Local gauge symmetries Lagrangians and equations of motion

In summary, the procedure of trying to find an invariant equation of motion is equivalent to imposing a symmetry on the EoM.
  • #1
FunkyDwarf
489
0
Hey gang,

I'm re-working my way through gauge theory, and I've what may be a silly question.

Promotion of global to local symmetries in order to 'reveal' gauge fields (i.e. local phase invariance + Dirac equation -> EM gauge field) is, as far as i can tell, always done on the Lagrangian, as the Lagrangian must be invariant under the symmetry imposed.

My question is: does this necessarily imply that the equation of motion is always invariant? If so, is the procedure of trying to find an invariant equation of motion equivalent? Is it just that due to the churning of the Lagrangian through the Euler Lagrange equations, the EoM is usually more complicated and so harder to see easy ways to make things invariant under certain operations?

Thanks,
-FD
 
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  • #2
What do you mean about the equation of motion being invariant?
for example a non-interacting spin-1/2 field EOM is just the simple Dirac Equation.
If you allow for the existence of the spin-1 "photonic" field, coming from the local U(1) gauge symmetry, then the Dirac equation changes (you apply the minimal coupling [itex]p^{\mu} \rightarrow p^{\mu} - q A^{\mu} [/itex]). In that sense, since your Lagrangian is invariant, then the EOM are also going to remain invariant (however they won't be the same for the 2 cases I mentioned). In the last case, both a transformed and not transformed Lagrangian are the same.
In addition invariances (in general) are mainly to keep the action invariant and not the Lagrangian (for example the last can change up to a total derivative, and yet yield the same EoM). Now if the Lagrangian happens to remain invariant, so does the action.
 
  • #3
Yes, I guess what I am asking is does an invariant Lagrangian imply an invariant EoM? If so why not apply the gauge symmetry to the EoM and search for a wavefunction that makes the EoM invariant under the symmetry?
 
  • #4
This question could also be asked in the classical mechanics...
What's the difference between working in the Lagrangian formalism and the 2nd Law of Newton (the EoM).
I think it's always easier to see the symmetries of the Lagrangian rather than the EoM.
 
  • #5
What do you mean by 'see' the symmetries, in this case the symmetry is imposed is it not? Do you mean see how to modify the lagrangian to make it invariant under the symmetry?
 
  • #6
yes and not only.
 
  • #7
Great, thanks!
 
  • #8
For example, at least as far as I've seen it, in the EoM you consider you take into account the electromagnetic interactions and thus you can apply the minimal coupling procedure... however in the Lagrangian you don't have to think of that. You just have to try turn a global symmetry that already exists into a local one, and the "electromagnetic" field appears as the connection into the covariant derivative. Also, in addition to that, the Lagrangian gives you the Maxwell equations, through the strength field tensor.
And I guess it's even more difficult to work with other symmetries (as for example SU(2) or SU(3)) from the equations of motion (I think you can't know the last).
 

1. What is a local gauge symmetry?

A local gauge symmetry is a type of symmetry in physics that allows for transformations of the fields in a theory to occur at each point in space and time. This means that the laws of physics remain unchanged under these transformations. Local gauge symmetries are important in the formulation of gauge theories, such as electromagnetism and the Standard Model of particle physics.

2. What is a Lagrangian in relation to local gauge symmetries?

A Lagrangian is a mathematical function that describes the dynamics of a system in terms of its generalized coordinates and their time derivatives. In theories with local gauge symmetries, the Lagrangian is constructed to be invariant under the transformations of the gauge symmetry. This allows for the equations of motion to be derived from the Lagrangian using the principle of least action.

3. How do local gauge symmetries affect the equations of motion?

Local gauge symmetries play a crucial role in determining the equations of motion in a theory. The invariance of the Lagrangian under these transformations leads to the presence of gauge fields, which mediate the interactions between particles. These gauge fields also contribute to the equations of motion, resulting in the existence of gauge bosons and the conservation of certain quantities, such as electric charge and color charge.

4. Can local gauge symmetries be broken?

Yes, local gauge symmetries can be broken in certain situations. This occurs when the ground state of a theory does not exhibit the full symmetry of the Lagrangian. In the Standard Model, for example, the Higgs mechanism breaks the local gauge symmetry of the electroweak force, giving mass to the W and Z bosons and leaving only the electromagnetic force as a long-range force.

5. How do local gauge symmetries relate to the fundamental forces of nature?

Local gauge symmetries are intimately connected to the fundamental forces of nature. In fact, the Standard Model of particle physics is based on the principle of local gauge symmetry, with different symmetries corresponding to the different fundamental forces. For example, the local gauge symmetry of the strong nuclear force is described by the SU(3) group, while the electroweak force is described by the combined SU(2) x U(1) symmetry.

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