A gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of *continuous symmetries* of these internal degrees of freedom, *i.e.* the predicted physics under any transformation in this group on the degrees of freedom. ...

learn more… | top users | synonyms

0
votes
0answers
17 views

about the Horizontal Lift in a Princiapal Bundle

I'm currently studying Fibre Bundle by Nakahara's book, and I'm a bit confused about the following: Imagine we have a Principal Bundle $P(M,G)$ with open chart {$U_i$} and a local section ...
1
vote
2answers
81 views

Why can the divergence of vector potential be anything?

Purcell in his book was deriving the vector potential $\bf A$ using $\text{curl}\;(\text{curl}\; \mathbf A)= \mu_0 \mathbf J\; .$ After some algebra, he came to this: $$-\frac{\partial^2 ...
1
vote
0answers
25 views

What exactly is the “diagonal embedding” in the supersymmetric topological twist?

Consider $\mathcal{N}=2$ pure SYM theory. If we want to put the theory in a 4-manifold we take its topological twist. The global symmetry group $$G= SU(2)_{+} \times SU(2)_{-} \times SU(2)_I \times ...
1
vote
0answers
26 views

References for the non-Abelian gauge covariant Laplace equation?

Is there a standard reference which discusses solutions to the non-Abelian gauge covariant Laplace equation $D_{\mu} D^{\mu} \phi = 0$, where $D_{\mu} \phi = \partial_{\mu} + ig[A_{\mu}, \phi]$? Note ...
3
votes
1answer
148 views

How can I understand instantons as sheaves?

In specific, instantons are considered or interpeted as torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two ...
1
vote
1answer
100 views

Gauge the symmetry $φ \to φ + a(x)$ for a free massless real scalar field

How does one alter the Lagrangian density for a real scalar field $$\frac{∂_μφ∂^μφ}{2}$$ such that is will be invariant under the gauge transformation $φ → φ + a(x)$? For a complex scalar field ...
1
vote
0answers
71 views

Unitary gauge transformation [closed]

I have the Hamiltonian of a charged particle without Spin in a time- independent magnetic field and an external Potential (no electric field). Also I have the standard Gauge-Transformation $A ...
2
votes
0answers
52 views

Why does a Gauge group have to be a compact Lie group? [duplicate]

In Topological Solitons by Nicholas Manton where he considers "compact Lie groups" to be the gauge groups for generalizing gauge theoretic concepts. But, he does not mention why that condition is ...
5
votes
2answers
278 views

Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
0
votes
0answers
34 views

Action functional of Born-Infeld model

I have a Born-Infeld action functional like this $$I[A,\phi]~=~\int b^2(\sqrt{1+(|\bigtriangledown\times A|^2)/b^2}-1)+|D_A\phi|^2 + b^2(1-\sqrt{(1-|\phi|^2)^2/b^2} ).$$ Have any books or notes talk ...
1
vote
0answers
39 views

$S^2$ and monopole [closed]

In mathematics, to describe a sphere $S^2$ we need two coordinate patches which we call the North semi-sphere and South semi-sphere. Between them there is a map which mathematicians called the ...
1
vote
1answer
87 views

Modified gauge fixing condition in Faddeev-Popov approach

Which gauge fixing conditions are allowed to choose for this approach? For example (following the steps of Peskin in 9.4) I can choose a "modified" lorenz gauge ( for a abelian theory): $$ ...
0
votes
0answers
19 views

When considering local phase transformations are we forced to use covariant derivatives?

When considering local phase transformations $e^{i\theta(x)}$ of the fields $\phi$ and $\phi^*$ corresponding to \begin{equation} \mathcal{L}=\partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi ...
2
votes
0answers
28 views

Construction of $\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$ [closed]

First, Consider a $\phi$ as a coordinates on a copy of $Z= C^N$ Then, I know \begin{align} |\phi_1|^2 + |\phi_2|^2 + \cdots |\phi_N|^2 = r \end{align} which describe $S^{2N-1}$. Implementing $U(1)$ ...
3
votes
2answers
145 views

Gauge symmetry for p-forms

It is well known that the Lorentz invariance of the S-matrix implies gauge redundancy for 1-forms or 'photons'. Does this argument go through to $p$-forms? That is, does Lorentz invariance of the ...
7
votes
4answers
566 views

Why gauge theories have such a success?

[This question was inspired by a identical question asked on a other forum] Note that we may morally include general relativity in the gauge theories. We may have several (some are deliberately ...
0
votes
2answers
174 views

Is my understanding of Gauge Symmetries correct?

I'm currently working on a project about Symmetry Breaking for my physics bachelor. Right now I'm trying to understand Gauge Symmetries (although I guess it's not much of a symmetry). And I've been ...
1
vote
0answers
41 views

Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
7
votes
1answer
104 views

Gauge group topology

The fundamental difference between spinors and tensors is that spinors are sensitive to the homotopy classes of paths through the rotation group $SO(3)$: \begin{equation} \pi_1(SO(3)) = \mathbb{Z}_2, ...
0
votes
0answers
17 views

Finding the gauge transformtation of a Lagrangian [duplicate]

I am asked to find the gauge symmetry of the following Lagrangian: $L = -\frac{1}{4} F^2_{\mu \nu} + (\partial_{\mu} \phi_1 - m_1 A_{\mu})^2 + (\partial_{\mu} \phi_2 - m_2 A_{\mu})^2$ Then I have to ...
0
votes
1answer
180 views

How can I prove that the axial gauge is a valid Gauge fixing condition?

I am studying classical electrodynamics and I have been introduced to the concept of gauge transformations and gauge fixing conditions. Right know I am trying to prove that some conditions are valid ...
2
votes
0answers
57 views

Equal-time commutation relations, Feynman propagator for gauge parameter $\lambda = 1$, physical meaning

Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F_{\mu\nu}F^{\mu\nu}\right),\text{ where }F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$The ...
2
votes
0answers
140 views

Intuition behind Nekreasov's instanton partition function. What do the partitions represent exactly?

I am struggling to understand many things behind Nekrasov's solution. Firstly I want to understand the following In this theory, $a$ represents VEVs the Higgs scalar. So, is the gauge field of the ...
6
votes
1answer
90 views

Feynman propagator for arbitrary values of the gauge parameter $\zeta$

For the choice $\zeta = 1$ the Lagrangian can be brought into a particularly simple form upon integration by parts in the action integral. Equation$$\mathcal{L}' = -{1\over4}F_{\mu\nu}F^{\mu\nu} - ...
1
vote
1answer
66 views

Three gauge bosons vertex

I was told that two $Z$ bosons could not decay to one (virtual) $Z$ boson at any loop level. Is it true? if so, why? Does it also hold for photons? Could we generalise the statement to "There cannot ...
1
vote
0answers
48 views

What is the importance of $2d$ supersymmetric gauge theory? [closed]

I have been working $2d$ $N=(2,2)$ theories focused on checking duality like, Seiberg duality (2d version: named "Hori-Tong" duality) and its extension. During the study i got $2d$ $N=(2,2)$ ...
13
votes
2answers
528 views

Why do we seek to preserve gauge symmetries after quantization?

Gauge symmetries do not give rise to conservation laws via Noether's theorem, and they represent redundancies in our description of the system. So why do we want to keep them after quantization? For ...
1
vote
2answers
117 views

Gauge covariant derivative of a creation operator

Suppose we define the (gauge) covariant derivative or as $$\tilde{\nabla}=\nabla+ie\textbf{A},$$ where the vector potential $\textbf{A}$ has a matrix structure where only the diagonal has nonzero ...
2
votes
1answer
88 views

What is a dual field?

Can you give me an intuitive, physical understanding of a "dual field"? For example, the Hodge dual of the gluon field strength matrix $F$ is $\tilde{F}_{\mu \nu}=\epsilon_{\mu \nu \alpha \beta} ...
4
votes
1answer
100 views

Yang and Mills' (and others') justification for local gauge invariance

In most physics textbooks, local gauge invariance is simply postulated---you start with a global symmetry, e.g. the global phase, then allow it to depend on the spacetime point, make the necessary ...
3
votes
1answer
270 views

Einstein-Yang-Mills Connections

I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations. Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = ...
1
vote
1answer
63 views

Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
3
votes
1answer
135 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) ...
1
vote
1answer
57 views

Phase diagram of gauge + matter theories

I am looking for some notes/reviews on confinement and Higgs phases suitable for Fermionic/Bosonic matter coupled to Abelian ($Z_2$ or $U(1)$ etc) gauge fields. The purpose is to understand issues ...
3
votes
1answer
115 views

Charge not conserved in scalar QED? [duplicate]

Since conservation of charge seems to be a well known concept, I am hoping that I am missing something and that the conclusion is incorrect. However, I have been unable to disprove this. Let me ...
4
votes
1answer
269 views

Polarization Sums in QCD for the calculation of parton model splitting functions

Before i state the actual problem, here's a premise. In the case of a Spin 1 massive particle it's possible to demonstrate that $$\sum_{\lambda=0,\pm1}\epsilon_{\lambda}^{* \ ...
3
votes
0answers
74 views

Do lattice gauge theories with discrete gauge groups have sensible continuum limits?

In lattice gauge theories the only gauge invariant observables are constructed from Wilson loops and local field strength observables are reconstructed as zero size limits of Wilson loops. Furthermore ...
2
votes
1answer
131 views

Dyon condensation and generalized Meissner effect

In section 2.B of Metlitski and Vishwanath's paper: "Generally condensation of a dyon with charges $(q,m)$ gives rise to an analogue of a Meissner effect for the gauge field combination ...
0
votes
1answer
27 views

Vector potential in gauge transformation

While applying Gauge transformation, $\psi\prime = U \psi$ , where $ U= e^{i q \lambda(x)}$ , transformation law for "Vector Potential" comes out to be : $$ A_{\mu}\prime= ...
2
votes
2answers
152 views

Is there any $SU(\infty)$ gauge theory in quantum field theory?

The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there ...
1
vote
1answer
82 views

A question to gauge fixing in nonabelian gauge theories

In quantum gauge theories it is usual to fix the gauge with the equation $\partial^\mu A_\mu = 0$ where $A_\mu$ is the gauge connection. From this gauge fixing condition the remaining gauge degree of ...
0
votes
0answers
62 views

About equivalence of two ways of “derivation” of Standard model

Two ways of SM derivation I know two methods of SM lagrangian "derivation". The first one, which I will call as Weinberg way, is based on approaches of SM as theory with spontaneusly broken ...
0
votes
0answers
35 views

Properties of vector potential

Given the definition of vector potential as, $ A_{\mu}= \sum_{k} \partial_{\mu}\theta F^{k}$ , where F are the generators and $\theta$ are the parameters of the symmetry group. I have two questions ...
2
votes
0answers
45 views

Gauge transformation and Special relativity

While explaining gauge theories, a book makes a comment that the U(1) transformation definition, $ U= e^{i q \lambda(x)}$ is analogous to a special relativity transformation in freely falling ...
4
votes
2answers
144 views

When do gauge theories have protected gapless excitations?

Goldstone's theorem states that a system in which a continuous symmetry is spontaneously broken necessarily has gapless excitations. (A hand-waving "proof" of Goldstone's theorem can be given by ...
1
vote
0answers
23 views

Expansion of non-abelian heat kernel operator $Q$

For abelain $U(1)$ gauge theory ($F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$) we can expand heat kernel operator $Q$ as \begin{align} Q &= -(\partial - iA)^2 + m^2 \\ & = - ...
1
vote
0answers
48 views

Effect of orbifolding on form ields

A paper by Lalak et al, entitled "Soliton Solutions of M-theory on an orbifold", considers the brane solutions of 11 dimensional supergravity on a space of the form $R^{10} \times S^1/\mathbb{Z}_2$. ...
1
vote
0answers
19 views

Are correlators constructed out of Wilson loops singular in pure Yang-Mills?

If I have some gauge invariant function of two Wilson loops (such as $\left<\text{Tr}W_1 \text{Tr}W_2\right>$) does the expectation value diverge when the loops coincide the same way ...
0
votes
0answers
57 views

$U(1)$ gauge symmetry in superfluid

The conventional superfluid phase in a Bose-Hubbard ground state has $U(1)$ symmetry. In the presence of spin-orbit coupling (SOC), the superfluid ground state has non-uniform phases. Why do people in ...
3
votes
2answers
256 views

How to include Berry connection in Hamiltonian?

When we calculate Berry connection, $A(R)=i<\psi(x,y)|\frac{d}{dR}|\psi(x,y)>\hat{R}$ corresponding to the Berry phase of any system, the gauge potential is related to the $R$ of the parameter ...