If quantum mechanics is universally correct, then one may try to apply it to the universe in order to find its wave function. This would allow us to find which events are probable and which are not. However, it often leads to paradoxes. For example, the essence of the Wheeler-DeWitt equation, which is the Shrödinger equation for the wave function of the universe, is that this wave function does not depend on time [ … ] Therefore if one would wish to describe the evolution of the universe with the help of its wave function, one would be in trouble: The universe as a whole does not change in time.
The resolution of this paradox by Bruce DeWitt is rather instructive. The notion of evolution is not applicable to the universe as a whole since there is no external observer with respect to the universe, and there is no external clock that does not belong to the universe. However, we do not actually ask why the universe as a whole is evolving. We are just trying to understand our own experimental data. Thus, a more precisely formulated question is why do we see the universe evolving in time in a given way. In order to answer this question one should first divide the universe into two main piece: (i) an observer with his clock and other measuring devices and (ii) the rest of the universe.
Then it can be shown that the wave function of the rest of the universe does depend on the state of the clock of the observer, i.e., on his “time.” [ … ] [W]without introducing an observer, we have a dead universe, which does not evolve in time. This example demonstrates an unusually important role played by the concept of an observer in quantum cosmology. John Wheeler underscored the complexity of the situation, replacing the word observer by the word participant, and introducing such terms as a “self-observing universe.”
Most of the time, when discussing quantum cosmology, one can remain entirely within the bounds set by purely physical categories, regarding an observer simply as an automaton, and not dealing with question whether he/she/it has consciousness or feels anything during the process of observation. This limitation is harmless for many practical purposes. But we cannot rule out the possibility that carefully avoiding the concept of consciousness in quantum cosmology may lead to an artificial narrowing of our outlook.
From “Inflation, Quantum Cosmology, and the Anthropic Principle,” in J.D. Barrow, P.C.W Davies and C.L. Harper, eds., Science and Ultimate Reality (Cambridge: 2004) pp. 449-451.