The Wheeler–DeWitt equation

The approaches to quantization described in chapters 5–7, although quite different, share one common feature. They are all ‘reduced phase space’ quantizations, quantum theories based on the true physical degrees of freedom of the classical theory. As we saw in chapter 2, not all of the degrees of freedom that determine the metric in general relativity have physical significance; many are ‘pure gauge’, describing coordinate choices rather than dynamics, and can be eliminated by solving the constraints and factoring out the diffeomorphisms. Indeed, we have seen that in 2+1 dimensions only a finite number of the ‘6 × ∞ 3 ’ metric degrees of freedom are physical. In each of the preceding approaches to quantization, our first step was to eliminate the nonphysical degrees of freedom, sometimes explicitly and sometimes indirectly through a clever choice of variables; only then were the remaining degrees of freedom quantized. An alternative approach, originally developed by Dirac, is to quantize the entire space of degrees of freedom of classical theory, and only then to impose the constraints. In Dirac quantization, states are initially determined from the full classical phase space; in quantum gravity, for instance, they are functionals ψ[ g ij ] of the full spatial metric. The constraints act as operators on this auxiliary Hilbert space, and the physical Hilbert space consists of those states that are annihilated by the constraints, acted on by physical operators that commute with the constraints.