Quantum Gravity

THE PROBLEM OF TIME 137parameter (an absolute element of the theory), whereas time in GR is dynamical.A consistent theory of quantum gravity should, therefore, exhibit a novel conceptof time. The history of physics has shown that new theories often entail a newconcept of space and time (Ehlers 1973). The same should happen again withquantum gravity.The absolute nature of time in quantum mechanics is crucial for its interpretation.Matrix elements are usually evaluated at fixed t, and the scalar product isconserved in time (‘unitarity’). Unitarity expresses the conservation of the totalprobability. ‘Time’ is part of the classical background which, according to theCopenhagen interpretation, is needed for the interpretation of measurements. Aswe have remarked at the end of Section 3.1, the introduction of a time operatorin quantum mechanics is problematic. The time parameter t appears explicitlyin the Schrödinger equation (3.14). Note that it comes together with the imaginaryunit i, a fact that finds an explanation in the semiclassical approximationto quantum geometrodynamics (Section 5.4). The occurrence of the imaginaryunit in the Schrödinger equation was already discussed in an interesting correspondencebetween Ehrenfest and Pauli; see Pauli (1985, p. 127). Pauli pointedout that the use of complex wave functions can be traced back to the probabilityinterpretation: 3I now turn to the initially asked question about the necessity of at least two scalarsfor the de Broglie–Schrödinger waves. I claim that this necessity and thus also theimaginary unit come into play through the search for an expression for the probabilitydensity W which satisfies conditions (1) and (2) and which does not contain thetemporal derivatives of ψ.Conditions (1) and (2) are the non-negativity of W and its normalization to one,respectively.In GR, space–time is dynamical and therefore there is no absolute time.Space–time influences material clocks in order to allow them to show propertime. The clocks, in turn, react on the metric and change the geometry. In thissense, the metric itself is a clock (Zeh 2001). A quantization of the metric canthus be interpreted as a quantization of the concept of time. Since the nature oftime in quantum gravity is not yet clear—the classical constraints do not containany time parameter—one speaks of the ‘problem of time’. One can distinguishbasically three possible solutions of this problem, as reviewed, in particular, byIsham (1993) and Kuchař (1992):1. choice of a concept of time before quantization;2. identification of a concept of time after quantization;3. ‘timeless’ options.3 Nun komme ich zur anfangs gestellten Frage über die Notwendigkeit von mindestens zweireellen Skalaren bei den de Broglie–Schrödinger-Wellen. Ich behaupte, diese Notwendigkeit unddamit auch die imaginäre Einheit kommt hinein beim Suchen nach einem Ausdruck für dieWahrscheinlichkeitsdichte W , der die Forderungen (1) und (2) befriedigt und der die zeitlichenAbleitungen der ψ nicht enthält.