Solution to nonlinear field equations

[wandering.the.cosmos]

In the usual way to do QFT, we find the Green functions to the quadratic part of the lagrangian (usually with Feynman boundary conditions), and use this in the computation of n point (usually time ordered) correlation functions. Suppose one manages to solve instead the full nonlinear equation, is there (1) some sort of Green function for non-linear equation we can construct out of these solutions? and (2) is it possible, and if so, how do we use these solutions instead of the quadratic Green functions to do perturbative QFT?

An example of this is, suppose one knows the solution to the classical massless $\phi^4$ equations:

$$\partial_\mu \partial^\mu \phi = -\frac{g}{3!} \phi^3$$

Instead of using

$$D_F(x-y) = \int \frac{d^dk}{(2\pi)^4} \frac{ie^{-ik(x-y)}}{k^2+i\epsilon}$$

can we use the full solution to the $\phi^4$ equations to do calculations? If so, how? Say I wish to now calculate the $\phi + \phi \to \phi + \phi$ scattering amplitude to 1 loop -- what's the difference between computing it using the usual Feynman propagator $D_F$ and using the full solution? Or more generally, how do we go about computing the generating functional

$$\int \mathcal{D}\phi \exp\left[ i\int d^4x \left( \frac{1}{2} \partial_\mu\phi \partial^\mu \phi - \frac{g}{4!}\phi^4 + J \phi \right) \right]$$

 

[azntoon]

using the full nonlinear solution?That is, can we use the full nonlinear solution instead of the usual Feynman propagator in perturbative QFT calculations, and if so, how?

 

[denian]

I would respond to this content by saying that finding a solution to the full nonlinear field equations is a significant achievement, and it opens up new possibilities for studying quantum field theory (QFT). This approach may provide a more complete understanding of the system, as it takes into account the nonlinear interactions that are often neglected in perturbative methods.

To answer the first question, it is possible to construct a "Green function" for the nonlinear equation using the solutions obtained. This can be done by considering the perturbation of the solution with respect to the initial conditions. This Green function can then be used in the computation of n-point correlation functions, similar to how the quadratic Green function is used in perturbative QFT.

To address the second question, using the full solution to the field equations instead of the quadratic Green function would change the perturbative calculations significantly. The difference lies in the fact that the full solution takes into account the nonlinear interactions, which can lead to different scattering amplitudes and a more accurate description of the system. However, the exact method for computing the generating functional in this case would depend on the specific form of the nonlinear field equations and the solutions obtained.

Overall, solving the full nonlinear field equations provides a powerful tool for studying QFT and has the potential to lead to new insights and understandings in the field. It may also open up new avenues for research and the development of more accurate and comprehensive theories. Further exploration and development of this approach could have significant implications for the field of quantum physics.