Kiefer C. Quantum gravity

134 QUANTUM GEOMETRODYNAMICS
and its momentum p cd (x) (or, in the approach of reduced quantization, a subset
of them, see Section 5.2 below). In the connection formulation of Section 4.3
one has the connection A i a (x) and the densitized triad Eb j (y), while in the loopspace
formulation one takes the holonomy U[A, α] and the flux of Ej b through a
two-dimensional surface.
In this chapter, restriction will be made to the quantization of the geometrodynamical
formulation, while quantum connection dynamics and quantum loop
dynamics will be discussed in Chapter 6. Application of (5.2) to (4.64) would
yield
[ĥab(x), ˆp cd (y)] = iδ(a c δd b) δ(x, y) , (5.3)
plus vanishing commutators between, respectively, the metric components and
the momentum components. Since p cd is linearly related to the extrinsic curvature,
describing the embedding of the three-geometry into the fourth dimension,
the presence of the commutator (5.3) and the ensuing ‘uncertainty relation’ between
intrinsic and extrinsic geometry means that the classical space–time picture
has completely dissolved in quantum gravity. This is fully analogous to
the disappearance of particle trajectories as fundamental concepts in quantum
mechanics and constitutes one of the central interpretational ingredients of quantum
gravity. The fundamental variables form a vector space that is closed under
Poisson brackets and complete in the sense that every dynamical variable can
be expressed as a sum of products of fundamental variables.
Equation (5.3) does not implement the positivity requirement deth >0of
the classical theory. But this could only be a problem if (the smeared version of)
ˆp ab were self-adjoint and its exponentiation therefore a unitary operator, which
could ‘shift’ the metric to negative values.
The second step addresses the quantization of a general variable, F ,ofthe
fundamental variables. Does the rule (5.2) still apply? As Dirac writes (Dirac 1958,
p. 87), 1
The strong analogy between the quantum P.B. ...and the classical P.B. ...leads us to
make the assumption that the quantum P.B.s, or at any rate the simpler ones of them,
have the same values as the corresponding classical P.B.s. The simplest P.B.s are those
involving the canonical coordinates and momenta themselves . . .
In fact, from general theorems of quantum theory (going back to Groenewald
and van Hove), one knows that it is impossible to respect the transformation
rule (5.2) in the general case, while assuming an irreducible representation of
the commutation rules; cf. Giulini (2003). In Dirac’s quote this is anticipated by
the statement ‘or at any rate the simple ones of them’. This failure is related to
the problem of ‘factor ordering’. Therefore, additional criteria must be invoked
to find the ‘correct’ quantization, such as the demand for ‘Dirac consistency’ to
be discussed in Section 5.3.
The third step concerns the construction of an appropriate representation
space, F, for the dynamical variables, on which they should act as operators. We
1 P.B. stands for Poisson bracket.