# Phi 3 scalar field theory

• tayyaba aftab
In summary, the problem is to write a matrix in second order perturbative theory for an interaction Hamiltonian with phi 4 and phi 3 contributions, with an initial state of 2 particles and a final state of 3 particles. The challenge is to determine the correct term for time ordering, given the condition of 2 incoming and 3 outgoing particles. One approach is to use Feynman's rules to draw diagrams and write down the terms, while another is to use Wick's theorem to transform the time-ordered product into a normal-ordered product.

#### tayyaba aftab

i have been given a problem for writing s matrix in second order perturbative theory for an interaction hamiltonian with phi 4 and phi 3 contributions.
it is also given that our initial state is of 2 particles and final state is of three particles.
now in solving that i have to take time ordering of the hamiltonian at two different points.
problem is that i am unable to guess the right term of time ordering for the cpndition of two in coming and three outgoing particles.
i have to submit my assignment after vacations
so its urgent

tayyaba

You don't have to guess here.
I will just cover the phi cubed case.
Are you allowed to use Feynman's rules?
If yes, just draw the 2+3 external lines and two vertices with three outgoing lines each and draw all topologically distinct diagrams. After that, you can write down the terms very easily.

If not, use Wick's theorem to transform the time-ordered into a normal-ordered product:

$$T\left[\phi^3(x)\phi^3(y)\right]=N\left[\phi^3(x)\phi^3(y)+\mathrm{all\, possible\, contractions}\right]=N\left[\phi^3(x)\phi^3(y)\right]+9D_F(x-y)N\left[\phi^2(x)\phi^2(y)\right]+18D_F^2(x-y)N\left[\phi(x)\phi(y)\right]+6D_F^3(x-y)$$

where $$D_F$$ denotes the Feynman propagator. Please check the combinatorial factors. After that, write out the normal-ordered products to see which terms remain after contracting them with the creation and annihilation operators.

## 1. What is Phi 3 scalar field theory?

Phi 3 scalar field theory is a quantum field theory that describes the behavior of a scalar field (a field with spin 0) in three-dimensional space. It is a theoretical framework used to understand the interactions between particles and the fundamental forces in nature.

## 2. How does Phi 3 scalar field theory differ from other quantum field theories?

Phi 3 scalar field theory differs from other quantum field theories in the number of dimensions it describes. While most quantum field theories are formulated in four dimensions, Phi 3 scalar field theory is specifically designed for three-dimensional space.

## 3. What is the significance of Phi 3 scalar field theory?

Phi 3 scalar field theory is important because it allows us to study the behavior of particles and their interactions in a simplified three-dimensional space. This can provide insights and predictions for more complex systems in higher dimensions.

## 4. How is Phi 3 scalar field theory applied in physics?

Phi 3 scalar field theory is applied in various areas of physics, such as particle physics, cosmology, and condensed matter physics. It is used to study the properties of particles, the behavior of matter at high energies, and the evolution of the universe.

## 5. What are some current research topics related to Phi 3 scalar field theory?

Some current research topics related to Phi 3 scalar field theory include the study of topological defects and their role in the early universe, the behavior of scalar fields in non-trivial backgrounds, and the application of the theory in understanding condensed matter systems and quantum information processing.

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