Questions tagged [gauge-invariance]

Invariance of a physical system (its action) under a continuous group of local transformations underlain by a global symmetry whose group parameters fixed in space-time have now been extended to vary in space-time instead. Use for buildup of the invariance, fixing the gauge, and accounting for the corresponding changes in the functional measure of the system.

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100
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9answers
24k views

What, in simplest terms, is gauge invariance?

I am a mathematics student with a hobby interest in physics. This means that I've taken graduate courses in quantum dynamics and general relativity without the bulk of undergraduate physics courses ...
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Noether's theorem and gauge symmetry

I'm confused about Noether's theorem applied to gauge symmetry. Say we have $$\mathcal L=-\frac14F_{ab}F^{ab}.$$ Then it's invariant under $A_a\rightarrow A_a+\partial_a\Lambda.$ But can I say that ...
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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 ...
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0answers
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How to apply the Faddeev-Popov method to a simple integral

Some time ago I was reviewing my knowledge on QFT and I came across the question of Faddeev-Popov ghosts. At the time I was studying thеse matters, I used the book of Faddeev and Slavnov, but the ...
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3answers
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What role does “spontaneously symmetry breaking” played in the “Higgs Mechanism”?

In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of ...
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To which extent is general relativity a gauge theory?

In quantum mechanics, we know that a change of frame -- a gauge transform -- leaves the probability of an outcome measurement invariant (well, the square modulus of the wave-function, i.e. the ...
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1answer
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Are there massless bosons at scales above electroweak scale?

Spontaneous electroweak symmetry breaking (i.e. $SU(2)\times U(1)\to U(1)_{em}$ ) is at scale about 100 Gev. So, for Higgs mechanism, gauge bosons $Z$ & $W$ have masses about 100 GeV. But before ...
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Why does charge conservation due to gauge symmetry only hold on-shell?

While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
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2answers
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Why do we seek to preserve gauge symmetries after quantization?

Gauge symmetries do not give rise to conservation laws via Noether's theorem, and they represent redundancies in our description of the system. So why do we want to keep them after quantization? For ...
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3answers
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Noether theorem, gauge symmetry and conservation of charge

I'm trying to understand Noether's theorem, and it's application to gauge symmetry. Below what I've done so far. First, the global gauge symmetry. I'm starting with the Lagragian $$L_{1}=\partial^{\...
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2answers
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Physical difference between gauge symmetries and global symmetries

There are plenty of well-answered questions on Physics SE about the mathematical differences between gauge symmetries and global symmetries, such as this question. However I would like to understand ...
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1answer
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How do symmetries “define” physical laws?

First of all, I do not have any problems concerning what symmetries are or how to describe them. However, I do not have any knowledge concerning how the reasoning for quantum field theory and thus ...
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1answer
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Faddeev-Popov Gauge-Fixing in Electromagnetism

Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. ...
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1answer
565 views

Conformal Field Theory in 1+1d Spontaneously Breaking Conformal Symmetry

Take any 1+1 dimensional conformal field theory on the plane. The Hamiltonian is invariant under the infinite-dimensional Virasoro algebra (with some central charge $c$), generated by $L_i$ ($i\in \...
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3answers
749 views

Coulomb gauge fixing and “normalizability”

The Setup Let Greek indices be summed over $0,1,\dots, d$ and Latin indices over $1,2,\dots, d$. Consider a vector potential $A_\mu$ on $\mathbb R^{d,1}$ defined to gauge transform as $$ A_\mu\to ...
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1answer
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What is a “free” non-Abelian Yang-Mill's theory?

I hope this question will not be closed down as something completely trivial! I did not think about this question till in recent past I came across papers which seemed to write down pretty much ...
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1answer
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Why mass terms are forbidden?

I would like to clarify my understanding on why mass terms in Lagrangians of gauge theories are forbidden. It's often repeated that particle masses are forbidden by electroweak symmetry because it is ...
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1answer
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Gauge invariance is just a redundancy. Why is massive abelian gauge field renormalizable but massive non-abelian gauge field nonrenormalizable?

For example, Kaku's QFT pp. 214-215: Massive vector theory with non-Abelian group is non-renormalizable. Massive vector Abelian theory is renormalizable. I heard about the following arguments,...
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1answer
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Can an observable be invariant under local $U(1)$ but not under global $U(1)$?

Consider a quantum field theory with two fields, a complex scalar field $\phi$ and a $U(1)$ gauge field $A$. Both fields are dynamic fields, not background fields. Suppose that spacetime is ...
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3answers
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What is the basis of gauge theory?

I’m learning about gauge concepts. I’ve always had the idea that by looking at a phenomenon from different viewpoints, that symmetries could be derived – in fact, that was what an equal sign signified....
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Understanding Elitzur's theorem from Polyakov's simple argument?

I was reading through the first chapter of Polyakov's book "Gauge-fields and Strings" and couldn't understand a hand-wavy argument he makes to explain why in systems with discrete gauge-symmetry only ...
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2answers
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Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

I am quite confused with the groups Diff$(M)$ and $GL(4,\mathbb{R})$ in the context of general relativity. I understand that the symmetries of GR are the transformations that leave the equations ...
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1answer
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Uniqueness of Yang-Mills theory

Question: Is there any sense of uniqueness in Yang-Mills gauge field theories? Details: Let's say we are after the most general Lagrangian Quantum Field Theory of (possibly self-interacting) $N$ ...
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If gauge symmetries are fake, then why do we care if they are anomalous?

My understanding is that gauge symmetries are fake in that they are only redundancies of our description of the system that we put in (either knowingly or unknowingly) see Gauge symmetry is not a ...
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2answers
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Gauge fields in Polyakov's treatment of renormalization for nonlinear sigma model

I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''. The action for ...
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2answers
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Why can't a real scalar couple to the electromagnetic field?

If we have a complex scalar $\phi$ we know that the gauge-invariant interaction with $A$ is given by $A^\mu J_\mu$, where $J$ is the Noether current of the $U(1)$ symmetry of the Lagrangian $$ J_\mu\...
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1answer
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What defines a large gauge transformation, really?

Usually, one defines large gauge transformations as those elements of $SU(2)$ that can't be smoothly transformed to the identity transformation. The group $SU(2)$ is simply connected and thus I'm ...
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3answers
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Why can't gauge bosons have mass?

Clearly, a mass term for a vector field would render the Lagrangian not gauge-invariant, but what are the consequences of this? Gauge invariance is supposed to be crucial for the renormalisation of a ...
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1answer
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**Group structure** in Chern-Simons theory?

A non-Abelian Chern-Simons(C-S) has the action $$ S=\int L dt=\int \frac{k}{4\pi}Tr[\big( A \wedge d A + (2/3) A \wedge A \wedge A \big)] $$ We know that the common cases, $A=A^a T^a$ is the ...
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Gauge invariant but not gauge covariant regularization

I'm not sure if someone's already asked this before, but I was wondering, in field theory, when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In ...
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1answer
485 views

How is the Chern-Simons action well-defined?

The Chern-Simons action $$ S = \int_M A \wedge \mathrm{d} A + \frac{2}{3}A \wedge A \wedge A $$ is not obviously gauge invariant. It is usually stated that under a gauge transformation, the action ...
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If Energy can be converted into mass, why can it not be converted into charge?

Probably a silly question, but something that came to mind yesterday. I couldn't find anything when searching. Why is there an Energy mass equivalence principle but not an Energy charge equivalence ...
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2answers
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Classical EM : clear link between gauge symmetry and charge conservation

In the case of classical field theory, Noether's theorem ensures that for a given action $$S=\int \mathrm{d}^dx\,\mathcal{L}(\phi_\mu,\partial_\nu\phi_\mu,x^i)$$ that stays invariant under the ...
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1answer
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Gauge fixing and degrees of freedom

Today, my friend (@Will) posed a very intriguing question - Consider a complex scalar field theory with a $U(1)$ gauge field $(A_\mu, \phi, \phi^*)$. The idea of gauge freedom is that two solutions ...
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1answer
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Local and Global Symmetries

Could somebody point me in the direction of a mathematically rigorous definition local symmetries and global symmetries for a given (classical) field theory? Heuristically I know that global ...
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1answer
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Is there a spontaneous $U(1)$ symmetry breaking in atomic BECs?

In the theory of Bose-Einstein condensation, one way to define the order parameter is by using the concept of spontaneous symmetry breaking. One says that, below the critical temperature, the ...
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1answer
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Invariance of Functional Integration Measure

Let us consider the functional integral: \begin{equation} \int \mathcal{D} A e^{iS[A]} \end{equation} where $S[A]$ is the action for $U(1)$ gauge field and \begin{equation} \mathcal{D}A\equiv \...
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1answer
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Why does local gauge invariance suggest renormalizability?

I'm reading Gauge Field Theories: An Introduction with Applications by Mike Guidry and this particular remark is not obvious to me: A tempting avenue is suggested by the QED paradigm, for if a ...
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1answer
325 views

Why not regard all large gauge transformations as genuine ones?

A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to ...
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3answers
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Gauge invariant Chern-Simons Lagrangian

I have to prove the (non abelian) gauge invariance of the following lagrangian (for a certain value of $\lambda$): $$\mathcal L= -\frac14 F^{\mu\nu}_aF_{\mu\nu}^a + \frac{k}{4\pi}\epsilon^{\mu\nu\...
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1answer
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Why do we solve the Wess-Zumino consistency condition using the method of descent?

Consider a quantum field theory in $d$ dimensions with a symmetry $G$. For the purpose of this discussion let's say that $d$ is even and $G$ is a compact, connected Lie group. We say that the symmetry ...
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2answers
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A question on gauge fixing

As I understand it, a physical theory that has a gauge symmetry is simply one that has redundant degrees of freedom in its description, and as such, is invariant under a continuous group of (in ...
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4answers
748 views

Can an Electromagnetic Gauge Transformation be Imaginary?

The Hamiltonian of a non-relativistic charged particle in a magnetic field is $$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2$$. Under a gauge transformation ...
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2answers
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Intuition for gauge parallel transport (Wilson loops)

I'm looking for a geometrical interpretation of the statement that "Wilson loop is a gauge parallel transport". I have seen QFT notes describe U(x,y) as "transporting the gauge transformation", and ...
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2answers
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The gauge covariant derivative and its substitution

I was wondering wether it would make a difference (in general) if I were to were introduce the gauge covariant derivative $$D_\mu=\partial_\mu+ieA_\mu$$ In the Lagrangian density and then derive the ...
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2answers
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Why does normal ordering violate the Ward identity?

It is well known that normal ordering the Lagrangian eliminates all Feynman diagrams with tadpoles$^{[1]}$. In the case of the photon self-energy in scalar QED, one of the diagrams is, in fact, a ...
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2answers
330 views

Definition of gauge freedom in electromagnetism and general relativity

The freedom we have in choosing the vector potential $\vec{A}$ in E&M is referred to as the gauge freedom, whereas in general relativity (GR), we refer to the freedom to choose any coordinate ...
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2answers
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Is internal symmetry the same as gauge symmetry?

This is more a terminology question. I have seen that some people differentiate between the two types of symmetry: internal symmetry and gauge symmetry (of a field theory). Is there a difference (in ...
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1answer
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What is a gauge theory?

Please note that I just read about 20 forum discussions, none of which answered my question. This question is related to my earlier question Is spacetime symmetry a gauge symmetry?. I am looking for ...
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1answer
825 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = (\partial_{\mu}+A_{\mu})\...

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