I am a physicist and I am pondering over a particular generalization of Stokes' theorem and Maxwell's equations. They apply to vector fields like the electric or magnetic one. However if the vectors aren't true elements of $R^3$ but elements of a Lie group, say SO(3) or SU(2), I wonder how to formulate Stokes' theorem (or Maxwell's equations). The derivative of a Lie group would be an element of the Lie algebra, thus one would put two incompatible quantities into one equation.

Let me add one more explanation why there should be a solution to the problem: For small values, elements of SO(3) (threedimensional rotations) correspond to "vectors": express the rotation axis as direction of the vector and the twisting angle as its length (the Euler axis representation). But for finite rotations, this does not work. Mind also that finite rotations do not commute, while vector addition does.

To summarize, I'd like generalize (or better formulate) Stokes' theorem and Maxwell's equations for elements of the Lie-group SO(3) (or its double cover SU(2)).

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    $\begingroup$ The appropriate generalization of Maxwell's equations are the Yang-Mills equations. $\endgroup$ – Steve Huntsman Mar 6 '13 at 12:29
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    $\begingroup$ The correspondence between "small" elements of $SO_3$ and vectors is called the exponential map for the Lie group. This is the function that sends a skew-adjoint matrix $A$ (3x3 in your case, so its determined by three real numbers) to $I + A + A^2/2 + A^3/3! + \cdots $. Provided $A$ is small, addition of matrices corresponds to matrix multiplication, at least, to first order. To second order you start seeing Lie brackets appear. $\endgroup$ – Ryan Budney Mar 6 '13 at 16:42

For a Lie group analogue of Stokes's theorem, perhaps look at Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9.

For Maxwell's equations, clearly Yang-Mills equations would be the natural generalization.

It might help if I had an idea what sort of generalizations you need.


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