Register to reply

Self-Dual Field Strength in complex coordinates

Share this thread:
earth2
#1
Apr25-12, 05:27 AM
P: 86
Hi guys,

I have to brush up my knowledge about self-dual Yang Mills and I'm reading an ancient paper by Yang about it...and of course I'm stuck...although Yang writes 'it is easy to see that'...

Ok, so the self-duality condition of the YM field strength tensor is defined as

[tex] 2F_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}[/tex].

If I know go to complex coords defined by

[tex]\sqrt{2}y=x_1+i x_2 \quad \sqrt{2}\bar{y}=x_1-i x_2[/tex]
and
[tex]\sqrt{2}z=x_3+i x_3 \quad \sqrt{2}\bar{z}=x_3-i x_4[/tex]

the metric transforms to
[tex]g_{y\bar{y}}=g_{\bar{y}{y}}=g_{z\bar{z}}=g_{\bar{z}{z}}=1[/tex]. So far i've understood everything. But then Yang says it's easy to see that the self-duality condition becomes

[tex]F_{yz}=0=F_{\bar{y}\bar{z}}[/tex]
[tex]F_{y\bar{y}}=F_{z\bar{z}}[/tex]

The question know is: how do i see the last two equations? Does the epsilon tensor somehow transform if i go to these complex coords?

Cheers,
earth2
Phys.Org News Partner Physics news on Phys.org
Engineers develop new sensor to detect tiny individual nanoparticles
Tiny particles have big potential in debate over nuclear proliferation
Ray tracing and beyond
Bill_K
#2
Apr25-12, 10:30 AM
Sci Advisor
Thanks
Bill_K's Avatar
P: 4,160
The self-duality condition says F12 = - F34, F13 = F24, F14 = - F23. So for example (ignoring √2's)

Fyz = F13 -i F23 -i F14 - F24 ≡ 0
earth2
#3
Apr25-12, 12:24 PM
P: 86
Ah cool, i didn't know that i could just plug in numbers back :) Nice, thank you!


Register to reply

Related Discussions
Calculating the field strength of an alternating magnetic field Engineering, Comp Sci, & Technology Homework 1
Magnetic Field Equation in Spherical Coordinates to Cartesian Coordinates Advanced Physics Homework 26
Calculating magnetic field strength (complex) General Physics 6
Dual field strength equals to itself Classical Physics 0
Parallel Plate Capacitor: Electric field strength, flux & magnetic field Introductory Physics Homework 0