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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Gauge invariance in General relativity [duplicate]
Suppose a theoretical physicist wants to construct a theory to explain some newly discovered phenomenon. The new theory is expected to follow certain rules or fundamental principles. There are four ...
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Are there two meanings of small gauge transformations and why must we restric ourselves to transformations small in the first sense?
For the sake of context suppose a euclidean pure Yang Mills theory with gauge group SU(2) for the rest of this question.
The terms large and small gauge transformations are used around in two ...
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Why does the boundary behavior of Lagrange multipliers matter for gauge vs global symmetries?
In Henneaux & Teitelboim (Quantization of Gauge Systems, p. 30), they discuss the variation of a dynamical variable
$$
\delta F = \int d^nx\, u(x)\,\{F, C(x)\}_{PB},\tag{1.62}
$$
where $C(x)$ is a ...
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Gauge Invariance and Degrees of Freedom [duplicate]
According to Wigner's classification, any massless particle (except for scalars) has 2 degrees of freedom i 4D. This reduction is usually understood in terms of gauge invariance. For instance, a ...
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$U(1)$: Gauge invariance
In Special Relativity, is it mathematically possible for a local, gauge-invariant field theory to have only one vector field $A_\mu$ and to have $U(1)$ symmetry, assuming the vector field $A_\mu$ and/...
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Could local charge conservation be violated by a moving point charge?
The charge density $\rho(\mathbf{x},t)$ and current density $\mathbf{j}(\mathbf{x},t)$ due to a point charge $q$ following a trajectory $\mathbf{r}(t)$ with velocity $\mathbf{v}(t)=d\mathbf{r}/dt$ is ...
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Why are classical gauge theories insensitive to large gauge transformations, while quantum gauge theories are affected by them?
In a previous question regarding large gauge transformations, one of the answers mentions that large gauge transformations are true redundancies for classical gauge theories, while they contain ...
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Why isn't gauge symmetry a symmetry while global symmetry is?
A typical opinion is that a gauge transformation doesn't change physical states, but a global transformation does. However, it's clear that global transformation is a subset of gauge transformation, ...
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Understanding the use of the Ward identities (in general and for an explicit example)
As the title suggests I am trying to understand the idea behind these identities and in order to do so, I will describe below, an example provided to us in the lecture. Once I understand how these ...
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Are there local transformations which leave the action invariant but are not gauge transformations?
For the sake of generality, I am not considering any specific scenario or field theory in particular.
Instead, if we consider some arbitrary field theory of choice and also global and local ...
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Unitary gauge and how to obtain a real representation from a complex one
I was reading an article by Weinberg that introduced the unitary gauge ("General Theory of Broken Local Symmetries", 1) at the classical level for a lagrangian $L$ of standard model-type, ...
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Explanation of statement concerning static gauge in flat metric
I am reading Tong's notes about string theory, the second chapter, and I encountered this part that I don't know how is derived. We are considering the worldsheet $(\tau,\sigma)$ whose gauge we set to ...
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Quantum Electrodynamics from local gauge symmetry of Dirac Equation
I'm taking at look at QED foundations, and started thinking about how it relates to Dirac's Equation.
Dirac spinors are invariant under a global phase transformation $\psi(x)\mapsto e^{i\alpha}\psi(x)$...
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What constitutes an admissable gauge?
Consider the Lorenz gauge condition
\begin{equation}
\partial_\mu A^\mu = 0. \tag{1}
\end{equation}
Suppose there exists a field configuration $B^\mu$ that satisfies the Lorenz gauge, and another ...
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Emergence of gauge invariance in QFT
This question is related to different motivations for the need of gauge invariance in QFT. I was introduced to gauge invariance in the following way.
Consider a vector field $A^\mu(x)$, with $x\equiv ...
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Invertibility of EOM in QED
Summary:
Equation of motion (EOM) for the electromagnetic 4-potential $A^\mu$ is non-invertible. Allegedly, this makes it impossible to determine the Green's function, and complicates quantisation of ...
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Physical insignificance of potential fields in classical electrodynamics
I am quoting the Aharonov-Bohm paper
"We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials
on charged particles"
Consider a region ...
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Gauge independence of photon attached to scalar loop
Schwartz shows in section 9.4 (on SCALAR QED) that the amplitude of diagram (9.41)
is not gauge invariant on its own, but once we add the amplitude of (9.43)
we get a gauge independent result. Now, ...
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In NRQM why are gauge equivalent electromagnetic potentials physically equivalent?
In the NRQM'al theory of a charged particle, if we gauge transform the potentials using some arbitrary gauge function $\Lambda(\textbf{X},t)$ then Schrodinger's equation implies that the quantum state ...
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In particle physics, what is the *motivation* to have invariance of a defined Lagrangian under a transformation of phase $e^{i\alpha}$ of the fields?
In particle physics, one checks the invariance of the Lagrangian under global transformations of the fields $\psi\rightarrow e^{i\alpha}\psi$ and local transformations $\psi\rightarrow e^{i\alpha(x)}\...
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Role of local gauge invariance in wavefunction self-interference
In quantum mechanics, we know that:
The global phase of a wavefunction does not have any physical effect, but the relative difference in phase ($\Delta \phi$) among different components of the same ...
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Transverse and longitudinal states in Gupta-Bleuler quantization
In page 109 of Weigand's QFT note (https://www.physics.umd.edu/grt/taj/624b/WeigandQFT.pdf), he defined a general polarization 4-vector
$$
\zeta^\mu=\sum_{\lambda,\lambda'}\alpha_\lambda\eta_{\lambda\...
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Physical meaning of gauge-invariant composite fields in a (lattice) gauge theory
I am reading a paper on gauge-invariant description of gauge theories on a lattice.
In p.$564$ formula $(6.1)$ of the paper, a complete set of gauge-invariant composite fields for an $SU(2)$ theory ...
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Meissner effect for BCS superconductor using number-conserving formalism with no global $U(1)$ phase breaking
The origin of Meissner effect is the Anderson Higgs mechanism (see page 82 https://www-thphys.physics.ox.ac.uk/people/SteveSimon/QCM2019/QuantumMatter.pdf), where the global $U(1)$ phase symmetry of ...
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Heuristic proof of $Z_1 = Z_2$ in Schwartz's "QFT and the SM"
I can't understand the heuristic proof of $Z_1 = Z_2$ in the beginning of section 19.5 in Schwartz's "QFT and the SM".
Basically, he considered a theory with an electron and a quark:
$$\!\!\!...
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Particles in Temporal/Weyl Gauge QED
So, in Weyl gauge (A^{0} = 0) all physical states must obey Gauss's law. But that even makes the vacuum state (which is the Dirac sea tensor with the free EM vacuum) unphysical. Then, how do you ...
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Is Lagrangian origin height invariance an example of gauge invariance?
When looking at What, in simplest terms, is gauge invariance? question, the OP in the last paragraph asks this:
In many high school physics calculations, you measure or calculate time, distance, ...
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Coordinate/Gauge transformations in GR beyond linear order
I'm trying to get my head around derivations of the PN and PPN order metric and Lagrangian in GR. Especially concerning the coordinate transformations to switch between the different gauges. I know ...
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What would an alternative to the Lorenz gauge mean? Difference vs Sum
How would the mathematics of electromagnetism and physical interpretation change if the gauge used a difference instead of a sum?
In other words, what would this gauge do:
$$\frac{1}{c}\frac{\partial \...
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Proving gauge transformation of non-abelian field strength
For a general Yang-Mills theory, we have the field strength
$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig [A_\mu, A_\nu]$$ I now want to prove that it transforms as $$F_{\mu\nu} \...
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"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
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Baumann's Cosmology: if adiabatic perturbations are generated by shifting conformal time, can't they just be gauged away?
In Baumann's Cosmology, he discusses a few aspects of adiabatic perturbations on Sec. 6.2.2. He states (p. 229)
Adiabatic perturbations can be created by starting with a homogeneous universe and ...
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Witten anomaly and product representation
The Witten anomaly restricts the number of fermion doublets in an $SU(2)$ theory.
If we have a gauge group $SU(2)_1 \times SU(2)_2$ and have only one fermion fundamental $\psi = (2,2)$, is this theory ...
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Impact of $U(1)$ gauge symmetry violation on quantum theories of Gravity?
QED and QFT take $U(1)$ gauge symmetry as fundamental and obtain extremely successful predictions at the scales and experiments physicists have been able to perform. This should not be taken lightly.
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Non-commutator of symmetries of a Lagrangian giving rise to gauge symmetry
If there are two symmetries of a Lagrangian, perhaps they are transformations, $A$ and $B$, and they don't commute $[A, B] \neq 0.$ Let this act on some field, then if $(BA)^{-1}AB$ does not return ...
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Gauge charge of the spin connection
When dealing with quantum field theory in curved spacetime, a spin connection field is introduced as a result of the Lorentz symmetry. I'm wondering what would be intuitively considered as "...
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About Aharonov-Bohm effect without path integral approach
I'm currently reading the book Quantum Paradoxes: Quantum Theory for the Perplexed by Aharonov and Rohrlich and in chapter 4 they show the existence of AB effect. I'm already familiar with the ...
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Why are the gauge symmetries important?
So the rule is that if you change the field by multiplying it by a phase factor that varies you get a different equation, so to compensate for this you change the derivative operator to get rid of the ...
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Solutions to system of PDEs, gauge invariance of Lagrangian
Let
$$L = \int \frac{J(a,b)}{2}\dot{x}^2-p(x_a^2+x_b^2-J(a,b)) da da$$
where $p$ is a Lagrange multiplier, $x=x(a,b,t)$ and $dx/dt = \dot{x}$. This Lagrangian corresponds to the following dynamics
$$ ...
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Infinitesimal transformation acting on einbein in Polchinski's String Theory [duplicate]
From Polchinski's String Theory:
Point-particle example
Let us consider an example, the point particles. Expanding out the condensed notation above, the local symmetry is coordinate ...
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Transitivity of the gauge action for quantum Yang-Mills theories
The basic definition of principal bundles involves free and transitive action of the group $G$.
And, it is well-known that classical Yang-Mills are modelled on principal bundles, where $G$ corresponds ...
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Worldsheet gauge fields in string theory?
Can we introduce a worldsheet gauge field in Polyakov action?
In superstring theory we need to be careful about anomaly cancellation, supersymmetry which fix the field content of worldsheet. Is there ...
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On the Gauge-fixing for the case of the Polyakov string action
In the book "String theory and M-theory" by Becker-Becker-Schwarz, the author says that
"reparametrization invariance of the string sigma-model action $$S_{\sigma}=\frac{-T}{2}\int d^2 ...
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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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