Quantum volume and length fluctuations in a midi-superspace model of   Minkowski space
  In a (1+1)-dimensional midi-superspace model for gravitational plane waves, a flat space-time condition is imposed with constraints derived from null Killing vectors. Solutions to a straightforward regularization of these constraints have diverging length and volume expectation values. Physically acceptable solutions in the kinematic Hilbert space are obtained from the original constraint by multiplying with a power of the volume operator and by a similar modification of the Hamiltonian constraint, which is used in a regularization of the constraints. The solutions of the modified Killing constraint have finite expectation values of geometric quantities. Further, the expectation value of the original Killing constraint vanishes, but its moment is non-vanishing. As the power of the volume grows the moment of the original constraint grows, while the moments of volume and length both decrease. Thus, these states provide possible kinematic states for flat space, with fluctuations. As a consequence of the regularization of operators the quantum uncertainty relations between geometric quantities such as length and its conjugate momentum do not reflect naive expectations from the classical Poisson bracket relations. 
Jeremy Adelman, Franz Hinterleitner, Seth Major
gr-qc