User dan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:31:04Z http://mathoverflow.net/feeds/user/1790 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7357/photon-propagator photon propagator Dan 2009-12-01T03:30:18Z 2012-10-01T12:16:10Z <p>I am reading Zee's book "QFT in a nutshell". I have a question on the photon propagator computation. For a massive photon, consider the Lagrangian $L = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2}m^2A_\mu A^\mu + A_\mu J^\mu$, then the path integral is $Z = \int dx ~L = \int dx ~{ \frac{1}{2}A_\mu[(\partial^2 + m^2)g^{\mu \nu} - \partial^\mu \partial^\nu]A_\nu + A_\mu J^\mu }$. From this we get that the photon propagator $D_{\mu \nu}$ satisfies $[(\partial^2 + m^2)g^{\mu \nu} -\partial^\mu \partial^\nu ] D_{\nu \lambda}(x) = \delta^\mu_\lambda \delta^{(4)}(x)$, and solving this, $$D_{\nu \lambda}(k) = \frac{-g_{\nu \lambda} + k_\nu k_\lambda/m^2}{k^2 - m^2}.$$ </p> <p>I can not see why the numerator has a term $k_\nu k_\lambda/m^2$. Any ideas?</p> http://mathoverflow.net/questions/40085/greens-function-in-dimension-two Green's function in dimension two Dan 2010-09-27T05:44:01Z 2012-03-14T13:20:30Z <p>Any hint to compute the Green's function:</p> <p>If $\Delta_z G(z,z') = 2\pi \delta^2(z-z')$, then</p> <p>$$G(z,z')=-2\pi \int \frac{d^2q}{4\pi^2}\frac{e^{iq(z-z')}}{q^2} = ln|\mu(z-z')|$$ where $\mu$ is some infrared cutoff at $q=0$.</p> <p>I can see the first step is Fourier transform and inverse Fourier transform but I don't know how to figure out the second step. Thank you.</p> http://mathoverflow.net/questions/42582/penrose-tilings-and-noncommutative-geometry/43008#43008 Answer by Dan for Penrose tilings and noncommutative geometry Dan 2010-10-21T04:53:02Z 2010-10-21T04:53:02Z <p>Chp 9 in "A WALK IN THE NONCOMMUTATIVE GARDEN" by ALAIN CONNES AND MATILDE MARCOLLI, and you can find the references there.</p> http://mathoverflow.net/questions/38651/compute-the-kahler-moduli-of-an-elliptic-curve compute the Kähler moduli of an elliptic curve Dan 2010-09-14T05:42:43Z 2010-09-14T18:43:13Z <p>Say given elliptic curve ${ (x,y) | y^2 = (x^2-1)(x^2-k^2) }$, what is the right form of the K$\ddot{a}$hler form and how to compute the K$\ddot{a}$hler moduli of this elliptic curve? Thank you. </p> http://mathoverflow.net/questions/37191/integration-involving-the-complete-elliptic-integral-of-the-first-kind-kk Integration involving the complete elliptic integral of the first kind K(k)? Dan 2010-08-30T20:36:11Z 2010-08-31T05:55:28Z <p>Is there any reference showing how to do definite integrals involving the complete elliptic integral of the first kind K(k)? Something like </p> <ol> <li>$\int_0^1 K(k) dk$</li> <li>$\int_0^1 k^nK(k) dk$ </li> <li>$\int_0^1 \frac{K(k)}{1+k} dk$</li> </ol> <p>etc...Thanks a lot.</p> http://mathoverflow.net/questions/38651/compute-the-kahler-moduli-of-an-elliptic-curve/38723#38723 Comment by Dan Dan 2010-09-16T21:57:48Z 2010-09-16T21:57:48Z Thanks a lot. It is very nice of you. http://mathoverflow.net/questions/38651/compute-the-kahler-moduli-of-an-elliptic-curve/38723#38723 Comment by Dan Dan 2010-09-16T01:35:33Z 2010-09-16T01:35:33Z Thank you for your comments. Let me put the question in another way, how can one verify some phenomenon is mirror symmetry of dimension one? I am reading &quot;motives from diffraction&quot; by Jan Stienstra <a href="http://arxiv.org/abs/math/0511485" rel="nofollow">arxiv.org/abs/math/0511485</a> http://mathoverflow.net/questions/38651/compute-the-kahler-moduli-of-an-elliptic-curve/38723#38723 Comment by Dan Dan 2010-09-14T21:55:14Z 2010-09-14T21:55:14Z I see what you mean, but I am still confused with the Fubini-Study like Kahler form here. We can choose a volume form (Kahler form) up to scaling, so the volume (Kahler moduli) will be a multiple of some area A. On the other side, the complex structure is normalised as $\tau$ ( say our lattice is $Z+Z\tau$). By the mirror map, $\tau \arrow \rho= b + iA$, my question is how can I fix $b$ and $A$?