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.

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5. A QED Formulation with Massive Photon Fields

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...
6. A Global vs. Local (gauge) Symmetry

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...
7. A Gauge invariance confusions: symmetry vs redundancy, active vs passive

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...
8. Lorentz and Gauge invariance of EM

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...

26. I Gauge invariance of momentum of charged particle

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}##?
27. A Gauge invariance and covariant derivative

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...
28. I Local Gauge Invariance Explained: Physics & Math Insight

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...
29. Gauge Invariance for field of *Uncharged* particles?

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...
30. Exploring the Relationship Between Electric Charge and Gauge Symmetry

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.
31. Electromagnetic gauge invariance with boundary conditions

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...
32. Why must the Higgs' gauge symmetry be broken?

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...
33. Please explain gauge invariance un-mathmatically

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.
34. Gauge fixing and gauge transformations

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...
35. Is the Sign in the Covariant Derivative Important for Local Gauge Invariance?

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...
36. What is the relationship between photons and electrons in quantum field theory?

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?
37. Gauge invariance is not normal invariance?

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...
38. Gauge Invariance of Weak Gravity Approximation

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...
39. Gauge Invariance (QED): How Does the Statement Hold?

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?
40. Gauge invariance of electroweak Lagrangian

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.}} =...
41. Gauge invariance of interaction lagrangian

Anyone can help me how to argue that interaction lagrangian is invariant under gauge transformation?
42. Asymptotic safety and local gauge invariance

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...
43. What is the Concept of Gauge Invariance in Physics?

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...
44. A manifesto on gauge invariance - how am I wrong?

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)...
45. Gauge invariance and conserved current in SU(N)

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...
46. General Gauge Invariance Problem

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...
47. Coulomb Gauge invariance, properties of Lambda

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...
48. Non-abelian Local Gauge Invariance in Field Theories

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
49. Physical significance of gauge invariance

I've read that gauge invariance leads to a fundamental phenomenon.What is that? Thanks
50. Gauge invariance of stress-energy tensor for EM field

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...