Linked Questions

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0answers
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Different derivations of first Noether's theorem [duplicate]

I'm my current studies in Noether's theorem, the two that I liked the most are joshphysics answer to this Phys SE. post, and the derivation in chapter $4$ of An Elementary Introduction to Classical ...
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5answers
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Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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2answers
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Invariance of Lagrangian in Noether's theorem

Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$. However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
4
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1answer
825 views

Noether's Current in QFT with position dependent variations?

Setup Consider a mapping $F$ that takes every point $x$ on the manifold $M$ to the point $x'$ on the same manifold. Under this mapping the field $\phi(x)$ evaluated at the point $x$ changes to $\phi'(...
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1answer
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Energy momentum tensor from generalized Noether current

Proceeding like here, let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a ...
3
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1answer
272 views

What is the most general expression of Noether's current in classical field theory?

Peskin & Schroeder's expression of the Noether current If a (quasi-)symmetry is defined as a transformation that changes the action by a surface term i.e. $$S\to S'=S+\int d^4x \partial_\mu K^\mu(\...
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1answer
213 views

How to use general expression for Noether's current to get the energy-momentum conservation law?

The most general form of the Noether's current (see here and here) is given by $$j^\mu(x)=\sum\limits_a\frac{\partial \mathscr{L}}{\partial(\partial_\mu\phi_a)}\delta\phi_a -\theta^{\mu\nu}\delta x_\...
3
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2answers
163 views

Why is there an extra term in definition of Noether current for spacetime translations?

I am reading Schwartz's Quantum Field Theory textbook. In chapter 3, Schwartz first defines the conserved current for a symmetry $\phi \rightarrow \phi + \delta \phi$ that depends on a parameter $\...
3
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1answer
167 views

Comparison between formulations of Noether's theorem

Version 1: An infinitesimal variation on the fields $\phi\mapsto\phi'$ is said to be a symmetry if $\delta \mathcal{L}:=\mathcal{L}(\phi',\partial\phi')-\mathcal{L}(\phi,\partial\phi)$ is a total ...
1
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0answers
81 views

2 kinds of Noether charges

When I looked at the Noether's theorem(here, we discuss the field case), there are two ways to derive it. One way is to assume that the variation of Lagrange $\delta L$ is exactly equal to derivative ...