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Questions tagged [gauge-theory]

A gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of *continuous symmetries* of these internal degrees of freedom, *i.e.* the predicted physics under any transformation in this group on the degrees of freedom. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.

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Transformation of electromagnetic potential under local U(1) transformation

Let $\mathcal{L}=-(\partial _{\mu} \Phi^*)(\partial ^{\mu} \Phi)$ With $\Phi , \Phi^*$ being complex fields. When looking at local U(1) transformations in class, we saw that $\mathcal{L}$ is not ...
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What's the meaning of “inequivalent quantizations”?

The notion "inequivalent quantizations" is regularly used when topological terms are discussed. From what I've gathered so far, "inequivalent quantizations" means that there are different quantum ...
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What happens to large gauge transformations in gauges different from the temporal gauge?

There are already several questions regarding the meaning and definition of large gauge transformations. Discussions of large gauge transformations typically only happen in the context of the ...
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Gauge transformation of wave function of a system of stationary charges

Let's say we have a system of $n$ stationary charges interacting via Coulomb potential. Let's ignore possible external electromagnetic fields. Moreover the system is quantum, and its wave function is $...
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198 views

Calculating $S$-matrix in string theory

To calculate string $S$-matrix, we mainly use Faddeev-Popov gauge fixing method, as in chapter 6 of Polchinsky's book 《string theory》. But in section 6.2, 'tree level amplitude', I didn't find ...
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longitudinal and transverse components in higher dimensions

I am familiar with the Helmholz decomposition of a vector field in three dimensions: $$\vec{V}=\vec{\nabla}\wedge\vec{A}+\vec{\nabla}\phi$$ But I am interested to show that something similar can be ...
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Reference for proof of renormalizability

I have been trying to truly understand the renormalizability of quantum (i.e., without anomalies) gauge theories (after which I will focus on the case with spontaneous symmetry breaking). The problem ...
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Why can the divergence of vector potential be anything?

Purcell in his book was deriving the vector potential $\bf A$ using $\text{curl}\;(\text{curl}\; \mathbf A)= \mu_0 \mathbf J\; .$ After some algebra, he came to this: $$-\frac{\partial^2 A_x}{\...
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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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Soft charges and infrared physics in standard gauge and gravitational physics [closed]

Recently, we have learned that there are an infinite number of possible soft charges in black hole physics. It turns that the commonly assumption of that infrared physics is completely independent of ...
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894 views

Reduction of Nambu-Goto action to true degrees of freedom

I) First consider the point particle $$S=m\int\sqrt{-\dot{X}^2}d\tau.$$ If you choose the static gauge $$\tau=X^0$$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau.$$ ...
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In what sense $Z_\mu^0$ is orthogonal to $A_\mu$?

I am reading Standard model. Please explain in what sense the $Z$-boson $$Z_\mu^0=(g^2+g^{\prime 2})^{-1/2}(g A^3_\mu-g^\prime B_\mu)$$ is an orthogonal linear combination of the photon $$A_\mu=(g^2+...
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Low dim physics: Examples of confinement-deconfinement phases of U(1) gauge theory in 2 dimensions

Please provide some examples of confinement-deconfinement phases of U(1) gauge theory in 2 spacetime dimensions (Low dimwnsional physics). U(1) gauge theory can be: pure U(1) gauge theory, or U(1) ...
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What is a gauge (for someone who has not studied gauge theory)? [duplicate]

I am taking a Quantum Mechanics II course and we were studying the relativistic corrections to the hydrogen atoms in perturbation theory. I was looking at the assignment, and a question is as follows: ...
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A question on supersymmetry variation of the Wilson loop in $\mathcal{N}=4$ SYM

The Wilson loop in $\mathcal{N}=4$ SYM is $$W=\frac{1}{N}tr P \exp \int ds (i A_\mu(x) \dot{x}^\mu+\Phi_i(x)\theta^i|\dot{x}|).\tag{2.3}$$ In order to check whether this operator is supersymmetric I ...
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Explicit counting of gauge field degrees of freedom

Consider a connection on a principal $U(1)$-bundle $A_\mu$ over the flat base manifold $M_4$. The action of the theory is described in terms of the curvatures of such connection coupled to some source ...
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How do interaction terms appear in the Lagrangian?

How does forcing the Lagrangian to be invariant under $U(1)$ group give rise to the electromagnetic interaction term?
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Computation of the Faddeev-Popov determinant

I am studying the Faddeev-Popov method from Pokorski's Gauge Theory book, and I am puzzled by what happens in the step below. He is writing the group element $g = 1 - i T_a\ \Theta^a(x)$ in a ...
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Can we do one-loop integrals in the unitary gauge?

$\hspace{5cm}$ Imagine we want ot compute one of the diagrams for the self-energy of the quark $u$, with external momentum $p$. Inside the loop, we would have a $W^+$ and a $d$-quark propagator, with ...
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Fields and gauge transformations vanishing at infinity

I find that, in field theory, it is very often assumed that the fields (classical) vanish at infinity. The same assumption is also applied to gauge transformations, for example, when saying that the ...
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199 views

Global anomaly for discrete groups

We know that: a global anomaly is a type of anomaly: in this particular case, it is a quantum effect that invalidates a large gauge transformations that would otherwise be preserved in the ...
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Landau levels in symmetric gauge

On Shankar’s Quantum Many body page 394 it says for one electron in a magnetic field, ignoring spin, $$H_0=\frac{(\bf{p}+e\mathbf{A})^2}{2m}$$ $$e\mathbf{A}=-\frac{\hbar}{2l^2}\hat{z}\times \mathbf{...
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Vector Potential of a rotating Spherical Shell

I have done all the calculation in finding the vector potential leading up to the equation 5.68. But then Prof. Griffiths goes back to the original figure and mentions that the coordinate of the point ...
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Gauge invariance or global invariance, which one makes theory renormalizable?

We know that gauge theory is renormalizable, due to the Ward-Takahashi identity (for non-Abelian theory, it is Slavnov-Taylor identity), which reflects the conserved current of gauge symmetry. But ...
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Does the existence of instantons imply non-trivial cohomology of spacetime?

Gauge theories are considered to live on $G$-principal bundles $P$ over the spacetime $\Sigma$. For convenience, the usual text often either compactify $\Sigma$ or assume it is already compact. An ...
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Is it possible to have a complex gauge field?

Beyond the obvious fact that the particles in the standard model described by gauge fields do not have an anti-particle pair, is there a reason why a complex gauge field is typically not considered? ...
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Derivation Of The Equation Of Motion Of String from Polyakov action

I'm stuck at a step in the derivation of the equations of motions of a string using the Polyakov action. In Polchinski's textbook in String Theory , Page 14 ; Equation ( 1.2.25 ) , Varying the ...
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Why is the Coulomb Gauge enough to fix extra degrees of freedom?

In classical electrodynamics, we have after the Coulomb gauge is applied: $$ \Delta U = -\frac{\rho}{\epsilon_0} $$ $$ \Box \vec{A} = \mu_0 \vec{j}-\frac{1}{c^2} \vec{\nabla} \frac{\partial U}{\...
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248 views

Is general relativity resulted from diffeomorphism invariance?

Any action expressed as the integral of a 4-form in 4-dimensional spacetime is diffeomorphism invariant. For example the following 4-form topological (Pontryagin) action $$ S = \int F\wedge F $$ is ...
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1answer
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Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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Inconsistency between $d_A = d + A \wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative. $d_A A=F$ --- @...
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Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
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Particle on a circle with magnetic flux$.$

I am trying to understand the model studied in 1905.09315 §2, to wit, a $0+1$ dimensional theory with target space $\mathbb S^1$ with non-trivial magnetic flux: $$ \mathcal L=\frac12m\dot q^2-\frac{i}{...
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Legal values of spin-1 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

For the spin-1/ boson field $A_\mu$, we may choose it to be a vector which needs to be real $\mathbb{R}$ usually for photon field. The field strength $F= dA$ is also real. Same for the nonabelian ...
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Custodial symmetry and Higgs-Kibble

In the context of Higgs mechanism only on $SU(2)_L$ model without the hypercharge, one writes the lagrangian with traces also for the Higgs, i.e. $$ \cdots+\text{Tr}[(D_\mu H)^\dagger D^\mu H)]-\frac{\...
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Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
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Relating the Yang-Mills field-strength to the Maxwell tensor in $SU(2)$ gauge theory

I'm studying topological monopoles in a $SU(2)$ Yang-Mills theory with spontaneous symmetry breaking, through the book "Topological Solitons", by Manton and Sutcliffe. In section 8.2, the authors ...
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Why is the relation $M_W=M_Z\cos\theta_W$ true only at tree-level?

In Glashow-Weinberg-Salam electroweak theory, the relation $$M_W=M_Z\cos\theta_W\tag{1}$$ is said to be remain true only at the tree-level; it receives corrections from the loop diagrams. See here. ...
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1answer
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Delta function conversion into gauge-fixing Lagrangian in the path integral

So, I am at the moment working on gauge-fixing a path integral. The procedure involves adding a delta function $\delta g$ to the path integral (together with the faddeev-popov determinant, but that is ...
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Particles and associated fields

When a particle is associated with a field. 1) It is said that the excitation of the field produces the particle, 2) it is also said that when the field is quantized, the quanta of the field is ...
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Reference request for Gribov ambiguity

I was hoping to find a reference (book or article) with a good introduction to the Gribov Ambiguity in non-abelian gauge theories. I’ve looked through QFT books by Schwartz and Srednicki, Rubakov’s ...
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Gribov's phenomenon

In the well known textbook by Itzykson-Zuber "Quantum Field Theory" there is a discussion of the Gribov phenomenon in non-abelian gauge theories (see Section 12-2-1). To my taste, the discussion ...
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Does a gauge group $G$ determine the Principal $G$-bundle?

I'm trying to understand the mathematical underpinnings of gauge theories in the language of principal $G$-bundles and associated vector bundles. Not long ago, I had assumed that the physical choice ...
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Anyon condensation – what's the precise definition?

Say I have an anyonic system modelled as a Chern-Simons system with group $G$. If the centre of $G$ is non-trivial, one may also study the system described by $G/\Gamma$, where $\Gamma$ is a discrete ...
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Polar representation of complex scalar field in spontaneous symmetry breaking

In Rubakov's "Classical Theory of Gauge Fields", in the chapter on Higgs mechanism, he mentions that you can switch to the polar form of the complex scalar field only for non-zero vacuum value of the ...
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Gauge invariance on Yang-Mills Lagrangian

How do I verify the invariance on Yang-Mills' Lagrangian: $$L = -\frac{1}{4} \sum_{a} \left(\partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^bA_\nu^c \right)^2$$ under the transformation:...
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Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source $$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$ My question is that what are the most simple form we can boil down this ...
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$U(N)$ & $SU(N)$ : What's the conceptual difference in Gauge Theory?

I know the mathematical difference that one means $ absolutevalue(det) = 1$ and one means det = 1 (rotation) and that ones the subgroup of the other and so on. But: has a local/gauged $SU(3)$ ...
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Classical Yang-Mills equation of motion with both electric and magnetic sources?

We know the classical Maxwell equation of motion (eom) with both electric and magnetic source can be written as: (1) Explicit form or more schematically as: (2) Differential form $$ d * F = * J_e $$...
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Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the $G$-...