Linked Questions

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vote
1answer
336 views

Conserved quantity of a relativistic free Lagrangian for a Lorentz boost

Let $$L~=~-mc^2\sqrt{1- \frac{|\textbf{v}|^2}{c^2} },$$ where $\textbf{v}$ is the usual velocity of the particle in a fixed inertial frame. Then, this is the Lagrangian for a relativistic free ...
23
votes
4answers
5k views

Galilean invariance of Lagrangian for non-relativistic free point particle?

In QFT, the Lagrangian density is explicitly constructed to be Lorentz-invariant from the beginning. However the Lagrangian $$L = \frac{1}{2} mv^2$$ for a non-relativistic free point particle is ...
2
votes
1answer
41 views

What is the definition of a symmetry of an action?

Symmetries of Lagrangians The definition of a symmetry of a theory is quite clear at the level of a Lagrangian. We say a Lagrangian $\mathcal{L}(\phi,\partial_\mu \phi)$ is symmetric under the ...
32
votes
5answers
8k views

Noether charge of local symmetries

If our Lagrangian is invariant under a local symmetry, then, by simply restricting our local symmetry to the case in which the transformation is constant over space-time, we obtain a global symmetry, ...
3
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2answers
83 views

Confusion about Noether's theorem

In my field theory class we recently derived Noether's theorem: We consider a infinitessimal transformation $\phi \to \phi + \epsilon \,\delta\phi$ of our field which preserves action i. e. $\delta S =...
8
votes
2answers
373 views

Why are symmetries in phase space generated by functions that leave the Hamiltonian invariant?

Hamilton's equation reads $$ \frac{d}{dt} F = \{ F,H\} \, .$$ In words this means that $H$ acts on $T$ via the natural phase space product (the Poisson bracket) and the result is the correct time ...
18
votes
2answers
3k views

Is there a kind of Noether's theorem for the Hamiltonian formalism?

The original Noether's theorem assumes a Lagrangian formulation. Is there a kind of Noether's theorem for the Hamiltonian formalism?
2
votes
2answers
103 views

Is this a gauge symmetry?

Imagine a hypothetical action: $$S=\int \left(\frac{\partial}{\partial t}\phi(x,t)\right)^2 d^3x dt$$ Now we have a symmetry of the action: $$\phi(x,t)\rightarrow \phi(x,t)+\chi(x).$$ At time $t$, $\...
4
votes
2answers
121 views

Do total derivatives have anything to do with central extensions?

I recently got interested in the Galilean group and its central extension and found a paper "Quantization on a Lie group: Higher-order Polarizations" by Aldaya, Guerrero and Marmo. Before asking my ...
5
votes
3answers
216 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 ...
1
vote
1answer
98 views

Symmetry modulo total derivative term in Noether's Theorem

I came across the proof of Noether's Theorem in David Tong's notes (page 14) on QFT. He writes something like, We say that the transformation $$\delta\phi(x) = \chi (\phi) \tag{1.34}$$ is a ...
2
votes
1answer
57 views

Why is an action built from superfields guaranteed to be supersymmetric?

Given a superfield (in 0+1 spacetime + 2 superspace coordinates) $$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$ and given the standard supercharges ...
3
votes
2answers
143 views

What's the name of the symmetry $ L \to L + \frac{d \Lambda}{dt}$?

In the Lagrangian formulation of Classical Mechanics, we have the freedom to add a total time derivative of an arbitrary function $\Lambda$ to the Lagrangian: $$ L \to L + \frac{d \Lambda}{dt} . $$ ...
0
votes
1answer
190 views

Conservation of energy for a class of Lagrangians with explicit time dependence

I have read in my book that if $\frac{\partial L}{\partial t}=0$, then the quantity $ L-\frac{\partial L}{\partial \dot{q}} \dot{q} $ is conserved, and we call it the energy of the system. But if ...
1
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1answer
41 views

How do we define the quantity $Q$, in the conservation of energy? And what does it rely on?

Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a ...

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