Quantum fields in toroidal topology

• Published in: Annals of physics Vol. 326; no. 10; pp. 2634 - 2657
• Main Authors: Khanna, F.C., Malbouisson, A.P.C., Malbouisson, J.M.C., Santana, A.E.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01-10-2011; Elsevier BV
• Subjects: [formula omitted]- and Lie algebra; BOGOLYUBOV TRANSFORMATION; CANONICAL TRANSFORMATIONS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPACTIFICATION; CONDENSATES; DECAY; DIFFERENTIAL EQUATIONS; ENERGY LEVELS; EQUATIONS; FIELD EQUATIONS; GAUGE INVARIANCE; Ground state; GROUND STATES; INTEGRALS; INVARIANCE PRINCIPLES; KLEIN-GORDON EQUATION; LIE GROUPS; Mathematical analysis; MATHEMATICAL SPACE; MATHEMATICS; MATRICES; MINKOWSKI SPACE; Modular; PARTIAL DIFFERENTIAL EQUATIONS; POINCARE GROUPS; PROPAGATOR; Quantum field theory; Quantum fields; Quantum physics; Representations; S MATRIX; SPACE; SPACE-TIME; SYMMETRY; SYMMETRY GROUPS; TOPOLOGY; Toroidal topology; TRANSFORMATIONS; WAVE EQUATIONS; Quantum fields; Toroidal topology; c ∗ - and Lie algebra; Compactification
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2011.07.005

The standard representation of
c
∗
-algebra is used to describe fields in compactified space–time dimensions characterized by topologies of the type
Γ
D
d
=
(
S
1
)
d
×
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, which follows as a generalization of the Fourier integral representation of the propagator at finite temperature. An important result is that the 2×2 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 initially analyze the free Klein–Gordon and Dirac fields, and then formulate non-abelian gauge theories in
Γ
D
d
. Using the
S
-matrix, the decay of particles is calculated in order to show the effect of the compactification.
►
C
∗
-algebra is used to describe fields in compactified space-time dimensions. ► The space–time is characterized by toroidal topologies. ► Representations of the Poincaré group are studied by using the modular operator. ► We derive non-abelian gauge theories in compactified regions of space–time. ► We show the compactification effect in the decay of particles using the
S
-matrix.