Holomorphic gauge theory

Abstract

A new invariant way of obtaining interactions from gauge freedom is explored. No use is made of Lagrangians. Instead, the starting point is a scalar quantity of immediate physical interest: the probability density ρ of the particle in phase space, as defined in references [3–6]. This theory is based not on space-time R⁴ but on the forward tube T, which is interpreted as an extended classical phase space. The probability density ρ is a positive function on T which can be expressed as the fiberwise inner product 〈f, f〉 of the wave function f with itself. Here f is a holomorphic section of the trivial holomorphic vector bundle T × C^S, and the inner product is with respect to a fiber metric h: 〈f, f〉 = f*hf. Conservation of probability, combined with holomorphy, leads to an equation for f which is closely related to the Klein-Gordon equation for a particle minimally coupled to a Yang-Mills field. The Yang-Mills potential is uniquely determined as the canonical connection of type (1,0) defined by h.