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
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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
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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. ...
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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}$$
in the book CFT by Di ...
- 3,599
2
votes
2
answers
875
views
Variation of the Lagrangian and the Noether current
In Schwartz’s book, QFT and Standard Model, section 8.3.1, he writes
if we then let $\alpha$ be a function of $x$, the transformed $\mathcal L_0$ can only depend on $\partial_\mu \alpha$. Thus, for ...
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3
votes
1
answer
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Noether's Theorem in Classical Field theory Confusion
Consider $N$ independent scalar fields $φ_i (x)$ in 4D space. Also consider a lagrangian density $$\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}...
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