# Seeking derivation of real scalar field Lagrangian

• snoopies622
In summary, this Lagrangian is a simple Lorentz invariant Lagrangian for a scalar field which describes particles with mass.
snoopies622
Here and there I come across the following formula for the Lagrangian density of a real scalar field, but not a deriviation.

$$\mathcal{L} = \frac {1}{2} [ \dot \phi ^2 - ( \nabla \phi ) ^2 - (m \phi )^2 ]$$

Could someone show me where this comes from? The m squared term in particular is a mystery.

The first two terms look like kinetic and potential energy, but the third one . . all that comes to mind is the mass-energy relationship of special relativity, but I thought this formula was from classical mechanics.

In what context are you studying this? Most often this comes up at the beginning of a quantum field theory course.

The Lagrangian is more a definition than anything; when we write down the Lagrangian we are saying, "Let's imagine a field whose dynamics are described by this Lagrangian. How does it behave?" And we can write down any Lagrangian we want.

However, there are some constraints we tend to impose on what Lagrangians we write down and study. One is Lorentz invariance: the Lagrangian should describe a field whose dynamics are consistent with special relativity. This limits what terms can appear in the Lagrangian, but doesn't determine it fully. If we ask for the simplest Lorentz invariant Lagrangian for a scalar field, we get the one you wrote down above. This is why this Lagrangian comes up often.

m is a free parameter. In quantum field theory, a scalar field with this Lagrangian has excitations which turn out to be particles of mass m.

Yes, I bumped into it studying quantum field theory. It's supposed to have the same form as the Klein-Gordon Lagrangian, yet come about without any quantum assumptions.

I assumed that there was some physical situation that corresponds to it, as

$$L = \frac {1}{2}m \dot {x}^2 - \frac {1}{2}kx^2$$

corresponds to a mass on a spring obeying Hooke's law.

why not.
the first two terms form a kinetic energy part
1/2(∂ψ/∂t)2-1/2(∇ψ)2 can be written as 1/2(∂μψ)2 which corresponds to kinetic energy and rest is of course some potential energy of spring type thing 1/2(mψ)2.

Oh I see — the first two terms are analogous to the Minkowski displacement vector with magnitude $ds^2 = (c dt) ^ 2 - dx ^2$ and the third term is the potential.

Yes, that makes some sense. Thanks, Andrien! :)

That's funny, I just noticed that it doesn't have the same form as the Klein-Gordon Lagrangian at all. One has second derivatives and the other second powers.

Now I'm wondering where I read that . .

Edit: Oops, one's a Lagrangian and one isn't. Moderators, feel free to delete this entry #7.

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## 1. What is a real scalar field?

A real scalar field is a mathematical concept used in physics to describe a quantity that has both magnitude and direction, but no inherent orientation. It is represented as a function of space and time, and each point in space and time has a specific value for the field.

## 2. What is a Lagrangian?

A Lagrangian is a mathematical function used in physics to describe the dynamics of a system. It takes into account the potential and kinetic energies of the system and is used to derive the equations of motion for the system.

## 3. How is the Lagrangian used to describe a real scalar field?

The Lagrangian for a real scalar field is used to describe its dynamics and interactions with other fields. It takes into account the potential energy of the field and its kinetic energy, along with any interactions with other fields, to derive the equations of motion for the field.

## 4. What is the importance of seeking a derivation of the real scalar field Lagrangian?

Seeking a derivation of the real scalar field Lagrangian is important because it allows us to understand the fundamental principles governing the behavior of the field. It also allows us to make predictions about the behavior of the field and how it interacts with other fields, which is crucial in many areas of physics, such as particle physics and cosmology.

## 5. What are some techniques used to derive the real scalar field Lagrangian?

There are a variety of techniques used to derive the real scalar field Lagrangian, including the Euler-Lagrange equations, Noether's theorem, and the path integral formulation. Each technique has its own advantages and is used in different contexts to derive the Lagrangian for the real scalar field.

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