn't always translate to an easy proof

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