Perturbation theory is used for approximate solutions from "summary" of The Quantum Theory of Fields by Steven Weinberg,Weinberg S
Perturbation theory is an essential tool in the physicist's arsenal. It allows us to tackle problems that are otherwise too difficult to solve exactly. The basic idea is straightforward: start with a system that we understand well, and then gradually introduce small changes, or perturbations, to see how the system responds. By treating these perturbations as small deviations from the known solution, we can make approximations that capture the essential physics without getting bogged down in the details. In quantum field theory, perturbation theory is particularly powerful. Here, we are dealing with systems that involve an infinite number of degrees of freedom, making exact solutions virtually impossible to find. Instead, we can start with a free theory, where interactions are turned off, and then systematically include terms that represent interactions between particles. These interactions are typically weaker than the free theory itself, justifying the perturbative approach. The success of perturbation theory hinges on the assumption that the perturbations are indeed small. If the interactions between particles are too strong, the approximations we make may no longer be valid, leading to incorrect predictions. This is why perturbation theory is often used in situations where the interactions are expected to be weak, such as in scattering processes or in systems where particles are far apart. Despite its limitations, perturbation theory has proven to be an invaluable tool in theoretical physics. It allows us to make predictions about the behavior of complex systems with remarkable accuracy, shedding light on the fundamental interactions that govern the universe. By treating small deviations from known solutions as perturbations, we can explore the intricate dance of particles and fields that make up the fabric of reality.
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