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.
So, recently I've been working through "Classical Theory of Gauge Fields" by Rubakov. I've more-or-less been able to do the exercises as they've come up, but every once in a while I feel like I'm symbol pushing to get the correct answer, or ignoring certain confusions I have in favour of doing...
I have a question about following statement about ghost fields in found here :
It states that introducing some ghost field provides one way to remove the two unphysical degrees of freedom of four component vector potential ##A_{\mu}## usually used to describe the photon field, since physically...
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...
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)...
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...
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...
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...
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
$$\delta S = \delta a E_a + \delta b E_b...
Hi Pfs
i am interested in spin networks (a pecular lattices) and i found two ways to define them. they both take G = SU(2) as the Lie group.
in the both ways the L oriented edges are colored with G representations (elements of G^L
the difference is about the N nodes.
1) in the first way the...
Lawrence Krauss, "The greatest story ever told ... so far", pp. 108-109. "Gauge symmetry in electromagnetism says that I can actually change my definition of what a positive charge is locally at each point of space without changing the fundamental laws associated with electric charge, as long...
Moderator's note: Spin off from previous thread due to advanced nature of topic.
There is classical field theory too, and GR is a relativistic classical field theory of the gravitational interaction. It's ironic that you fight for a geometrical-interpretation-only point of view and at the same...
Edward G. Timoshenko
PhD, MSc, EurPhys, CPhys MInstP, CChem MRSC
Web site: https://www.EdTim.live
Bio:
2011- Researcher, TEdQz Research after an early retirement from UCD
2005 - 2011 Senior Lecturer in Physical Chemistry, School of Chemistry and Chemical Biology, UCD
1997 College Lecturer...
I tried as first step to find Z_q the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon A^a_\mu when a is representation index a \in {1,2,...,N^2-1} such that A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}...
I am taking a course on General Relativity. Recently, I was given the following homework assignment, which reads
> Derive the following transformation rules for vielbein and spin connection:
$$\delta e_a^\mu=(\lambda^\nu\partial_\nu e_a^\mu-e_a^\nu\partial_\nu\lambda^\mu)+\lambda_a^b e_b^\mu$$...
1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##.
The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...
I’m reading Lancaster & Blundell, Quantum field theory for the gifted amateur (even tho I”m only an amateur...) and have a problem with their explanation of symmetry breaking from page 242. They start with this Lagrangian:
##
\mathcal{L} =
(\partial_{\mu} \psi^{\dagger} - iq...
Hopefully, I am in the right forum.
I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as:
Wμ = Wμata
Gauge field = Gauge group x generators in the adjoint...
If anyone is familiar with the calculation of scattering amplitudes using momentum twistors. I am working through the book "Scattering Amplitudes in Gauge Theory and Gravity" by Elvang and Huang.
I am completely stumped by one step that should be simple. My question is about Eq. (5.45). My...
Hello everyone,
I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is$$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)} $$
I have rewritten this term using...
Reading the interesting book "Groups_and_Manifolds__Lectures_for_Physicists_with_Examples_in_Mathematica", in the introduction it is stated:
(...) we have, within our contemporary physical paradigm, a rather simple and universal scheme of interpretation of the Fundamental Interactions and of...
This is not a technical question. I'd like to have a more conceptual discussion about what - if anything - gauge invariance tells us about reality. If we could, please try to keep the discussion at the level of undergrad or beginning grad.
To focus my questions and keep things elementary, I'd...
Hi PhysicsForums,
I have a pretty basic question about extracting physical parameters from lattice QCD simulations. As described in "Quantum Chromodynamics on the Lattice" by Gattringer and Lang, it seems we should be able to extract the static quark/anti-quark potential by considering the...
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...
I have been reviewing GR lately because as a mentor I find myself now answering more of those questions. I learned GR years ago from Wald and other sources, but since then have been exposed to the symmetries of the Standard Model. What struck me during this review is I now have a different...
This is a bit of a philosophical/conceptual question. I've done tons of reading on it, of course, but haven't found anything that makes me go 'ah ha'!
I am working steadily through the mathematical formalism of differential geometry, but am struggling to grasp how the things we say in this...
In a thesis, I found double sided arrow notation in the lagrangian of a Dirac field (lepton, quark etc) as follows.
\begin{equation}
L=\frac{1}{2}i\overline{\psi}\gamma^{\mu}\overset{\leftrightarrow}{D_{\mu}}\psi
\end{equation}
In the thesis, Double sided arrow is defined as follows...
Dear All
Can anyone explain for me what is meant by gravitational anomalies in gauge theory?
What is the difference between it and gauge anomalies?
Thank you
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...
Hi there,
I have just read that the gauge field term Fμν is proportional to the commutator of covariant derivatives [Dμ,Dν]. However, when I try to calculate this commatator, taking the symmetry group to be U(1), I get the following:
\left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( {...
I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold.
My question is for the Physicists in the room, why do you want to know the number of...
Hi everyone,
So I recently read a chapter in a math book that vaguely describe how connections on bundles occur in particle physics, but they are very cryptic about the physics part and I just want to know a little bit more about it. So I'll tell you what I read and then follow up with some...
Hi,
is correct to say that there is no interaction between four photons because the gauge group of QED is U (1) while there are interactions of four gluons or four W's because the gauge group of QCD is SU (3) and EW's one is SU (2) xU (1)?
I know that the interaction between four photons is not...
Currently working through some exercises introducing myself to quantum field theory, however I'm completely lost with this problem.
Let $$L$$ be a Lagrangian for for a real vector field $$A_\mu$$ with field strength $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$ gauge parameter...
Dear all
I have a question is the dual of the field strength ( of abelian gauge theory) in 3 dimensional space the same as the gauge field?
I have a formula for the dual field strength and am trying to bring that of gauge field!
Thank you
Dear all
I am trying to prove that the action resulting from studying conformal gauge theory is invariant under SO(2,4)*diff. Can anyone give me a hint to start from thank. I considering several papers: E.A.Ivanov and J.Niederie and others...
Homework Statement
I'm having a bit of trouble evaluating the following commutator
$$ \left[T^{+},T^{-}\right] $$
where T^{+}=\int_{M}d^{3}x\:\bar{\nu}_{L}\gamma^{0}e_{L}=\int_{M}d^{3}x\:\nu_{L}^{\dagger}e_{L}
and...
Is there a physical reason why all gauge groups considered in SM and especially beyond are always semisimple? [+ U(1)] What would happen if they were solvable?
Although I have a good understanding of how to do calculations in gauge field theory, I am still dissatisfied with my understanding of why we use them in the first place.
From a philosophical point, it should be possible to characterize a gauge theory in terms of observables only. I suppose one...
In Griffiths, it seems that the conceptual introduction of the magnetic vector potential to electrodynamics was justified based on the fact that the divergence of a curl is zero; so we can define a magnetic field as the curl of another vector A and still maintain consistency with Maxwell's...
Why is ##\bar{\psi}=e^{i\theta}\psi##, where ##\theta## is a real number, used as the global gauge transformation? Why ##e^{i \theta}##; what's the physical significance or benefit?
Why is ##\bar{\psi} = e^{i \theta(x)} \psi## the local gauge transformation? What does ##\theta## being a...
Hello everyone, I'm trying to write down a Lagrangian invariant under local ISO(3) (rotations+shifts) transformations. I'm working at classical level and there will be no quantization of any kind so the theory shouldn't have any ghost pathology.
However, I found that, out of the 6 gauge fields...
Hello everyone. Does anyone know if it is possible to build a gauge theory with a local ISO(3) symmetry (say a Yang-Mills theory)? By ISO(3) I mean the group composed by three-dimensional rotations and translations, i.e. if ##\phi^I## are three scalars, I'm looking for a symmetry under:
$$...
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?!
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...
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.
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...