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1h
comment Question about scalar product of 2 four-vectors
Ah yes, that's the standard way of writing it. Glad it helped!
1h
revised Question about scalar product of 2 four-vectors
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5h
revised Question about scalar product of 2 four-vectors
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5h
revised Question about scalar product of 2 four-vectors
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7h
answered Question about scalar product of 2 four-vectors
1d
comment Elliptic genus; What is it within string/M-theory?
Chapter 10 discusses the relationship between the Euler characteristic and the Witten index (pages 206-211) and also the general properties of supersymmetric theories. The elliptic genus is a generalization of these results, so it's good (almost necessary) to have these concepts down first.
May
22
comment Fermionic path integral on the disk - Recovering the vacuum state
Thanks for your patience, Lubos.
May
22
comment Fermionic path integral on the disk - Recovering the vacuum state
I'm afraid I might be misunderstanding some of what you're saying. Why would integrating over the boundary configurations confirm our answer? And since the ground state, like you said, is independent of half of the Fourier modes wouldn't integration give zero? Are you suggesting that since an explicit path integral can't give the answer the answer in above form therefore, I should verify it by other means (integration?). To reiterate my confusion, what I'm looking for is a sketch as to how one would recover the expression of the ground state using the path integral. Is that possible to do so?
May
22
comment Fermionic path integral on the disk - Recovering the vacuum state
Dear @LuboŇ°Motl. Please see the edits.
May
22
revised Fermionic path integral on the disk - Recovering the vacuum state
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May
22
comment Fermionic path integral on the disk - Recovering the vacuum state
Dear Lubos, I'm not sure I follow. I have the wave function expression for the ground state which I got independently from the operator formalism, and which depends on infinitely many Grassmann variables. I'm now just trying to see whether the path integral gives me the same result. The classical action giving zero (I think) implies that the ground state doesn't depend on the boundary conditions, which isn't true once we look at the wave functional expression.
May
22
revised Fermionic path integral on the disk - Recovering the vacuum state
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May
21
comment Fermionic path integral on the disk - Recovering the vacuum state
Thanks Lubos. Yes the issues you point out all need to be addressed (they mainly come from using the action on the cylinder rather than the disk), however they are not what's troubling me. I can't see what to plug in for the $\psi$'s and integrate over. The classical solution with boundary conditions would be simply that the $\psi's$ are either left of right moving. But plugging this into the action gives me zero.
May
21
asked Fermionic path integral on the disk - Recovering the vacuum state
May
14
comment Why two different Lagrangians to derive geodesic equations?
Sorry for the late response, but I just mean that it follows from applying $\frac{dx^{\mu}}{d\lambda} = \frac{dx^{\mu}}{d\tau}\frac{d\tau}{d\lambda}$. If further clarification is required, please ask it as a separate question and I'll be happy to go into more detail there.
May
13
comment Why two different Lagrangians to derive geodesic equations?
One simply applies the chain rule for that, if I understand your confusion correctly.
May
13
comment Why two different Lagrangians to derive geodesic equations?
I'm not familiar with the source you're using, but the clearest (IMO) way to get the affine geodesic equations from the square root Lagrangian, is to first get the non-affine equations, and then switch your arbitrary parameter to an affine one. As far as having a source for the derivation, I don't know of any which derives the non-affine equation directly (although it's not difficult). Carroll in his notes starts with the square-root Lagrangian and then changes to an affine parameter in the middle of his derivation. Look at page 15 in: preposterousuniverse.com/grnotes/grnotes-three.pdf
May
12
answered Why two different Lagrangians to derive geodesic equations?
May
12
answered Elliptic genus; What is it within string/M-theory?
May
11
comment $su(1,1) \cong su(2)$?
@LuboŇ° Thanks for the clarification. It was the first time I had seen the $SU(n,m)$ notation being used.