Kiefer C. Quantum gravity

114 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
The term in parentheses in the first integral has to be added to H g ⊥
, while the
term p φ φ ,a (which we have already encountered in (4.14)) must be added to Ha g.
For the electromagnetic field, starting from
L = − 1 4
√ −gg µρ g νσ F µν F ρσ , (4.75)
one gets for the Hamiltonian
∫ (
√ )
H EM = d 3 h ab h
xN
2 √ h pa p b +
2 hab B a B b
∫
∫
+ d 3 xN a (∂ a A b − ∂ b A a ) p b − d 3 xA 0 ∂ a p a , (4.76)
where p a = ∂L/∂Ȧa is the electric field and B a =(1/2)ɛ abc F bc , the magnetic
field. Note that the term in parentheses in the second integral is just the ¯H a from
(4.28), and variation with respect to the Lagrange multiplier A 0 yields Gauss’
law (4.30). If fermionic degrees of freedom are present, one must perform the
3+1 decomposition with respect to the vierbein instead of the four-metric; see for
example, Ashtekar (1991). The classical canonical formalism for the gravitational
field as discussed up to now was pioneered by Peter Bergmann, Paul Dirac,
‘ADM’, and others in the 1950s. For a historical account and references, see
for example, Bergmann (1989, 1992) and Rovelli (2004). We finally want to
remark that the canonical quantization of higher-derivative theories such as R 2 -
gravity can also be performed (cf. Boulware 1984). The formalism is then more
complicated since one has to introduce additional configuration-space variables
and momenta.
4.2.3 Discussion of the constraints
The presence of the constraints derived in the last subsection means that only
part of the variables constitute physical degrees of freedom (cf. Section 3.1.2).
How may one count them? The three-metric h ab (x) is characterized by six numbers
per space point (often symbolically denoted as 6×∞ 3 ). The diffeomorphism
constraints (4.70) generate coordinate transformations on three-space. These are
characterized by three numbers, so 6−3 = 3 numbers per point remain. The constraint
(4.69) corresponds to one variable per space point describing the location
of Σ in space–time (since Σ changes under normal deformations). In a sense,
this one variable therefore corresponds to ‘time’, and 2 ×∞ 3 degrees of freedom
remain. Baierlein et al. (1962) have interpreted this as the ‘three-geometry
carrying information about time’.
The gravitational field thus seems to be characterized by 2 ×∞ 3 intrinsic
degrees of freedom. This is consistent with the corresponding number found in
linear quantum gravity—the two spin-2 states of the graviton (Section 2.1). One
can alternatively perform the following counting in phase space: the canonical
variables (h ab (x),p cd (y)) are 12 ×∞ 3 variables. Due to the presence of the four