The Dirac Lagrangian is invariant under the transformation of Ψ that is shown below. This transformation is known as U(1), and gauging it yields QED.pic.twitter.com/nq7fPmX13Z
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The Quantum Electrodynamics Lagrangian is obtained by gauging the U(1) symmetry of the Dirac Lagrangian. This process is also known as "minimal gauge coupling".
Ψ is a Dirac fermion field
Aμ is the photon field
Fμν is the EM field strength tensorpic.twitter.com/ILEo8YkzpG
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"Gauging" the Lagrangian refers to making the transformation parameter spacetime dependent α --> α(x), and imposing that the Lagrangian be invariant under this. To do this it's necessary to add the photon field that transforms under the gauge transformation.
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It is not frequent to see someone bring out Lagrangian of QED for a spin. Some part of me wonder if one day someone will bring out Lagrangian of the standard model of particle physics. Probably not, that will need too many footnote/explanation, RGEs, non-Abelian gauge theories.
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Probably the most concise explanation you'll findpic.twitter.com/hcY9hS0NVK
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I like this equation
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Just learned this in QFT last week! Pretty mind blowing.
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Trivial
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Introducing a local gauge transformation i.e. a gauge field A_mu i.e. introducing a covariant derivative "partial derivative_mu +i e A_mu", to establish local gauge invariance. (A'_mu=A_mu-partial_mu Labda(x))
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