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Questions tagged [noethers-theorem]

A theorem that relates continuous symmetries (continuous transformations that don't affect the value of the lagrangian) to quantities conserved in time.

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How could I derive the Noether charge for a real scalar field?

I know for a (free) complex scalar field $\psi$ the Lagrangian is: $$ \mathcal{L} = \partial^\mu \psi^\ast\partial_\mu \psi$$ and that Noether's theorem from the $U(1)$ symmetry of the system gives a ...
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What is a coordinate-free version of Noethers theorem? [closed]

What are some examples and derivations of some basic symmetries (not coordinate symmetries)? For example I remember a sufficient condition for being a symmetry of the lagrangian system is being an ...
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Infinitesimal transformations that leave the action invariant

I have the Klein-Gordon Lagrangian for three scalar fields and I want to find three independent infinitesimal transformations that leave the action invariant. I suppose that these three ...
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Why is the Lagrangian for space-like geodesics equal to 1?

In Schwarzschild spacetime, the Lagrangian can be defined as $$ L = -\left( 1 - \frac{2M}{r} \right) \dot{t}^2 + \left( 1- \frac{2M}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta ...
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Conserved currents in quantum electrodynamics

A general Noether theorem in fields theory says that an infinitesimal symmetry of the action leads to a conserved current $j^\mu$, i.e. $\partial_\mu j^\mu=0$. Below I would like to consider a minor ...
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Getting a Conserved Quantity from a Lagrangian [duplicate]

So I've been messing around with the implications of Noether's theorem, and though I conceptually get what it's saying, I'm having a hard time actually using it to retrieve a conserved quantity from a ...
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Can Noether's theorem be applied to cyclic time symmetries? [duplicate]

Noether's theorem relates continuous symmetries in the time evolution of a system to a conserved value. Many conserved values, such as conservation of momentum, can be described via this theorem. I'...
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Conservation of magnetic charge

It is well known that the electric charge of a system can be thought of as the Noether charge associated with isotropic large gauge transformations. That is, given Einstein-Maxwell theory $$S=\frac{1}...
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Problems of Klein Gordon equation

Consider the Klein-Gordon equation $$(\square+m^2)\varphi=0.$$ People usually claim that $\varphi^* \varphi$ cannot be interpreted as a probability density because $\int d^3\vec{x}\varphi(t,\vec{x})^*...
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Is there a higher dimension analogue of Noether's theorem?

So I have recently read the proof of Noether's theorem from the book variation calculus by Gelfand. Basically, what I have already seen is that for any single integral functional, if we have a ...
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Can there be conservation laws without a corresponding symmetry? [duplicate]

Noether's theorem implies that if there is a continuous symmetry in the Lagrangian of a system, this necessarily implies a corresponding conservation law. But does the theorem also imply the reverse? ...
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Boundary terms and Symmetries

Consider Maxwell-Chern-Simons theory in 2+1 dimension, with Lagrangian $$L = -(1/4)F_{\mu v}F^{\mu v} + (m^2/4) \epsilon_{\mu v \rho}A^\mu F^{v \rho},$$ when I make a gauge transformation $A_\mu \to ...
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How can we derive from $\{G,H\}=0$ that $G$ generates a transformations which leaves the form of Hamilton's equations unchanged?

In the Hamiltonian formalism, a symmetry is defined as transformation generated by a function $G$ is a symmetry if $$\{G,H\}=0 ,$$ where $H$ denotes the Hamiltonian. On the other hand, a symmetry is ...
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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 ...
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Calculating Noether Current for Electromagnetic Current interacting with a Dirac Fermion

I'm trying to confirm that the conserved current of the Lagrangian $$ {L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} -j^{\mu}A_{\mu}+\bar\psi(i\gamma^{\mu}\partial_{\mu}-m)\psi $$ associated with the ...
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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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What goes wrong, theoretically, when we reverse time?

(Please bear with me if this is a stupid question; I'm not a physicist, just a curious student.) I know that Noether's Theorem links symmetries to conserved quantities: the fact that the laws of ...
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Why are some symmetries invisible to the configuration space Lagrangian $L(q, \dot q,t)$?

Usually, when people talk about Lagrangians they are talking about a function of configuration space variables $q_i$ and their time derivatives $\dot q_i$. This is a function $L = L(q_i, \dot q_i,t)$. ...
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What does a symmetry that changes the Lagrangian by a total derivative do to the Hamiltonian $H$?

A tiny symmetry transformation may change the Lagrangian $L$ by a total time derivative of some function $f$. This is a basic fact used in the proof of Noether's theorem. How can we see the effect of ...
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About “conserved quantities” in a diffeomorphism-invariant theory by Wald and Zoupas

In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism. Let $\xi^a$ be a vector ...
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Momentum density of the EM field - Classical field theory

The Lagrangian density of the EM field is given by $$ \mathcal{L} = \frac{1}{8\pi}\left(E^2-B^2\right) $$ Let $\vec{A}$,$\phi$ be such that $$ \vec{E} = -\frac{1}{c}\frac{\partial\vec{A}}{\partial t} -...
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Exercise about symmetry in the Lagrange equations [closed]

This question was asked during a classical mechanics exam (no solutions were given afterwards). Suppose a free particle in $\mathbb{R}^n$ with the following Lagrangian: $$L = \frac{m}{2}\sum_{i=...
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Higher form fermionic conserved currents

Higher form conserved currents have already been defined, such as those seen in Klebanov and Polyakov's work in 2002. There, the authors studied the $\text{AdS}_4$/CFT correspondence -- more ...
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Noether's theorem for scale invariance

When we have the Lagrangian $$\mathcal{L} = \frac{1}{2} \partial _\mu \phi\partial^\mu \phi \tag{1} $$ We have a symmetry given by $$x^\mu\mapsto e^\alpha x^\mu, \qquad\phi\mapsto e^{-\alpha} \phi.\...
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Noether's conserved charges: why are there not “spatially” conserved charges? [duplicate]

Noether theorem implies that there is a conserved current $j^\mu$ for every continuous symmetry of the action, i.e. $$\partial_\mu j^\mu=0 $$ to each conserved current we can associate a conserved ...
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Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)

In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has: $$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
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Where do I go from here to show that linear momentum is conserved under all instances of translation symmetry?

I've worked through a simple derivation of symmetries implying conservation laws from an invariant Lagrangian. Namely a quantity $Q$ is conserved in the equation below, where $i$ is a degree of ...
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Isometries and coordinate transformations in the context of Noether's Theorem

If I have a theory defined on some manifold, my understanding is that the dynamical objects in the theory should carry a representation of the isometry group of that manifold. Moreover, the action $S$...
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Problem using Noether's theorem in time-dependent lagrangian

I have some problems calculating the conserved quantity for a lagrangian of the form $$ L = \frac{1}{2}m\dot{q}^2 - f(t) a q, $$ because I found the general problem too abstract, I tried at first ...
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Why are infinitesimal shifts sufficient to prove that a symmetry holds

Why are infinitesimal shifts in the Lagrangian sufficient to prove that a symmetry holds? Couldn't a lot of things happen at higher orders? Especially when I am introducing an infinitesimal shift of a ...
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Conserved charge: partial or total derivative?

I want to obtain some clarification on the concept of Noether charge. Given conserved current $J^\mu$ e.g. in free scalar field theory in $(n+1)$ dimensional Minkowski spacetime $M$, i.e. $\partial_\...
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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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Does Noether's theorem apply to constrained system?

The Lagrangian of a constrained system will be $$L-\lambda_1f_1-\lambda_2f_2-...\lambda_kf_k.$$ If a transformation will not affect the constrained Lagrangian, the there is some corresponding ...
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Question about the concepts of Noether charge and Noether current

I read that a noether current occurs when the lagrangian assume vector values. Well, what are noether current and noether charge in comparison to elementary classical mechanics notions of Noether's ...
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Noether charges of spacetime translation in KG field

When applying a spacetime translation $x^\mu\rightarrow x^\mu+a^\mu$ the KG lagrangian density changes by - $$\mathcal{L} \rightarrow \mathcal{L} + a^\nu \partial_\mu \delta^\mu_{\;\nu} \mathcal{L}$$...
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Elementary argument for conservation laws from symmetries *without* using the Lagrangian formalism

It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and ...
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Understanding Noether's second theorem

Wikipedia writes that "if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by $k$ arbitrary functions and their derivatives up to order $m$, ...
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How to calculate Noether current in quantum field theory?

I'm studying particle physics with an experimental approach. I have still few theory lectures including QFT. However, I'm lost about calculating Noether current. I saw this formula for example in my ...
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On-shell and off-shell transformations in Noether's theorem

For any transformation of the fields, $$\varphi\to\varphi'=\varphi+\delta\varphi$$ the change in the Lagrangian can be written as $$\delta\mathcal L = \text{EoM} + \partial_\mu j^\mu\tag{1}$$where "...
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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 ...
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In the derivation of Noether theorem, why do we subtract the same quantity to obtain Noether current? [closed]

In a general approach derivation of Noether theorem, we have $$ \alpha \mathcal{L} = \alpha \partial _{\mu} \left( \frac{\partial \mathcal{L}}{\partial \left( \partial _{\mu} \phi \right)} \Delta \...
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Conserved currents in Yang-Mills theory: gluon current vs. quark current

In Yang-Mills theory there are two currents we can construct. There is the well-known quark current related to the global $SU(3)_C$ symmetry, $$j{}^{\mu\,A}_\text{quark} = \overline{\psi}{}^i \gamma{}^...
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Noether (quasi)symmetry with jet coordinates?

In classical mechanics, there is a symmetry definition for a lagrangian as invariance under $$L\rightarrow L+\dfrac{dF(x)}{dt}$$ or even $$L\rightarrow L+\dfrac{dF(x,\dot{x})}{dt}$$ But, what is the ...
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Fabri-Picasso theorem: Why can we assume that the charge commutes with momentum?

To prove the Fabri-Picasso theorem, we assume that the charge $Q$ is translationally invariant, i.e. it commutes with the momentum operator: $$ [Q,P^\mu]=0 $$ Why can we assume this?
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Infinitely many conserved currents in any QFT?

So I have the following curiosity: Consider for example, in QED, the quantity $$ j^\mu\equiv\partial_\nu (\lambda(x) F^{\mu \nu}) $$ where $\lambda(x)$ is an arbitrary scalar function of spacetime, ...
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Difference between conserved quantities and constants of motion?

In Hamiltonian mechanics, consider extended phase space, the trajectory followed by a particle in that space is formed by an intersection of different 2n dimensional surfaces, all of these surfaces ...
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Gauge transformations with varying phase give us conservation of the charge density. Hence charged particles cannot move?

I stumbled upon the following paragraph in Quark confinement and Topology of gauge theories by Polyakov "Gauge invariance with constant phase $\Psi \to e^{i \alpha}$ lead to conservation of the ...
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What happens to the global $U(1)$ symmetry in alternative formulations of Quantum Mechanics?

The global $U(1)$ symmetry in Quantum Mechanics corresponds to the freedom to shift the phase of the wave function $$ \Psi \to e^{i\varphi} \Psi \, $$ and can be used to understand the conservation ...
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Does a source term for electric charge necessarily break global $U(1)$ symmetry?

The conservation of electric charge in, e.g., quantum electrodynamics $$\mathcal{L} = -\frac{1}{4}F^2 - A \cdot J + \mathcal{L}_\mathrm{matter}(J)$$ can be derived using the invariance under global $U(...
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What is the Noether charge associated with the the color $SU(3)$ symmetry of QCD?

A version of the Noether's theorem applies to local gauge symmetries. What is the Noether's charge associated with a non-abelian gauge symmetry such as the color $SU(3)$ and how is that derived? I ...