Quantum field theory on toroidal topology: Algebraic structure and applications

Abstract

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 particles 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 matted physics. The theory on a torus Γ d D = (S 1 ) d × R D−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 focussing on its topological structure, the torus Γ 1 4 . 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 field 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.

Casimir effect for scalar current densities in topologically nontrivial spaces

“…[1][2][3][4][5][6][13][14][15][16][17][18]). In higher-dimensional models the Casimir energy of bulk fields induces an effective potential for the compactification radius.…”

Vacuum currents in braneworlds on AdS bulk with compact dimensions

“…with 50) and is obtained from (2.37) by the replacement Ω (R) (z) → Ω (L) (z). This contribution should be added to the right-hand side of (2.43).…”

A model to study finite-size and magnetic effects on the phase transition of a fermion interacting system

“…The methodology to deal with this effect was improved in Refs. [21][22][23][24] inspired on the imaginary-time formalism, the so-called Matsubara formalism [25]. Before presenting size restriction on the model as a quantum field theory on a toroidal topology, let us briefly recall the imaginary-time formalism.…”

“…Then, as argued in [22,24], L can be interpreted as the separation between two parallel planes orthogonal to the z direction. Thus, after performing these two compactifications, we get a system in thermal equilibrium in the form of a film.…”