Hopf algebras for physics at the Planck scale

Abstract

Applies ideas of non-commutative geometry to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime. The reformulation maintains an interesting symmetry between observables and states in the form of a Hopf algebra structure. In the simplest example both the dynamics and quantum mechanics are completely determined by the Hopf algebra consideration. The simplest example is a two-parameter algebra-co-algebra KAB which is the unique Hopf algebra extension, such as is possible, of the self-dual Hopf algebra of functions on flat phase space C(R*R). In the limit (A=0,B) the author recovers functions on a curved phase space with curvature proportional to B2 and in another limit (A= infinity , B= infinity ), A/B=h(cross), the author recovers quantum mechanics on R>or=0 with an absorbing wall at the origin. The algebra in this way corresponds to a toy model of quantum mechanics of a particle in one space dimension combined with gravity-like forces. It has an interesting Z2 symmetry interchanging A to or from B, and thereby, in some sense, the quantum element with the geometric element. The compatibility conditions that are solved are a generalisation of the classical Yang-Baxter equations.