Non-relativistic quantum field theory

Rainer Dick

Rainer Dick

Rainer Dick

Τ. Padmanabhan

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 non-relativistic 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 relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.

Jean Zinn‐Justin

Abstract This chapter introduces the relativistic quantum field theory (QFT) of the neutral scalar boson field. It is a local, relativistic invariant, theory for a real field extension of the non-relativistic field theory of the Bose gas. Locality is a property that plays a central role in most of this work. The QFT is discussed both from the viewpoint of real-time evolution and statistical physics. The holomorphic formalism leads to representations of the S-matrix in terms of field integrals. The S-matrix elements are related to the continuation to real time of various kinds of Euclidean correlation functions. It is argued that the massive φ4 QFT has the quantum Bose gas with a pair potential, in the grand canonical formulation, as a non-relativistic limit.

A. C. Manoharan

Because the subject of relativistic quantum field theory (QFT) contains all of non-relativistic quantum mechanics, we expect quantum field computation to contain (non-relativistic) quantum computation. Although we do not yet have a quantum theory of the gravitational field, and are far from a practical implementation of a quantum field computer, some pieces of the puzzle (without gravity) are now available. We consider a general model for computation with quantum field theory, and obtain some results for relativistic quantum computation. Moreover, it is possible to see new connections between principal models of computation, namely, computation over the continuum and computation over the integers (Turing computation). Thus we identify a basic problem in QFT, namely Wightman's computation problem for domains of holomorphy, which we call WHOLO. Inspired by the same analytic functions which are central to the famous CPT theorem of QFT, it is possible to obtain a computational complexity structure for QFT and shed new light on certain complexity classes for this problem WHOLO.