A Clifford algebra approach to bosonic strings with fermionic boundaries

The Brauer-Weyl theorem (Brauer, Weyl. 1935 Am. J. Maths. 57, 425-449. (doi:10.2307/2371218)) states that the algebra of tensor products of spinors is isomorphic to the Clifford (geometric) algebra, in any number of space-time dimensions. The algebraic relation between fermions and bosons seen in the standard model agrees with the Brauer-Weyl theorem, but not with supersymmetry. It is commonly argued that string theory requires supersymmetry, on the grounds that non-supersymmetric bosonic string theory (i) does not admit fermions, and (ii) has an unstable tachyonic ground state. This paper rebuts those standard objections, pointing out that bosonic string theory admits fermions lying on a D-brane boundary of open strings, and that the open-string tachyon has all the properties of a Higgs field. The original 1970's formulation of bosonic string theory, re-envisaged as a theory of all the forces, not just the strong force, fits the standard model beautifully. This article is part of the theme issue 'Modern applications of geometric algebra'.