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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
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
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
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
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.
May
11
comment $su(1,1) \cong su(2)$?
What exactly do you mean by $su(1,1)$? Is it the same as $so(1,1)$ by any chance?
Mar
21
comment Issues with the Operator to State map using Path Integral
I don't think one needs to invoke FQFT to answer my questions. It seems that FQFT is simply a formulation of QFT which uses the above properties as axioms in order to proceed. What I am looking for is why the above properties should hold, since in traditional QFTs, they are properties, not axioms.
Nov
22
comment Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?
Alright, so limits and integrals don't commute in general. But then I guess the OP's asking how does one decide what value to assign to the integral above, since there are many different non-equivalent choices, as he has demonstrated.
Nov
22
comment Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?
@DavidZ I think Peter's point is that both integrands are equal to $e^{ipx}$ when you let $\varepsilon \rightarrow 0$.
Sep
9
comment Does one really need classical physics in order to understand quantum physics?
You don't necessarily need to know whats usually covered in "freshman physics". Instead, given your good math background, I would learn a few essential topics of classical mechanics mainly Lagrangians and Hamiltonians, from a sophisticated textbook. Landau and Lifschitz or Goldstein might be decent as they are commonly used in physics, though I haven't looked at either in a lot of detail. Another very concise but good book which covers classical mechanics, QM and QFT from a mathematician's perspective is by Dimock, called Quantum Mechanics and QFT, a mathematical primer.
Jan
30
comment Curvature of Spacetime
Also, relativity can be especially confusing and misleading without the math. Things like the twin paradox are essentially the result of the confusion caused by muddling conceptual phrases like "a moving clock works slower" and the precise mathematical equations. There is no paradox if you formulate the problem and subsequently solve it using just the precise mathematics.
Jan
28
comment Interpretation of the 1D transverve field Ising model vacuum state in a spin-language
... and if you get to that, do post it here as an answer, as I'd also be curious to see what it looks like. If you have any questions about this particular process, don't hesitate to ask.
Jan
28
comment Interpretation of the 1D transverve field Ising model vacuum state in a spin-language
Also note that since your Hilbert space $2^N$ dimensional, a general state could have just as many terms. So even for small $N$, your ground state could look very terrible and have lots of terms. I say this since it's often a misconception when doing many spins that a state is decomposable or has a simple form where some spins are up and others are down. If you really are bent on it, I'd try working it out explicitly for some low dimensional cases like $N = 2$, $3$ etc, in which case you could just write out a matrix for the Hamiltonian and use a computer to find your ground state....
Jan
20
comment Proving Lorentz invariance of Maxwell equations
Well firstly, it would be difficult to teach the most sophisticated version to people who are just starting out their undergraduate careers in Physics. Seeing how the electric and magnetic fields behave physically is also much clearer if you write them in terms of vector calculus notation.
Jan
20
comment Proving Lorentz invariance of Maxwell equations
Yes. You can write Maxwell's equations in terms of differential forms as well. This equation is simply $dF = 0$ in that notation.
Jan
20
comment Proving Lorentz invariance of Maxwell equations
Webb, the equation of the OP isn't derived from that Lagrangian however. That's the Bianchi identity and holds as soon as you say that your field strength $F$ is derived from a potential. $\mathcal{L}_{EM}$ gives you the other equation $\partial_{\mu}F^{\mu \nu} = 0$, if I'm not mistaken.