# Classical electrodynamics as an $\mathrm{U}(1)$ gauge theory

Preface: I haven't studied QED or any other QFT formally, only by occasionally flipping through books, and having a working knowledge of the mathematics of gauge theories (principal bundles, etc.).

As far as I am aware, the status of electrodynamics as an $\mathrm{U}(1)$ gauge theory comes from quantum-mechanical considerations, namely the ability to rotate the complex phase of a wave function describing a charged particle, however the "classical" gauge freedom $A\mapsto A+\mathrm{d}\chi$ follows naturally from this, and so does the $F$ electromagnetic field strength tensor as the $\mathrm{U}(1)$ connection's curvature, and this $F$ is the same $F$ as it is in classical ED.

My question is regarding if it is possible to formulate purely classical electrodynamics as an $\mathrm{U}(1)$ gauge theory, including motivation to do so (eg. not just postulating out of thin air that $A$ should be a $\mathrm{U}(1)$ connection's connection form, but giving a reason for it too)?

• Did you check Lagrangian formulation of classical electromagnetism, e.g. here and here? – Qmechanic Jan 12 '16 at 0:51
• @Qmechanic Yes. I am specifically interested in a well-motivated principal bundle formalism for classical ED. – Bence Racskó Jan 12 '16 at 1:16
• A little self-promotion: you may want to check this notes of mine where I explain the above question using classical bundle theory: www.theorie.physik.uni-goettingen.de/~tedesco/files/connections.pdf – gented Jan 12 '16 at 8:29
• @Uldreth: Consider focusing the question. E.g. if you already know Lagrangian formulation of classical electromagnetism, there is no need for potential answerers to repeat that. – Qmechanic Jan 13 '16 at 0:22
• @GennaroTedesco I had the same question as the OP and would be very interested in the document you refer to. When I try to open it however it says that I don't have permission to do so. Would you be willing to make it available elsewhere? – Anonymous Feb 3 '16 at 18:08