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 cutoff 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).
Highlights
In many quantum field theories (QFTs), the formulas that one obtains for the Hamiltonian contain terms for the creation and annihilation of particles that are ultraviolet (UV) divergent
The key element of the approach is a new type of boundary condition that we call an interior–boundary condition (IBC) because it relates the values of ψ on the boundary of configuration space Q to the values in the interior of Q, as we will explain presently
We describe examples of IBCs and how they help define a Hamiltonian, results about the rigorous existence and self-adjointness of the Hamiltonians, and how these Hamiltonians are related to some known cases in which a UV cutoff can be removed, making it plausible that the IBC Hamiltonians are physically relevant and not merely mathematical curiosities
Summary
In many quantum field theories (QFTs), the formulas that one obtains for the Hamiltonian (by means of quantization or other heuristics) contain terms for the creation and annihilation of particles that are ultraviolet (UV) divergent. After a condition equivalent to an IBC had been considered as early as 1930 [7], this approach was not followed further; instead, much research took for granted that the Hamiltonian of a QFT is the sum of two self-adjoint operators, the free Hamiltonian and the interaction Hamiltonian. This is not so in the IBC approach, where the free Hamiltonian and the interaction Hamiltonian each map square-integrable functions to distributions and are not defined as self-adjoint operators in Hilbert space.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
