EM Field strength and plane waves

[parton]

Hi !

I've a question. Where is the connection between the (kinetic) Lagrangian $$- \dfrac{1}{4} F_{\mu \nu} F^{\mu \nu}$$ and a plane wave of the form $$\vec{\varepsilon} exp(i \vec{k} \cdot \vec{x}) / \sqrt{V}$$ (the epsilon is a polarization vector) confined in a box with a finite volume V ? I should somehow "motivate" the factor $$- \dfrac{1}{4}$$ occurring in the Lagrangian by such plane waves. But I absolutely dont't have a clue how to do that. Does anyone have an idea? I hope somebody could help me.

 

[lbrits]

Blegh. It's a helluva long road. Quickest thing to do is to fix a gauge like the Lorenz gauge, and then find out what the Hamiltonian is. The factor of -1/4 will give you a canonical Hamiltonian that is something like $$\int\!dx\, \left(\frac{1}{2} \mathbf{E}^2 + \frac{1}{2}|\boldsymbol\nabla \mathbf{A}|^2\right)$$ where $$E \sim \dot{A}$$. The Laplacian is a vector Laplacian, in case you're wondering. I have also dropped factors of c and $$\epsilon_0$$, since I can work in units in which they are 1.

Ok, where were we? You can Fourier expand your vector potential as follows:
$$\mathbf{A} = \sum_\lambda \int\!\frac{dk}{(2\pi)^3} a_\lambda(\mathbf{k}) \boldsymbol{\epsilon}(\lambda)e^{i \mathbf{k}\cdot \mathbf{x} }$$,
where the polarization vectors are chosen such that the gauge fixing condition is satisfied (i.e., only two transverse polarizations). If you're doing this inside a box, it's a plain Fourier series, instead of an integral. If you write your Hamiltonian in terms of these, you'll obtain something like
$$H = \int\!dk\, \left(\frac{1}{2} p_\lambda^2 + \frac{1}{2} k^2 a_\lambda^2\right)$$,
where $$p$$ is the momentum conjugate to $$a$$ i.e., in the Lagrangian terminology, $$p = \dot{a}$$. Now you get a bunch of simple harmonic oscillators.

The factors of two are just convention, really, and keep the equations nice. The original factor of 4 comes about because when you obtain the Euler Lagrange equations, you take a derivative of a square. So to "motivate" all this, you could say that each plane wave is a separate harmonic oscillator. But you'll have to do some math =)

 

[parton]

Thank you! But is it somehow possible to explain the 1/4 factor by considering the dimensions?

I've seen something like $$\left[ F_{\mu \nu} F^{\mu \nu} \right] = 4$$, but I don't really understand why it is equal to 4.
But if one use this, isn't it possible to somehow justify the 1/4 factor? I need a simple explanation without (much) calculations.