most recent 30 from http://mathoverflow.net 2013-05-22T19:22:38Z http://mathoverflow.net/feeds/question/82413 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82413/simple-question-on-the-foundations-of-spin-foam-formalism Pedro 2011-12-01T22:17:24Z 2011-12-02T10:37:49Z

<p>To make it simple, take the spin foam formalism of ($SU(2)$) 3D gravity. My question is about the choice of the data that will replace the (smoothly defined) fields $e$ (the triad) and $\omega$ (the connection) on the disretized version of space-time $\mathcal{M}$: the 2-complex $\Delta$: why choose to replace $e$ by the assignment of elemnts $e\in su(2)$ to each 1-cell of $\Delta$, and elements $g_{e}\in SU(2)$ to each edge in the dual 2-complex $\mathcal{J}_{\Delta}$? I mean, these are both $su(2)$-valued 1-forms, thus, roughly speaking, assigning elements of $su(2)$ to vectors of the tangent bundle $\mathcal{TM}$, in other terms assigning elements of $su(2)$ to infinitesimal displacements represented by the 1-cells of $\Delta$. I can understand the choise for $e$, but not for $\omega$, why this is so? Why it is not the inverse choice?</p>