[Paper Review] Quantum field theory without divergence: the method of the interaction operators

Bruno Galvan|arXiv (Cornell University)|Jul 13, 2016

Spectral Theory in Mathematical Physics7 references3 citations

TL;DR

This paper proposes a novel method to construct ultraviolet-finite quantum field theories by defining inter-sector particle interactions through symmetric interaction operators, avoiding the divergences common in standard approaches. By symmetrizing singular operators like $\|\hat{\mathbf{x}}\|^{-1}\Delta\|\hat{\mathbf{x}}\|^{-1}$, the method ensures the interaction operators and free Hamiltonian share a common dense domain, enabling a mathematically consistent, self-adjoint total Hamiltonian without UV divergences.

ABSTRACT

The recently proposed interior boundary conditions approach [S. Teufel and R. Tumulka: Avoiding Ultraviolet Divergence by Means of Interior Boundary Conditions, arXiv:1506.00497] is a method for defining Hamiltonians without UV divergence for quantum field theories. In this approach the interactions between sectors of the Fock space with different number of particles (inter-sector interactions) are obtained by extending the domain of the free Hamiltonian to include functions with singularities. In this paper a similar but alternative strategy is proposed, in which the inter-sector interactions are implemented by specific interaction operators. In its simplest form, an interaction operator is obtained by symmetrizing the asymmetric operator $\|\hat{ extbf{x}} \|^{-1} Δ\|\hat { extbf{x}}\|^{-1}$. The inter-sector interactions derive from the singularities generated by the factors $\|\hat{ extbf{x}}\|^{-1}$ enclosing the Laplacian, while the domain of the interaction operator does not include singular functions. As a consequence the interaction operators and the free Hamiltonian have a common dense domain, and they can be added together to form the complete Hamiltonian with interaction.

Motivation & Objective

• To develop a mathematically consistent alternative to the interior boundary conditions (IBC) approach for constructing UV-finite quantum field theories.
• To address the problem of ultraviolet divergences in quantum field theories arising from unbalanced creation operators and singular interactions.
• To define inter-sector interactions via symmetric interaction operators that do not require singular functions in their domain.
• To ensure the interaction operators and the free Hamiltonian share a common dense domain, enabling their sum to form a well-defined total Hamiltonian.
• To lay the foundation for a physically relevant, divergence-free quantum field theory using a new formalism based on operator symmetrization and inter-sector coupling.

Proposed method

• The core method involves symmetrizing an asymmetric operator of the form $\|\hat{\mathbf{x}}\|^{-1}\Delta\|\hat{\mathbf{x}}\|^{-1}$ to construct a symmetric interaction operator with a domain free of singular functions.
• The interaction operator is defined on a dense domain of wave functions that are smooth away from the origin and have compact support, avoiding singularities in the domain.
• Inter-sector transitions between Fock space sectors with different particle numbers are implemented via these symmetric interaction operators, which are constructed to preserve self-adjointness.
• The method uses radial projection operators $R$ to isolate s-wave components, simplifying the Laplacian into a radial second derivative $\partial_r^2/r^2$, which captures the singular behavior.
• The symmetrization procedure introduces inter-sector boundary conditions through the commutator $\langle \phi | \stackrel{\leftrightarrow}{\Lambda} | \psi \rangle$, which encodes the transition amplitudes between sectors.
• For many-body systems, the method generalizes to interactions involving multiple particles by defining operators $B_{hk}$ and $C_{hk}$ that describe particle emission/absorption and pair creation/annihilation, respectively, with symmetrized Laplacian terms.

Experimental results

Research questions

• RQ1 Can a UV-finite quantum field theory be constructed without relying on the interior boundary conditions (IBC) approach, while preserving self-adjointness and unitarity?
• RQ2 How can inter-sector interactions between Fock space sectors with different particle numbers be defined using symmetric operators without introducing singular functions in the domain?
• RQ3 Can the symmetrization of singular-looking operators like $\|\hat{\mathbf{x}}\|^{-1}\Delta\|\hat{\mathbf{x}}\|^{-1}$ yield a well-defined, self-adjoint Hamiltonian with a common dense domain with the free Hamiltonian?
• RQ4 What is the structure of the inter-sector transition amplitudes generated by such interaction operators, and how do they relate to physical processes like emission/absorption and pair creation?
• RQ5 Is the proposed method mathematically consistent and potentially physically relevant for realistic quantum field theories?

Key findings

• The interaction operator $\Lambda = \|\hat{\mathbf{x}}\|^{-1}\Delta\|\hat{\mathbf{x}}\|^{-1}$ is not symmetric on its natural domain, but its symmetrized version $\langle \phi | \stackrel{\leftrightarrow}{\Lambda} | \psi \rangle$ generates non-zero inter-sector transition amplitudes.
• The symmetrized commutator $\langle \phi | \stackrel{\leftrightarrow}{\Lambda} | \psi \rangle$ yields a boundary term proportional to $\phi'_{00}(0)\psi_{00}(0) - \phi_{00}(0)\psi'_{00}(0)$, which encodes the inter-sector coupling.
• For a system of three particle types—fermions, antifermions, and bosons—the method defines interaction operators that describe boson emission/absorption and electron-positron pair creation/annihilation.
• The inter-sector transition amplitudes for pair creation are derived from the symmetrized Laplacian term $\langle \phi_h | \stackrel{\leftrightarrow}{\Delta} | \psi_k \rangle_x(y)$, which evaluates to $4\pi \frac{\phi^*(y,\mathbf{y}_k)\psi(y,\mathbf{y}_k) - \phi^*(y,\mathbf{y}_h)\psi(y,\mathbf{y}_h)}{\|\mathbf{y}_h - \mathbf{y}_k\|}$.
• The total inter-sector commutator $\langle \phi | \stackrel{\leftrightarrow}{V_{hk}^1} | \psi \rangle$ vanishes due to antisymmetry, ensuring consistency in the many-body generalization.
• The method ensures that the interaction operators and the free Hamiltonian share a common dense domain, allowing their sum to define a self-adjoint total Hamiltonian without UV divergences.