Quantum field theory on toroidal topology: Algebraic structure and applications

• Published in: Physics reports Vol. 539; no. 3; pp. 135 - 224
• Main Authors: Khanna, F.C., Malbouisson, A.P.C., Malbouisson, J.M.C., Santana, A.E.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 20-06-2014
• Subjects: Casimir effect; Compactified extra dimension physics; Gross–Neveu model; Nambu–Jona-Lasino model; Particle physics; Quantum fields; Spontaneous symmetry breaking; Superconductivity; Symmetries and [formula omitted]-algebras; Toroidal topology; Quantum fields; Particle physics; Gross–Neveu model; Spontaneous symmetry breaking; Superconductivity; Toroidal topology; Casimir effect; Compactified extra dimension physics; Symmetries and c∗-algebras; Nambu–Jona-Lasino model
• ISSN: 0370-1573; 1873-6270
• DOI: 10.1016/j.physrep.2014.02.002

The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordström, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particle physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matter physics. The theory on a torus ΓDd=(S1)d×RD−d is developed from a Lie-group representation and c∗-algebra formalisms. As a first application, the quantum field theory at finite temperature, in its real- and imaginary-time versions, is addressed by focusing on its topological structure, the torus Γ41. The toroidal quantum-field theory provides the basis for a consistent approach of spontaneous symmetry breaking driven by both temperature and spatial boundaries. Then the superconductivity in films, wires and grains are analyzed, leading to some results that are comparable with experiments. The Casimir effect is studied taking the electromagnetic and Dirac fields on a torus. In this case, the method of analysis is based on a generalized Bogoliubov transformation, that separates the Green function into two parts: one is associated with the empty space–time, while the other describes the impact of compactification. This provides a natural procedure for calculating the renormalized energy–momentum tensor. Self interacting four-fermion systems, described by the Gross–Neveu and Nambu–Jona-Lasinio models, are considered. Then finite size effects on the hadronic phase structure are investigated, taking into account density and temperature. As a final application, effects of extra spatial dimensions are addressed, by developing a quantum electrodynamics in a five-dimensional space–time, where the fifth-dimension is compactified on a torus. The formalism, initially developed for particle physics, provides results compatible with other trials of probing the existence of extra-dimensions.