I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html, with notes at http://arxiv.org/pdf/1110.5013.pdf), and I'm now on to the spin 1/2 part of the course. We've gone through all the mechanics of constructing irreducible representations [itex]D^{(s1,s2)}[/itex] of the Lorentz group, and have begun to build Lagrangians out of [itex]D^{(1/2,0)}[/itex] and [itex]D^{(0,1/2)}[/itex], the right- and left-handed Weyl spinors.(adsbygoogle = window.adsbygoogle || []).push({});

One of the requirements of a Lagrangian density is that it transform like a scalar, so he turns to the task of finding objects that can be built out of spinors that transform like a scalar. He notes that if you have a spinor field [itex]u(x)[/itex], then you can construct a field which transforms like a vector by doing [itex]V^\mu(x) = u^*(x)\sigma^\mu u(x)[/itex], where [itex]\sigma^\mu = (1, \stackrel{\rightarrow}{\sigma})[/itex].

Seeing that, I would expect us to go on to construct a scalar field by taking [itex]\partial_\mu\cdot V^\mu(x) = \partial_\mu\cdot(u^*(x)\sigma^\mu u(x))[/itex], since taking the four-divergence is normally how one constructs a scalar field out of a vector. However, he instead constructs the object [itex]u^*(x)\sigma^\mu\cdot\partial_\mu u(x)[/itex], and uses that to build a Lagrangian, which when minimized produces the Weyl Equation. I'm having trouble understanding how this works--is that equivalent to putting the derivative outside like I did above, or is it something different? And if it's different, why would one want to do it that way, and how does one show that it transforms like a scalar?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Building a Lagrangian out of Weyl spinors

Loading...

Similar Threads - Building Lagrangian Weyl | Date |
---|---|

I QCD Lagrangian | Feb 28, 2018 |

A Building on QP from 5 reasonable axioms | Dec 7, 2017 |

I Build Your Own Quantum Entanglement Experiment? | May 21, 2017 |

B How do you build a quantum suicide machine? | Mar 12, 2017 |

B Build a "full" wave function without data in simple problems | Dec 15, 2016 |

**Physics Forums - The Fusion of Science and Community**