# What is Gauge symmetry: Definition and 46 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.

View More On Wikipedia.org
1. ### I Gauge and Phase Symmetries in Quantum Systems: Exploring the Confusion

This is about the paper by Greiter: https://arxiv.org/pdf/cond-mat/0503400.pdf Greiter argues that local electromagnetic gauge symmetry cannot change the state of a quantum system. On the other hand, in QED charge or particle conservation (if energy is too low to produce particle-antiparticle...
2. ### A How can gauge fields be associated with real particles?

When we make our lagrangian invariant by U(1) symmetry we employ the fact that nature doesn't care how I describe it, but, how come that I can associate the real physical particles with the coordinates I use to describe? Even though gauge symmetry is not a physical Symmetry,

26. ### U(1) Gauge Symmetry: What Informs Its Choice?

So, I have a basic/general question here. I understand that, for example, the QED Langrangian has U(1) gauge symmetry. I also understand that this means (when you have written the Lagrangian with the covariant derivative) that if you transform the wavefunction (\psi \rightarrow e^{i \theta (x)}...
27. ### Trouble explaining Gauge Symmetry

I'm currently attempting to explain the concept of Gauge Symmetry to a friend. Copied and pasted pretty much directly from MathIM, (And the same applies for any other potential field, such as gravitational potential.) Would this be correct? I've tried explaining Gauge Symmetry multiple...
28. ### Gauge Symmetry in Quantum Mechanics (QM I) Explained

Hi all, I'm taking graduate level QM I and trying to wrap my head around the notion of gauge symmetry. For some reason I've struggled with this concept more than others. I don't really have a specific question; I'm more looking to see if someone has a succinct explanation of the relevant...
29. ### Photon helicity: Wigner's unitary rep. of Poincare group and gauge symmetry

1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f. 2) For gauge theories with...
30. ### Noether currents for local gauge symmetry

hi everyone, I have been trying to understand gauge theory. I am familiar with the Noether's theorem applied in the context of simpler textbook cases like poincare invariant Lagrangians. This is my question: Are there Noether currents corresponding to the local gauge symmetries too and would...
31. ### Spontaneously broken gauge symmetry

I have read 2 arguments that a gauge symmetry cannot be spontaneously broken. 1. Wen's textbook says a gauge symmetry is a by definition a "do nothing" transformation, so it cannot be broken. 2. Elitzur's theorem, eg.http://arxiv.org/abs/hep-ph/9810302v1 The first argument seems sound...
32. ### [Holography] Global symmetry in boundary corresponds to gauge symmetry in bulk?

I hear the statement that global symmetries in the boundary field theory corresponds to gauge symmetries in the bulk. 1) Is this a generic statement that is expected to hold for all holography pairs? (Maldacena states this towards the end of his first lecture at PiTP2010, which was supposed to...
33. ### Gauge symmetry and renormalization

Here and then I read gauge symmetry makes theories renormalizable. Unfortunately I could not find a satisfactory explanation why that so is. Could someone shed some light? thanks
34. ### Gauge symmetry in EM by inspection

Hello: I was under the impression that gauge symmetry was a property of the Lagrange density. Here is the Lagrangian for EM written out in its components: \begin{align*} \mathcal{L}_{EM} &= J\cdot A +\frac{1}{2}\left(B^2-E^2\right) \quad eq.~1\\ &=\rho \phi - Jx Ax - Jy Ay - Jz Az \\...
35. ### Why are there several gauge fixing choice for gauge symmetry fields?

Please teach me this: For gauge symmetry fields,only one of any elementary subconfiguration of the whole configuration covers the all physics of the field.So we need to cut off the redundant configuration.It seem to me,in a loose sense,there is only one way to cut off the redundancy(the gauge...
36. ### Gauge symmetry for spin 1/2 fields

Why is there no gauge symmetry for spin-1/2 fields? Has gauge symmetry to be related to spin-1 fields/ particles? thanks
37. ### Does Noether theorem apply to gauge symmetry?

Basically, the title says it all. I've never heard of Noether charge corresponding to gauge symmetry of the Lagrangian. Is it because gauge symmetry isn't the "right type" of symmetry (one parameter continuous symmetry) so the Noether theorem doesn't apply to it?
38. ### Deriving the needed wavefunction transformation for gauge symmetry?

Homework Statement Take the Schrodinger equation for a point particle in a field: i\hbar \frac{\partial \Psi}{\partial t} = \frac{1}{2m}(-i\hbar\nabla - q\vec{A})^2\Psi + q\phi\Psi I'm supposed to determine what the transformation for Psi is that corresponds to the gauge transformation...
39. ### Exploring the Meaning of Gauge Symmetry

I would like to hear an original explanation of gauge symmetry. What gauge symmetry really means and why it is needed to describe nature. I am more or less familiar with the standard treatment of electromagnetism and Yang Mills theories from QFT texts, but feel still unsatisfied since I have...
40. ### Kaluza Klein and gauge symmetry breaking.

In standard, old-fashioned, Kaluza Klein theory we have new dimensionful parameters, the size of the compact dimensions, but they become dimensionless after quotient against the Plank size, so they become the adimensional coupling constants of the gauge groups associated to the symmetry of the...
41. ### Exact Gauge Symmetry of the Standard Model

Hello: The gauge symmetry of the standard model is written in authoritative places like wikipedia :-) as U(1)xSU(2)xSU(3). This would have 12 elements in its Lie algebra corresponding to one photon, W+, W- and W0 or Z, and the 8 gluons. I recall reading discussions that such a...
42. ### Understanding Local Gauge Symmetry

Local Gauge Symmetry ?? Trying to understand local gauge symmetry ================================ I have an undergraduate degree in physics, so I know basic quantum mechanics, but that's all. Still, I'm trying to understannd the concept of local gauge symmetry. I would appreciate if...
43. ### Strings Chan-Paton U(N) gauge symmetry fractional winding number

I understand why in the presence of a constant vector potential A=-\frac{\theta}{2 \pi R} along a compactified dimension (radius R) the canonical momentum of a -e charged particle changes to P=p-eA. Due to the single valuedness of the wavefunction [itex]\propto e^{iPX}[/tex] P should be...
44. ### Gauge symmetry and symmetry breaking

How would one know in general, whether an original gauge symmetry in the theory is still gauge symmetrical after symmetry breaking? I mean is there a theorem or something like that? And the other way around, is there a general way of knowing whether there is the possibility of a hidden, i.e...
45. ### Definition for the term gauge symmetry

Could somebody please give me a definition for the term gauge symmetry in contrast to any other symmetry? Is the decisive difference that a gauge symmetry is local i.e. a function of the coordinates in contrast to being constant? I would also appreciate it if it could be pointed out how the...
46. ### How does non-abelian gauge symmetry affect quark interactions?

How does a non-abelian gauge symmetry lead to asymptotic freedom for quarks?