Lagrangian invariance under infinitesimal transformations

In summary, the Noether current associated with the finite U(1) transformation has the formJ^{\mu} = i \epsilon (J_{1}^{\mu} + \epsilon J_{2}^{\mu}).This is what the professor meant by "anomalies" - the higher order terms in the Taylor expansion of the transformation may not be conserved and can lead to inconsistencies in the theory. However, this only applies to other symmetry groups, as the Taylor expansion of matrices is not straightforward and can lead to further complications.
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ShayanJ
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This is my second term in my master's and one of the courses I've taken is QFT1 which is basically only QED.
In the last class, the professor said the Klein-Gordon Lagrangian has a global symmetry under elements of U(1). Then he assumed the transformation parameter is infinitesimal and , under the assumption that the first order transformation leaves the Lagrangian invariant, derived a conserved current.
Now it seems to me that because the Lagrangian is invariant under a finite transformation, and the terms in the Taylor expansion with different orders of parameter are linearly independent, the Lagrangian should also be invariant under each term of the Taylor expansion separately and there should be a conserved current associated to every order.
But when I explained the above line of reasoning to the professor, he said there may be anomalies or something like this. It was really vague actually and I didn't understand what he said. But I figured the above line of reasoning can't go wrong in the case of U(1) so whatever he meant must apply to other symmetry groups. Because for those groups the transformations are done by matrices and the Taylor expansion of matrices is not very straightforward. But still I can't understand what can go wrong.
Can anybody explain further?
Thanks
 
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Shyan said:
This is my second term in my master's and one of the courses I've taken is QFT1 which is basically only QED.
In the last class, the professor said the Klein-Gordon Lagrangian has a global symmetry under elements of U(1). Then he assumed the transformation parameter is infinitesimal and , under the assumption that the first order transformation leaves the Lagrangian invariant, derived a conserved current.
Now it seems to me that because the Lagrangian is invariant under a finite transformation, and the terms in the Taylor expansion with different orders of parameter are linearly independent, the Lagrangian should also be invariant under each term of the Taylor expansion separately and there should be a conserved current associated to every order.
But when I explained the above line of reasoning to the professor, he said there may be anomalies or something like this. It was really vague actually and I didn't understand what he said. But I figured the above line of reasoning can't go wrong in the case of U(1) so whatever he meant must apply to other symmetry groups. Because for those groups the transformations are done by matrices and the Taylor expansion of matrices is not very straightforward. But still I can't understand what can go wrong.
Can anybody explain further?
Thanks

No, in local field theory, you don’t get any new current. Consider, the K-G field and [itex]U(1)[/itex] transformation [itex]e^{i\epsilon}[/itex]. The “currents” associated with odd powers of [itex]\epsilon[/itex] coincide with the usual Noether current associated with first order power, while those associated with even powers of [itex]\epsilon[/itex] are not conserved: To see that look at the transformations to second order:
[tex]\delta \phi = ( i \epsilon - \frac{1}{2} \epsilon^{2} ) \phi .[/tex]
[tex]\delta \phi^{\dagger} = (- i \epsilon - \frac{1}{2} \epsilon^{2} ) \phi^{\dagger} .[/tex]
Look at the signs of the second order terms, they are the same, and this prevents you from having conserved “current”. Indeed, if you substitute the above transformations in the [itex]U(1)[/itex] Noether current of the complex KG Lagrangian, you obtain
[tex]J^{\mu}(\epsilon) = i \epsilon \left( \phi \partial^{\mu} \phi^{\dagger} - \phi^{\dagger} \partial^{\mu} \phi \right) + \frac{1}{2} \epsilon^{2}\left( \phi \partial^{\mu} \phi^{\dagger} + \phi^{\dagger} \partial^{\mu} \phi \right) [/tex]
Now, if you write the current as
[tex]J^{\mu}(\epsilon) = i \epsilon J_{1}^{\mu} + \frac{1}{2} \epsilon^{2} J_{2}^{\mu}[/tex]
You can identify
[tex]J_{1}^{\mu}(x) = \phi \partial^{\mu} \phi^{\dagger} - \phi^{\dagger} \partial^{\mu} \phi ,[/tex]
as the conserved Noether current of the [itex]U(1)[/itex] symmetry. But
[tex]J_{2}^{\mu}(x) = \phi \partial^{\mu} \phi^{\dagger} + \phi^{\dagger} \partial^{\mu} \phi ,[/tex]
is not conserved.
 
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What is Lagrangian invariance under infinitesimal transformations?

Lagrangian invariance under infinitesimal transformations is a fundamental principle in physics that states that the equations of motion for a system should remain unchanged under small, continuous changes in the variables that describe the system.

Why is Lagrangian invariance important in physics?

Lagrangian invariance allows us to simplify the mathematics of a system and make predictions about its behavior without having to solve complex differential equations. It also provides a deeper understanding of the symmetries and laws of nature.

What is an infinitesimal transformation?

An infinitesimal transformation is a small, continuous change in the variables that describe a system. It is typically represented by a parameter, such as time or position, that is allowed to vary slightly.

How is Lagrangian invariance related to Noether's theorem?

Noether's theorem states that for every continuous symmetry in a physical system, there exists a corresponding conservation law. Lagrangian invariance is a manifestation of this symmetry, and therefore, Noether's theorem can be used to derive conservation laws for the system.

What are some practical applications of Lagrangian invariance?

Lagrangian invariance is used extensively in various fields of physics, including classical mechanics, quantum mechanics, and field theory. It is also applied in engineering, such as in the design of control systems and robots, and in the study of fluid dynamics and electromagnetism.

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