"Gauge theory"의 두 판 사이의 차이
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| − | + | <h5 style="margin: 0px; line-height: 2em;">gauge field tensor</h5> | |
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| + | * electromagnetic field tensor <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math><br> | ||
| + | * general gauge fields <math>G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}</math><br> | ||
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<h5 style="margin: 0px; line-height: 2em;">encyclopedia</h5> | <h5 style="margin: 0px; line-height: 2em;">encyclopedia</h5> | ||
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* http://en.wikipedia.org/wiki/principal_bundle | * http://en.wikipedia.org/wiki/principal_bundle | ||
* [http://en.wikipedia.org/wiki/Connection_%28vector_bundle%29 http://en.wikipedia.org/wiki/Connection_(vector_bundle)] | * [http://en.wikipedia.org/wiki/Connection_%28vector_bundle%29 http://en.wikipedia.org/wiki/Connection_(vector_bundle)] | ||
2011년 2월 3일 (목) 14:03 판
meaning of the gague invariance
- gauge = measure
- gauge invariance 란 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 \(G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}\)
examples of renormalizable gauge theory
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
- http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
- http://www.google.com/search?hl=en&rls=IBMA,IBMA:2008-50,IBMA:en&q=brief+introduction+to+principal+bundles+connections&aq=f&oq=&aqi=
encyclopedia
- http://en.wikipedia.org/wiki/principal_bundle
- http://en.wikipedia.org/wiki/Connection_(vector_bundle)
- http://viswiki.com/en/
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- 다음백과사전 http://enc.daum.net/dic100/search.do?q=
books
- An elementary primer for gauge theory
- 찾아볼 수학책
- http://gigapedia.info/1/gauge
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
articles
- Quantum field theory and the Jones polynomial
- Edward Witten, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399
- http://www.zentralblatt-math.org/zmath/en/