Lagrangian for electromagnetism

1. Nov 19, 2007

jdstokes

In A. Zee's quantum field theory in a nutshell he assumes familiarity with Maxwell's lagrangian $\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ with A the vector potential.

Although I've seen the magnetic vector potential, I've never seen the lagrangian formalism in either electrodynamics or lagrangian/hamiltonian dynamics courses.

Could anyone point me in the direction of a suitable reference to allow me to familiarise myself with this?

Thanks

2. Nov 19, 2007

Avodyne

3. Nov 19, 2007

siddharth

Try chapter 12 in Griffiths, for a brief introduction into the field tensor and the four-vector potential. For an introduction to the lagrangian formalism in electrodynamics, try Goldstein's book on classical mechanics.

4. Nov 20, 2007

jdstokes

I find Zee's notation a little bit confusing here. It seems like he is writing $\partial_\mu$ to mean $(\partial_t,\nabla)$ and at the same time writing e.g. $A_\mu = (V,-\mathbf{A})$ and thus $A^\mu = (V,\mathbf{A})$. Is this standard or am I misunderstanding his notation?

This is the only way I could get Maxwell's equations out of

$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$.

$F^{0i} = \partial^0 A^i - \partial^i A^0 = -E^i$. etc

Last edited: Nov 20, 2007
5. Nov 20, 2007

jdstokes

After checking in another QFT text by Ryder it seems like this is indeed standard notation.

6. Nov 20, 2007

nrqed

yes, for $\partial_\mu$ the sign is opposite to the other vectors. That's because
$$\partial_\mu \equiv \frac{\partial}{\partial x^\mu}$$

7. Jul 16, 2009

jacobrhcp

never mind I'll make my own topic

8. Jul 16, 2009

turin

The ultimate E&M reference: J. D. Jackson, "Classical Electrodynamics", 3rd ed., Chap. 12, Sec. 7.
R. Shankar, "Principles of Quantum Mechanics", 2nd ed., Chap. 18, Sec. 5, Subsec. "Field Quantization".
(more advanced) Peskin & Shroeder, "An Introduction to Quantum Field Theory", Chap. 15.

9. Jul 16, 2009

turin

It is perhaps standard, but it is certainly just a convention. For example, in the "East Coast Metric" (η=diag(-1,+1,+1,+1)), that could be changed to $A_\mu = (-V,\mathbf{A})$, and for implicit metric: $\partial_\mu=\partial^\mu=(\nabla,ic\partial_t)$, $A_\mu=A^\mu=(\mathbf{A},icV)$.