- #1

joneall

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- TL;DR Summary
- Is it all U(1)?

I am trying to get a foothold on QFT using several books (Lancaster & Blundell, Klauber, Schwichtenberg, Jeevanjee), but sometimes have trouble seeing the forest for all the trees. My problem concerns the equation of QED in the form

$$

\mathcal{L}_{Dirac+Proca+int} =

\bar{\Psi} ( i \gamma_{\mu} \partial^{\mu} - m ) \Psi

+ g A_{\mu} \bar{\psi} \gamma^{\mu}\psi

-\frac{1}{2} (

\partial^{\mu} A^{\nu} \partial_{\mu} A_{\nu} -

\partial^{\mu} A^{\nu} \partial_{\nu} A_{\mu}

)

$$

consisting of a Klein-Gordon scalar term, an interaction term and a massless vector. The interaction term may be found by insisiting that the scalar term be invariant under a local U(1) transformation ## e^{i\alpha (x)} ## . One then notes that The EM vector potential A must have internal symmetry $$

A_{\mu} \rightarrow A_{\mu}^{\prime} =

A_{\mu} + \partial_{\mu}a(x)

$$,

which can be done explicitly or by including it in a so-called covariant derivative.

So I have two questions:

(1) Is not the transform of A a separate requirement from U(1) symmetry for the KG Lagrangian? It has nothing to do with U(1)?

(2) Is there any correspondence between this covariant derivative and the one in GR with Christoffel symbols?

My goal is to understand what this is all about, not just to be able to derive the equations.

Thanks in advance for any help. Please note that my level of expertise in this domain is as an advanced beginner.

$$

\mathcal{L}_{Dirac+Proca+int} =

\bar{\Psi} ( i \gamma_{\mu} \partial^{\mu} - m ) \Psi

+ g A_{\mu} \bar{\psi} \gamma^{\mu}\psi

-\frac{1}{2} (

\partial^{\mu} A^{\nu} \partial_{\mu} A_{\nu} -

\partial^{\mu} A^{\nu} \partial_{\nu} A_{\mu}

)

$$

consisting of a Klein-Gordon scalar term, an interaction term and a massless vector. The interaction term may be found by insisiting that the scalar term be invariant under a local U(1) transformation ## e^{i\alpha (x)} ## . One then notes that The EM vector potential A must have internal symmetry $$

A_{\mu} \rightarrow A_{\mu}^{\prime} =

A_{\mu} + \partial_{\mu}a(x)

$$,

which can be done explicitly or by including it in a so-called covariant derivative.

So I have two questions:

(1) Is not the transform of A a separate requirement from U(1) symmetry for the KG Lagrangian? It has nothing to do with U(1)?

(2) Is there any correspondence between this covariant derivative and the one in GR with Christoffel symbols?

My goal is to understand what this is all about, not just to be able to derive the equations.

Thanks in advance for any help. Please note that my level of expertise in this domain is as an advanced beginner.