# Gauge invariance Definition and 17 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. ### A Massive QED

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...
2. ### 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...
3. ### How does gauge invariance determine the nature of electromagnetism?

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...
4. ### A Measurements and electroweak gauge invariance/transformations

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...
5. ### Gauge Invariance in Hamiltonian

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
6. ### Special relativity - Gauge invariance

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...
7. ### 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...
8. ### 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...
9. ### Counting degrees of freedom in field theory

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...

17. ### How a photon is created?

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?