English

BRST quantization of the massless minimally coupled scalar field in de Sitter space (zero modes, euclideanization and quantization)

General Relativity and Quantum Cosmology 2010-04-21 v1 High Energy Physics - Theory

Abstract

We consider the massless scalar field on the four-dimensional sphere S4S^4. Its classical action S=12S4dV(ϕ)2S={1\over 2}\int_{S^4} dV (\nabla \phi)^2 is degenerate under the global invariance ϕϕ+constant\phi \to \phi + \hbox{constant}. We then quantize the massless scalar field as a gauge theory by constructing a BRST-invariant quantum action. The corresponding gauge-breaking term is a non-local one of the form SGB=12αV(S4dVϕ)2S^{\rm GB}={1\over {2\alpha V}}\bigl(\int_{S^4} dV \phi \bigr)^2 where α\alpha is a gauge parameter and VV is the volume of S4S^4. It allows us to correctly treat the zero mode problem. The quantum theory is invariant under SO(5), the symmetry group of S4S^4, and the associated two-point functions have no infrared divergence. The well-known infrared divergence which appears by taking the massless limit of the massive scalar field propagator is therefore a gauge artifact. By contrast, the massless scalar field theory on de Sitter space dS4dS^4 - the lorentzian version of S4S^4 - is not invariant under the symmetry group of that spacetime SO(1,4). Here, the infrared divergence is real. Therefore, the massless scalar quantum field theories on S4S^4 and dS4dS^4 cannot be linked by analytic continuation. In this case, because of zero modes, the euclidean approach to quantum field theory does not work. Similar considerations also apply to massive scalar field theories for exceptional values of the mass parameter (corresponding to the discrete series of the de Sitter group).

Keywords

Cite

@article{arxiv.0911.2064,
  title  = {BRST quantization of the massless minimally coupled scalar field in de Sitter space (zero modes, euclideanization and quantization)},
  author = {Antoine Folacci},
  journal= {arXiv preprint arXiv:0911.2064},
  year   = {2010}
}

Comments

This paper has been published under the title "Zero modes, euclideanization and quantization" [Phys. Rev. D46, 2553 (1992)]