# What is Gauge symmetries: Definition and 14 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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1. ### I Noether's second theorem: two questions

A technical subject, well above my level it seems (I'm still learning about quantum physics and special relativity), but one about which I absolutely must get some clear ideas as soon as possible. From what I 'understand', Noether's second theorem applies to infinite-dimensional symmetry...
2. ### A Basic Question about Gauge Transformations

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...
3. ### I Would infinite entropy break all symmetries?

If the Universe could somehow reach a state of infinite entropy (or at least a state of extremely high entropy), would all fundamental symmetries of the physical laws (gauge symmetries, Lorentz symmetry, CPT symmetry, symmetries linked to conservation principles...etc) fail to hold or be...
4. ### I Physicists who propose that symmetries are emergent?

I know of some physicists (e.g Holger B Nielsen, Grigory Volovik or Edward Witten) who have proposed that all symmetries (Local gauge symmetries associated with forces and dynamics and global symmetries associated with conservation laws) are emergent rather than fundamental. Are there any other...
5. ### A Structure of Matter in Quantum Field Theory

This is a topic I've mentioned a few times before. Essentially the structure of matter in quantum gauge field theories is unclear to me. I have no clear question here, just some initial discussion points. So at the first level, it seems a particle based view of quantum field theory is difficult...
6. ### I ##A_\mu^a=0## in global gauge symmetries ?

Hi, this question is related to global and local SU(n) gauge theories. First of all, some notation: ##A## will be the gauge field of the theory (i.e: the 'vector potential' in the case of electromagnetic interactions) also known as 'connection form'. In components: ##A_\mu## can be expanded in...
7. ### Mathematical Quantum Field Theory - Gauge symmetries - Comments

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Gauge symmetries Continue reading the Original PF Insights Post.
8. ### A Any good idea how non-abelian gauge symmetries emerge?

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...
9. ### I Difference between global and local gauge symmetries

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...
10. ### Local gauge symmetries Lagrangians and equations of motion

Hey gang, I'm re-working my way through gauge theory, and I've what may be a silly question. Promotion of global to local symmetries in order to 'reveal' gauge fields (i.e. local phase invariance + Dirac equation -> EM gauge field) is, as far as i can tell, always done on the Lagrangian...
11. ### What are the proposed gauge-symmetry groups for Grand Unified Theories?

I first thought of posting on cataloguing various Grand Unified Theory proposals, but that would be an enormous task, so I decided on something simpler: cataloguing proposed GUT gauge-symmetry groups. The unbroken Standard-Model symmetry is SU(3)C * SU(2)L * U(1)Y QCD: SU(3)C -- color...
12. ### What Are the Gauge Symmetries of This Theory?

Homework Statement I want to derive Gauge symmetries of the following gauge theory: S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} F_{\mu\nu}^{\;\;IJ} Where B is an antisymmetric tensor of rank two and F is the curvature of a connection A i.e: F=dA+A\wedge A...
13. ### Spontaneous symmetry breaking of gauge symmetries

hello all gauge symmetries are redundencies of the description of a situation. Therefore they are not real symmetries. So in what sense does it mean to spontaneously break a gauge symmetry? ian
14. ### Searching for Gauge Symmetries and Their Application in Physics

I'm searching informations about the Gauge simmetries and their application in physics; where can i search in internet and where on books? thanks for answers