Quantum Gravity

12 WHY QUANTUM GRAVITY?above, torsion may play a significant role in the early Universe. It was estimatedthat torsion effects should become important at a density (Hehl et al. 1976)ρ = m eλ e lP2 ≈ 9 × 10 48gcm 3 ≪ ρ P ,where m e is the electron mass and λ e its Compton wavelength. Under the assumptionof a radiation-dominated universe with critical density, the densitywould already be higher than this value for times earlier than about 10 −22 s,which is much later than the time scale where the inflationary phase is assumedto happen.We mention that this framework is also of use in the study of a ‘generalized’Dirac equation to parametrize quantum tests of general relativity (Lämmerzahl1998) and to construct an axiomatic approach to space–time geometry,yielding a Riemann–Cartan geometry (see Audretsch et al. 1992). A detailed reviewof the interaction of mesoscopic quantum systems with gravity is presentedin Kiefer and Weber (2005).In concluding this subsection, we want to discuss briefly one important occasionon which GR seemed to play a role in the foundations of quantum mechanics.This is the discussion of the time–energy uncertainty relations by Bohr and Einsteinat the sixth Solvay conference, which took place in Brussels in 1930 (cf.Bohr 1949).Einstein came up with the following counter-argument against the validity ofthis uncertainty relation. Consider a box filled with radiation. A clock controlsthe opening of a shutter for a short time interval such that a single photon canescape at a fixed time t. The energy E of the photon is, however, also fixedbecause it can be determined by weighing the box before and after the escapeof the photon. It thus seems as if the time–energy uncertainty relation wereviolated.In his response to Einstein’s attack, Bohr came up with the following arguments.Consider the details of the weighing process in which a spring is attachedto the box; see Fig. 1.3. The null position of the balance is known with an accuracy∆q. This leads to an uncertainty in the momentum of the box, ∆p ∼ /∆q.Bohr then makes the assumption that ∆p must be smaller than the total momentumimposed by the gravitational field during the time T of the weighingprocess on the mass uncertainty ∆m of the box. This leads to∆p