# Tagged Questions

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### Conservation of kinetic energy on a moving inertial frame

The velocity of an object differs from the point of views of two different inertial observers standing at two different frame of reference. Assuming no gravity and acceleration = 0 for the object and ...
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### Conserved charge from conserved current associated with translational invariance

(c.f Di Francesco, 'Conformal Field Theory' P.45) Di Francesco calls the conserved charge arising from the conserved current associated with a translation invariant theory the 'four momentum'. While ...
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### 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 ...
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### Translations and Noether's Theorem

I'm fine with $U(1)$ symmetry and Noether's Theorem, but struggling with the translations of the field; namely $$\phi'(x^{\mu})=\phi(x^{\mu}-a^{\mu}),$$ where $a^{\mu}$ constant four-vector ...
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### Intuition Behind Conservation of Angular Momentum

I'm having a fairly hard time understanding the intuition behind Noether's derivation of the conservation of angular momentum from the rotational invariance of the Lagrangian, though I do understand ...
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### Noether's Theorem For Functionals of Several Variables

My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Does Noether's theorem, for functionals of a single independent variable, just say ...
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### Easy proof of Noether's theorem? [duplicate]

Where could I find an easy proof of Noether's theorem? I mean I know that the variation must be $0=\delta S = (EULER-LAGRANGE)+ (CONSERVED\, \, \, CURRENT)$ for the case of a particle $q(t)$. I ...
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### What is the symmetry associated with electric charge conservation [duplicate]

Is there a kind of symmetry that yields the conservation of charges? and if so , how it works for both type of charges? Electrical Charges
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### How are symmetries defined mathematically? [duplicate]

I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding ...
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### Nonlinear Klein Gordon equation

For the Klein Gordon nonlinear equation, $$u_{tt}- \Delta u +f(u)=0,$$ how could I use Noether's theorem to prove that there is a conserved quantity? I.e., $$(\Pi _{k} )_{t} - \rm div(j_{k})=0$$ ...
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What's a good book (or other resource) for an advanced undergraduate/early graduate student to learn about symmetry, conservation laws and Noether's theorems? Neuenschwander's book has a scary review ...
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### Does the action and Lagrangian have identical symmetries and conserved quantities?

From the book Introduction to Classical Mechanics With Problems and Solutions by David Morin, page 236 states: Noether's Theorem: For each symmetry of the Lagrangian, there is a conserved ...
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### Symmetry of Minkowksi Metric -> Conserved Current?

My understanding of the Minkowski Metric is that we have the freedom to choose whether to place the negative sign on the time-component or on the spatial-components. That is, either basis should ...
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### How is the current for the Dirac equation derived?

Why is it that the derivative of the current $j^\mu$ is the difference between the Dirac equation and its adjoint?
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### Where do the conservation laws come from?

I know the conservation of energy comes from Noether's theorem via the time-translational symmetry, and if I remember correctly, the conservation of momentum comes from space-translational symmetry. ...
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### Energy-momentum conservation without translation symmetry?

As I checked, the energy-momentum tensor defined as ${T^\mu}_\nu=\frac{\partial {\cal L}}{\partial(\partial_\mu \phi)}\partial_\nu \phi-{\cal L}{\delta^\mu}_\nu$ at the solution $\phi$ of equation of ...
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### Noether's current expression in Peskin and Schroeder

In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we ...
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### Constants of motion vs. integrals of motion

Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
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### What conservation law corresponds to this local $U(1)$ symmetry of the CCR?

It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by ...
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### Does a constant factor matter in the definition of the Noether current?

This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is: Consider a field Lagrangian with only ...
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### Noether theorem and classical proof of electric charge conservation

How to prove conservation of electric charge using Noether's theorem according to classical (non-quantum) mechanics? I know the proof based on using Klein–Gordon field, but that derivation use ...
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### How to apply Noether's theorem

Say I have a point transformation: $$x' ~=~ (1 +\epsilon)x,$$ $$t' ~=~ (1 +\epsilon)^2t,$$ and Lagrangian $$L ~=~ \frac{1}{2}m\dot{x}^2 - \frac{\alpha}{x^2}.$$ How do I go out about showing ...
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### Why has the trace of the energy-momentum tensor to vanish for conserved scaling currents to exist?

In this paper, the authors say that the trace of the energy-momentum tensor has to vanish to allow for the existence of conserved dilatation or scaling currents, as defined on p 10, Eq(22)  ...
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### Improved energy-momentum tensor

While still dealing with this issue, I've stumbled upon this answer to a question asking about the conserved quantity corresponding to a scaling transformation. It mentions that in accordance with ...