  We consider the quantum mechanics of Einstein gravity linearised about flat
spacetime. The two transverse-traceless components of the metric perturbation
are the true physical degrees of freedom. They appear in the quantum theory as
free quantum fields. Like the full Einstein action, the Euclidean action for
linearised gravity is unbounded below. It is therefore not possible to use that
action to represent the ground state wave function as a Euclidian functional
integral over exp{[-(action) /\hbar]}. However, it is possible to represent the
ground state as a Euclidian integral over the (deparametrised) action involving
only the true physical degrees of freedom. Starting from this integral
representation of the ground state and using the techniques of Faddeev and
Popov we show how to construct a Euclidean functional integral for the ground
state wave function. The integral explicitly exhibits the theory's gauge
symmetry, locality, and O(4) invariance. The conformal factor appears naturally
rotated into the complex plane. Other representations of the ground state are
exhibited.
