# Tagged Questions

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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### Noether Current when the Lagrangian depends on second derivative of the fields

Let a Lagrangian density for a field theory of $N$ fields $\left\{\phi_i\right\}_{i=1}^N$ be given. Assume that the Lagrangian density depends on the fields, their spacetime derivatives, and their ...
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### On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
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### Lorentz transformation and symmetries of the Lagrangian [duplicate]

Since the Lagrangian of our quantum field theories is covariant under Lorentz transformations I'm asking myself if there is any link to some symmetries (like that we get from gauge transformations ...
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### Rigorous derivation of general relativity from first principles

What is the minimal set of axioms required to derive the mathematical formulation of General Relativity from first principles? What are these first principles? What are good references that detail ...
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### Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
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### What's the interpretation of Feynman's picture proof of Noether's Theorem?

On pp 103 - 105 of The Character of Physical Law, Feynman draws this diagram to demonstrate that invariance under spatial translation leads to conservation of momentum: To paraphrase Feynman's ...
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The equation of motion of a damped oscillator $$\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x=0$$ which is invariant under time-translation $t\rightarrow t+a$, but not under time reversal $t\... 1answer 94 views ### Energy-momentum tensor transformation [closed] I've been trying to find how the energy-momentum tensor changes if we add a total derivative to the lagrangian: $$L\to L+\mathrm d_\mu X^\mu.\tag{1}$$ From the answer key: $$T^{\mu\nu}\to T^{\mu\nu}... 4answers 1k views ### When can a global symmetry be gauged? Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be ... 4answers 17k views ### Why can't energy be created or destroyed? My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed. Why is that? Is there a theory or scientific evidence that proves his statement true or ... 0answers 65 views ### how are the infinitesimal generators of translation related to the lagrangian? In studying analytical mechanics (or it's quantum analog), one will come across statements such as:$$f(x^{i}+\delta x^{i})=f(x^{i})+\delta f(x^{i})=f(x^{i})+\frac{\partial f(x^{i})}{\delta x^{i}}\... 0answers 27 views ### Scale invariance and stress energy tensor I have seen in a paper [1] that in a quantum field theory scale invariance takes place provided the stress energy tensor is traceless. How this is true? References: "INFINITE CONFORMAL SYMMETRY IN ... 2answers 126 views ### Conservation Laws and Symmetry The toughest of topics in physics, like Quantum Mechanics, Relativity, String theory, can be explained in layman words and many have done so. Though there is no substitute to the understanding a ... 0answers 27 views ### Lorentz invariance & Noether theorem of classical ED I want to check invariance of the action under Lorentz boosts for classical electrodynamics. The action is $$S = \int \mbox{d}^4x F_{\alpha \beta} F^{\alpha \beta}$$ I assumed that the fields ... 1answer 383 views ### Conservation of BRST current in QED I am trying to understand the conservation of the BRST current in QED but am having some trouble. This is what I have so far, QED lagrangian density in Lorenz gauge is, $$L = \frac{1}{4}F_{\mu\nu}F^{\... 1answer 204 views ### Understanding Noether's theorem rigorously I've known about Noether's theorem for some time and reading some things about it recently I've realised I haven't completely understood it. In that case I've been trying to understand a more rigorous ... 2answers 136 views ### Is the Noether charge always a Hermitian operator? Noether's theorem tells us that to every continuous symmetry of the Lagrangian there corresponds a conserved current j^\mu. From the time component of this current, we can then define the Noetherian ... 4answers 296 views ### Noether's theorem for space translational symmetry Imagine a ramp potential of the form U(x) = a*x + b in 1D space. This corresponds to a constant force field over x. If I do a classical mechanics experiment with a particle, the particle behaves ... 0answers 52 views ### Expression of Noether current corresponding to Lorentz transformation of scalar fields I find the derivation of Noether current given in the post http://physics.stackexchange.com/a/56905/45429 very clear. However, following a similar logic, I am having problems deriving the Noether ... 0answers 14 views ### Isolated system and mutual interaction potential We know that the total linear momentum of a closed (isolated) system is conserved due to homogeneity of space (Landau and Liftshitz, page 15, Mechanics). Hence for an isolated system of two bodies ... 0answers 27 views ### SU(2) symmetry and conservation law in condensed matter systems [closed] My question has a few parts, I know from Noether that if there is a symmetry in a Hamiltonian, there is a conservation law. What would be the conservation law associated with SU(2) symmetry? ... 1answer 88 views ### Deeper principles in classical mechanics While teaching introductory physics, my professor explained that the conservation of linear momentum, conservation of energy and conservation of angular momentum are based on deeper principles in ... 1answer 79 views ### What is the reason behind why energy must always be conserved, apart from observation? [duplicate] I know that we see in experiments (physical and thought) that energy is always transformed into something else, but what propels our universe to behave this way? What is happening at small levels that ... 0answers 25 views ### How can intuitively guess what conserved quantities has the system that I am studying? I'm taking a course in Classical Electrodynamics and in one problem my teacher introduced us to a triplet of fields (\phi^a) invariant under internal rotations, i.e. transformations like:$$\phi'^... 0answers 46 views ### Recovering a symmetry transformation from a conserved charge I'm going through some notes on how to apply the Hamiltonian formalism to systems with gauge invariance and I found a derivation of Noether's theorem I had never seen before. The idea is roughly that ... 0answers 38 views ### Free Complex scalar field and conservation principle In a free complex scalar field, the difference between the number of Particles and antiparticles is conserved. This constarint can be satisfied with a simultaneous creation of equal number of ... 1answer 500 views ### Does time invariance conclude conservation of energy? [closed] I find it hard to understand that time-translation invariance necessarily implies conservation of energy. As I understand it, Noether's theorem says that there is an energy conservation because the ... 1answer 100 views ### What symmetry gives you charge conservation? This is a popular question on this site but I haven't found the answer I'm looking for in other questions. It is often stated that charge conservation in electromagnetism is a consequence of local ... 2answers 71 views ### Variation of a Lagrange density Symmetries So I am reading Goenner's Spezielle Relativitästheorie and I am currently in chapter §4.9.1 Variation under Inclusion of Coordinates p. 129. So basically we have: $$\delta W_\zeta=\int d^4x' \mathcal{... 0answers 53 views ### What kind of conservation law is energy conservation in thermodynamics? As I understand it, Noether's theorem is an important result that allows us to show when certain kinds of conservations arise. Is energy conservation in thermodynamics a result of Noether's theorem? ... 1answer 95 views ### Are there conserved quantities in field theory which don't arise from Noether's Theorem? In some QFT texts one writes down the number operator N for free theories, such that when acting on an n-particle state |n\rangle we have$$N|n\rangle=n|n\rangle$$In free theories this is a ... 1answer 64 views ### Is spin angular momentum conserved? According to the Noether theorem, we only have the conserved quantity$$J+S,$$where J is the orbital angular momentum and S is the spin angular momentum. But I am always impressed that the spin ... 2answers 183 views ### Pass to globally conserved currents from locally conserved currents in curved spacetime Let us begin with a Lagrangian of the form$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$where$$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$... 1answer 67 views ### Einstein tensor as a conserved current? As is well-known, the traditional" conserved quantities (energy, momentum...) are Noether currents whose conservation depends on the existence of various Killing fields in Minkowski space. In ... 2answers 86 views ### How to describe time-shifts in Noether's theorem in Hamiltonian formalism As was described in, for example, this post, one can formulate Noether's Theorem also in Hamiltonian Mechanics. Symmetries are then represented by vector fields generated by observables whose Poisson ... 2answers 821 views ### A kind of Noether's theorem for the Hamiltonian formalism How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ... 1answer 61 views ### Conserved currents from Noether's theorem I'm not sure if I understand the concept correctly. Given an infinitesimal transformation$$\phi \rightarrow \phi + \alpha \Delta\phi$$the change in the Lagrangian density \mathcal{L}(\phi,\... 2answers 299 views ### What symmetry is associated with conservation of Lipkin's zilch? The 'zilch' of an electromagnetic field is the tensor$$ Z^{\mu}_{\ \ \ \nu\rho}=^*\!\!F^{\mu\lambda}F_{\lambda\nu,\rho}-F^{\mu\lambda}\,{}^*\!F_{\lambda\nu,\rho} \tag1$$given in terms of the ... 3answers 3k views ### 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 ... 0answers 110 views ### Noether's theorem and translations I'm a bit confused about Noether's theorem (or about calculus of variations in general) when it comes to the translational symmetry$x^\mu\mapsto {x'}^\mu=x^\mu-a^\mu$. My professor just wrote that if ... 1answer 70 views ### How is it possible to vary time without affect the coordinates or their derivatives? In the context of Noether's theorem , the Hamiltonian is the constant of motion associated with the time-translational invariance of the Lagrangian. Time-translational invariance is equivalent to the ... 1answer 60 views ### Why Conserved Current Should Not Need Renormalization? May be this is trivial but I need to understand why the renormalization of conserved current is not necessary ? As for example, in this paper, they demand (2$^{nd}$paragraph of the 2$^{nd}\$ column in ...
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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 ...