Gauge theory

introduction

• 틀:수학노트

meaning of the gague invariance

• gauge = measure
• gauge invariance = measurement에 있어서의 invariance를 말함
• Lagrangian should be gauge invariant.

gauge symmetry and measurement

• symmetry implies the existence of something unmeasurable.
• phase is one example

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 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

• Yang-Mills 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

• principal bundles
• topology and vector bundles

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

related items

• differential geometry

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

• Connections, Gauges and Field Theories

• WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG

articles

• Weatherall, James Owen. ‘Fiber Bundles, Yang-Mills Theory, and General Relativity’. arXiv:1411.3281 [gr-Qc, Physics:hep-Th, Physics:math-Ph, Physics:physics], 12 November 2014. http://arxiv.org/abs/1411.3281.
• Quantum field theory and the Jones polynomial Edward Witten, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399[1]