Recent content by ismaili

Dear guys, I'm also interested in the definition of spin in various dimensions. For the usual 4D QFT, the definition of spin comes from the other Casimir operator W_\mu \equiv \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}P^\nu J^{\rho\sigma} However, in higher dimensions, we can have defined a...

aha! How stupid I was! Just let U be Y . Thank you a lot. :shy:

I am beginning to read about the topology, I met a problem puzzled me for a while. If Y is a topological space, and X\subset Y, we can make the set X to be a topological space by defining the open set for it as U\cap X, where U is an open set of Y. I would like to show that this indeed...

Thank you very much. This looks reasonable, so it is valid generically because it's from Lorentz transformation property. But, this leads to a peculiar result: I am calculating the self-energy diagram of photon in 2D QED, The divergent part of the integral is: \int...

When we are calculating the scattering amplitudes in QFT, we often encounter something like \int \frac{d^Dp}{(2\pi)^D} \frac{p^\mu p^\nu}{(p^2+\Delta)^n} and we often make the substitution for the numerator p^\mu p^\nu \rightarrow \frac{g^{\mu\nu}p^2}{D} It looks like...

Sorry, I forgot to say the F in the first post is the field strength of the the gauge field. Scalar field usually involves square of D? why scalar field has something to do with the number of covariant derivative D? I am reading about the instanton in gauge theory. Those two formulas are...

I don't get your first point. Even for rank-2 tensors, we have \partial_\mu(\sqrt{-g}F^{\mu\nu}) = \sqrt{-g}D_\mu F^{\mu\nu} , so that we can always use Stoke's theorem for covariant derivative in GR. I don't get the second point either, for that 2nd integral in my last post, we have...

I was worrying about this too. For GR, we have corresponding Stoke's theorem for covariant derivative, i.e. \int \sqrt{-g} d^4x ~D_\mu v^\mu = \int \sqrt{-g} d^3S_\mu ~ v^\mu because, we have \partial_\mu(\sqrt{-g}v^\mu) = \sqrt{-g}D_{\mu} v^\mu However, we have no corresponding...

Is the following formula correct? Suppose we work in a 4D Euclidean space for a certain gauge theory, \int d^4x~ \text{tr}\Big(D_i(\phi X_i )\Big) = \oint d^3S_i~ \text{tr}(\phi X_i) and, \int d^4x~\partial_j \text{tr}(\phi F_{mn}\epsilon_{mnij}) = \oint d^2S_j~ \text{tr}(\phi...

I was told that the existence of higher spin fields whose spin is higher than 2 is forbidden by "equivalence principle" of GR(general relativity). But after considering about it, I can't understand why equivalence principle could imply the nonexistence of higher spin fields (>2). Could...

I found another example. For 4D Euclidean, say, SU(n) gauge theory, if we look for finite action field configuration, the asymptotic form of gauge potential would be A(x) \sim g(x)\partial g^{-1} , i.e. in the pure gauge. In this way, for a spacetime point x at infinity, we have a...

I don't understand the link from soliton solution of QFT to the homotopy group. The argument goes like following: Consider the field configuration such that the action is finite, therefore we must require the field vanishes at spacetime infinity, hence, we defined a map from the...

Supposed we are given a set of SUSY transformation law, the way to get the BPS equation is by requiring that \delta \psi = 0 where \psi is a fermion field. Could somebody explain why this is the BPS equation? Thanks!

hmm... so you mean mathworld writes \partial_\mu A_\nu = \frac{\partial}{g^{\mu\alpha} \partial x_\alpha} A_\nu But mathworld's metric is lower-indexed, which is NOT upper-indexed, this is contradiction. Moreoever, I still don't get why their eq(61), i.e. D_\theta A_\theta has the...

Sorry, I don't get it. For example, their eq(61) should correspond to D_\theta A_\theta in 2D polar coordinate. However, for the partial derivative term of the covariant derivative, why is it \frac{1}{r} \frac{\partial A_\theta}{\partial \theta} ? I think the usual covariant...