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Symmetry modulo total derivative term in Noether's Theorem

I came across the proof of Noether's Theorem in David Tong's notes (page 14) on QFT. He writes something like, We say that the transformation $$\delta\phi(x) = \chi (\phi) \tag{1.34}$$ is a ...
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1answer
17 views

Srednicki chapter 22: continuous symmetries and conserved current

In Srednicki's book he says that: The Noether current plays a special role if we can find a set of infinitesimal field transformations that leaves the lagrangian unchanged, or invariant. In this ...
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1answer
84 views

Noether's theorem for arbitrary coordinate transformations

I have been reading Introduction to Conformal Field Theory by Blumenhagen and Plauschinn. Equation (2.19) on page 19 states that if our theory is invariant under a general conformal transformation $x^\...
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1answer
45 views

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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1answer
52 views

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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0answers
23 views

Interpretation of vanishing Noether charge

I was told that Gauge symmetries are redundancies because the Noether charge of a gauge symmetry vanishes, i.e. that there exist no observable quantities that would allow you to distinguish two ...
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1answer
48 views

Interpreting the conserved charge in scalar QED

In scalar QED, applying Noether's theorem for internal global symmetries results in a Noether current that is dependent on the gauge because of the presence of the covariant derivative. $$j_\mu=-i(\...
2
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2answers
113 views

Does it make sense to speak in a total derivative of a functional? Part III

In this third part of the series, I will continue the deduction of Noether's theorem initiated in the previous post - Does it make sense to speak in a total derivative of a functional? Part II. ...
2
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2answers
179 views

Does it make sense to speak in a total derivative of a functional? Part II

I am trying to derive the Noether theorem from the following integral action: \begin{equation} S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}% \phi_{r},x\right) , \tag{II.1}\...
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1answer
149 views

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 ...
1
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2answers
125 views

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 ...
2
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1answer
154 views

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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2answers
103 views

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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2answers
65 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} -...
2
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1answer
151 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.\...
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0answers
51 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 ...
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0answers
41 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 $S$...
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2answers
62 views

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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0answers
49 views

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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0answers
73 views

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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2answers
294 views

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, ...
3
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1answer
119 views

Redefinition of spacetime coordinates for Noether's Theorem

In the derivation of Noether's theorem some authors consider not only redefinitions of the fields \begin{equation} \phi(x) \rightarrow \phi'(x) = \phi(x) +\delta\phi(x) \end{equation} but also ...
3
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2answers
152 views

Transformation of coordinates in Noether's Theorem

I am confused, in the proof of Noether's theorem, by the change of boundary in the action integral during the transformation of coordinates. I have seen on Wikipedia that along with the change of ...
2
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2answers
350 views

Energy-momentum tensor of transformed Dirac Lagrangian

Consider the standard Dirac Lagrangian, $\mathcal{L}=\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi$, and a transformed one differing by a total derivative $$ \mathcal{L}'=\mathcal{L}-...
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2answers
101 views

Trying to check current conservation under symmetry transformation

Consider a simple scalar field and its Lagrangian $L=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$. Then say you have the following transformation $$x^{\mu}\rightarrow e^{\omega}x^{\mu},\tag{1}...
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1answer
157 views

Translational invariance and the derivation of the energy-momentum tensor

Consider a field $\phi$ which transforms as $\phi\rightarrow\phi+\delta\phi$ and say $X\left(\phi\right)=\delta\phi$ is a symmetry of your Lagrangian, which under that transformation changes by a ...
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0answers
43 views

How to find $F^\mu$ in Noether's current

According to David Tong's lectures on QFT, Noether's current is given by $$ J^\mu = \left( \sum_{\forall\text{ fields }\phi_{i}} \delta \phi_{i} \frac{\partial L}{\partial(\partial_{\mu}\phi_{i})} \...
0
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1answer
58 views

Missing a factor in the conserved current equation for QFT

I am currently reading some notes in QFT and I came across the conserved current equation for a complex scalar field that has a transformation given by $$ \hat{\phi}\longrightarrow\hat{\phi}+i\theta\...
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5answers
124 views

Help with D. Tong example on Noether in QFT

In this lectures, example 1.3.2 on page 14 concludes that the Noether current is But how can the current be a two index object when it is defined in eq. (1.38), which is as a one index object? ...
3
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2answers
411 views

Global $U(1)$ symmetry of 2+1D Abelian-Higgs Model

In the Abelian-Higgs model, $$S=\int d^{3}x\left\{-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+|D\phi|^{2}-a|\phi|^{2}-b|\phi|^{4}\right\}\tag{5.34}$$ there is a $U(1)$ gauge symmetry. In David Tongs' ...
1
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0answers
68 views

2 kinds of Noether charges

When I looked at the Noether's theorem(here, we discuss the field case), there are two ways to derive it. One way is to assume that the variation of Lagrange $\delta L$ is exactly equal to derivative ...
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0answers
40 views

How is the total energy invariant under the addition of a total derivative? [duplicate]

I am trying to answer problem 3.3 part b in Schwartz's QFT book: (b) Show that the total energy $Q=\int \mathcal{T}^{00} d^3 x$ is invariant under the addition of a total derivative to the ...
3
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2answers
93 views

Understanding the total spin as the Noether's charge and rotation generator of the Heisenberg model

Consider the Heisenberg model where the Hamiltonian $$H= J\sum_{\langle i,j\rangle}\textbf{s}_i\cdot \textbf{s}_j$$ has continuous rotational symmetry. Since $\textbf{s}_i\in\mathbb{R}^3$, the ...
1
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1answer
61 views

What is the physical intuition of Noether current?

What is the intuition behind Noether current $$J^{\mu}_N={\Pi}^{\mu}D{\phi}-W^{\mu}$$ where $J^{\mu}_N$ is the Noether current. $D\phi$ is the change of the field $\phi$ with respect to some ...
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3answers
106 views

Proving that the 3-current density corresponding to the global phase invariance vanishes at infinity

The components $j^i$ of the 3-current density $\textbf{j}$ corresponding to the global phase invariance of the action of a complex scalar field $\phi$ i.e., $\phi\to e^{-iq\theta}\phi$ is given by $$...
2
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2answers
396 views

What is the “surface term”?

In Peskin's quantum field theory book, There is a sentence in page 17: ... More generally, we can allow the action to change by a surface term, since the presence of such a term would not ...
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1answer
89 views

Expansion of $\mathcal{L}(\phi'(x'),\partial_\mu'\phi'(x'),x')$ when looking at Noether's current

I just asked this question concerning the application of Noether's theory. Think about this got me wondering about the following. In the usual derivation of the Noether current the assumption is made ...
3
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1answer
424 views

Noether's Current in QFT with position dependent variations?

Setup Consider a mapping $F$ that takes every point $x$ on the manifold $M$ to the point $x'$ on the same manifold. Under this mapping the field $\phi(x)$ evaluated at the point $x$ changes to $\phi'(...
3
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1answer
625 views

Derivation of Noether's Theorem

I have a question concerning the derivation of Noether's theorem in Peskin & Schroeders introduction to QFT. Below is a picture of the first part of it. The proof then proceeds by saying that one ...
1
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1answer
260 views

Variation of the Action under infinitesimal arbitrary transformations and Noether's Theorem

Let's consider an arbitrary infinitesimal transformation of the fields and their coordinates : $$x'^{\mu}= x^{\mu} + \delta x^{\mu} = x^{\mu} + \frac{\delta x^{\mu}}{\delta{\omega}^a}{\omega}^a\tag{1}...
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2answers
179 views

Noether's theorem with infinite parameters

I'm trying to understand something regarding Noether's theorem - and with the given situation, my question isn't that much of a question, I'm rather just seeking confirmation whether I'm thinking ...
5
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3answers
647 views

Conserved charges and generators

For the Klein Gordon field, the conserved charge for translation in space is given by: $$\vec{P}=\frac{1}{2}\int d^{3}k \, \vec{k}\{a^{\dagger}_{k}a_{k}+a_{k}a^{\dagger}_{k}\}$$ If we were to find ...
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0answers
130 views

A problem with Lagrangian density variation in Noether theorem derivation

I am reading a book on classical field theory and have a problem with understanding of one moment. Let the action be $$S = \int_{\Omega} d^4x\mathcal{L}$$ and some continuous transformation $$ \{x \...
1
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1answer
98 views

How do I tell if a symmetry is gauge or not?

I understand the interpretation given in topics like this one that gauge symmetries are "fake" in the sense that they do not represent an actual difference in physical states. I also know that gauge ...
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3answers
425 views

Scalar field in curved spaces

How do we find the energy momentum tensor as Noether charge for translations in curved spaces. This should still exist since the action is still an integral over space such that it it invariant under ...
8
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4answers
398 views

What constitutes a symmetry for Noether's Theorem?

I have some confusion over what exactly constitutes a symmetry when trying to apply Noether's theorem. I have heard both that a symmetry in the action gives a conserved quantity, and that a symmetry ...
5
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1answer
649 views

Big puzzle about Noether's theorem of coordinate transformation (spacetime symmetry)

For Noether theorem with only internal symmetry, I've found there has been a very clear proof. But I still struggle with the proof of coordinate transformation. Because there are so many different ...
1
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1answer
424 views

Understanding this phrasing of a conserved current in QFT

I'm trying to derive a version of Noether's theorem. I found this online and I'm really confused as to what's going on on page 202 (the second slide in the link). I'll start off by saying I am ...
5
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1answer
736 views

How to find the continuous transformations which leave the action invariant?

Assume one has a continuous transformation of fields, and also of coordinates - in case if we consider coordinate transformations as well. Global internal symmetries, rotations, translations, ...
1
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2answers
132 views

Hilbert space operator associated to gauge transformation

Suppose we have a Lagrangian that with fields that are acted on by a symmetry group, e.g. $$\mathcal{L} = \partial_{\mu}\phi \partial^{\mu}\phi^* - m^2 \phi \phi^*$$ with $G=U(1)$ (i.e. $\phi \to e^{i ...