Classical electromagnetism via relativity: An alternative by William Geraint Vaughan Rosser
By William Geraint Vaughan Rosser
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Additional info for Classical electromagnetism via relativity: An alternative approach to Maxwell's equations
Incidentally, one immediately recognizes how to recover the spin network boundary states of the Ponzano-Regge model: through the harmonic expansion of ϕ in terms of Fourier modes labeled by triplets of spins. 28) where dμC is a Gaussian measure defining the notion of propagation of boundary data, and Sint is the interaction part of an action, encoding the non-Gaussian part of the dynamics. Note that, for reasons which will become clear shortly, the kinetic part of the action is encoded in the Gaussian measure dμC , together with the (illdefined) Lebesgue measure on L 2 (SU(2)3 ).
There is an important subtlety however, lying in the fact that smooth connections are replaced by a generalized space of connections before the quantization. It can be given an inductive definition in terms of holonomy functionals associated to closed embedded graphs , which play a central role in the definition of the kinematical Hilbert space. As far as we know, alternative constructions might be proposed at this level, relying on slightly different generalizations of smooth connections. This concerns for example regularity properties of the graphs (usually assumed to be piecewise analytic), as was already explored in .
Solving for the equations of motion the variation of λ provides therefore gives back the Holst action (in one sector). The main interest of the HolstPlebanski action principle is that BF theory can be rigorously quantized by spin foam techniques. This consists in two steps: (1) define a discretization of the B-field and the curvature on some adequate cell complex, which thanks to the topological nature of BF, captures all its dynamical features; (2) quantize the discrete theory through path-integral methods.