Higher Order Measures, Generalized Quantum Mechanics and Hopf Algebras

Abstract

We study Sorkin's proposal of a generalization of quantum mechanics and find that the theories proposed derive their probabilities from $k$-th order polynomials in additive measures, in the same way that quantum mechanics uses a probability bilinear in the quantum amplitude and its complex conjugate. Two complementary approaches are presented, a $C^{\ast }$ and a Hopf-algebraic one, illuminating both algebraic and geometric aspects of the problem.

14 pages, no figures