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Description
Quantum field theory is the universal framework underlying particle physics, critical phenomena, gravity, and much of modern theoretical physics, yet large parts of its strongly interacting landscape remain analytically out of reach. In this colloquium I present a semiclassical canovaccio for quantum field theory: a framework in which suitably chosen classical configurations provide the mathematical skeleton for extracting nonperturbative information and lead to predictions for infinite order perturbative computations. After introducing conformal field theories as especially tractable yet highly nontrivial corners of the QFT landscape, I focus on the spectrum of composite operators, with particular emphasis on the Ising conformal field theory. I show how heavy composite operators admit a semiclassical description via the state-operator correspondence, periodic classical motion on the cylinder, and Bohr-Sommerfeld quantization. This approach reorganizes the problem in a way that overcomes the breakdown of ordinary perturbation theory at large operator number and yields analytic control over scaling dimensions beyond leading order. I will discuss how this framework connects free theory intuition, interacting conformal dynamics, and modern precision studies of the 3d Ising CFT, while suggesting a broader strategy for tackling difficult sectors of quantum field theory.
