electromagnetism
nlab article on the electromagnetic field
Mathematical model from physical input
- The electromagnetic field is a circle bundle with connection
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Experimental input
- Maxwell’s equations imply the fields strength of electromagnetism is a closed differential 2-form on spacetime
- The Dirac charge quantization argument shows that in order for the electromagnetic field to serve as the background gauge field to which a charged quantum mechanical particle couples, it must be the curvature 2-form of a circle bundle with connection
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Maxwell’s equations
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The electromagnetic field strength is a closed differential 2-form on spacetime named the Faraday tensor in \[\Omega^2 (U)\]
- \[F = E \wedge dt + B\]
- Where \[ E \wedge dt = E_1 dx^1 \wedge dt + E_2 dx^2 \wedge dt + E_3 dx^3 \wedge dt\]
- and \[B = B_1 dx^2 \wedge dx^3 + B_2 dx^3 \wedge dx^1 + B_3 dx^1 \wedge dx^2 \]
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The electric charge density and current density combine to a differential 3-form in \[\Omega^3(U)\]
- \[j_{el} = j \wedge dt * \rho dx^1 \wedge dx^2 \wedge dx^3\]
- where \[j \wedge dt = j_1 dx^2 \wedge dx^3 \wedge dt + j_2 dx^1 \wedge dx^1 \wedge dt + j_3 dx^1 \wedge dx^2 \wedge dt \]
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Such that
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\[ dF = 0\]
- the magnetic Gauss law \[\nabla \cdot B = 0\]
- Faraday’s law \[d_t B + \nabla \times E = 0\]
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\[d \star F = j_{el}\]
- Gauss’ law \[\nabla \cdot D = \rho\]
- Ampere’s law \[-d_t D + \nabla \times H = j_{el}\]
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- where \[d\] is the de Rham differential operator and \[\star\] the Hodge star operator
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