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.
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
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,
a. The resistance thermometer bridge circuit shown in FIGURE 1 has a designed maximum temperature of 200°C, ignoring the effects of connecting wire resistance. If the connecting loop is 250 m determine the smallest gauge (swg) of copper wire which must be used if the indicated maximum...
In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...
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
$$\delta S = \delta a E_a + \delta b E_b...
I am given an initial vector potential let's say:
\begin{equation}
\vec{A} = \begin{pmatrix}
g(t,x)\\
0\\
0\\
g(t,x)\\
\end{pmatrix}
\end{equation}
And I would like to know how it will transform under the Lorenz Gauge transformation. I know that the Lorenz Gauge satisfy...
In this video( ) it's explained what is gauge pressure.
Can someone please explain to me what does atmospheric pressure acting on a tube(in video at 3:51) has to do with displacement of a tube?
I understand that the atmospheric pressure acts on the tube, but in the open space that does not...
In Schutz 8.3, while proving that a Lorentz gauge exists, it is stated that
$$\bar h^{(new)}_{\mu\nu} = \bar h^{(old)}_{\mu\nu} - \xi_{\mu,\nu} - \xi_{\nu,\mu} + \eta_{\mu\nu}\xi^\alpha_{,\alpha}$$
where ##\bar h## is the trace reverse and ##\xi^\alpha## are the gauge functions. Then it follows...
If I put this in technically correct terms, to enable a local symmetry related to electron phase change, we need to introduce a spin 1 field which is identified as electromagnetic field.
Yet there are other spin 1/2 particles. Say, what about neutrino? As it does not couple with electromagnetic...
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...
My article has been published in Quantum Reports.
Expanded abstract:
There is currently no consensus on the interpretation of quantum theory, so this article may be of interest as it contains a review and new results on some relevant mathematical models emulating well-known quantum theories...
You see in the literature that the vector potentials in a gauge covariant derivative transform like:
A_\mu \rightarrow T A_\mu T^{-1} + i(\partial_\mu T) T^{-1}
Where T is not necessarily unitary. (In the case that it is ##T^{-1} = T^\dagger##)
My question is then if T is not unitary, how is...
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...
Free Kindle via Amazon, both part 1 and part 2
Part 1:
https://www.amazon.com/dp/B00OD49Z96/?tag=pfamazon01-20
Part 2:
https://www.amazon.com/dp/B00OD49Z00/?tag=pfamazon01-20
Enjoy
Hi.
(not a native English speaker, so apologies in advance for inadequate techical terms)
220V AC, Europe.
There is a cable with 5 wires (2.5mm2 crosssection each) that I would like to cut and make a junction box in the attic to connect another cable to it, to get another outlet. The wires are...
The EM field seems to be required for for local gauge symmetry of the electron matter field under local phase variation. Following is a description (not my verbiage):
There is a symmetry in physics which we might call the Local Phase Symmetry in quantum mechanics. In this symmetry we change...
Wanted to check with you guys that I'm not going crazy...
Exercise 19: Let ##\phi : \mathbf{R}^2 \rightarrow \mathbf{R}^2## be a counterclockwise rotation by angle ##\theta##. Let ##\partial_x, \partial_y## be the coordinate vector fields on ##\mathbf{R}^2##. Show, at any point of...
Hi, I'd like to clarify the following terminology
(Fradkin, Quantum Field Theory an integrated approach)
"carry the quantum numbers of the representation of the gauge group":
Does the author basically mean that the wilson loop is a charged operator, in a sense that it transforms non-trivially...
Happy Holidays! I saw your posting on airflow thru mesh screen and went to Perry's looking for answers. I haven't found what I'm looking for yet and was wondering if you can help?
Our problem concerns bees coming down the chimney and how to stop them.
We installed a Lymance chimney damper with...
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...
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...
In p.385 of Griffiths QM the vector potential ##\textbf{A} = \frac{\Phi}{2\pi r}\hat{\phi}## is chosen for the region outside a long solenoid. However, couldn't we also have chosen a vector potential that is a multiple of this, namely ##\textbf{A} = \alpha \frac{\Phi}{2\pi r} \hat{\phi}## where...
I want to show that the action staying the same action after taking ##A^\mu \to A^\mu + \partial ^\mu \chi##, for the first term I suceeded in showing the invariance using the fact ##[\partial ^ \mu , \partial ^\nu]=0## but for the second term I'm getting: ##\epsilon^{\alpha\mu\nu}A_\alpha...
A transformer supplies 240VAC to a 20A circuit breaker as shown in the attached image. I believe I can use 12AWG wire for wires X1, X4, 2 & 3. However, I'm not sure what gauge wire the neutral should be. Maybe 12AWG or something larger?
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...
Symmetry transformations in physics can be either passive or active. Symmetries in field theory can be either global or local. But only the local ones, the so called gauge symmetries, are fundamental. Except that local transformations cannot be active (despite the fact that diffeomorphisms are...
I have been reading the book of Chris Quigg, Gauge theories, Chapter 3, sec 3.3 in which he explains how local rotations transform wave function and variations in Schrodinger equation forces us to introduce the electromagnetic interaction between the particles. I need a bit deep concept of the...
Show that the Feynman amplitude for Compton scattering ##\mathcal{M} = \mathcal{M}_a + \mathcal{M}_b## is gauge invariant while the individual contributions ##\mathcal{M}_a## and ##\mathcal{M}_b## are not, by considering the gauge transformations
$$\varepsilon^{\mu} (\vec k_i) \rightarrow...
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...
Hi everyone!
How do I go about solving this problem?
I tried working out the gauge pressure using this but I have a few unknowns which won't make this possible such as what is the length of x which I labelled in the figure
Any help would be appreciated! Thanks
In linearized gravity we define the spatial traceless part of our perturbation ##h^{TT}_{ij}##. For some reason this part of the perturbation should be gauge invariant under the transformation $$h^{TT}_{ij} \rightarrow h^{TT}_{ij} - \partial_{i}\xi_{j} - \partial_{j}\xi_{i}$$ Which means that...
I have introduced the Lorentz gauge on my perturbed metric ## \gamma_{\alpha\beta} ## given by ##\partial^{a}\gamma_{\alpha\beta}##. However, there remains the freedom to make further gauge transformations $$\gamma_{\alpha\beta} \rightarrow \gamma_{\alpha\beta} + \partial_{\alpha}\xi_{\beta} +...
I see that this procedure helps to get rid of the two extra degrees of freedom (due to the scalar and longitudinal photons) one firstly encounters while writing the electromagnetic field theory in a Lorentz-covariant way; it indeed shows that modifying the allowed admixtures of longitudinal and...
Hi
I have read that the propane gas tank gauges that rely on pressure are useless because as you use the tank part of the liquid propane turns to gas raising the pressure again inside the tank.
What id like to know is then how do these gauges work?
And also I would like more detail on how...
Least count of the screw gauge = Pitch÷No. of divisions on circular scale=1.5÷100 mm =0.015mm
According to me,in this case the main scale reading should be taken as 2 mm because it is the one which is visible and circular scale reading should be 76.
So, Diameter=2 mm + 0.015×76 mm
= 2 mm + 1.14...
Hi everybody, this is my 1st post as I am a little stuck.
I am putting together a short presentation for a job application and have built a Tool similar to the attached ( basic ) schematic,
My question is
What is the Purpose of the Active Inverted Magnetron ( AIM ) - ( on the chamber ) and the...
I'm going through the "Advanced Lectures on General Relativity" by G. Compère and got stuck with solving one set of conditions on the subject of asymptotic flatness. Let ##(M,g)## be ##4##-dimensional spacetime and ##(u,r,x^A)## be a chart such that the coordinate expression of ##g## is in Bondi...
How do we verify whether a condition on the magnetic vector potential A constitutes a possible gauge choice ?
Specifically, could a relation in the form A x F(r,t) be a gauge , where F is an arbitrary vector field?
The Klein-Gordon equation is based on the relation
(E-eΦ)2-(pc-eA)2=m2^2c2, which is the magnitude of the difference between the momentum four-vector and the four-potential.
Since the magnitude of the momentum four-vector is given by
E2-p2c2=m2c4, does it follow that the magnitude of the...
I'm setting up a Faraday/Lenz Law lab and was wondering if anyone had a suggestion on what gauge of wire I should use to get the best results. We don't have any wire here so I can't test it myself.
Thanks.
This is from QFT for Gifted Amateur, chapter 14.
We have a Lagrangian density: $$\mathcal{L} = (D^{\mu}\psi)^*(D_{\mu}\psi)$$
Where $$D_{\mu} = \partial_{\mu} + iq A_{\mu}(x)$$
is the covariant derivative.
And a global gauge transformation$$\psi(x) \rightarrow \psi(x)e^{i\alpha(x)}$$
We are...
My answer : Both pressures are equal, i.e. ##\boxed{P_A = P_B}##.
Reason : (1) The block of wood displaces an amount (mass) of liquid equal to its weight (archimedes' principle for floating bodies, or law of floatation). Hence we can imagine removing the block in the second case and filling it...
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...
Assuming water to flow out of the pipe with the same speed as inside and the thickness of water column ##h_{ab} = h_{cd} = h##, my answer would be ##\mathbf{(P_b = P_c) > (P_a = P_d)}##.
My reasoning is as follows : at positions ##a\; \text{and}\; d## the gauge pressure is 0 and the total...
Hello all
I was wondering someone could help clear up my understanding about the difference between Absolute and Gauge Pressure.
After some reading i have been told that the Absolute Pressure is pressure taken at 0 relative to a vacuum.
I am trying to understand what this actually means...
Hello!
I have a volume of 50 liters which I pressurize with air so that I read 1 bar on the manometer.
But there is a leakage in the volume so after 30 sec the manometer shows 0,5 bar.
What is then the air flow ( liter / min) of the leakage?