# 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 to check for invariance in Lagrangian after gauge transformation?

If I have the Lagrangian density: $$\mathcal{L}=\left(\partial_{\mu} \phi^{*}\right)\left(\partial^{\mu} \phi\right)-m^{2} \phi^{*} \phi$$ How can I show it is invariant under the following gauge ...
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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, ...
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### Noether current Lorentz rotation massive vector field

I'm considering a massive vector field in classical field theory. With the Lagrangian density $$\mathscr{L}=-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}+\frac{1}{2}m^2V^{\mu}V_{\mu}.$$ I want to prove from the ...
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### How to calculate the conserved energy $E$ from the Lagrangian?

I am reading a PhD thesis that considers the Lagrangian $$\mathcal{L}=\partial_\mu\phi\partial^\mu\phi^\star-U(|\phi^2|)$$ where $\phi$ is a complex scalar field and $U(|\phi|^2)$ is an arbitrary ...
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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? ...
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 $A_\mu \to ... 3answers 114 views ### How can we derive from$\{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 ... 2answers 137 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 ... 1answer 84 views ### 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 ... 1answer 33 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 ... 2answers 129 views ### 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 ... 1answer 73 views ### 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$L = L(q_i, \dot q_i,t)$. ... 3answers 113 views ### 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 ... 2answers 65 views ### 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 ... 2answers 62 views ### 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 $$\vec{E} = -\frac{1}{c}\frac{\partial\vec{A}}{\partial t} -... 2answers 54 views ### Exercise about symmetry in the Lagrange equations [closed] This question was asked during a classical mechanics exam (no solutions were given afterwards). Suppose a free particle in \mathbb{R}^n with the following Lagrangian:$$L = \frac{m}{2}\sum_{i=... 0answers 23 views ### 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 ... 1answer 148 views ### Noether's theorem for scale invariance [duplicate] When we have the Lagrangian $$\mathcal{L} = \frac{1}{2} \partial _\mu \phi\partial^\mu \phi \tag{1}$$ We have a symmetry given by $$x^\mu\mapsto e^\alpha x^\mu, \qquad\phi\mapsto e^{-\alpha} \phi.\... 0answers 50 views ### Noether's conserved charges: why are there not “spatially” conserved charges? [duplicate] Noether theorem implies that there is a conserved current j^\mu for every continuous symmetry of the action, i.e.$$\partial_\mu j^\mu=0 $$to each conserved current we can associate a conserved ... 0answers 79 views ### Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5) In Weinberg's QFT textbook, using a gauge transformation$$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_{... 1answer 57 views ### 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 ... 0answers 40 views ### 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\$...
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 ...