Kiefer C. Quantum gravity

102 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
Using these general properties, the behaviour of various fields under coordinate
transformations generated by H a can be studied and the detailed form of
H a and H ⊥ be derived. The first case is a scalar field φ. Under an infinitesimal
coordinate transformation x ′a = x a − δN a (x), it transforms as
δφ(x) ≡ φ ′ (x) − φ(x) ≈ ∂φ
∂x a δNa ≡L δN φ, (4.13)
where L denotes the Lie derivative. This is generated by
H a = p φ φ ,a . (4.14)
Comparison with (4.9) shows that b ab =0=b ba , so that (4.10) is fulfilled and
ultralocality holds.
For a vector field, A a (x), the transformation is
which is generated by
Comparison with (4.9) shows that
δA a = A a,b δN b + A b δN b ,a ≡ (L δN A) a
, (4.15)
H a = −p b ,b A a +(A b,a − A a,b )p b . (4.16)
b b
aC= −A a δ b C . (4.17)
Therefore, the condition for ultralocality (4.10) is not fulfilled for vector fields.
Its restoration will lead to the concept of gauge theories (Section 4.1.3).
For a covariant tensor field of second rank (not necessarily symmetric), t ab (x),
one has
δt ab = t ab,c δN c + t ac δN c ,b + t cbδN c ,a ≡ (L δNt) ab
, (4.18)
which is generated by
H a = t bc,a p bc − (t ab p cb ) ,c − (t ca p cb ) ,b . (4.19)
It turns out that in order for (4.10) to be fulfilled, one must have
t ab = f(x)h ab (4.20)
with an arbitrary function f(x), that is, the tensor field must be proportional
to the metric itself. Choosing in particular t ab = h ab , one finds for the generator
(4.19) the expression
Ha g = −2pa,c+2Γ c d acpd
c ≡−2D b pa
b ≡−2pa b
|b . (4.21)
The last two terms denote the covariant derivative in three dimensions (recall
that p ab is a tensor density of weight one).