Feynman Path Integral Formulation

110 4 Hamiltonian and Wheeler-DeWitt Equation
helicity states of the linearized gravitational field, which describes a massless spin
two particle.
4.4 Orthogonal Decomposition in the Linearized Theory
The linearized gravitational field case is the easiest to work out. As usual one assumes
boundary conditions such that some or all the field vanish at infinity, where
space is assumed to be flat. One needs an orthogonal decomposition of the metric
into trace part, longitudinal part and transverse-traceless part, which is achieved by
writing for any symmetric tensor
f ij = f TT
ij
+ f T
ij + ∂ i f j + ∂ j f i , (4.41)
which similar to what is done in electromagnetism, where the vector potential A is
written as a transverse and a longitudinal part. Here one writes
f i =(1/∂ 2 ) [ ∂ j f ij − 1 2 (1/∂ 2 )∂ i ∂ j ∂ k f kj
]
f T = f ii − (1/∂ 2 )∂ i ∂ j f ij
= f ij − fij[ T f ] − ∂ i f j [ f ] − ∂ j f i [ f ] , (4.42)
f TT
ij
for the longitudinal, trace and transverse-traceless part respectively, with the quantity
fij T defined by
fij T ≡ 1 [
2 δij f T − (1/∂ 2 )∂ i ∂ j f T ] . (4.43)
In the above expressions ∂ 2 ≡ ∂ i ∂ i . To this order one can show that both g T and
π i vanish. Furthermore π T and g i also can be eliminated by a choice of coordinate
condition, such as
t =(−1/2∂ 2 ) π T , (4.44)
giving N i = g 0 i = 0 everywhere, as well as N = 1.
This finally leaves g TT
ij and π ijTT as the only two remaining canonically conjugate
variables in the linearized theory, with fundamental equal time Poisson bracket
{
}
g TT
ij (x), π kl TT (x ′ ) = δ kl ij(x − x ′ ) , (4.45)
and all other equal time Poisson brackets equal to zero. The modified Dirac δ function
on the r.h.s. ensures that the transversality constraint on the fields appearing on
the l.h.s. is not violated. The use of a fundamental Poisson bracket then allows a
straightforward transition to a quantum mechanical treatment via the usual replacement
{q, p} →(1/i¯h)[ˆq, ˆp].