- #1

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- I'd like clarification of how the covariant derivative fits into the invariance of the Dirac Lagrangian

This is from Griffiths particle physics, page 360. We have the full Dirac Lagrangian:

$$\mathcal L = [i\hbar c \bar \psi \gamma^{\mu} \partial_{\mu} \psi - mc^2 \bar \psi \psi] - [\frac 1 {16\pi} F^{\mu \nu}F_{\mu \nu}] - (q\bar \psi \gamma^{\mu} \psi)A_{\mu}$$

This is invariant under the joint transformation:

$$\psi \rightarrow \exp(-\frac{iq\lambda(x)}{\hbar c})\psi \ \ \text{and} \ \ A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \lambda$$

Then, we have the covariant derivative:

$$\mathcal D_{\mu} = \partial_{\mu} + \frac{iq}{\hbar c} A_{\mu}$$

And he says "in the original free Lagrangian we replace every derivative ##\partial_{\mu}## by the covariant derivative ... and the invariance of ##\mathcal L## is restored".

There are a lot of different Lagrangians floating around in this chapter, so I'm not sure which one he means. It would be good to understand what the Lagrangian looks like with the covariant derivative in it. What does this covariant Lagrangian look like?

The second question. Once we have this Lagrangian, is it invariant under the transformation $$\psi \rightarrow \exp(-\frac{iq\lambda(x)}{\hbar c})\psi$$

Or, do we still need the additional transformation:

$$A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \lambda$$

Thanks.

$$\mathcal L = [i\hbar c \bar \psi \gamma^{\mu} \partial_{\mu} \psi - mc^2 \bar \psi \psi] - [\frac 1 {16\pi} F^{\mu \nu}F_{\mu \nu}] - (q\bar \psi \gamma^{\mu} \psi)A_{\mu}$$

This is invariant under the joint transformation:

$$\psi \rightarrow \exp(-\frac{iq\lambda(x)}{\hbar c})\psi \ \ \text{and} \ \ A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \lambda$$

Then, we have the covariant derivative:

$$\mathcal D_{\mu} = \partial_{\mu} + \frac{iq}{\hbar c} A_{\mu}$$

And he says "in the original free Lagrangian we replace every derivative ##\partial_{\mu}## by the covariant derivative ... and the invariance of ##\mathcal L## is restored".

There are a lot of different Lagrangians floating around in this chapter, so I'm not sure which one he means. It would be good to understand what the Lagrangian looks like with the covariant derivative in it. What does this covariant Lagrangian look like?

The second question. Once we have this Lagrangian, is it invariant under the transformation $$\psi \rightarrow \exp(-\frac{iq\lambda(x)}{\hbar c})\psi$$

Or, do we still need the additional transformation:

$$A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \lambda$$

Thanks.