What is Gauge transformation: Definition and 35 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.
I understand that adding gradient of a scalar to the electromagnetic field potential keeps the evolution equation of the four potential invariant. But how does that give us the freedom to choose the gauge?
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...
Hello!
I'm starting to study curved QFT and am slightly confused about the invariance of the Klein Gordon Lagrangian under a linear diffeomorphism.
This is $$L=\sqrt{-g}\left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi-\frac{m^2}{2}\phi^2\right),$$
I don't see how ##g^{\mu\nu}\to...
In General Relativity, "gauge" transformations are basically coordinate transformations which preserve length.
In Electroweak and the gauge forces like EM.. what are being preserved? I forgot my lessons before and would like to refresh.
Homework Statement
For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else?
Homework EquationsThe Attempt at a Solution
The equation of motion for a charged particle with mass ##m## and charge ##q## in a static magnetic field is:
##\frac{d}{dt}[m{\dot{\vec{r}}}]=q\ \dot{\vec{r}}\times \vec{B}##
From this, we can see that ##\frac{d}{dt}[m\dot{\vec{r}}-q \vec{r}\times \vec{B}]=0##
and so the following quantity is...
1. Problem
##g_{uv}'=g_{uv}+\nabla_v C_u+\nabla_u C_v##
If ##g_{uv}' ## is given by ##ds^2=dx^2+2\epsilon f'(y) dx dy + dy^2##
And ##g_{uv}## is given by ##ds^2=dx^2+dy^2##, Show that ## C_u=2\epsilon(f(y),0)##?
Homework Equations
Since we are in flatspace we have ##g_{uv}'=g_{uv}+\partial_v...
In some Yang-Mills theory with gauge group ##G##, the gauge fields ##A_{\mu}^{a}## transform as
$$A_{\mu}^{a}
\to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$
$$A_{\mu}^{a}
\to A_{\mu}^{a} \pm...
Usually, one defines large gauge transformations as those elements of ##SU(2)## that can't be smoothly transformed to the identity transformation. The group ##SU(2)## is simply connected and thus I'm wondering why there are transformations that are not connected to the identity. (Another way to...
Hello!
I will be attending a course on condensed matter physics with emphasis on geometrical phases and I was wondering if the are any good books on gauge transformations, gauge symmetry and geometrical phases that you know of.
Thanks in advance!
Based on this lecture notes http://www.helsinki.fi/~hkurkisu/CosPer.pdf
For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system...
Spinors in $N=2, D=4$ supergravity can be simplified using gauge transformation and thus canonical spinors can be found. In the case of $N=2, D=4$ supergravity the gauge transformation Spin (3,1) is used. My question is how do we know which transformation can be used in a certain theory in order...
I asked this question to PhysicsStackExchange too but to no avail so far.
I'm trying to understand the way that the Higgs Mechanism is applied in the context of a U(1) symmetry breaking scenario, meaning that I have a Higgs complex field \phi=e^{i\xi}\frac{\left(\rho+v\right)}{\sqrt{2}}
and...
I've having trouble understanding one of the consequences of using the length gauge.
The length gauge is obtained by the gauge transformation ##\mathbf{A} \rightarrow \mathbf{A} + \nabla \chi## with ##\chi = - \mathbf{r} \cdot \mathbf{A}##. Starting from the Coulomb gauge, we have
$$...
Homework Statement
The lagrangian is given by:
L = -\frac{1}{4} F^2_{\mu \nu} + (\partial_{\mu} \phi_1 - m_1 A_{\mu})^2 + (\partial_{\mu} \phi_2 - m_2 A_{\mu})^2
Homework Equations
Find the gauge transformation of the fields that corresponds to a symmetry.
Find the combination of scalar...
Gauge transformation can be written as:
##\psi(\vec{r},t)\rightarrow e^{-i \frac{e}{\hbar c}f(\vec{r},t)}\psi(\vec{r},t)##
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html
Does it have any sense that we choose such function ##f##, that all right side is constant in time. Is this...
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...
It's well known when if we are working on problems related to particles in presence of an electromanetic field, the way we state the problem can be done using the next Hamiltonian:
H=\dfrac{(p-\frac{e}{c}A)^2}{2m} +e \phi where the only condition for A is: \vec{\nabla } \times \vec{A} =\vec{B}...
I have a question about this classical invariance problem I'm working on. I'm almost done, and I understand the theory I think, so my question may seem a bit more math-oriented (it's been a few years since crunching equations). I have found that under a gauge transformation for a single particle...
Hi All,
I'm working through the theory of the strong interaction and I roughly follow it. However I have some questions about the meaning of the terms.
The book I use gives the gauge transformation as: \psi \rightarrow e^{i \lambda . a(x)} \psi
First question ... What are the a(x)...
I understand ##\vec A\rightarrow\vec A+\nabla \psi\;## as ##\;\nabla \times \nabla \psi=0##\Rightarrow\;\nabla\times(\vec A+\nabla \psi)=\nabla\times\vec A
But what is the reason for
V\;\rightarrow\;V+\frac{\partial \psi}{\partial t}
What is the condition of ##\psi## so
\nabla...
Hi. I'm reading about non-abelian theories and have thus far an understanding that a gauge invariant Lagrangian is something to strive for. I previously thought that the Yang-Mills gauge boson free field term ##-1/4 F^2 ## was gauge invariant, but now after realizing that the field strength...
I understand the conceptual meaning of gauge transformation which "can be broadly defined as any formal, systematic transformation of the potentials that leaves the fields invariant". I understand for example the U(1) and S(3) gauge symmetry in Gauge Theory. But what is this got to do with...
I was wondering if anyone could explain to me where the 2nd order terms in the gauge transformation
h_{\mu\nu}\rightarrow h_{\mu\nu}-\xi_{\mu ,\nu}-\xi_{\nu, \mu}-\xi^{\alpha}h_{\mu\nu, \alpha}-\xi^{\alpha}_{,\mu}h_{\alpha\nu}-\xi^{\alpha}_{,\nu}h_{\mu\alpha}[/itex]
come from. The...
In the weak field approximation,
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}
If we make a coordinate transformation of the form
[itex]x^{\mu'}=x^{\mu}+\xi^{\mu}(x)[\itex]
it changes [itex]h_{\mu\nu}[\itex] to
[itex]h'_{\mu\nu}=h_{\mu\nu}+\xi_{\mu,\nu}+\xi_{\nu,\mu}+O(\xi^{2})[\itex]
I...
What is the meaning of the local gauge transformation exactly??
These days I'm studying. [D.J. Griffiths, Introduction to Elementary Particles 2nd Edition, Chapter 10. Gauge Theories] Here the Section 3. Local Gauge Invariance, the author gives the Dirac Lagrangian, \mathcal{L}=i \hbar c...
Though I've learned gauge transformation for a while, I can't figure out why it is significance in describing fields? For example, why electromagnetic tensor has to be gauge invariant? What does it physically mean?
Hi there!
Few weeks ago I came upon the following problem:
Let B be a vector field derivable from a vector potential A (on a simply connected topological space, smooth enough and everything well established so that mathematicians do not have to care about), i.e. \vec B=rot \vec...
Given a set of a scalar function V and a vector function, how does one recognize that it is a coulomb gauge or lorentz gauge transformation?
Actually there is a method that i use but i am not sure if it is always true:
what i do is to make an electric field (from that set and using known gauge...
Dear All,
I'd be grateful for a bit of help with the following problems:
Consider the Lagrangian:
\displaystyle \mathcal{L} = (\partial_{\mu} \phi) (\partial^{\mu} \phi^{\dagger}) - m^2 \phi^{\dagger} \phi
where \phi = \phi(x^{\mu})
Now making a U(1) gauge transformation...
[SOLVED] invariance of maxwell's equations under Gauge transformation
Homework Statement
Show that the source-free Maxwell equations \partial_{\mu} F^{\mu\nu}=0 are left invariant under the local gauge transformation
A_{\mu}(x^{\nu})\rightarrow...
For electromagnetic field we usually use the Lagrange's density
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},\quad\quad\quad\quad\quad\quad\quad(1)
but we could also use a simpler Lagrange's density
-\frac{1}{2} (\partial_{\mu} A_{\nu})(\partial^{\mu} A^{\nu}),\quad\quad\quad\quad\quad(2)
which...