Clifford Algebra As a Bridge Between Discrete and Continuous Worlds

Abstract

There are several topics of Clifford algebra where discrete and continuous worlds are deeply intertwined, one for all the Cartan theorem that expresses continuous automorphisms by a succession of reflections, but they thrive separately an do not fit in a comprehensive picture. We will focus on some cases where Clifford algebra manifestly shows discrete features with the hope that shedding light on some details may help a more complete scenario to come out of darkness. In particular we will tackle two themes: the first is the ubiquitous role played by the base two expansion of integers in Clifford algebra: for example the interplay between the signature of the vector space and the ’spinorial chessboard’ or the determination of row and column indices of the isomorphic matrix algebra. The second theme is the contribution that could give the Abelian subalgebra of the idempotents to the solution of the SATisfiability problem, the mother of all combinatorial problems.