I was trying to prove all those little things you spend long as the local invariance in the free Lagrangian of electroweak interaction.(adsbygoogle = window.adsbygoogle || []).push({});

Taking into account the appropriate SU(2) transformations (without covariant derivatives), came to the following expression

[tex]\mathcal{L}_{\text{ferm.}} = i\bar{\Psi}^Lexp\left [-ig\tau_j\frac{\omega_j(x)}{2}\right] \gamma^\mu \left [\frac{ig\tau_j}{2}\partial_\mu \omega_j(x)exp\left [ig\tau_j\frac{\omega_j(x)}{2} \right]\Psi^L+...\right] [/tex]

Where: [itex]\Psi^L[\itex]: left leptonic doublet, [itex]\tau_j[\itex]: Pauli matrices

Well, i'm stuck with this: the right exponential expression can commute with the real [itex]\partial_\mu\omega_j(x)[\itex] and tha matrix [itex]\tau_j[\itex] but it can't commute with [itex]_gamma^\mu[\itex] because the superindex [itex]\mu[\itex] doesn't need the same as j. T This is necessary to cancel the exponential terms.

What's missing?

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# Gauge invariance of electroweak Lagrangian

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