Rend. Mat. Appl. (7). Volume 39 (2018), 217 – 228 RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI A note on the infrared problem in model field theories Massimo Bertini, Diego Noja and Andrea Posilicano Dedicated to Gianfausto Dell’Antonio on the occasion of his 85th birthday Abstract. In this note we critically re-examine the usual procedure of quantization of classical wave equations with static sources. We point out that the origin of infrared difficulties in the so called van Hove model is related to the complex Hilbert space structure one puts on the classical phase space and the corresponding unitarization of the classical symplectic evolution. Whereas in the usual framework the condition of infrared regularity forces the total charge of the external source to vanish, in our setting the infrared regularity condition is equivalent to having a source with a finite (electrostatic) energy. A similar analysis could be applied to models of field-particle interaction such as the Nelson model. 1 Introduction The van Hove model is the name by which it is known the most elementary exam- ple of a quantum field beyond the free one. Basically it is given by the quantization of the d’Alembert wave equation with a constant fixed source. It was proposed by Leon van Hove [34] in the early fifties as a simplified model where some typical phenomena appearing in Quantum Field Theory, namely the infrared and ultra- violet divergencies, display themselves with manifest evidence. These phenomena were well known to the theoreticians at a heuristic level, in particular in the grand example of Quantum Electrodynamics, but the overwhelming difficulty of that theory prevented (and largely prevents nowadays) a secure comprehension of the fundamental aspects of the problems, at least from a rigorous mathematical point of view. In the van Hove model things are simpler, or at least simple enough to allow some definite statements and their rigorous proofs. Let us begin by giving a short summary of the main features of the van Hove model as it is usually de- scribed in the mathematically oriented presentations, and with an emphasis on the infrared problem. A fairly complete analysis is given in [12] and in the recent treatise [2, Chapter 13]. The classical hamiltonian of the van Hove model is given by H(ϕ,π)= 1 2  ‖(−Δ) 1/2 ϕ‖ 2 + ‖π‖ 2  + g〈ρ,ϕ〉 , (1.1) 2010 Mathematics Subject Classification: 81Q10, 47B25. Keywords: Quantum fields, Infrared divergence. c  The Author(s) 2018. This article is an open access publication.