Validity of Scalar Field Lagrangian with Linear and Quadratic Terms

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• Gaussian97
In summary, the most general Lagrangian of a single scalar field up to two fields and two derivatives is usually given as $$\mathscr{L} = \phi \square \phi + c_2 \phi^2$$ but the question arises whether the Lagrangian $$\mathscr{L} = \phi \square \phi + c_1 \phi + c_2 \phi^2$$ is valid. However, with the linear term, the energy is not bounded from below, making it problematic for perturbation theory. To solve this issue, one can shift the field by a certain amount and the linear term disappears. This theory is still valid as long as the quadratic term has the right sign. For
Gaussian97
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TL;DR Summary
Is a term proportional to ##\phi## valid in a scalar Lagrangian?
Hi, if I want to construct the most general Lagrangian of a single scalar field up to two fields and two derivatives, I usually see that is
$$\mathscr{L} = \phi \square \phi + c_2 \phi^2$$ i.e. the Klein-Gordon Lagrangian.
My question is, would be valid the Lagrangian
$$\mathscr{L} = \phi \square \phi + c_1 \phi + c_2 \phi^2$$
?

With a linear term, the energy is not bounded from below. So this is usually not considered. This is a problem for applying perturbation theory, this indicates that one is expanding around a field configuration that cannot be used as a vacuum. So one then shifts ## \phi \rightarrow \phi - \frac{c_1}{2c_2} ## which gets rid of the linear term.

nrqed said:
With a linear term, the energy is not bounded from below.
A linear term shifts the location of the minimum of the potential and shifts the overall potential by a finite constant. (speaking loosely about potential densities as potentials)

With the linear term and the quadratic term the energy is still bounded from below provided ##c_2## has the "right sign". So it's a fine theory (as ##\phi^3## theory is not though it's treated at length in some textbooks to have a simple model to explain perturbative renormalization theory, e.g., in Collin's Renormalization; adding a ##\phi^4## term makes it again a theory with the Hamiltonian bounded from below).

For the free field here you can just introduce a new field shifted field as explained in #2, and you are back at the usual free theory.

dextercioby
Ok! Thank you!

What is a general scalar Lagrangian?

A general scalar Lagrangian is a mathematical function that describes the dynamics of a scalar field in a physical system. It is used in the field of theoretical physics to study the behavior of particles and fields, and is an important tool in understanding the fundamental laws of nature.

What is the purpose of a general scalar Lagrangian?

The purpose of a general scalar Lagrangian is to describe the dynamics of a scalar field in a physical system. It allows scientists to study the behavior of particles and fields and make predictions about their interactions and properties.

How is a general scalar Lagrangian used in theoretical physics?

In theoretical physics, a general scalar Lagrangian is used to derive the equations of motion for a system and to study its symmetries and conservation laws. It is also used to calculate the energy and momentum of a system and to make predictions about its behavior.

What are the key components of a general scalar Lagrangian?

A general scalar Lagrangian typically includes the kinetic energy term, potential energy term, and interaction terms. It may also include terms for external forces and constraints on the system. The specific components depend on the specific system being studied.

How does a general scalar Lagrangian relate to the Lagrangian formalism?

The Lagrangian formalism is a mathematical framework used to describe the dynamics of a physical system. A general scalar Lagrangian is a specific type of Lagrangian that is used to study scalar fields. It is a key component of the Lagrangian formalism and is used to derive the equations of motion for a system.

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