Obtaining the non-relativistic quantum mechanics from quantum field theory: issues, folklores and facts

Abstract

Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles, reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and nonrelativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot, in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special a e-mail: paddy@iucaa.in relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.

Probing the Planck scale: the modification of the time evolution operator due to the quantum structure of spacetime

“…Their definition, through their expansion in the momentum basis, is rigorous and unambiguous but it is not clear what they physically mean; this is again because we do not have a notion of position operator. (In spite of several attempts in the literature, it has not been possible to define a conceptually sensible single particle position operator in QFT -and there are excellent reasons for this failure; see e.g., [6].) Third, the occurrence of |t| in the evolution operator (and the propagator) is vital for the consistent interpretation of the theory with particles and antiparticles.…”

Fakeons, quantum gravity and the correspondence principle

“…For an analysis of nontrivial issues concerning the nonrelativistic limit of quantum field theory, see ref. [19].…”

“…In other words, the correspondence principle fails to explain why the gauge group of the standard model is the product of the three simplest groups, U(1), SU(2) and SU(3), instead of anything else. For example, we cannot say why factors such as SU (13), SU (19), etc., are absent.…”

Manifestly covariant canonical quantization of the scalar field and particle localization

“…with the Hamiltonian (24) or (27). Inserting the multiparticle state (30) into the Schrödinger equation (34), we obtain the following equations for the wave packet profiles: i ∂g(s,p 1 ,p2, ...,p r ∂s ) = r k=1 ωp k g(s,p 1 ,p 2 , ...,p k , ...p r ),…”