[Paper Review] Photons and Fermions in Spacetime with a Compactified Spatial Dimension

Efrain J. Ferrer, Vivian de la Incera|ArXiv.org|Aug 20, 2004

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates quantum electrodynamics (QED) and gauged-NJL models in a spacetime with one compactified spatial dimension (S¹×R³), showing that photon propagation becomes anisotropic with massive modes and a superluminal transverse mode. The compactification suppresses chiral symmetry breaking, increasing the critical four-fermion coupling needed for condensation, and leads to effective (2+1)-dimensional photon confinement at small radii.

ABSTRACT

The effects of a nonsimply connected spacetime with the topology of $S^{1} imes R^{3}$ in the vacua of QED and gauged-NJL theories are investigated. It is shown that the polarization effects of twisted and untwisted fermions in QED are equivalent, once the corresponding stable vacuum solution of each fermion class is taken into account. The photon propagation in QED is found to be anisotropic and characterized by several massive photon modes and a superluminal transverse mode. At small compactification radius the masses of the massive modes increase as the inverse of the radius, while the massless photon mode has a superluminal velocity that increases logarithmically with that distance. At low energies the photon masses lead to an effective confinement of the gauge fields into a $(2+1)-$dimensional manifold transverse to the compactified direction. In the gauged-NJL model, it is shown that for both twisted and untwisted fermions, the smaller the compactification radius, the larger the critical four-fermion coupling needed to generate a fermion-antifermion chiral symmetry breaking condensate.

Motivation & Objective

• To analyze the effects of nontrivial spacetime topology (S¹×R³) on QED and gauged-NJL theories.
• To determine how fermion boundary conditions (twisted vs. untwisted) affect vacuum stability and photon propagation.
• To investigate the influence of compactification on chiral symmetry breaking and fermion-antifermion condensation.
• To clarify whether vacuum polarization can distinguish between twisted and untwisted fermions when stable vacua are properly accounted for.
• To explore the emergence of effective lower-dimensional physics in compactified extra dimensions.

Proposed method

• Analyzing one-loop effective potentials in QED and gauged-NJL models on a spacetime with compactified spatial dimension S¹.
• Using dimensional regularization and momentum space integration with discrete momentum quantization along the compact dimension.
• Computing vacuum polarization effects via path integral methods and considering both twisted (antiperiodic) and untwisted (periodic) fermion boundary conditions.
• Deriving the effective potential for the gauged-NJL model by summing over discrete momenta and incorporating the stable vacuum solution for the gauge field.
• Applying the relation between finite-temperature field theory and compactified spatial dimensions to map results from thermal field theory to spatial compactification.
• Solving for the minimum of the effective potential to determine critical coupling and symmetry breaking conditions under compactification.

Experimental results

Research questions

• RQ1 How does the nontrivial topology S¹×R³ affect photon propagation in QED, particularly in terms of mass generation and velocity anisotropy?
• RQ2 Can vacuum polarization distinguish between twisted and untwisted fermions when their respective stable vacua are considered?
• RQ3 What is the impact of compactification on the critical four-fermion coupling required for chiral symmetry breaking in the gauged-NJL model?
• RQ4 How does the compactification radius influence the effective dimensionality of photon propagation?
• RQ5 Does the compactified dimension restore chiral symmetry in the gauged-NJL model, and if so, how?

Key findings

• In QED, photon propagation becomes anisotropic with multiple massive modes and one superluminal transverse mode, whose velocity increases logarithmically with the compactification radius.
• The masses of the massive photon modes scale inversely with the compactification radius, increasing as the radius shrinks.
• At low energies and small compactification radii, gauge fields are effectively confined to a (2+1)-dimensional manifold, where photons propagate superluminally.
• For both twisted and untwisted fermions in the gauged-NJL model, the critical four-fermion coupling required for chiral symmetry breaking increases as the compactification radius decreases.
• The stable vacuum solution for untwisted fermions in QED corresponds to a constant gauge field configuration with A₃ = π/(ea), which is gauge-equivalent to a nontrivial flux.
• The effective potentials for twisted and untwisted fermions coincide when their respective stable vacua are used, implying vacuum polarization cannot distinguish between the two fermion classes.