# Deriving classical potentials from tree diagrams

1. Sep 19, 2007

### A/4

I'm looking for a good book for graduate students that indicates how one can obtain a (classical) potential from a tree-level Feynman process. For example, how can one go from the S-matrix of e.g. Compton scattering and derive the Coulomb potential. Any help would be appreciated.

2. Sep 19, 2007

### meopemuk

I can recommend section 83 in

V. B. Berestetskii, E. M. Livs h itz, L. P. Pitaevskii, "Quantum electrodynamics"

where they derived the Breit potential (Coulomb+ magnetic + spin-orbit + spin-spin) of the electron-positron interaction from the 2nd order S-matrix element of the electron-positron scattering.

The Compton scattering S-matrix elements would yield the electron-photon potential. I have never seen such a derivation.

Eugene.

3. Sep 20, 2007

### meopemuk

By the way, one can build a "classical" potential representation of QFT beyond the tree level, i.e., including loops etc. In this case, particle-number-changing potentials should be allowed as well. Then in addition to the Coulomb and electron-photons potentials one can get "potentials" for annihilation, bremsstrahlung, etc. This is the idea of the "dressed particle" approach, which also provides a consistent way of dealing with divergences. You can find more info in

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory" Nuovo Cim. 8 (1958), 378

A. V. Shebeko, M. I. Shirokov, "Unitary transformations in quantum field theory and bound states" Phys. Part. Nucl. 32 (2001), 15; http://www.arxiv.org/abs/nucl-th/0102037

E. V. Stefanovich, "Relativistic quantum dynamics" http://www.arxiv.org/abs/physics/0504062.

Eugene.

4. Sep 20, 2007

### arivero

Perhaps some of the books of Sakurai, too?

For tree level, most books should have it under the topic, or the keyword index, of Born Transformation.

It should be interesting if someone can name a textbook containing examples both for spin 1, spin 0, and spin 2 forces. Or at least for spin 1 and spin 0 forces.

5. Sep 20, 2007

### mjsd

but Sakurai uses that old $$ict$$ notation (so if you don't like it, like me these days, try Mandl & Shaw, Peskin & schroeder, see also Merzbacher, Quantum Mechanics)

according to Sakurai, to construct an effective 3-dim potential, once you know the non-relativistic limit of your covariant matrix element (the M-matrix), just Fourier Transform the lowest-order (ie. tree-level) M-matrix element. Something like:
$$V=\frac{1}{(2\pi)^3}\int\, \mathcal{M}_{i\rightarrow f} \;e^{i\vec p \cdot \vec x} d^3 p$$

Last edited: Sep 20, 2007