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5 votes
2 answers
630 views

Confusion about Noether's Theorem

In classical mechanics, a transformation $q \rightarrow q + \delta q$ is a symmetry if the resultant change in the Lagrangian is a total derivative, $$ \delta L = \frac{dF}{dt}.$$ If we derive the ...
Bilge K. Aksebzeci's user avatar
3 votes
0 answers
69 views

Consequences for symmetries of the equations of motion in QFT

In general, if a Quantum Field Theory is described by a Lagrangian $\mathcal{L}$, the symmetries of $\mathcal{L}$ lead to classically conserved currents along the equations of motion and Ward ...
Marcosko's user avatar
  • 382
2 votes
1 answer
221 views

Noether’s second theorem: about the action principle

Noether's second theorem is supposed to show that the invariance of the Lagrangian by the Lie group (infinite in dimension) of certain theories necessarily implies that the field equations proper to ...
Husserliana's user avatar
3 votes
2 answers
118 views

Can the $\eta_{\mu\nu}\mathcal{L}$ term in canonical energy–momentum tensor be omitted?

From Noether theory we can define the canonical energy–momentum tensor as \begin{equation} T_{\mu\nu}\equiv\frac{\partial\mathcal{L}}{\partial(\partial^\mu\phi)}\partial_\nu\phi-\eta_{\mu\nu}\mathcal{...
Kernifan's user avatar
1 vote
1 answer
101 views

Why the classical Euler-Lagrange equation is assumed when deriving the Noether's conserved current?

As known, in QFT, the conserved currents, such as the energy-momentum tensor, can be derived from the Noether's theorem and expressed as the product of the field operators. These conserved currents ...
dudulu's user avatar
  • 163
3 votes
2 answers
599 views

Deriving conserved charges from the equations of motion

It is very well established how to derive conserved charges associated to the symmetries of Lagrangian using the Noether's theorem. Also in the Hamiltonian formulation, we know how to derive the ...
Ali Seraj's user avatar
  • 1,140
5 votes
3 answers
425 views

In what sense are the equations of motion conserved by symmetries?

I am studying variational principles and I have been reading this set of notes by Townsend. In the first paragraph of Section 9, Townsend defines what it means for a transformation to be a symmetry of ...
MB10000's user avatar
  • 51
1 vote
2 answers
669 views

Showing the invariance of the equations of motion

It is strange to me that for a symmetry which involves $\dot{x}$, there seems to always appear a term with $\dddot{x}$ in the variation of the equations of motion, which doesn't makes much sense. I ...
rsaavedra's user avatar
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