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seen Dec 16 at 16:28

A mathematician, a programmer, etc. etc.


Mar
3
awarded  Revival
Mar
3
answered Superspace as the Hilbert Space for Quantum Gravity
Mar
3
comment Superspace as the Hilbert Space for Quantum Gravity
Well, no, you never do that in -- say -- using geometric quantization for the canonical formalism for electromagnetism (or, a simpler example, a scalar field). You don't do it when geometrically quantizing a mechanical system either. The symplectic manifold $\mathcal{M}$ is a necessary ingredient for constructing the Hilbert space (or Fock space for field theories), but not sufficient to qualify as the Hilbert space (resp. Fock space) for the full quantum theory. NB: you can use phase space path integrals, but this is an irrelevant fun fact rather than pertinent information.
Mar
3
comment Superspace as the Hilbert Space for Quantum Gravity
You might want to read Woodhouse's Geometric Quantization. You don't turn $\mathcal{Riem}$ into a Hilbert space: you use it (well, technically $\mathcal{Riem}/\mathcal{Diff}$) as the configuration space, then construct an infinite-dimensional symplectic manifold $\mathcal{M}$ which you use as the underlying manifold for a complex line bundle. The space of sections on this line bundle then in (insert magical full quantization step here) and you've got a Fock space.
Feb
20
revised Identity $\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$
Modified title, updated some TeX
Feb
20
suggested approved edit on Identity $\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$
Feb
20
comment How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?
Hint: consider $g(x)=xF(x)$ and Differentiate under the Integral Sign
Feb
20
comment Simple proof exercise recommendation, with full answers
And play with the first equation. The $\sqrt{ab}$ strongly hints at squaring both sides of the equation, just to see what happens...
Feb
20
comment Simple proof exercise recommendation, with full answers
"It's 2am, and I've spent more time than I really want to think about on an exercise which was clearly not meant to cause someone this much distress." You might be pushing too hard. Instead of forcing yourself to think about the problem, try taking a walk. I've noticed it help considerably, and many mathematicians recommend it (e.g., Alain Connes).
Feb
14
comment Symbolic math engines barf on this ostensibly tractable integral.
The trick is to expand this out using $2\cos(x)=\exp(ix)+\exp(-ix)$, consolidate the terms as $\int\exp(Ct)\,\mathrm{d}t$ then execute it.
Feb
13
comment integral of 1/(sqrt(e^x)) from 0 to infinity(Improper integral)
For 3, consider $\int (1+e^{ix})e^{-x}dx$ since $\exp(ix)=\cos(x)+i\sin(x)$ we take the imaginary part of the resulting integral...
Feb
12
comment Symbolic math engines barf on this ostensibly tractable integral.
What is the variable you're integrating? (There's no $dx$ factor in the integrand)
Feb
12
comment Vector Delta Function Identity
These are all simple roots, too, so none of them would have a multiplicity greater than 1...
Feb
12
comment Applications of Operator Algebras to modern physics
@Jeff Since you're at Caltech (I gather), go find Matilde Marcolli and speak with her. She's an expert on Noncommutative geometry, which basically amounts to applying operator algebras to...everything...
Feb
4
comment Is computer science a branch of mathematics?
@CarlMummert: true, but I figured everyone else would give rigorous books related to the formal field "the theory of computation", like the ones you noted. So, I wanted to just note there's a programming book that uses the axiomatic method :)
Feb
2
comment How must I understand concepts equations of physics?
It might be wise to study (a) differential geometry, and (b) partial differential equations. Differential geometry for the obvious reasons. But (b) because classical field theory, for the most part, boils down to PDEs [or, depending on your outlook, some "abstract nonsense" involving sections of fiber bundles ;)].
Jan
29
comment If 1 $\leq x$, then $\sqrt{x} \leq x $
Just a style thing, I would begin the proof writing "Proof. Assume for contradiction $x<\sqrt{x}$. Square both sides of $x\geq1$ gives us $x^{2}\geq x$. But squaring both sides of our assumption gives us $x^{2}<x$ which is a contradiction. We reject our assumption, and conclude $\sqrt{x}\leq x$." Just to make crystal clear which assumption we are trying to contradict. But I'm a windbag, so...
Jan
24
answered Is computer science a branch of mathematics?
Jan
24
comment Does $\frac12+\frac14+\frac18+\dots$ equal $1$?
Wait, there's a typo in your answer: $\sum^{n}_{k=1}2^{-k}=1-2^{-n-1}$ (you wrote the RHS as $1-2^{-k-1}$).
Jan
24
comment Metric space not a vector space
@SanathDevalapurkar: well, yes and no. Yes, because tangential thought is always helpful in speculative reasoning. Ostensibly, if you were interested in this direction, you would then ask about inverses and automorphisms, then about the diffeomorphism group's structure. That's a good approach, since the "distinct" Riemannian metrics would be $\mathbf{Riem}/\mathbf{Diff}$ (the space of all Riemann metrics modulo the diffeomorphism group). But it won't help coerce $\mathbf{Riem}$ into a vector space :(