"Gauge theory"의 두 판 사이의 차이
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| 9번째 줄: | 9번째 줄: | ||
| − | <h5> | + | <h5>gauge field</h5> |
* a gauge field is defined as a four-vector field with the freedom of gauge transformation, and it corresponds to massless particlas of spin one<br> | * a gauge field is defined as a four-vector field with the freedom of gauge transformation, and it corresponds to massless particlas of spin one<br> | ||
| 19번째 줄: | 19번째 줄: | ||
| − | <h5 | + | <h5>Gauge invariance of the QED Lagrangian</h5> |
<math>\mathcal{L} = \bar{\psi} (i\gamma^\mu \partial_\mu -m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e\bar{\psi}\gamma^\mu \psi A_\mu</math> | <math>\mathcal{L} = \bar{\psi} (i\gamma^\mu \partial_\mu -m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e\bar{\psi}\gamma^\mu \psi A_\mu</math> | ||
| 33번째 줄: | 33번째 줄: | ||
| − | + | <h5>gauge field tensor</h5> | |
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| − | <h5 | ||
* electromagnetic field tensor <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math><br> | * electromagnetic field tensor <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math><br> | ||
2012년 8월 26일 (일) 12:00 판
meaning of the gague invariance
- gauge = measure
- gauge invariance = measurement에 있어서의 invariance를 말함
- Lagrangian should be gauge invariant.
gauge field
- a gauge field is defined as a four-vector field with the freedom of gauge transformation, and it corresponds to massless particlas of spin one
- one example is the electromagnetic field
Gauge invariance of the QED Lagrangian
\(\mathcal{L} = \bar{\psi} (i\gamma^\mu \partial_\mu -m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e\bar{\psi}\gamma^\mu \psi A_\mu\)
Now we have a Lagrangian with interaction terms.
- local phase transformation of fields
\(\psi(x) \to e^{i\alpha(x)}\psi(x)\) - gauge transformation of electromagnetic field
\(A_{\mu}(x) \to A_{\mu}(x)+\frac{1}{e}\partial_{\mu}\alpha(x)}\) - Look at the QED page
gauge field tensor
- electromagnetic field tensor \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!\)
- general gauge fields tensor \(G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}\)
examples of renormalizable gauge theory
- QED
- QCD
- renormalization
Abelian gauge theory
- abelian gauge theory has a duality
Non-Abelian gauge theory
differential geometry formulation
- manifold \(\mathbb R^{1,3}\) and having a vector bundle gives a connection
- connection \(A\) = special kind of 1-form
- \(dA\) = 2-form which measures the electromagnetic charge
- Then the Chern class measures the magnetic charge.
Principal G-bundle
3d Chern-Simons theory
- 3d Chern-Simons theory on \(\Sigma\times \mathbb R^{1}\) with gauge choice \(A_0=0\) is the moduli space of flat connections on \(\Sigma\).
- analogy with class field theory
- replace \(\Sigma\) by \(spec O_K\)
- then flat connection on \(spec O_K\) is given by Homomorphism group the absolute Galois group Gal(\barQ/K)->U(1)
- Now from An's article,
메모
encyclopedia
- http://en.wikipedia.org/wiki/principal_bundle
- http://en.wikipedia.org/wiki/Connection_(vector_bundle)
books
- The Geometry of Physics: An Introduction
- An elementary primer for gauge theory
- 찾아볼 수학책
expositions
- On the Origins of Gauge Theory , Callum Quigley, April 14, 2003
- WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG
articles
- Quantum field theory and the Jones polynomial Edward Witten, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399[1]