# What is Gauge theory: Definition and 93 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 Problem evaluating an anticommutator in supersymmetric quantum mechanics

I am trying to reproduce the results of a certain paper here. In particular, I'm trying to verify their eqn 5.31. The setup is N = 4 gauge quantum mechanics, obtained by the dimensional reduction of N = 1 gauge theory in 4 dimensions. ##\sigma^i## denotes the ith pauli matrix. ##\lambda_{A...
2. ### A What exactly does 'Locality' in Gauge Theory mean?

What means exactly the principle of 'locality' in context of gauge theory? Motivation: David Tong wrote in his notes on Gauge Theory (p 115): "their paper (the 'original' paper by Yang & Mills introducing their theory) suggests that global symmetries of quantum f ield theory– specifically SU(2)...
3. ### A What does it mean when the eom of a field is trivially satisfied?

If a Lagrangian has the fields ##a##, ##b## and ##c## whose equations of motion are denoted by ##E_a, E_b## and ##E_c## respectively, then if \begin{align} E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c \end{align} where ##f_1## and ##f_2## are some functions of the fields, if ##E_b## and ##E_c## are...
4. ### Gauge invariance in a non-abelian theory SU(2)xU(1)xU(1)

I believe what is asked is impossible. Here is why. The U(1) factors are abelian, so V and T commute with each other and with U, so i can just try to build a term containing and even number of T-s,V-s and U-s. From the transformation laws we see that a bilinear term in the Weyl fermions must...
5. ### A Hodge decomposition of a 1-form on a torus

I was reading Dunne's review paper on Chern-Simons theory (Les-Houches School 1998) and I don't get how he decomposes the gauge potential on the torus. My own knowledge of differential geometry is sketchy. I do know that the Hodge decomposition theorem states that a differential form can be...

44. ### Understanding the Weak Interaction: What Causes It & What is Its Range?

Recently it struck me that I'm not sure I understand the weak interaction at all. What causes it to happen? I know that its mediated by the W and Z bosons and has a short range as a result of the large mass these bosons posses, but what does that range refer to? Range from what?!
45. ### Theory of Everything (TOE) Without a Grand Unified Theory (GUT)

Normally, we think about a grand unified theory (GUT) that unifies the standard model forces and particles into an overarching unified framework, as a pre-requisite to a theory of everything (TOE) which adds quantum gravity to a GUT. But, developments of both beyond the Standard Model physics...
46. ### Explaining Electroweak Theory Decomposition to a Beginner

I have come across physicists representing electroweak theory as some kind of decomposition (i.e. U(1)xSU(2)). I was wondering if someone could explain this 'crossing' to me a little further. Fair warning, my understanding of group/gauge theory is v rudimentary at this point.
47. ### Order Parameter in a Gauge Theory, Can we find one in a Gauge Theory(like QCD)?

Hello Community! I can't find a good answer(if there is) to my question. When in statistical mechanics we can define the order parameter for to study some phase transition. we need to define a order parameter. Now, I want to know if we can to define/find some "order parameter" for to...
48. ### Is this an example of a gauge theory? How?

In the presence of a magnetic field with vector potential \vec A and an electric field, the Schrodinger equation for a charged particle with charge q and mass m becomes: \frac{1}{2m} (\frac{\hbar}{i} \vec \nabla-q\vec A)^2 \psi =(E-q \phi)\psi Another fact is that, Schrodinger equation...
49. ### Chiral gauge theory and C-symmetry

Hi, I have a question in Srednicki's QFT textbook. In p.460 section 75(about Chiral gauge theory), it says "In spinor electrodynamics, the fact that the vector potential is odd under charge conjugation implies that the sum of these diagrams(exact 3photon vertex at one-loop) must vanish."...
50. ### Really quick explanation for gauge theory?

Could anyone give a really quick explanation for gauge theory to me? Or a link, or a book is perfectly fine. I just completely don't understand SU symmetry breaking and etc. etc. I also have a question, is everyone who lurks around here a college professor on quantum physics or something...