# QFT Nonlinearity: Interactions & Maxwell's Eqns

• gentsagree
In summary, the linearity of Maxwell's equations does not mean that they do not predict interactions. While they do not have self-interactions, they can still interact with other fields. However, the presence of interactions in QFT does not necessarily mean that they come from nonlinearity. There is no example of a linear interaction, but it is a mathematical fact that the statement "A=>B does not imply B=>A." In the classical theory, two electromagnetic waves can pass through each other without interacting, and in the quantum theory, photons can also pass through each other.
gentsagree
Nonlinearlity of QFT produces interactions, or so I was told today. Maxwell's eqns, though, are perfectly linear. Does that mean that Maxwell's eqns don't predict interactions?

Thanks

Free field theories in which no interaction is present(of course) are linear,you add a nonlinear term to account for interaction like in $\phi^4$ theory in which a self interacting term is present.But that does not mean that interaction term is always nonlinear but nature does not chose one.Maxwell's eqns are linear but they still have interaction.

The Maxwell equations don't lead to self-interaction. That is perfectly fine, as there is no direct (without other fields involved) self-interaction of electromagnetic fields.

In addition, "Nonlinearlity of QFT produces interactions" does not mean the reverse ("Interactions in QFT have to come from nonlinearity") has to be true.

mfb said:
In addition, "Nonlinearlity of QFT produces interactions" does not mean the reverse ("Interactions in QFT have to come from nonlinearity") has to be true.
Please, can you give an example of an interaction which is linear?

I cannot, and I don't think there is one, but the statement "A=>B does not imply B=>A" is a mathematical fact.

gentsagree said:
Does that mean that Maxwell's eqns don't predict interactions?

Correct. Two classical electromagnetic waves will simply pass through each other without affecting each other at all. This is what we mean by a "noninteracting" or "free" field theory.

Maxwell's equations with no source have no interactions. In the classical theory we have waves which pass straight through each other and in the quantum theory we have photons passing right through each other.

## 1. What is QFT nonlinearity?

QFT nonlinearity refers to the nonlinearity of quantum field theory, which describes the interactions between particles at a quantum level. Nonlinearity occurs when the interactions between particles cannot be described by a simple linear equation, and instead involve complex and nonlinear equations.

## 2. How are interactions described in QFT?

In QFT, interactions are described using Feynman diagrams, which represent the probability amplitudes for particles to interact and exchange energy and momentum. These diagrams take into account the nonlinear nature of interactions and allow for precise calculations of physical quantities.

## 3. What are Maxwell's equations in QFT?

Maxwell's equations in QFT are the fundamental equations that describe the behavior of electromagnetic fields. They consist of four equations that relate the electric and magnetic fields to their sources, such as charges and currents. These equations are nonlinear and play a crucial role in understanding the behavior of particles and their interactions.

## 4. How does QFT nonlinearity affect the behavior of particles?

QFT nonlinearity can lead to phenomena such as particle creation and annihilation, where particles can spontaneously appear and disappear due to the interactions between fields. It also affects the scattering and decay of particles, as well as the energy and momentum transfer between them.

## 5. What are the practical applications of studying QFT nonlinearity?

Studying QFT nonlinearity has many practical applications, such as in the development of new technologies in quantum computing, particle accelerators, and quantum communication. It also helps us understand the behavior of matter at a fundamental level, and has implications in fields such as cosmology and high-energy physics.

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