Gauge symmetry for massless Klein-Gordon field

In summary, the task at hand is to gauge the symmetry of the Lagrangian L=\partial_\mu\phi\partial^\mu\phi under the transformation \phi \rightarrow \phi + a(x). The first step is to determine how the Lagrangian transforms under this gauge transformation. Then, a term containing the gauge field A_\mu must be added to the Lagrangian in a Lorentz invariant and dimension 4 form. Finally, the gauge field must be transformed in a way that keeps the total Lagrangian invariant. Further guidance and reading on this topic may be necessary in order to successfully complete the task.
  • #1
Plaetean
36
0

Homework Statement


I need to gauge the symmetry:

[tex]
\phi \rightarrow \phi + a(x)
[/tex]

for the Lagrangian:
[tex]
L=\partial_\mu\phi\partial^\mu\phi
[/tex]

Homework Equations

The Attempt at a Solution


We did this in class for the Dirac equation with a phase transformation and I understood the method, but can't seem to figure out a way to apply the same approach to this field. All I have done so far is plug the transformation into the Lagrangian, but what I'm not sure about is what form the gauge field [tex]A_\mu[/tex] needs to take, or even where in the Lagrangian to put it! I have a load of QFT books too but the only mention of gauge invariance in them is very brief, not at all comprehensive, and is way further into the book than I am currently. Any guidance on where to start or where to read further would be really appreciated. Thanks as always!
 
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  • #2
Plaetean said:

Homework Statement


I need to gauge the symmetry:

[tex]
\phi \rightarrow \phi + a(x)
[/tex]

for the Lagrangian:
[tex]
L=\partial_\mu\phi\partial^\mu\phi
[/tex]

Homework Equations

The Attempt at a Solution


We did this in class for the Dirac equation with a phase transformation and I understood the method, but can't seem to figure out a way to apply the same approach to this field. All I have done so far is plug the transformation into the Lagrangian, but what I'm not sure about is what form the gauge field [tex]A_\mu[/tex] needs to take, or even where in the Lagrangian to put it! I have a load of QFT books too but the only mention of gauge invariance in them is very brief, not at all comprehensive, and is way further into the book than I am currently. Any guidance on where to start or where to read further would be really appreciated. Thanks as always!
First: under that gauge transformation, how does the Lagrangian
[itex]
L=\partial_\mu\phi\partial^\mu\phi
[/itex] transform?

Next, you have to add a term that contains [itex] A_\mu[/itex]. It will have to be Lorentz invariant and be of dimension 4. What form can it take? Then you will have to see how the gauge field transforms to make the total Lagrangian invariant.
 

Related to Gauge symmetry for massless Klein-Gordon field

What is gauge symmetry for massless Klein-Gordon field?

Gauge symmetry for massless Klein-Gordon field is a mathematical concept in quantum field theory that describes the relationship between the physical properties of a particle and its underlying symmetries. It states that the laws of physics should be invariant under a certain transformation, known as a gauge transformation.

Why is gauge symmetry important in quantum field theory?

Gauge symmetry is important in quantum field theory because it helps to explain the fundamental interactions between particles and their interactions with fields. It also allows for the unification of different theories, such as electromagnetism and the weak nuclear force, into a single theory known as the Standard Model.

How does gauge symmetry relate to the massless Klein-Gordon field?

Gauge symmetry is a fundamental principle in quantum field theory that applies to all fields, including the massless Klein-Gordon field. It describes the behavior of the field under transformations and helps to determine the physical properties of particles that interact with the field.

What are some real-world applications of gauge symmetry for massless Klein-Gordon field?

The concept of gauge symmetry has been successfully applied in various areas of physics, including particle physics, condensed matter physics, and cosmology. It has helped to explain the behavior of elementary particles and their interactions, as well as the properties of materials in different states.

Are there any challenges or controversies surrounding gauge symmetry for massless Klein-Gordon field?

While gauge symmetry has been successful in explaining many phenomena in physics, there are still ongoing debates and challenges surrounding its application to certain theories, such as gravity. Some physicists also question the necessity of gauge symmetry in describing the fundamental properties of particles and fields.

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