Hamiltonians Without Ultraviolet Divergence for Quantum Field Theories

Abstract

We propose a way of defining Hamiltonians for quantum field theories without any renormalization procedure. The resulting Hamiltonians, called IBC Hamiltonians, are mathematically well-defined (and in particular, ultraviolet finite) without an ultraviolet cut-off such as smearing out the particles over a nonzero radius; rather, the particles are assigned radius zero. These Hamiltonians agree with those obtained through renormalization whenever both are known to exist. We describe explicit examples of IBC Hamiltonians. Their definition, which is best expressed in the particle-position representation of the wave function, involves a kind of boundary condition on the wave function, which we call an interior-boundary condition (IBC). The relevant configuration space is one of a variable number of particles, and the relevant boundary consists of the configurations with two or more particles at the same location. The IBC relates the value (or derivative) of the wave function at a boundary point to the value of the wave function at an interior point (here, in a sector of configuration space corresponding to a lesser number of particles).Comment: 27 pages LaTeX, 1 figure. The old version v1 has been (revised and) split into two papers, the first of which is v2 of this post, and the second of which is available as arXiv:1808.06262. v3, v4, v5: minor improvements, updated references, corrected prefactor in Eq. (58