Φ43 THEORY FROM MANY-BODY QUANTUM GIBBS STATES

We derive the Φ4 3 measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the Φ4 3 measure as a semiclassical limit which captures the formation of Bose–Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the Φ4 3 measure through a Hartree measure with nonlocal interaction, achieved by paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models.

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