Backreaction on background fields: A coherent state approach

There are many situations in which a strong electromagnetic field may be approximated as a fixed background. Going beyond this approximation, i.e. accounting for the backreaction of quantum processes on the field, is however challenging. Here we develop an approach to this problem which is a straightforward extension of background field methods. The approach follows from the observation that scattering in an on-shell background is equivalent to scattering between coherent states; we show that by deforming these states one can model backreaction. Focussing on intense laser-matter interactions, we provide examples which model beam depletion and, furthermore, introduce an extremization principle with which to determine the level of depletion in a given scattering process.

Figure 1

Feynman diagrams for the Furry picture expansion of correlation functions, in which the interaction between matter and the background field is, in principle, treated exactly. The double line is the background-dressed fermion propagator (2).

Figure 2

The physical situation of interest; an initial state containing matter (not shown), photons (in a number state) and a coherent state of photons (representing a strong field) scatters into a different state also comprising matter, emitted photons, and a coherent state of typically reduced amplitude representing depletion of the initial strong field.

Figure 3

Illustrative relation between two descriptions of the same process, in which an initial coherent state $z_i$ is completely depleted. On the left, the coherent state is expanded as a sum of incoming photons, none of which survive to the final state. On the right, the same expression is rewritten in terms of an infinite series of corrections to the background field amplitude, in which the corrections remove background field diagrams with photons in the final state.

Figure 4

Position-space Feynman rules in the Furry picture; the interaction between the matter fields and the coherent states (described by $A_D$) is accounted for exactly by using the propagator $G_{\varphi }$ in (34).

Figure 5

On the left, a photon is absorbed by the external source $J(x)$. This source mocks up the matter current on the right, in which a (scalar) photon produces a (scalar) pair in the Yukawa theory. The triple line indicates that the coherent states can in principle be accounted for exactly in the modified Furry picture discussed in Sec. 3d. (The blob represents loop corrections.)