Eur. Phys. J. C (2021) 81:931 https://doi.org/10.1140/epjc/s10052-021-09742-0 Regular Article - Theoretical Physics From quantum field theory to quantum mechanics Nuno Barros e Sá 1,2,a , Cláudio Gomes 1,3 1 DCFQE, Faculdade de Ciências e Tecnologia da Universidade dos Açores, 9500-801 Ponta Delgada, Portugal 2 OKEANOS, Faculdade de Ciências e Tecnologia da Universidade dos Açores, 9901-862 Horta, Portugal 3 Centro de Física das Universidades do Minho e do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal Received: 30 July 2021 / Accepted: 10 October 2021 © The Author(s) 2021 Abstract The purpose of this article is to construct an explicit relation between the field operators in Quantum Field Theory and the relevant operators in Quantum Mechanics for a system of N identical particles, which are the symmetrised functions of the canonical operators of position and momen- tum, thus providing a clear relation between Quantum Field Theory and Quantum Mechanics. This is achieved in the con- text of the non-interacting Klein–Gordon field. Though this procedure may not be extendible to interacting field theo- ries, since it relies crucially on particle number conserva- tion, we find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton–Wigner posi- tion operator. We make some clarifications on the interpreta- tion of Newton–Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as ten- sors, rather in a manner that preserves the canonical commu- tation relations. From a complex Klein–Gordon field, posi- tion and momentum operators can be constructed for both particles and antiparticles. Introduction Since its inception, Quantum Field Theory suffered from a number of problems, perhaps most notably: it inherited the conceptual problems already present in the interpretation of Quantum Mechanics; the theory is plagued with infinities – despite the successes of renormalization, the situation is uncomfortable – ; and the theory is not “dynamical”, i.e., it a e-mail: nuno.bf.sa@uac.pt (corresponding author) does not provide for a picture of the spacial and temporal evolution of a system. None of these problems have had up to now a satisfactory resolution. Nevertheless, no one questions the validity of both theories, Quantum Mechanics and Quantum Field Theory, as their successes have been so overwhelming, and many people believe that the resolution of these problems shall come someday. Even if these theories are not “correct”, it is legitimate to believe that they must be correct in “some limit”, as their validity has been solidly proven experimentally in many domains. Given the proximity between Quantum Mechanics in first and second quantisation, it would be sensible too to find the “limit” when Quantum Field Theory can be approximated by Quantum Mechanics, that is, Quantum Mechanics should be contained in Quantum Field Theory, because the former deals with quantum systems with fixed number of particles while the latter deals with arbitrary numbers of particles. While there are some methods that allow one to recover single parti- cle Quantum Mechanics from Quantum Field theory (see [1] for an extensive review on propagator methods and [2, Chap. 33], for approximative methods in the context of effective field theories), it would be interesting to construct an explicit relation between the two. Such a task seems formidable for interacting field theories. Here we propose a method by which from a theory for a free quantum field (the Klein–Gordon field, for simplicity) one can construct all the quantum mechanical systems with fixed numbers of particles in first quantization. In particu- lar, we construct the algebra of operators in first quantisation (that is, the x ’s and p’s, or rather their symmetrized combi- nations, for systems with more than one particle) from the field operators in second quantisation. We find it rewarding that, at least for free theories, the reduction of Quantum Field Theory to Quantum Mechanics can be done - and can be done for any number of particles. 0123456789().: V,-vol 123