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 ...