  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).
