# Does QED reproduce classical electrodynamics? How?

1. Oct 9, 2012

### Feynlee

It seems to be a dumb question. But I haven't seen anyone making this connection between QED and Classical EM in a complete fashion. The only example I've seen is the connection between two particle scattering amplitude calculation in QED (Peskin's book), and the amplitude of a particle scattering of a potential in non-relativistic Quantum Mechanics. By making that connection, you can reproduce the equivalent "potential" from QED. In this case, the coulomb potential (see Peskin P121~P125).

But what about the full solution to Maxwell's equations. For example, how would one reproduce the Lienard-Wiechert potential, or equivalently the electric field distribution of a randomly moving charge from QED's calculation? It seems to me, in QED there is no real concept of "potential", all are amplitudes. But there must be some way to connect this QED picture to the Classical relativistic potentials that already worked so well.

Is there any way to do that? Any reference would be greatly appreciated!

2. Oct 9, 2012

### Dickfore

Classical ED corresponds to tree-level processes and is obtained within the saddle-point approximation. Similarly, the motion of a particle according to classical mechanics is obtained within the WKB approximation, which is similar in spirit to the saddle point approximation.

3. Oct 9, 2012

### Feynlee

Thanks for your reply! Is there any reference you can recommend that shows how exactly this works?

4. Oct 9, 2012

### tom.stoer

The relation is visible in the canonical formulation. One piece of the gauge fixed QED Hamiltonian is

$$\int_{\mathbb{R}^3 \otimes \mathbb{R}^3} d^3x\,d^3y \frac{\rho(x)\,\rho(y)}{|x-y|}$$

with

$$\rho = j^0 = \psi^\dagger \psi$$

By expanding the charge density operators as

$$\rho = \rho_0 + \tilde{\rho}$$

one obtains a quantum theory of fluctuations on top of a classical background charge distribution. Such a separation as classical fields (determined by Maxwell and Dirac equation) + fluctuations is possible for other field operators as well.

5. Oct 9, 2012

### Jano L.

Mario Bacelar Valente wrote an interesting thesis on the relation between the classical and quantum electrodynamics:

philsci-archive.pitt.edu/8764/1/PhD.Bacelar.pdf

6. Oct 11, 2012

### A. Neumaier

For the optical sector, you should look at the book on quantum optics by Mandel and Wolf. You cannot fail to see the connection.