Quantum Gravity

QUANTUM THEORY AND THE GRAVITATIONAL FIELD 9ω describes the influence of the terrestrial rotation on the interference pattern(‘neutron Sagnac effect’). It yields∆β Sagnac = m ∮i(ω × r)dr = 2m iωA , (1.14)where A denotes the normal area vector of the loop ABDC.Of main interest here is the gravitational part of the phase shift. Since thecontributions of the sides AC and DB cancel, one has∆β g = m i∮vdr ≈ m i(v 0 − v 1 )AB , (1.15)where v 0 and v 1 denote the absolute values of the velocities along AB and CD,respectively. From energy conservation one gets√v 1 = v 01 − 2∆Vm i v 2 0≈ v 0 − m ggh 0 sin θm i v 0,where ∆V = m g gh 0 sin θ is the potential difference, h 0 denotes the perpendiculardistance between AB and CD, and the limit 2∆V/m i v02 ≪ 1 (about 10−8 inthe experiment) has been used. The neutrons are prepared with a de Brogliewavelength λ =2π/p ≈ 2π/m i v 0 (neglecting the ω part, since the Sagnaceffect contributes only 2% of the effect), attaining a value of about 1.4 Åintheexperiment. One then gets for the gravitational phase shift the final result∆β g ≈ m im g gλA sin θ2π 2 , (1.16)where A denotes the area of the parallelogram ABDC. This result has beenconfirmed by ‘COW’ with 1% accuracy. The phase shift (1.16) can be rewrittenin an alternative form such that only those quantities appear that are directlyobservable in the experiment (Lämmerzahl 1996). It then reads∆β g ≈ m gm igGTT ′ , (1.17)where T (T ′ ) denotes the flight time of the neutron from A to B (from A toC), and G is the reciprocal lattice vector of the crystal layers (from which theneutrons are scattered in the beam splitter). Now m g and m i appear as in theclassical theory as a ratio, not as a product. The ‘COW’ experiment has alsoconfirmed the validity of the (weak) equivalence principle in the quantum domain.Modern tests prefer to use atom interferometry because atoms are easierto handle and the experiments allow tests of higher precision (Lämmerzahl 1996,1998). There the flight time is just the time between laser pulses, that is, theinteraction time with the gravitational field; T is chosen by the experimentalist.For example, Peters et al. (2001) have used atom interferometry to measure gwith a resolution of ∆g/g ∼ 10 −10 .