Quantum Field Theory of 2+1 Dimensional BTZ Gravity: Graviton Self-Energy, Axion Interactions, and Dark Matter in the Ultrahyperfunction Framework

We present a comprehensive quantum field theory treatment of gravitons and axions in 2+1 dimensions using the novel mathematical framework of Ultrahyperfunctions theory. Building upon the foundational work of Gupta and Feynman, we derive exact quantum relativistic expressions for graviton and axion self-energies that overcome the traditional challenges of non-renormalizable quantum gravity. Our approach employs Ultrahyperfunctions, the generalizations of Schwartz tempered distributions to the complex plane in which naturally resolve the infinities that plague conventional quantum field theories. This mathematical framework, developed by Bollini et al. and extended over 25 years, transforms the Ultrahyperfunction space into a ring with zero divisors, enabling well-defined products and eliminating the need for regularization or renormalization procedures. We quantize Einstein gravity using the rigorous Schwinger-Feynman variational principle rather than functional integral methods, as our Lagrangian contains derivative couplings that render standard approaches inapplicable. By incorporating a novel constraint into the Einstein Lagrangian obtained by Gupta, we achieve a unitary S-matrix and eliminate the need for ghost fields while maintaining gauge invariance. In our 2+1-dimensional formulation, we carefully preserve graviton propagation by constraining the real physical space coordinates rather than naively reducing spacetime dimensions. This ensures the existence of gravitons while maintaining the essential physics of the BTZ black hole background. Our calculations reveal that graviton self-energy exhibits rich momentum and mass dependencies, with distinct behaviors in bradyonic (spacelike) and tachyonic (timelike) momentum regimes. The interaction with axions, an hypothetical dark matter candidates that significantly modifies graviton propagation properties, inducing measurable but subtle changes to the graviton mass and dressed propagator structure. The results demonstrate finite, well-behaved quantum corrections across all momentum scales, with the Ultrahyperfunctions approach successfully handling what would otherwise be intractable divergences. Our numerical analysis reveals complex scaling relationships and provides the first comprehensive graphical study of graviton self-energy in 2+1 dimensions, both in pure gravity and in the presence of axion interactions. This work establishes Ultrahyperfunctions theory as a powerful tool for quantum gravity, potentially offering new pathways for understanding gravitational quantum effects and dark matter interactions while maintaining mathematical rigor and physical consistency.