Quantum fields in toroidal topology

Abstract

The standard representation of c*-algebra is used to describe fields in compactified space-time dimensions characterized by topologies of the type ${\Gamma }_D^d=({\mathbb{S}}^1{)}^d\times {\mathbb{M}}^{D-d}$. The modular operator is generalized to introduce representations of isometry groups. The Poincaré symmetry is analyzed and then we construct the modular representation by using linear transformations in the field modes, similar to the Bogoliubov transformation. This provides a mechanism for compactification of the Minkowski space-time, that follows as a generalization of the Fourier-integral representation of the propagator at finite temperature. An important result is that the $2\times 2$ representation of the real time formalism is not needed. The end result on calculating observables is described as a condensate in the ground state. We analyze initially the free Klein-Gordon and Dirac fields, and then formulate non-abelian gauge theories in ${\Gamma }_D^d$. Using the S-matrix, the decay of particles is calculated in order to show the effect of the compactification.

34 pages, LATEX, no figures

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• Magnetic effects on spontaneous symmetry breaking/restoration in a toroidal topology
• Lorentz Symmetry and IR Structure of The BFSS Matrix Model
• Gravitational Casimir effect at finite temperature
• Spontaneous symmetry restoration in a field theory at finite chemical potential in a toroidal topology
• Lorentz violation and Gravitoelectromagnetism: Casimir effect and Stefan-Boltzmann law at Finite temperature
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• Casimir effect in Very Special Relativity at finite temperature
• Quantum forces in cavities
• On Stefan-Boltzmann law and the Casimir effect at finite temperature in the Schwarzschild spacetime
• On the Temperature of Gravitation in the de Sitter Space-Time
• On Gravitational Stefan-Boltzmann Law and Casimir Effect in FRW Universe
• Casimir effect and Stefan-Boltzmann law at finite temperature in a Friedmann-Robertson-Walker universe