# Computing the spectrum of a Lagrangian in field theory

• snypehype46
In summary, the conversation discusses a lagrangian density that is invariant under a chiral symmetry and has a conserved current. The question at hand asks about the spectrum of the theory when the potential is changed to ##V(\sigma^2 + \pi^2) = \lambda(\sigma^2 + \pi^2 - c^2)^2##. The goal is to determine the particle content of the theory, which involves identifying stable equilibrium points and calculating masses and interactions between particles by expanding the lagrangian around an equilibrium configuration of fields. The concept of spontaneous breaking of a global symmetry is important in this context.
snypehype46
Homework Statement
What is the spectrum of theory given by this lagrangian?
Relevant Equations
$$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi - g\bar{\psi}(\sigma + i\gamma^5\pi)\psi + \frac{1}{2}(\partial_\mu \sigma)^2+ \frac{1}{2}(\partial_\mu \pi)^2 -V(\sigma^2 + \pi^2)$$
I have the following lagrangian density:

$$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi - g\bar{\psi}(\sigma + i\gamma^5\pi)\psi + \frac{1}{2}(\partial_\mu \sigma)^2+ \frac{1}{2}(\partial_\mu \pi)^2 -V(\sigma^2 + \pi^2)$$

where $\pi$ and $\sigma$ are scalar fields.

I have show that this lagrangian density is invariant under a chiral symmetry and has a conserved current.
Now the question I'm attempting asks the following:

> What is the spectrum of theory when ##V(\sigma^2 + \pi^2) = \lambda(\sigma^2 + \pi^2 - c^2)^2##?

Now I don't quite understand how to do this problem. I assume that by spectrum the question means what kind of particle you obtain in that specific condition. Am I right? If so, how do I go about computing the spectrum? I thought about plugging the value in the lagrangian and then working out the EOMs but not sure how that would help anything.

It's a bit unclear to me, what the problem statement really wants us to calculate. My guess is they mean to determine the "particle content" of the theory. To that end you first have to think about what's the stable equilibrium point given your potential. Then you can figure out the masses and interactions between the physical "particles" by expanding the Lagrangian around an arbitrarily choosen equilibrium configuration of the fields (note that the ground state is degenerate here; usually you choose the vacuum expectation value along the ##\sigma## direction).

The very important fundamental concept here is the spontaneous breaking of a global symmetry (in this case chiral symmetry).

It means to identify the particles predicted by the theory and their masses.

vanhees71

## 1. What is a Lagrangian in field theory?

A Lagrangian in field theory is a mathematical function that describes the dynamics of a physical system. It is used to calculate the equations of motion and predict the behavior of the system.

## 2. How is the spectrum of a Lagrangian computed?

The spectrum of a Lagrangian is computed by first solving the equations of motion derived from the Lagrangian. This results in a set of solutions, or eigenvalues, which make up the spectrum of the system.

## 3. Why is computing the spectrum of a Lagrangian important?

Computing the spectrum of a Lagrangian is important because it allows us to understand the behavior and properties of a physical system. It can also help us make predictions and test the validity of our theories.

## 4. What is the relationship between the spectrum of a Lagrangian and the energy levels of a system?

The spectrum of a Lagrangian is directly related to the energy levels of a system. Each eigenvalue in the spectrum corresponds to a specific energy level of the system.

## 5. Are there any practical applications of computing the spectrum of a Lagrangian?

Yes, there are many practical applications of computing the spectrum of a Lagrangian. It is used in various fields such as quantum mechanics, particle physics, and condensed matter physics to study and understand the behavior of physical systems.

Replies
1
Views
889
Replies
0
Views
801
Replies
1
Views
1K
Replies
1
Views
616
Replies
30
Views
5K
Replies
2
Views
1K
Replies
10
Views
1K
Replies
2
Views
689