All Questions
Tagged with group-theory field-theory
113
questions
0
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38
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Free fields in Weinberg QFT vol.1
Background:
In section 5.1 Weinberg discusses free fields. He had shown that for interaction of the form, $V(t) = \int{d^3x \mathscr{H}(\mathbf{x},t)}$ if $$U_0(\Lambda,a) \mathscr{H}(x) U_0^{-1}(\...
- 56
0
votes
1
answer
57
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Does the Wigner little group classification of particles have consequences for classical field theory?
Does the Wigner little group classification of particles have consequences for classical field theory? In particular, I'm curious whether it can be used to predict the two propagating modes for ...
- 2,292
3
votes
0
answers
59
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Global properties of the gauge group
In this very good P.E. answer, it is explained precisely what it means for a quantum system/theory to have a symmetry group $G$ (where $G$ is a Lie group): going back to first principles, it means ...
- 5,989
2
votes
1
answer
132
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Confusion about tensors in $SU(3)$
I have some confusion regarding the notion of tensors in $SU(3)$ (or some other matrix Lie group, but let's keep the discussion to $SU(3)$).
For concreteness, I will refer to Peskin and Schroeder's ...
- 119
7
votes
1
answer
658
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How can a scalar field have components and how do I interpret these components?
From lecture notes$^\zeta$ I've been reading that:
Consider a real three-component scalar field
$$\phi=\begin{pmatrix}\phi_1 \\\ \phi_2 \\\ \phi_3\end{pmatrix}\tag{a}$$
with Lagrangian
$$\mathcal{L}=\...
- 156
3
votes
1
answer
80
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How to find the full global symmetry group of a Lagrangian of $N$ complex scalars?
I have the Lagrangian
$$ \mathcal{L} = \frac{1}{2}D_\mu \Phi^\dagger D^\mu \Phi - \frac{m^2}{2} \Phi^\dagger \Phi - \frac{\lambda}{4}(\Phi^\dagger \Phi)^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$
where $\...
1
vote
0
answers
83
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QFT visual interpretation of $U(1)$
Anyone has (even a "pictorial") way of visualize what the group $U(1)$ does on the fields in the QFT framework?
I know that $U(1)$ can be seen as a circle and the operation of the groups is ...
- 41
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134
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Finite dimensional irreducible projective representations of $SO^+(1,3)$
Conventions:
Take the Minkowski metric tensor to have signature $(+, -, -, -)$. Use Hermitian (instead of skew-symmetric) generators of rotations $J_i$ and anti-Hermitian (instead of symmetric) ...
- 2,676
2
votes
1
answer
158
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Srednicki 36.5 symmetry question
This is from the intro to a problem 36.5 in Srednicki and not part of the problem itself. I am having trouble proving that $$\mathcal{L}=i\psi_j^\dagger\sigma^\mu\partial_\mu\psi_j$$
Has $U(N)$ ...
- 1,713
0
votes
1
answer
64
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CFT In Embedding Space
I am trying to figure out how a translation or a conformal transformation explicitly look like in embedded space.
Given a CFT in Euclidian (or Minkowski) coordinates $x^\mu$ we can embedded them in $d+...
- 194
15
votes
3
answers
3k
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Why do fields have to form a representation of the Lorentz group?
It is often claimed in quantum field theory texts that to have a sensible Lorentz invariant theory, the fields introduced must be in representations of the Lorentz group. This fact has always seemed ...
- 307
0
votes
0
answers
29
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Unclear passage in Lorentz generators derivation
It's not clear to me a passage, in the extraction of the generators of Lorentz's group acting on the Minkowksi's space points: we have
\begin{equation*}
\begin{split}
x^{' \alpha} & = \Lambda^{\...
- 361
1
vote
1
answer
51
views
Why do we look for the representations of $\mathfrak{so}(1,3)$ when looking for projective representations of $SO^+(1,3)$?
In a relativistic quantum field theory, one classifies quantum fields by looking for finite dimensional projective representations of the restricted Lorentz group $SO^+(1,3)$ over the target space $V$ ...
- 2,676
0
votes
1
answer
56
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Confusion on the helicity formalism
I'm studying group representation theory from a more mathematical point of view but I don't understand the link between helicity formalism and the "classical one". They should be both ...
- 191
0
votes
1
answer
122
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Chiral transformations
In Gerard Ecker's book 'Chiral Perturbation Theory' he states that if we have a symmetry group $G$, an element $g \in G$ induces a transformation in $u \in G/H$
\begin{equation}\tag{1}
u(\phi) \...
- 778