In the Curci–Ferrari gauge for perturbative quantum gravity on flat spacetime, Faizal (2011) showed at one-loop semiclassical order that a nonlinear ghost-sector interaction drives a Faddeev–Popov ghost condensate ⟨ϕa⟩=±νa≠0\langle\phi^a\rangle=\pm\nu^a\neq 0 through a Hubbard–Stratonovich linearization, and that the modified BRST/anti-BRST nilpotency is restored on shell. Whether this mechanism survives beyond its original setting is open. The source's conclusion explicitly poses one piece of future work: the generalization to curved, in particular maximally symmetric de Sitter and anti-de Sitter, backgrounds, including whether a condensate forms there and whether it spontaneously breaks de Sitter invariance.

Background

Perturbative quantum gravity treats the metric as a small fluctuation gμν=ημν+κhμνg_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu} around a fixed background and quantizes hμνh_{\mu\nu} by gauge fixing, in close analogy with Yang–Mills theory. Gauge fixing in a path integral requires Faddeev–Popov ghosts — unphysical anticommuting scalar fields, here denoted cac^a and cˉa\bar c^a (the index aa is a spacetime vector index, since the diffeomorphism gauge parameter is a vector), which cancel the unphysical metric degrees of freedom — and the gauge-fixed action retains a residual rigid symmetry, the BRST and anti-BRST symmetries (Becchi–Rouet–Stora–Tyutin transformations), whose nilpotency encodes gauge consistency at the quantum level. The Curci–Ferrari gauge is a nonlinear, covariant gauge in which the sum of the gauge-fixing and ghost terms is expressible as a combination of BRST and anti-BRST variations of habhab−iα cˉacah^{ab}h_{ab}-i\alpha\,\bar c^a c_a, so that the graviton quadratic couples to a ghost bilinear, with α\alpha a gauge parameter. In gravity, modifying the BRST and anti-BRST transformations by the nonlinear terms characteristic of this gauge generates an additional quartic, derivative ghost self-interaction α2cˉb∂bca cˉc∂cca\frac{\alpha}{2}\bar c^b\partial_b c^a\,\bar c^c\partial_c c_a. Faizal (J. Phys. A: Math. Theor. 44 (2011) 402001) linearized this quartic ghost term by a Hubbard–Stratonovich transformation (a standard Gaussian linearization that replaces a quartic interaction by an auxiliary field coupled bilinearly), α2cˉb∂bca cˉc∂cca=−12αϕaϕa−iϕacˉb∂bca\frac{\alpha}{2}\bar c^b\partial_b c^a\,\bar c^c\partial_c c_a=-\frac{1}{2\alpha}\phi^a\phi_a-i\phi^a\bar c^b\partial_b c_a, introducing a hermitian auxiliary field ϕa\phi^a of vanishing ghost number. Summing the one-loop ghost diagrams with an arbitrary number of external ϕa\phi^a legs gives, after regularization, an effective potential of schematic form V[ϕ]=ϕaϕa[12α+132π2(log⁡∣ϕ∣4πμ2+C)]V[\phi]=\phi^a\phi_a\left[\frac{1}{2\alpha}+\frac{1}{32\pi^2}\left(\log\frac{|\phi|}{4\pi\mu^2}+C\right)\right], where CC is a regularization-dependent constant and μ\mu a renormalization scale; the published formula is itself schematic (the argument of the logarithm is not dimensionally explicit), and any rigorous treatment must first clean it up. Besides the trivial stationary point ϕa=0\phi^a=0, the potential is asserted to admit nontrivial stationary points ϕa=±νa\phi^a=\pm\nu^a; in the semiclassical treatment the field is shifted as ϕa→ϕcla+ϕ~a\phi^a\to\phi^a_{\mathrm{cl}}+\tilde\phi^a and the vacuum value ⟨ϕa⟩=±νa≠0\langle\phi^a\rangle=\pm\nu^a\neq 0 is identified with the classical field. Because ϕa\phi^a carries a Lorentz vector index and the potential depends only on the invariant ϕaϕa\phi^a\phi_a, a nonzero vacuum value spontaneously selects a preferred spacetime direction, which is the sense in which Lorentz symmetry is spontaneously broken. The condensate modifies the BRST and anti-BRST transformations of the ghost sector (new ghost and antighost fields ψa\psi^a and ψˉa\bar\psi^a enter), and although the modified transformations fail to be nilpotent, their nilpotency is restored on shell. This construction is the gravity analogue of an earlier, more developed Yang–Mills result: Dudal, Verschelde, Lemes, Sarandy, Sorella and Picariello (Ann. Phys. 308 (2003) 62) analyzed the on-shell BRST-invariant gluon–ghost condensate ⟨AμaAaμ+α cˉaca⟩\langle A^a_\mu A^{a\mu}+\alpha\,\bar c^a c^a\rangle of mass dimension two in the Curci–Ferrari gauge of Yang–Mills theory, using the local composite operator technique (promoting the composite operator to an elementary field coupled through an auxiliary term, so that its effective potential can be renormalized) and algebraic renormalization, and established gauge-parameter independence of the condensate-induced vacuum energy. It should not be confused with the separate, large literature on the scalar 'ghost condensate' of Arkani-Hamed, Cheng, Luty and Mukohyama, which concerns a physical scalar field with wrong-sign kinetic term and its cosmological consequences. The source's companion on BRST symmetries in curved spacetime (Faizal, Found. Phys. 41 (2011) 270) and one further paper (Upadhyay, Ann. Phys. 340 (2014) 110) address the same gauge-fixed gravity system, and the companion de Sitter program (Sofi–Akhoon–Rather–Maini; Faizal–Upadhyay–Mandal) developed mode expansions and two-point functions for perturbative quantum gravity on de Sitter spacetime (the maximally symmetric spacetime with positive cosmological constant; anti-de Sitter spacetime has negative cosmological constant) without treating the condensate or its effective potential. The source paper's conclusion explicitly poses the curved-background generalization as future work and speculates on Lorentz-violating phenomenology (polarization of the cosmic microwave background (CMB), high-energy astrophysics, matter/antimatter spectral differences, a possible link to the cosmological constant), while the one-loop analysis is presented with schematic, regulator-dependent constants and no renormalization-group or higher-loop discussion.

Problem Statement

Generalize the Curci–Ferrari ghost-condensation construction — the nonlinear ghost sector, the Hubbard–Stratonovich linearization introducing ϕa\phi^a, the one-loop determinant, and the effective potential — to perturbative quantum gravity on curved, in particular maximally symmetric de Sitter and anti-de Sitter backgrounds, and decide whether a nonzero ghost condensate ⟨ϕa⟩≠0\langle\phi^a\rangle\neq 0 forms there, whether it spontaneously breaks de Sitter/anti-de Sitter invariance, and what becomes of the on-shell modified BRST/anti-BRST nilpotency. This is the open question explicitly posed in the conclusion of Faizal (2011), whose flat-space one-loop analysis is the baseline; settling it means deciding whether ghost-induced spontaneous Lorentz breaking in perturbative quantum gravity survives on curved backgrounds or is an artifact of the flat-background treatment.

Two related directions are recorded here as context, not as parts of the posed problem, and settling them is not required for a solution: (a) beyond-one-loop robustness — the renormalization-group behavior of the regulator-dependent constant CC and scale μ\mu, higher-loop corrections, and the quantitative determination of the condensate magnitude νa\nu^a — on which the source is silent, presenting only the one-loop semiclassical effective potential; and (b) physical content — whether the condensate, which lives in the gauge-fixing (BRST-exact) sector, so that gauge-independence would have to be demonstrated as it was for the Yang–Mills analog, yields genuine, gauge-independent Lorentz-violating observables, and whether any physical signal translates into predicted observable effects (for example in CMB polarization, high-energy astrophysical tests, or matter/antimatter spectral differences) comparable with experimental bounds — which the source raises only as speculative concluding remarks. A solution of the curved-background problem would naturally inform both.

References

  1. Mir Faizal, 'Spontaneous Breaking of Lorentz Symmetry by Ghost Condensation in Perturbative Quantum Gravity', Journal of Physics A: Mathematical and Theoretical 44 (2011) 402001, DOI 10.1088/1751-8113-44-40-402001, arXiv:1108.2853, https://arxiv.org/abs/1108.2853
  2. Mir Faizal, 'BRST and Anti-BRST Symmetries in Perturbative Quantum Gravity', Foundations of Physics 41 (2011) 270-277, DOI 10.1007/s10701-010-9511-6, arXiv:1010.1143, https://arxiv.org/abs/1010.1143
  3. D. Dudal, H. Verschelde, V. E. R. Lemes, M. S. Sarandy, S. P. Sorella, M. Picariello, 'Gluon-ghost condensate of mass dimension 2 in the Curci-Ferrari gauge', Annals of Physics 308 (2003) 62-77, DOI 10.1016/S0003-4916(03)00131-3, arXiv:hep-th/0302168, https://arxiv.org/abs/hep-th/0302168
  4. Ashaq Hussain Sofi, Shabir Ahmad Akhoon, Asloob Ahmad Rather, Anil Maini, 'Perturbative Quantum Gravity on de Sitter Spacetime', Open Access Library Journal 2 (2015) 1422, DOI 10.4236/oalib.1101422, https://doi.org/10.4236/oalib.1101422
  5. Asloob Ahmad Rather, Ashaq Hussain Sofi, Shabir Ahmad Akhoon, Anil Maini, 'Gauge Theories on de Sitter Spacetime', arXiv:1408.4329, https://arxiv.org/abs/1408.4329
  6. Mir Faizal, Sudhaker Upadhyay, Bhabani Prasad Mandal, 'IR finite graviton propagators in de Sitter spacetime', European Physical Journal C 76 (2016) 189, DOI 10.1140/epjc/s10052-016-4008-x, arXiv:1604.00390, https://arxiv.org/abs/1604.00390
  7. Sudhaker Upadhyay, 'Finite field-dependent symmetries in perturbative quantum gravity', Annals of Physics 340 (2014) 110-118, DOI 10.1016/j.aop.2013.10.012, arXiv:1310.8579, https://arxiv.org/abs/1310.8579