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The electromagnetic field on a spacetime $X$ is mathematically modeled by a circle bundle with connection $\nabla$ on $X$. If the underlying bundle is trivial, or else on local coordinate patches $\mathbb{R}^n \hookrightarrow X$ over which it is so, this connection is equivalently a differential 1-form $A \in \Omega^1(\mathbb{R}^n)$.
This is then called the electromagnetic potential of the electromagnetic field (sometimes: “vector potential” or “gauge potential of the electromagnetic field”).
is the actual field strength of the electromagnetic field.
On a 4-dimensiona Minkowski spacetime with its canonical coordinates $\{t,x^1, x^2, x^3\}$, the electromagnetic potential $A$ is naturally expanded into corredinate components, traditionally written as
Here
$\phi$ is the electric potential
$\vec A = [A_1, A_2, A_3]$ is the magnetic potential
(for this choice of coordinates).
The identification of gauge potentials with connections on fiber bundles is due to:
See also at fiber bundles in physics.
History:
A. C. T. Wu, Chen Ning Yang, Evolution of the concept of vector potential in the description of the fundamental interactions, International Journal of Modern Physics A 21 16 (2006) 3235-3277 [doi:10.1142/S0217751X06033143]
Chen Ning Yang, The conceptual origins of Maxwell’s equations and gauge theory, Phyics Today 67 11 (2014) [doi:10.1063/PT.3.2585, pdf]
Review:
Last revised on February 8, 2024 at 10:15:14. See the history of this page for a list of all contributions to it.