Saturday, March 21, 2015

notation - Gauge-covariance of the Yang-Mills field strength Famunu


Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that


Dμ=μigAμ,


you can built the kinetic term for the gauge potential Aμ as



LA=12tr{FμνFμν},Fμν=ig[Dμ,Dν].


The action of the covariant derivative transforms under a local group tranformation Ω in the following way:


DμψΩDμψ.


And the gauge field as


AμΩAμΩ+igΩμΩ.


Introducing Eq. (4) in Eq. (1) you get that under the group tranformations, the covariant derivate transforms as,


DμμigΩAμΩ+ΩμΩ=ΩDμΩ+μ.


Eq. (5) into definition of strength tensor in Eq. (2) gives


FμνΩFμνΩ+ig(μ(ΩDνΩ)ν(ΩDμΩ)).


Taking Ω1gθiTi, with {Ti} the set of generators of the group, θiR and fijk the structure constants:



FμνΩFμνΩ+ig(μAkννAkμ)Tk+ig2fajkTk[μ(θjAaν)ν(θjAaμ)].


In Particle Physics books, it is said that FμνΩFμνΩ,

but I don't get that in my calculus (Eq. (7)). What am I doing wrong? This result is important because if Fμν doesn't transforms as books say, the kinetic term in Eq. (2) isn't gauge invariant.



Answer



Ok, I think I found the answer. Eq. (5) has to be written as:


Dμμ+Ω(μΩ)igΩAμΩ


As you can deduce from Eq. (4) that is a consequence of imposing DμψΩDμψ. Nevertheless, you can re-write Eq. (A) as:


Dμ(Ωμ)ΩigΩAμΩ=ΩDμΩ


After considering,


(Ωμ)Ω=Ω(μΩ)+ΩΩμ=Ω(μΩ)+μ


So with this, everything makes sense and the strength tensor is perfectly gauge-covariant, keeping in mind Eqs. (B)-(C) when interpreting OP's Eq. (8).



No comments:

Post a Comment

classical mechanics - Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that: The moment of a force about a given axis (or Torque) is defined by the equation: $M_X = (\vec r \times \...