Time evolution in various interpretations of QM
Consider a closed system in a state with definite energy. (A Universe described by Wheeler-DeWitt equation is just a special case, in which the definite energy turns out to be zero.) How time evolution is possible in such a system? Unfortunately, an interpretation-independent answer to this question does not exist. Here I briefly review different answers provided by different interpretations.
``Copenhagen''-collapse interpretation
According to this interpretation proposed by von Neumann, everything, including the observer, is described by the wave function. However, the time-evolution of the wave function is not always governed by the Schrodinger equation. Instead, the act of observation is associated with a wave-function collapse. The collapse introduces an additional time-evolution in the system, not present in the evolution by the Schrodinger equation. In this interpretation the act of observation plays a fundamental role, but the concept of observation itself is not described by physics.
``Copenhagen'' interpretation with classical macro-world
According to this interpretation, usually attributed to Bohr, quantum mechanics can be applied only to the micro-world, not to the macro-world. The macro-world is described by classical mechanics, so the time evolution in the macro-world is not governed by a Schrodinger equation. In a closed system a quantum micro-subsystem interacts with a classical macro-subsystem, so that the time-dependence of the latter induces a time dependence of the former.
Modern instrumental ``Copenhagen'' interpretation
This is a widely-used practically oriented interpretation of QM (see e.g. the book by Peres), in which QM is nothing but a tool used to predict the probabilities of measurement outcomes for given measurement preparations. The measurement preparations are freely chosen by experimentalists. The experimentalists themselves are not described by QM. The free manipulations by experimentalists introduce additional time-dependence in the system not described by the Schrodinger equation. Within such an interpretation, the concept of wave function of the whole Universe does not make sense.
Objective collapse
In this interpretation the Schrodinger equation is modified by adding a stochastic term due to which the wave function collapses independently on any observers. The best known example of such a modification is the GRW theory.
Hidden variables
In this class of interpretations, the physical objects observed in experiments are not the wave functions, but some other time-dependent variables ##\lambda(t)##. Even if the wave function governed by the Schrodinger equation is time-independent, the ``hidden'' variable ##\lambda(t)## may depend on time. The best known and most successful model of such variables is given by the Bohmian interpretation.
Statistical ensemble
According to this interpretation, the wave function is only a property of a statistical ensemble of similarly prepared systems and tells nothing about properties of individual physical systems. So if a wave function is time-independent, it does not mean that individual systems do not depend on time. This interpretation can be thought of as an agnostic variant of the hidden-variable interpretation, in the sense that the existence of hidden variables is compatible (and perhaps even natural) with the statistical-ensemble interpretation, but the statistical-ensemble interpretation refrains from saying anything more specific about them.
Consistent histories
In this interpretation, the wave function is a tool to assign a probability to a given time-dependent history of the physical system. In this sense, it is similar to hidden-variable interpretations. However, to avoid non-localities typically associated with normal hidden-variable theories, the consistent-histories interpretation replaces the classical propositional logic with a different kind of logic.
Many worlds
According to the many-world interpretation, the Universe as a whole is nothing but a wave function evolving according to the Schrodinger equation. So, if wave function of the Universe is a state with definite total energy, at first sight it seems impossible to have any nontrivial time-dependence in the system. Nevertheless, a non-trivial time dependence can be introduced in a rather subtle way, by redefining the concept of time itself. Even if ##\psi(x_1,\ldots,x_N)## does not depend on an evolution parameter ##t##, some of the configuration variables ##x_1,\ldots,x_N## may represent readings of a physical clock, on which ##\psi(x_1,\ldots,x_N)## still depends. In such an interpretation of QM all probabilities are interpreted as conditional probabilities.
Of course, none of these interpretations is without difficulties. However, to avoid controversy and keep neutrality, the difficulties will not be discussed. I expect that critical readers will immediately recognize some difficulties with most of these interpretations, even without my assistance.
