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arXiv:1701.06542 · Journal of Physics A Mathematical and Theoretical

Non relativistic limit of integrable QFT with fermionic excitations

Alvise Bastianello, Andrea De Luca, Giuseppe Mussardo

Published May 12, 2017• 11 citations
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Abstract

Abstract The aim of this paper is to investigate the non-relativistic limit of integrable quantum field theories with fermionic fields, such as the O ( N ) Gross–Neveu model, the supersymmetric Sinh–Gordon and non-linear sigma models. The non-relativistic limit of these theories is implemented by a double scaling limit which consists of sending the speed of light c to infinity and rescaling at the same time the relevant coupling constant of the model in such a way to have finite energy excitations. For the general purpose of mapping the space of continuous non-relativistic integrable models, this paper completes and integrates the analysis done in Bastianello A et al (2016 J. Stat. Mech . 123104) on the non-relativistic limit of purely bosonic theories.

DOI
10.1088/1751-8121/aa6f69

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Abstract The aim of this paper is to investigate the non-relativistic limit of integrable quantum field theories with fermionic fields, such as the O ( N ) Gross–Neveu model, the supersymmetric Sinh–Gordon and non-linear sigma models. The non-relativistic limit of these theories is implemented by a double scaling limit which consists of sending the speed of light c to infinity and rescaling at the same time the relevant coupling constant of the model in such a way to have finite energy excitations. For the general purpose of mapping the space of continuous non-relativistic integrable models, this paper completes and integrates the analysis done in Bastianello A et al (2016 J. Stat. Mech . 123104) on the non-relativistic limit of purely bosonic theories.
Abstract In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously adjusting the coupling constant g of the quantum field theories in such a way to keep finite the energies of the various excitations. The QFT considered here are Toda field theories and the O ( N ) non-linear sigma model. In both cases the resulting non-relativistic integrable models consist only of Lieb–Liniger models, which are fully decoupled for the Toda theories while symmetrically coupled for the O ( N ) model. These examples provide explicit evidence of the universality and ubiquity of the Lieb–Liniger models and, at the same time, suggest that these models may exhaust the list of possible non-relativistic integrable theories of bosonic particles with local interactions.
We show that the matrix elements of integrable models computed by the Algebraic Bethe Ansatz can be put in direct correspondence with the Form Factors of integrable relativistic field theories. This happens when the S-matrix of a Bethe Ansatz model can be regarded as a suitable non-relativistic limit of the S-matrix of a field theory, and when there is a well-defined mapping between the Hilbert spaces and operators of the two theories. This correspondence provides an efficient method to compute matrix elements of Bethe Ansatz integrable models, overpassing the technical difficulties of their direct determination. We analyze this correspondence for the simplest example in which it occurs, i.e. the Quantum Non-Linear Schrodinger and the Sinh-Gordon models.
We introduce a new family of integrable theories with NNN bosons and NNN freely adjustable mass parameters. These theories restrict in particular limits to the ``generalized supersymmetric'' sine-Gordon models, as well as to the flavor anisotropic chiral Gross Neveu models (studied recently by N. Andrei and collaborators). The scattering theory involves scalar particles that are no bound states, and bears an intriguing resemblance wih the results of a sharp cut-off analysis of the Thirring model carried out by Korepin in (1980). Various physical applications are discussed. In particular, we demonstrate that our theories are the appropriate continuum limit of integrable quantum spin chains with mixtures of spins.
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.
Extending previous work on scalar field theories, we develop a quantum algorithm to compute relativistic scattering amplitudes in fermionic field theories, exemplified by the massive Gross-Neveu model, a theory in two spacetime dimensions with quartic interactions. The algorithm introduces new techniques to meet the additional challenges posed by the characteristics of fermionic fields, and its run time is polynomial in the desired precision and the energy. Thus, it constitutes further progress towards an efficient quantum algorithm for simulating the Standard Model of particle physics.
The class of relativistic spin particle models reveals the `quantization' of parameters already at the classical level. The special parameter values emerge if one requires the maximality of classical global continuous symmetries. The same requirement applied to a non-relativistic particle with odd degrees of freedom gives rise to supersymmetric quantum mechanics. Coupling classical non-relativistic superparticle to a `U(1) gauge field', one can arrive at the quantum dynamical supersymmetry. This consists in supersymmetry appearing at special values of the coupling constant characterizing interaction of a system of boson and fermion but disappearing in a free case. Possible relevance of this phenomenon to high-temperature superconductivity is speculated.