Linked Questions
30 questions linked to/from On a trick to derive the Noether current
24
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6
answers
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Hamiltonian for relativistic free particle is zero
One possible Lagrangian for a point particle moving in (possibly curved) spacetime is
$$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$
where a dot is a derivative with respect to a parameter $\...
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16
votes
5
answers
7k
views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
- 1,273
15
votes
3
answers
12k
views
Energy-momentum tensor from Noether's theorem
In the book "Quantum Field Theory" by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p. 22):
Assume a Lagrangian density depending on the spacetime ...
- 4,156
11
votes
2
answers
3k
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Why is the Hamiltonian zero in relativity?
I'm trying to understand something with the lagrangian and the hamiltonian formalisms in relativity theory, and why the following result cannot be the same in classical (non-relativistic) mechanics. ...
- 7,677
17
votes
1
answer
5k
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Trick for deriving the stress tensor in any theory
In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case ...
- 929
8
votes
1
answer
2k
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How to calculate an axial anomaly in 1+1 dimensions?
As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like
$$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{...
- 2,678
4
votes
2
answers
5k
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Prove energy conservation using Noether's theorem
I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
- 1,155
7
votes
3
answers
1k
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Derivation of the Noether current
(Cf. Di Francesco et al, Conformal Field Theory, pp. 40-41)
I am trying to derive eqn. (2.142) or
$$\delta S = \int d^d x ~\omega_a~\partial_{\mu}j^{\mu}_a \tag{2.142}$$
in the book CFT by Di ...
- 3,599
2
votes
2
answers
875
views
Variation of the Lagrangian and the Noether current
In Schwartz’s book, QFT and Standard Model, section 8.3.1, he writes
if we then let $\alpha$ be a function of $x$, the transformed $\mathcal L_0$ can only depend on $\partial_\mu \alpha$. Thus, for ...
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3
votes
1
answer
2k
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Noether's Theorem in Classical Field theory Confusion
Consider $N$ independent scalar fields $φ_i (x)$ in 4D space. Also consider a lagrangian density $$\mathcal{L} = \mathcal{L}(φ_i, \partial_μφ_i).$$
Suppose we perform the following infinitesimal ...
- 354
3
votes
2
answers
306
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Variation of action in terms of energy-momentum tensor
In Di Francesco, Mathieu, and Sénéchanl, Conformal Field theory section 4.2.2 it is stated that under an arbitrary diffeomorphism $x\rightarrow x+\epsilon$ the action transforms like
$$\delta S=\int d^...
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1
vote
3
answers
1k
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How did we find the Noether current $j^\mu(x) = \bar\psi(x)\gamma^\mu\psi(x)$ for Dirac equation?
The Dirac Lagrangian reads:
\begin{equation*}
\mathcal{L} = \bar\psi(i\not\partial-m)\psi.\tag{1}
\end{equation*}
It's invariant under the transformation $\psi(x)\rightarrow e^{i\alpha}\psi(x)$. Now ...
- 1,853
1
vote
2
answers
614
views
Noether's theorem under arbitrary coordinate transformation
Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
Suppose our action is of the form
$S = \int d^4x\, \mathcal{L}(\...
- 485
2
votes
1
answer
493
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Subtlety in derivation of Noether's theorem by Di Francesco
In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter $\...
- 3,599
3
votes
1
answer
1k
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Variation of the Action under infinitesimal arbitrary transformations and Noether's Theorem
Let's consider an arbitrary infinitesimal transformation of the fields and their coordinates :
$$x'^{\mu}= x^{\mu} + \delta x^{\mu} = x^{\mu} + \frac{\delta x^{\mu}}{\delta{\omega}^a}{\omega}^a\tag{1}...
- 95
4
votes
1
answer
553
views
Conserved currents in Noether Theorem with varying parameter
I have a continuous transformation on the field $\phi$ of the form
$$\phi(x)\rightarrow \phi'(x)=\phi(x)+\alpha\Delta\phi(x),\tag{1}$$
where $\alpha$ is a constant infinitesimal parameter and $\...
- 1,724
3
votes
0
answers
808
views
Global and local symmetries in Noether's theorem. And also Stress-Energy tensors
Noether's theorem for fields is usually given as follows:
Given a field theory with action $S=\int\mathcal{L}(\phi,\partial\phi)d^4x$, and given a one-parameter variation of the fields $\phi_\epsilon$...
- 11.6k
6
votes
1
answer
515
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Comparison between formulations of Noether's theorem
Version 1:
An infinitesimal variation on the fields $\phi\mapsto\phi'$ is said to be a symmetry if $\delta \mathcal{L}:=\mathcal{L}(\phi',\partial\phi')-\mathcal{L}(\phi,\partial\phi)$ is a total ...
- 3,975
5
votes
1
answer
215
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What is the meaning of the parameter in Noether's theorem?
According to the explanation of Noether's theorem in Peskin & Schroeder's QFT book, pp. 17-18,
If the Lagrangian $\mathcal{L}(x)$ change to $$\mathcal{L}(x)+\alpha\partial_\mu\mathcal{J}^\mu\tag{...
- 497
1
vote
0
answers
457
views
Noether's theorem - Making a global symmetry local (via the SEM tensor) [closed]
We are doing Lorentz-invariant Lagrangian field theory in Minkowski spacetime, and I'm now considering only the form of Noether's theorem where the fields are varied.
Assume that $\delta \phi$ is a ...
- 11.6k
4
votes
1
answer
255
views
Deriving conserved currents by promoting parameter
I currently reading Tong's text on String Theory.
In Chapter 4.1.1 he alludes to a technique to derive conserved currents
Recall that we can usually derive conserved
currents by promoting the ...
- 3,902
1
vote
1
answer
156
views
In a simple case of a particle in a uniform gravitational field, do we have translation invariance or not?
Consider a system where a particle is placed in a uniform gravitational field $\vec{F} = -mg\,\vec{e}_{z}$. The dynamics of this are clearly invariant under translations. When we take $z\rightarrow z+...
- 8,776
3
votes
1
answer
248
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A Simple question on Noether's first theorem
I am trying to understand Noether's first theorem in field theory and have read several references on the subject. They are all pretty clear except on one issue that all but one share:
These ...
- 221
4
votes
1
answer
166
views
What is the role of the classical equations of motion in the derivation of the Noether current?
I am trying to understand a very fundamental statement from the Book: Condensed Matter Field Theory from A.Altland and B.Simons:
Suppose we have a transformation:
$$x^\mu \to (x^{\prime})^{\mu} = x^\...
- 155
2
votes
1
answer
72
views
Change of action after a transformation with space-time dependent parameters
I've been following David Tong's lecture on introduction to quantum field theory. In his lecture notes page number 19 (and his video class on Youtube), he talks about global transformation that ...
1
vote
0
answers
87
views
Recovering a symmetry transformation from a conserved charge
I'm going through some notes on how to apply the Hamiltonian formalism to systems with gauge invariance and I found a derivation of Noether's theorem I had never seen before. The idea is roughly that ...
- 732
2
votes
1
answer
87
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Lorentz/rotational invariance parameter doesn't vanish on boundaries
As I know Stress-Energy tensor is defined as Noether current under arbitrary coordinate transformations $\boldsymbol{x} \rightarrow \boldsymbol{x}+\epsilon(\boldsymbol{x})$.
$$
\delta_\epsilon S=\int_{...
- 93
0
votes
1
answer
65
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Masslessness of Goldstone modes
Suppose we have a $G$-invariant action $S$ of a field $\phi$, where $G$ is a Lie group; let then exist a non-zero value $v$ of $\langle\phi\rangle$ such that the $G$-symmetry of the action is broken, ...
- 670
0
votes
0
answers
36
views
Deriving conserved currents from variation of action
I am reading An Modern Introduction to Quantum Field Theory by Maggiore. I have difficulty following the calculation of $\delta ( d^4 x)$ and $\delta (\partial_\mu \phi_i)$. Also, wonder whether the ...
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