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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Weak Equality & Strong Equality?
I have been trying to understand the meaning of these concepts: Weak $(\approx)$ and Strong $(=)$ Equality in the Dirac-Bergmann Algorithm for Hamiltonian Constrained Systems. I have already read ...
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Definitions of different types of symmetries
I'm a math student and I started studying physics last year. I'm sorry if this question has been asked before but I'm completely confused about it. In page 30 of the book "String theory and M-...
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Can we do Wilson RG in gauge theory in a gauge invariant fashion?
As we all know, cutoff regularization doesn’t work for gauge theory. So won’t the traditional Wilson RG work in gauge theory since it involves explicit cutoff for the same reason.
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Gauge covariance vs gauge invariance
For Bloch electrons $\psi_{nk}=e^{ikr}u_{nk}$, with $k$ the crystal momentum and $n$ the band index, I would like to know if the integral ($m \neq n, k^\prime \neq k$)
$$
\langle u_{nk}|u_{nk^\prime}\...
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Gauge choice in showing Landau level degeneracy via the algebraic method
I'm trying to understand the algebraic method of formulating the Landau level problem better. I'm referring to David Tong's notes on the Quantum Hall effect for this (but not exactly following his ...
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Is First-Class Constraint Generator of matter Gauge Symmetry in EM example?
In EM theory, we can find first-class primary constraint,
$$\Pi^{0}(x) = 0\tag{1}$$
and first-class secondary constraint,
$$\partial_{i} \Pi^{i}(x) = 0\tag{2}$$
with Lagrangian $$\mathcal{L} = -(1/4)F^...
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Independence of $S$-matrix of $\xi$-gauge in QED
On page 298 in Peskin and Schroeder, the authors attempt to argue that the $S$-matrix should be independent of the $\xi$-gauge in QED. However, I don't understand their argument, in particular the ...
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Analog between Electromagnetism and Gravity
Feynman makes an analogy between EM field and gravity field in his Feynman's Lectures on Gravitation. The vector field representing EM potential would couple to the current source(vector) in the ...
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Writing gauge transformation of the gauge fields explicitly in terms of coordinates
Let $G$ be a non-Abelian simple compact gauge group and $\{ t^\alpha\}$ be a normalized set of generators for its Lie algebra $\mathfrak{g}$. Let $C^{\alpha \beta}_\gamma$ be the coupling constant for ...
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Gauge invariance of Hadamard point-splitting renormalization procedure
The current density of a charged complex scalar field in a background electromagnetic field $A_\mu$ is given by
$$ j^\mu = ie( (D^\mu \phi)^*\phi - \phi^*D^\mu \phi)$$
with $D_\mu = \partial_\mu + ...
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How to find a covariant gauge derivative from a field transformation
For reference: I'm self-studying from Peskin's Particle Physics 2019, which tries to sweep all QFT under the rug.
Consider an SU(3) gauge theory; I am told a $3\times 3$ scalar field $\Phi$ transforms ...
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Uniqueness of Maxwell Lagrangian: Why does it not include the term $c_3 (\partial_\mu j_\nu)F^{\mu \nu}$?
In the textbook Condensed Matter Field Theory by Altland and Simons, it is said that the Maxwell Lagrangian $\mathcal{L}$ coupled to a four-current $j^\mu$ satisfying $\partial_\mu j^\mu = 0$ is the ...
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Understanding the Gaussian weight and the parameter $\xi$ when quantizing gauge theories
In section 9.4 of Peskin & Schroeder's textbook on quantum field theory, when applying the Faddeev Popov procedure to quantize an Abelian gauge theory, they obtain the following functional ...
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Extracting a gauge-invariant variable from a given Wilson line? (NOT Wilson loop)
Let $W[x_i,x_f]$ be the Wilson line as defined here.
Under a local gauge transform $g(x)$, it transforms as
\begin{equation}
W[x_i,x_f] \to g(x_f)W[x_i,x_f] g^{-1}(x_i)
\end{equation}
which is shown ...
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Does LQG break gauge invariance?
So, I'm working with another researcher on a possible connection between Loop Quantum Gravity (LQG) and String Theory (ST). My colleague is proposing and insisting on a action that is not WS ...
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Commutation in the Local Gauge Transformations
Let's say that I have a Unitary Local Gauge Transformation $U$, in which the Lie Generators are $T$:
$$ \partial_{\mu} U = \partial_{\mu} e^{-i T^{a} \alpha_{a}(x)} = U \partial_{\mu} \left( -i T^{a} \...
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Dirac field coupling to gauge fields
I've seen in couple sources that the gauge invariant Lagrangian for the Dirac field being written as follows:
$$\mathcal{L} = \frac{i}{2}[\bar{\psi}\gamma^{\mu}D_{\mu}\psi-(\bar{D}_{\mu}\bar{\psi})\...
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Masses of $SU(2)$ gauge bosons
I'm currently learning quantum field theory and I'm wondering one thing.The way I understood it is that in the $SU(2)$ Yang-Mills theory, all gauge bosons have the same mass due to the spontaneous ...
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Number of independent reparametrization gauge invariances of the 'world $(n+1)$-manifold action' of $n$-dimensional objects
As a generalization of point particle dynamics, one can conceive of a theory of $n-$dimensional objects with 'world-manifold' action given by
$$ S[X] = -\frac{T}{2} \int d^{n+1}\sigma \sqrt{h} h^{\...
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Non-invertible symmetries: Half gauging and 't Hooft lines
In (2.27) of https://arxiv.org/abs/2205.05086, when performing a gauge transformation of the background gauge field $B \to B +d \Lambda $, the 't Hooft line $H(\gamma)$ transforms as
\begin{equation}
...
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Is work done by a charged particle not gauge invariant?
Work done by a charged point particle with charge $q$ in an external electric field derived from a scalar potential $\phi$ is given by $$W=q \phi.$$ Even if we add a magnetic field the definition ...
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Connection between Noether's theorem for gauge theories and 1-form symmetries?
Applying Noether's theorem to a gauge theory, one can show that the conserved current is generically of the form
$$J^\mu=\partial_\nu k^{[\mu\nu]},$$
such that the conserved charge is really ...
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How is Wald deriving this Gauge condition: $\partial^b\, \overline{\gamma}_{ab} = 0$?
R. Wald in Section#4.4 of his book General Relativity derives the EFE in the case of a weak gravitational field by taking the curved spacetime metric $g_{ab}$ to be a "small" perturbation $\...
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What is a gauge transformation? How does it relate to Cauchy intial value problem and second functional derivative of the action?
I am having conceptual problems about 'gauge transformation'. I have well heard that gauge trnasformation is a 'local symmetry' and 'fake symmetry', but I want to understand it more precisely.
I am ...
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Relation between the number of curvature functions and dimensions in GR
I am reading Weinberg's Gravitation and Cosmology. On page 10, it reads
In $D$ dimensions there will be $D(D+1)/2$ independent metric functions $g_{ij}$, and our freedom to choose the $D$ coordinates ...
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Class of on-shell and gauge equivalent potentials in Chern-Simons theory
Let $(P, M, \pi, G)$ be a principal bundle with three dimensional manifold $M$ and compact, connected, simply-connected, and simple structure group $G$. We define a Lie algebra valued connection $1$ ...
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Why semi-simple and compact Gauge Group in YM Theory? [duplicate]
I'm studying the Yang-Mills theory, with the Action:
$$
S=-\frac{1}{2}\int\mathrm{tr}_{\rho}(\mathcal{F}\wedge\star\mathcal{F})
$$
where $\mathcal{F}:=\mathrm{d} \mathcal{A}+\frac{1}{2}[\mathcal{A},\...
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The Abelian versus the non-Abelian commutator of covariant derivatives in field theory
In the case of Abelian symmetry, the covariant derivative is defined as $D_\mu\equiv \partial_\mu + ieA_\mu$, where $e$ is an arbitrary constant and the vector field, $A_\mu$ is a called a gauge field....
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Discrepance between gauge symmetry and Noether's first theorem
In QFT we're interested in the symmetries of our theory (encoded in the invariance of the Lagrangian under symmetries) because they let us study conserved currents of the theory by Noether's theorem.
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How are the gauge transformations of $\epsilon(\mu)$ and $A^\mu$ related?
To find a local field description of massless spin-1 particles that is Lorentz invariant, we can identify $\epsilon^\mu_{\pm}(k)$ with $\epsilon^\mu_{\pm}(k)+\alpha(k)k^\mu$. As $A^\mu$ and $\epsilon^\...
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Physical motivation behind gauging a global symmetry
Consider complex scalar field Lagrangian
$$\mathcal{L}=(\partial^\mu\psi)^\dagger(\partial_\mu\psi) - m^2\psi^\dagger\psi\tag{1}$$
Which exhibits $U(1)$-invariance, i.e $\psi\mapsto e^{i\alpha}\psi$. ...
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Silly confusion about gauge invariance in supersymmetric Lagrangians - in particular, in the ${\cal N}=1$ superfield formulation of ${\cal N}=4$ SYM
Hoping to resolve a simple confusion I have about supersymmetric gauge theory, one which I ran into while trying to understand the ${\cal N}=1$ superfield formulation of ${\cal N}=4$ supersymmetric ...
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How to derive the gauge invariance of Yang-Mills action with external source?
In the Faddeev-Popov procedure of path integral of
$$
Z[J] = \int [DA] e^{iS(A,J)},
\quad S(A,J)= \int d^4x [-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu} + J^{a\mu}A_{a\mu} ]
$$
we have used that $S(A,J)$ is ...
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Gauge invariance using equations of motion [duplicate]
I am working with a lagrangian on a homework problem. I expect it to have some gauge invariance. I can show that the Lagranian is invariant under those (gauge) tansformations but I have to use ...
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Why is it valid to only consider linear-order gauge transformation when quantizing non-Abelian gauge theory?
To quantize the non-Abelien gauge theory. We multiply the path integral by:
$f[A]=\int \mathcal{D}\pi exp[-i\int d^4 x {1\over \xi}(\partial_\mu D_\mu \pi^a)^2]$
then we can shift the argument in the ...
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Noether's first Vs Noether's Second theorem
I am reading about the first and second Noether's theorem from https://arxiv.org/abs/2112.05289. In the text, there is this piece, which I am not sure I entirely understand.
Let us reflect briefly on ...
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In general relativity, is gauge invariance the same as coordinate invariance?
I always understood that gauge invariance of general relativity comes from the fact that the physical observables and states are the same regardless of the coordinates we choose to express them in. I ...
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Parity of a 1d Ising model, and with higher order terms
I don't know if this should be asked here or in a math stack exchange, but I'll try here first.
Consider the classical 1d Ising model with periodic boundary condition:
\begin{equation}
H_2 (\vec{\...
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Gauge redundancy and Gauge fixing
Take any gauge invariant theory, for instance QED. The QED Lagrangian is invariant under
$$A_{\mu}(x)\rightarrow A'_{\mu}(x)=A_{\mu}(x)+\partial_{\mu}. \alpha(x)$$
I have chosen a local gauge ...
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Infinitesimal transformation of the Yang-Mills field
I am trying to obtain the infinitesimal transformation for the Yang-Mills field $A_{\mu}$. I want to show that
$$ A^{\prime a}_\mu=A_\mu^a-\partial_\mu \theta^a-g_s f^{a b c} \theta^b A_\mu^c $$
For ...
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How to take the second-order gauge covariant derivative in quantum field theory?
I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field.
(1) The first way is to write the second order gauge ...
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Gauge symmetries, isometries of spacetime and asymptotic symmetries
I am having a hard time understanding the physical meaning of asymptotic symmetries in General relativity. I think I understand the mathematics, but the meaning eludes me. I'll try to write the things ...
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Why impose constraints in (Path Integral) Quantization of Proca action?
I was reading the Wikipedia page on Proca Action. To summarize, it is almost like Maxwell action, but with a mass term because of which Proca action does NOT have gauge invariance. From the equation ...
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Does the gauge transformation rule on the gauge fields satisfy the definition of group action?
According to definition group action, it is required that action of a group $G$ on a set $X$ must satisfy the compatibility condition:
\begin{equation}
g \cdot (h \cdot x) = (gh) \cdot x \text{ for ...
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For these gauge transformations in electromagnetism, $\phi\to \phi-\partial_t \lambda$ and $\vec A\to \vec A+\nabla\lambda$, why do the signs differ?
I was looking at this question on Mathematics S.E, as I would like to know the origin of the signs in the gauge transformations of the scalar and vector potentials components, $\phi$ and $\vec A$, of ...
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Can the derivative of a gauge-invariant quantity be gauge-dependent?
I am wondering whether it is possible for derivatives of a gauge-invariant quantity to be gauge-dependent. Certainly, the converse is true; taking the curl of a gauge-dependent quantity (the vector ...
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Does 2-form curvature $\Omega \in \Omega^2(P,\mathfrak{g})$ represent a physical quantity in gauge theory?
In gauge theory, all measurable physical quantities remain invariant under a gauge transformation. I have always seen that the curvature 2-form $\Omega \in \Omega^2(P,\mathfrak{g})$ associated to a ...
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Expression for the gravitational-wave energy-momentum tensor without choosing a gauge
While studying section 7.6 of Carroll's introduction to general relativity, I encountered difficulties deriving equation 7.165 for the gravitational-wave energy-momentum tensor. Unfortunately, I was ...
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Perturbative expansion and renormalization of non-abelian Yang-Mills theory solely in terms of gauge-invariant quantities?
In standard QFT, each term in the perturbative expansion for a gauge theory is not necessarily gauge-invariant. Only the whole sum of Feynman diagrams is guaranteed so.
However, at least for QED, ...
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Gauge theories, boundaries and Wilson lines
My understanding of Wilson loops
Let's work with classical electromagnetism. The 4-potential $A_\mu$ determines the electric and magnetic fields, which are the physical entities responsible for the ...
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