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Explore a groundbreaking mathematical approach to quantum field theory computation in this seminar lecture from Harvard CMSA's New Technologies in Mathematics series. Learn about tropicalized quantum field theory, an innovative intermediate framework between free and interacting field theories that addresses the computational challenges of traditional perturbative approaches in quantum field theory. Discover how this method transforms the superexponentially growing complexity of Feynman diagrams into a tractable computational problem through exact solutions manifested as non-linear recursion equations for quantum effective action expansion coefficients. Examine the geometric interpretation of these recursions as volume computations of moduli spaces of metric graphs, drawing parallels to Mirzakhani's volume recursions on moduli space of curves. Understand the polynomial-time Monte Carlo algorithm that samples points from graph moduli spaces proportional to their perturbative contributions, potentially revolutionizing the computational complexity of perturbative quantum field theory from exponential to polynomial time. See practical applications through concrete examples of massive scalar quantum field theories in 3D φ³ and 4D φ⁴ systems, with computations reaching unprecedented loop orders of 20-50 using proof-of-concept implementations that would be impossible with conventional methods.
