What is Gauge invariance: Definition and 76 Discussions
In physics, a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups.
The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.
Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).
Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.
Gauge theories are also important in explaining gravitation in the theory of general relativity. Its case is somewhat unusual in that the gauge field is a tensor, the Lanczos tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of general covariance with a true gauge principle with new gauge fields.
Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.
Consider two non-Abelian gauge fields ##A_\mu^a## and ##A_\mu^{'a}## belonging to the same symmetry group. An example could be the SM electroweak isospin fields and another exotic SU(2) hidden sector where ##a=1, \dots 3##.
Is the kinetic mixing of the following form gauge-invariant?
$$...
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...
Suppose we have an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. We denote the variation of ##S## wrt to a given field, say ##a##, i.e. ##\frac{\delta S}{\delta a}##, by ##E_a##.
Then ##S## is gauge invariant when
$$\delta S = \delta a E_a + \delta b E_b...
I was reading Diagrammatica by Veltman and he treats the photon field as a massive vector boson in which gauge invariance is disappeared and the propagator has a different expression than in massless photon. After some googling, I found that this is one way to formulate QED which has the...
Gauge symmetry is highly confusing, partly because many definitions differ in the literature. Strictly speaking gauge symmetry should be called gauge redundancy since you are mapping multiple representations to the same physical state.
What is your favourite definition of what "large" gauge...
Symmetry transformations in physics can be either passive or active. Symmetries in field theory can be either global or local. But only the local ones, the so called gauge symmetries, are fundamental. Except that local transformations cannot be active (despite the fact that diffeomorphisms are...
I have been reading the book of Chris Quigg, Gauge theories, Chapter 3, sec 3.3 in which he explains how local rotations transform wave function and variations in Schrodinger equation forces us to introduce the electromagnetic interaction between the particles. I need a bit deep concept of the...
Show that the Feynman amplitude for Compton scattering ##\mathcal{M} = \mathcal{M}_a + \mathcal{M}_b## is gauge invariant while the individual contributions ##\mathcal{M}_a## and ##\mathcal{M}_b## are not, by considering the gauge transformations
$$\varepsilon^{\mu} (\vec k_i) \rightarrow...
In linearized gravity we define the spatial traceless part of our perturbation ##h^{TT}_{ij}##. For some reason this part of the perturbation should be gauge invariant under the transformation $$h^{TT}_{ij} \rightarrow h^{TT}_{ij} - \partial_{i}\xi_{j} - \partial_{j}\xi_{i}$$ Which means that...
Given the schrodinger equation of the form $$-i\hbar\frac{\partial \psi}{\partial t}=-\frac{1}{2m}(-i\hbar \nabla -\frac{q}{c}A)^2+q\phi$$
I can plug in the transformations $$A'=A-\nabla \lambda$$ , $$\phi'=\phi-\frac{\partial \lambda}{\partial t}$$, $$\psi'=e^{-\frac{iq\lambda}{\hbar c}}\psi$$...
In an earlier question I asked if the EM field was truly a separate field from the matter field in QFT, as it's field structure is naturally complementary to phase changes in the matter field in just the right way to restore gauge invariance (poorly formed question, but hopefully you get the...
In his book, "The greatest story ever told", Lawrence Krauss states: "Gauge invariance ... completely determines the nature of electromagnetism."
My question is simple: How?
I have gone back thru the math. Gauge invariance allows us to use the Lorenz gauge with the vector and scalar potentials...
Most gauge transformations in the standard model are easy to see are measurement invariant. Coordinate transformations, SU(3) quark colours, U(1) phase rotations for charged particles all result in no measurable changes. But how does this work for SU(2) rotations in electroweak theory, where...
I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric.
Something is unclear when we try to find fake vs. real perturbation of the metric.
We use an arbitrary small vector field to have the effect of a chart transition map or...
This is not a technical question. I'd like to have a more conceptual discussion about what - if anything - gauge invariance tells us about reality. If we could, please try to keep the discussion at the level of undergrad or beginning grad.
To focus my questions and keep things elementary, I'd...
Homework Statement
Hello Everyone
I'm wondering, why in below product rule was not used for gradient of A where exponential is treated as constant for divergent of A and only for first term of equation we used the product rule?
Homework Equations
https://ibb.co/gHOauJ
The Attempt at a Solution
Homework Statement
In an inertial reference frame ##S## is given the four-potential:
$$A^\mu=(e^{-kz}, e^{-ky},0,0)$$
with ##k## a real constant.
##A^\mu## fullfills the Lorentz gauge? And the Coulomb gauge?
Which is the four-potential ##A'^\mu## in a reference frame ##S'## which is moving...
I'm reading a book on gauge symmetry, and in the discussion of the Aharanov-Bohm effect, the author says the following:
But a paragraph later, he goes on to say:
It seems to me like there is a contradiction here (indicated by phrases in bold). How can the a change in potential be...
The problem:
$$\mathcal{L} = F^{\mu \nu} F_{\mu \nu} + m^2 /2 \ A_{\mu} A^{\mu} $$
with: $$ F_{\mu \nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu} $$
1. Show that this lagrangian density is not gauge invariance
2.Derive the equations of motion, why is the Lorentzcondition still...
I think the story where abelian, i.e. U(1), gauge symmetry comes from is pretty straight-forward:
We describe massless spin 1 particles, which have only two physical degrees of freedom, with a spin 1 field, which is represented by a four-vector. This four-vector has 4 entries and therefore too...
Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity:
In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This means that local correlation functions like ##\langle...
The mantra in theoretical physics is that global gauge transformations are physical symmetries of a theory, whereas local gauge transformations are simply redundancies (representing redundant degrees of freedom (dof)) of a theory.
My question is, what distinguishes them (other than being...
I'm having a bit of trouble with counting the number of physical ("propagating") degrees of freedom (dof) in field theories. In particular I've been looking at general relativity (GR) and classical electromagnetism (EM).
Starting with EM:
Naively, given the 4-potential ##A^{\mu}## has four...
Hi,
I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdfEven the easy abelian one (third row)
which is
$$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$
doesn't match my calculation
Since
$$D_\rho B_{\mu\nu}=(\partial_\rho+i g...
I know that, in the presence of a magnetic field, the momentum of a charge particle changes from ##p_{i}## to ##\pi_{i}\equiv p_{i}+eA_{i}##, where ##e## is the charge of the particle.
I was wondering if this definition of momentum is gauge-invariant?
How about ##\tilde{\pi}_{i}=p_{i}-eA_{i}##?
Consider the covariant derivative ##D_{\mu}=\partial_{\mu}+ieA_{\mu}## of scalar QED.
I understand that ##D_{\mu}\phi## is invariant under the simultaneous phase rotation ##\phi \rightarrow e^{i\Lambda}\phi## of the field ##\phi## and the gauge transformation ##A_{\mu}\rightarrow...
Hello! Can someone explain to me what exactly a local gauge invariance is? I am reading my first particle physics book and it seems that putting this local gauge invariance to different lagrangians you obtain most of the standard model. The math makes sense to me, I just don't see what is the...
A complex classical field Φ of particles is, by itself, invariant under global phase changes but not under local phase changes. It is made gauge invariant by coupling it with the EM potential, A, by substituting the covariant derivative for the normal partial derivative in the Lagrangian. But...
Does the property of electric charge of an elementary or composite particle exist only within the context of gauge symmetry - of the complex phase of the wave function, i.e. does gauge symmetry define electric charge?
Thanks in advance.
Hello. I'm trying to wrap my head around how Lagrangians work in classical field theory.
I have a book that is talking about the gauge invariance of the Lagrangian: \mathscr{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^\mu A_\mu. It shows that we can replace A^\mu with A^\mu+\partial^\mu\chi for...
The part I understand:
I understand that the spontaneous symmetry breaking of the Higgs produces the 'Mexican hat' potential, with two non-zero stable equilibria.
I understand that as the Higgs is a complex field, there exists a phase component of the field. Under gauge transformations of...
please explain what gauge symmetry is, gauge transformation is, gauge invariance is, and also how gauge invariance deletes the timelike polarization of a massless vector boson. without fancy math and formulas.
If a theory is gauge invariant and one chooses to fix a particular gauge, having done this is it then possible to make a gauge transformation from this chosen gauge to another gauge, or have we already "spent" the gauge symmetry?
Apologies if this is a really basic question, but I've got myself...
Homework Statement
Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., ##\psi## is understood to be a three-component vector in colour space whilst...
I hear that the interaction between a photon and an electron is introduced by the local gauge invariance in the quantum field theory. On the other hand, I know that an decelerated electron emits a photon. Are these two saying the same thing? Or how these two are related?
I recently learned that with (local) gauge invariance, functional quantization needs to factor out volume factor(Faddeev-Popov procedure).
Why does this has to be done?Just to remove infinity? As far as I am concerned, ##\phi^4## theory contains invariance(for example ##\phi\to\phi\cdot e^{i...
Hey guys,
So I have a question about the gauge invariance of the weak field approximation. So if I write the approximation as
\Box h^{\mu\nu} -\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha})+\partial^{\mu}\partial^{\nu}h=0
then this is invariant under the gauge...
My book says that in this case $$e^+e^- \rightarrow \gamma \gamma $$ gauge invariance requires that $$k_{1\nu}(A^{\mu\nu} + \tilde{A}^{\mu\nu})=0=k_{2\mu}(A^{\mu\nu} + \tilde{A}^{\mu\nu})$$ Please see attachment. My question is how does this statement hold?
I was trying to prove all those little things you spend long as the local invariance in the free Lagrangian of electroweak interaction.
Taking into account the appropriate SU(2) transformations (without covariant derivatives), came to the following expression
\mathcal{L}_{\text{ferm.}} =...
Hi folks -- does anyone know of a good survey article on the topic of whether local gauge invariance is a requirement of a fundamental theory within QFT -- hence of an asymptotically safe theory?
I only have a few scattered remarks to this effect (by F. Wilczek mostly), so any good...
Definition/Summary
Gauge invariance is a form of symmetry.
An experiment here today will work the same way over there tomorrow and with the apparatus pointing in a different direction.
This is called "global invariance" … the laws of physics are invariant under translations, both in...
If gauge symmetries are really just redundancies in our description accounting for nonphysical degrees of freedom, then how does one explain the deep and powerful fact that if one begins with, say, just fermions and no gauge field in one's theory (and no interactions & essentially no dynamics)...
Hi, so I'm trying to derive the charge conservation law for a general SU(N) gauge field theory by using gauge invariance. For U(1) this is trivial, but for the more general SU(N) I seem to be stuck... So if anyone sees any flaws in my logic below please help!
Starting with the Lagrangian...
Hi!
I have to prove that the amplitude of the process
\gamma \gamma \to W^+ W^-
does not depend on the gauge we will choose, R_{\xi}.
So I use the most general expressions for the propagators and vertices. I find 5 diagrams. One that involves only the 4 fields and a vertex, 1 t and...
Homework Statement
A gauge transformation is defined so as to leave the fields invariant. The gauge transformations are such that \vec{A}=\vec{A'}+\nabla\Lambda and \Phi=\Phi'-\frac{\partial\Lambda}{\partial t}. Consider the Coulomb Gauge \nabla\cdot\vec{A}=0. Find out what properties the...
These are notes I made when I was studying the subject 20 years ago. They seem fine considering that I was student then. I believe they can be useful for those who are studying Yang-Mills and other related material.
Sam
For free EM field:
L=-\frac{1}{4}FabFab
Then the stress-energy tensor is given by:
Tmn=-Fml∂vAl+\frac{1}{4}gmnFabFab
The author then redefines Tmn - he adds ∂lΩlmn to it,
where Ωlmn=-Ωmln.
The redefined tensor is:
Tmn=-FmlFvl+gmv\frac{1}{4}FabFab
It is gauge invariant and still satisfies...