The Cosmological Constant of One-Dimensional Matter Coupled Quantum Gravity Is Quantised

Abstract

Coupling any interacting quantum mechanical system to gravity in one (time) dimension requires the cosmological constant to belong to the matter energy spectrum and thus to be quantised, even though the gravity sector is free of any quantum dynamics. Furthermore, physical states are also confined to the subspace of the matter quantum states for which the energy coincides with the value of the cosmological constant. These general facts are illustrated through some simple examples. The physical projector quantisation approach readily leads to the correct representation of such systems, whereas other approaches relying on gauge fixing methods are often plagued by Gribov problems in which case the quantisation rule is not properly recovered. Whether such a quantisation of the cosmological constant as well as the other ensuing consequences in terms of physical states extend to higher dimensional matter-gravity coupled quantum systems is clearly a fascinating open issue.Dedicated to John R. Klauder on the occasion of his 70 th birthday * Permanent address.

Quantum gravity and the cosmological constant: Lessons from two-dimensional dilaton gravity

“…To this purpose let us recall that it is natural for physical gauge invariant states to carry a dependence on the values of the free parameters. This can be seen in two simple examples discussed in detail in [8]. If we consider the case of 0+1 gravity with a cosmological constant coupled with a generic matter sector it can be easily shown that the cosmological constant value must coincide with the energy of the solution at the classical level.…”

“…In a generalization of the one-dimensional and the two-dimensional Liouville gravity cases [5,8] it is interesting to look at the case of two-dimensional dilaton gravity. Two-dimensional generalized dilaton theories provide a very interesting class of models, widely studied in the last two decades.…”

World-line quantization of a reciprocally invariant system

“…Here, is a real constant of the same physical dimension as L (and H), playing the rôle of a cosmological constant on the world-line [16].…”

“…The associated Noether symmetry, the generator of local world-line gauge transformations, is the extended Hamiltonian and becomes a first class constraint. Following the Dirac quantisation procedure, imposition of the first class constraint on physical states enforces the condition that the cosmological constant term, allowed by the reparametrisation invariant coupling of the system to the world-line metric, must be identified with a discrete fixed eigenvalue of the quadratic Casimir invariant of the quaplectic algebra, in line with well understood features of Hamiltonian quantisation [14]. The decomposition of such irreducible representations of the full quaplectic group with respect to the physical Poincaré group IO(D−1, 1) ∼ = O(D−1, 1)⋉T (D) is discussed, where the generators P µ are identified with the standard energy-momentum operators, the generators of spacetime translations.…”

Constraint quantization of a worldline system invariant under reciprocal relativity: II

“…The state space of the system (before imposition of the constraint) corresponds to the direct product of two independent D-dimensional Heisenberg algebras -one generated by the conserved generators X µ , P ν of translations in the original 'position' and 'momentum' worldline coordinates, and a second auxiliary, non-conserved set X µ , P ν (both sets have the same central extension, the conserved θ-momentum, Π θ ). Imposition of the first class constraint, the generator of local time reparametrisations, on physical state space enforces identification [10] of the world-line cosmological constant Λ in the model, with a fixed value of the quadratic Casimir of the quaplectic symmetry group. However, both sets of Heisenberg generators provide the material for the construction of the (conserved) generators of the homogeneous U (D−1, 1) component of the quaplectic algebra,…”