Why Lagrangian is Used in Quantum Field Theory

[lonewolf219]

Is the lagrangian used in QFt because its the only information of motion we can obtain about a system at relativistic speeds? Does the lagrangian reflect the conservation of energy ? Is this why the lagrangian must be invariant... Meaning that it must be constant... Meaning that energy is conserved, which is how other values of the system can be determined?

 

[stevendaryl]

Is the lagrangian used in QFt because its the only information of motion we can obtain about a system at relativistic speeds? Does the lagrangian reflect the conservation of energy ? Is this why the lagrangian must be invariant... Meaning that it must be constant... Meaning that energy is conserved, which is how other values of the system can be determined?

The nice things about formulating physics in terms of a Lagrangian are:

1. It makes it easy to see that the physics has the appropriate symmetries (translations, rotations, lorentz transformations, time translations, etc.)
2. The symmetries automatically imply corresponding conservation laws (via Noether's theorem)
3. Newton's "equal and opposite" rule (and its generalization) is automatically enforced.

The last point is an amazingly powerful tool in developing theories of physics. For example, if you start with the free particle Lagrangian, and then add a term to reflect the effect of the electromagnetic field on charged particles, you automatically get the corresponding term describing how charged particles affect the electromagnetic field.

 

[physwizard]

Lagrangian for theories compatible with relativity can be written in a manifestly Lorentz-invariant way whereas the hamiltonian formalism requires an explicit choice of the time direction and separation of the spatial coordinates and time. Energy may be conserved but it is not invariant.

Is this why the lagrangian must be invariant... Meaning that it must be constant...

Saying that something is invariant and saying that it is a constant are two different things.

 

[lonewolf219]

Thanks for the correction physwizard...
Stevendaryl, so we need symmetries to invoke Noether's theorem, so we can invoke conservation laws? Is that right? For convenience? And why do we need to perform translations and rotations? Are we constructing a vector space?

 

[stevendaryl]

Thanks for the correction physwizard...
Stevendaryl, so we need symmetries to invoke Noether's theorem, so we can invoke conservation laws? Is that right? For convenience? And why do we need to perform translations and rotations? Are we constructing a vector space?

If a theory is not invariant under rotations, then that implies that there is a perferred direction in space. If a theory is not invariant under translations, then that implies that there is a preferred location in space. Turning those around, if we believe that there are no preferred directions in space, and there are no preferred locations in space, then our theories should be invariant under rotations and translations.

 

[lonewolf219]

Thanks for your help stevendaryl! I'm trying to understand SU(2)... Is it that the Pauli matrices need to be invariant under these transformations? I am trying to see the chain that takes us from the basis vectors (I think are the generators?) all the way to Noether's theorem... Maybe I don't understand the relationship between the Lie Group, the Lie Algebra and the Lagrangian... But maybe this is a different question?