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Results tagged with gauge-theory
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user 37286
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. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.
3
votes
1
answer
229
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What's the meaning of "inequivalent quantizations"?
The notion "inequivalent quantizations" is regularly used when topological terms are discussed.
From what I've gathered so far, "inequivalent quantizations" means that there are different quantum s …
- 10.1k
0
votes
Why are transformations that only change something within a finite region redundancies?
Physical systems are described by differential equations plus suitable boundary conditions. (Only when we combine differential equations with proper boundary conditions, we can expect unique solution …
- 10.1k
1
vote
0
answers
40
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Are there more general gauge transformations than simple phase shifts?
Usually, in the context of a $U(1)$ gauge theory, we only consider gauge transformations of the form
\begin{equation}
\Psi(x) \to \mathrm{e}^{i\epsilon(x)} \Psi(x) \, .
\end{equation}
Are there m …
- 10.1k
4
votes
Gluon have colour-anticolour; what about weak bosons?
My answer to my own question here may be helpful.
No, there is no anti-isospin as there is no anti-spin, because $SU(2)$ has no complex representations. In contrast, $SU(3)$ has complex representatio …
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3
votes
1
answer
405
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How to check if some term in the Lagrangian involving gauge bosons is gauge invariant withou...
Normally (for fermions and scalars) we can simply use the decomposition of tensor products of gauge group representations to find invariant terms that we can write into the Lagrangian.
For example f …
- 10.1k
1
vote
1
answer
513
views
Global SU(2) invariance of QED Lagrangian
I'm having problems seeing the global SU(2) invariance of the QED Lagrangian.
My specific problem is seeing why
\begin{equation}
e^{-i a_i \sigma_i} \gamma_\mu e^{i a_i \sigma_i} = \gamma_\mu
\end{e …
- 10.1k
3
votes
1
answer
98
views
Why are transformations that only change something within a finite region redundancies?
I'm trying to build some intuition for a very particular definition of the notions global and local gauge symmetries. The definition goes as follows and appears, for example, in "Quantum Field Theory …
- 10.1k
3
votes
0
answers
238
views
Are mass terms forbidden in the Lagrangian because of parity violation or because fermions l...
Normally one argues that we can't write down Lorentz AND gauge invariant mass terms, because of parity violation, i.e. l-chiral and r-chiral fields transform differently. This means that mass terms ar …
- 10.1k
0
votes
1
answer
94
views
Why do we require that functions which parametrize gauge transformation are smooth?
A local $U(1)$ transformation is given by
\begin{equation}
f(x) = e^{i\epsilon(x)} \qquad \text{with} \qquad \epsilon(x) \in C^\infty \, .
\end{equation}
Why do we require that the functions in …
- 10.1k
3
votes
2
answers
228
views
What's the physical meaning of the gauge invariant quantity $\partial_\mu \varphi(x) - A_\mu...
A famous locally gauge invariant quantity is
$$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , $$
which is interpreted as the measurable electric and magnetic field strengths.
Now, anoth …
- 10.1k
1
vote
0
answers
138
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Is it possible to derive the correct QED Lagrangian without demanding local gauge invariance?
Usually, the correct interaction term $A_\mu \bar{\Psi} \gamma_\mu \Psi$ in the Lagrangian is derived by demanding local gauge invariance.
Is there any other argument that fixes the form of the inter …
- 10.1k
13
votes
1
answer
3k
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Do gauge bosons really transform according to the adjoint representation of the gauge group?
Its commonly said that gauge bosons transform according to the adjoint representation of the corresponding gauge group. For example, for $SU(2)$ the gauge bosons live in the adjoint $3$ dimensional re …
- 10.1k
6
votes
1
answer
1k
views
Why does Coulomb gauge condition $\partial_i A_i =0$ pick exactly one configuration from eac...
There are infinitely many configurations of a vector field $A_\mu$ that describe the same physical situation. This is a result of our gauge freedom
$$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + \ …
- 10.1k
0
votes
Do the equations of motion simply tell us which degrees of freedom are superfluous?
The answer to the question is: Yes!
Electromagnetic interactions are mediated by photons. Since a photon is massless it only has 2 physical degrees of freedom (there is no longitudinal polarization). …
- 10.1k
5
votes
4
answers
1k
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Why does Lorenz gauge condition $\partial_\mu A^\mu =0$ pick exactly one configuration from ...
For a vector field $A_\mu$, there are infinitely many configurations that describe the same physical situation. This is a result of our gauge freedom
$$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + …
- 10.1k