# The scalar field lagrangian

• qft1
In summary, the Lagrangian density for a scalar field in Quantum Field Theory books is often stated as being in the form of 1/2(d\phi)^2 - V(\phi), where V(\phi) is a polynomial in \phi. This is due to the conditions of Lorentz invariance and a maximum of two time derivatives, which restrict the kinetic energy term to be in this form. This is also related to dimensional analysis and the requirement for the action to be dimensionless. Any interactions beyond (phi)^3 or (phi)^4 would result in negative mass dimension coupling constants and a nonrenormalizable theory. It is important to note that this is not a requirement for a maximum of two time derivatives

#### qft1

Hi,

I have a question about a statement I've seen in many a Quantum Field Theory book (e.g. Zee). They say that the general form of the Lagrangian density for a scalar field, once two conditions are imposed:
(1) Lorentz invariance, and
(2) At most two time derivatives,
is:

L = 1/2(d\phi)^2 - V(\phi)

where V(\phi) is a polynomial in \phi.

Why is this? I can understand how the conditions restrict the kinetic energy term to being what it is, but I don't understand why V has to be _polynomial_ in \phi.

Good question.

The reason is dimensional analysis. The action must be dimensionless to be a lorentz invariant, so the lagrangian has to have dimension mass^4 .

So you can simply power count your fields to tabulate all renormalizable interactions. You've probably seen this before.. You know the spinor field has dimension 3/2, scalar fields dimension 1 etc

So for scalar fields you can only have a (phi)^3, or b phi^4 where a is dimension 1 and b is dimensionless.. Anything higher than that would lead to negative mass dimension coupling constants and a badly nonrenormalizable theory.

It's not a requirement for max.2 time-derivatives.Think about Weyl gravity.The Hamiltonian formalism can be externded to an arbitrary # of time derivatives in the "kinetic" term...

Daniel.

## 1. What is a scalar field lagrangian?

A scalar field lagrangian is a mathematical formulation used in theoretical physics to describe the dynamics of a scalar field. It is derived from the lagrangian density, which is a function that describes the potential energy of a field at a given point in space and time.

## 2. What is the role of scalar fields in physics?

Scalar fields play a crucial role in various areas of physics, including quantum field theory, cosmology, and particle physics. They are used to describe the forces and interactions between particles, as well as the properties of spacetime itself.

## 3. How is the scalar field lagrangian used in practice?

The scalar field lagrangian is used to derive the equations of motion for a given system, which can then be used to make predictions about the behavior of that system. It is an essential tool in theoretical physics and has been used to make significant advancements in our understanding of the universe.

## 4. What are some examples of scalar fields?

Scalar fields are present in various physical phenomena, such as the Higgs field, which gives particles their mass, and the inflaton field, which is thought to have driven the rapid expansion of the universe during the period of inflation. Other examples include the electric potential field and the temperature distribution in a room.

## 5. How does the scalar field lagrangian relate to other lagrangians?

The scalar field lagrangian is a special case of the more general lagrangian formulation, which is used to describe the dynamics of fields and particles in classical and quantum physics. It is closely related to the lagrangians used to describe other types of fields, such as vector and tensor fields, but differs in its mathematical form and physical interpretation.