# Psi^4 Theory: A Fermion with Quartic Self-Interaction

• jdstokes
In summary, the conversation discusses the analysis of a toy theory called psi^4 theory, which involves a fermion with quartic self-interaction. The interaction can describe contact processes and has a Feynman rule of 4i\lambda. However, the process \psi + \bar{\psi} \to \psi \bar{\psi} has a Feynman rule of zero and zero scattering amplitude due to the cancellation of four operator contributions after normal ordering. The speaker wonders why this might be expected physically, considering the unphysicality of the Lagrangian, and compares it to the scalar theory Phi^4. They also mention the possibility of antisymmetry after exchange of external legs, but are unsure of its impact.

#### jdstokes

I've been analysing the following toy theory which I've called psi^4 theory for want of a better name.

$\mathcal{L} = :i\bar{\psi}\gamma^\mu\partial_\mu \psi - m\bar{\psi}\psi + \lambda (\bar{\psi}\psi)^2:$.

Ie a fermion with quartic self-interaction. This interaction can describe contact processes such as $\psi + \bar{\psi} \to \psi + \bar{\psi}$ whose Feynman rule I derived to be $4i\lambda$.

Interestingly, the process $\psi + \bar{\psi} \to \psi \bar{\psi}$ has a Feynman rule of zero and consequently zero scattering amplitude. This comes about because if one expands the interaction Lagrangian in positive and negative frequency parts, there are four operator contributions which cancel after normal ordering. Does anyone know why this might be expected physically (ignoring the obvious unphysicality of the Lagrangian).

Phi^4 is a scalar theory. I think this is more like a four-Fermi interaction. I was unaware that the amplitude were zero. I would think that it is equal to $$4i\lambda$$, after renormalization, but I haven't actually done the calculation. It might have to do with antisymmetry after exchange of external legs, but I'm not sure how.

I find this toy theory intriguing and potentially useful in understanding certain physical processes. The inclusion of a quartic self-interaction term in the Lagrangian allows for the description of contact processes, which are important in understanding particle interactions. The Feynman rule derived for this interaction provides a clear understanding of the mathematical representation of these processes.

The observation that the process \psi + \bar{\psi} \to \psi \bar{\psi} has a Feynman rule of zero is interesting and warrants further investigation. It is possible that this result is a consequence of the normal ordering of the operators, which may lead to a cancellation of contributions. However, it is important to consider the physical implications of this result and whether it is consistent with experimental observations.

In conclusion, the psi^4 theory presents an interesting and potentially useful framework for understanding certain physical processes. Further research and analysis of this theory may lead to a better understanding of particle interactions and their underlying mechanisms.

## 1. What is Psi^4 theory?

Psi^4 theory is a mathematical framework used to describe the behavior of a fermion (a particle with half-integer spin) with quartic self-interaction, meaning the particle can interact with itself through four different paths.

## 2. What is the significance of the quartic self-interaction in this theory?

The quartic self-interaction in Psi^4 theory allows for a more accurate description of the behavior of fermions, as it takes into account interactions that were previously neglected in other theories. It also has important applications in fields such as condensed matter physics and particle physics.

## 3. How does Psi^4 theory differ from other theories in particle physics?

Psi^4 theory differs from other theories in its inclusion of the quartic self-interaction, which allows for a more complete understanding of fermion behavior. It also has a more mathematically rigorous framework, making it a preferred choice for many scientists studying fermions.

## 4. What are some applications of Psi^4 theory?

Psi^4 theory has various applications in both theoretical and experimental physics. It has been used to study the behavior of fermions in condensed matter systems, such as superconductors and superfluids. It also has important implications in particle physics, particularly in the study of interactions between particles in high energy collisions.

## 5. Are there any limitations to Psi^4 theory?

Like any scientific theory, Psi^4 theory has its limitations. It is primarily applicable to fermions and does not account for interactions with other types of particles. It also has limitations in its ability to accurately predict certain properties of fermions, particularly at high energies. However, it is still a valuable tool for understanding the behavior of fermions in various systems.