Kiefer C. Quantum gravity

132 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
is then defined as starting from P, going through the loop α, then through β,
and ending at P.
Of particular interest is the quantity
∫
E[S,f] ≡ dσ a Ei a f i , (4.148)
S
where S denotes a two-dimensional surface in Σ and f = f i τ i .Thevariable
(4.148) describes the flux of E a i through the two-dimensional surface; it is the
variable conjugate to the holonomy U[A, α]. Note that both the holonomy and the
flux are of a distributional nature because they have support on one-dimensional
and two-dimensional submanifolds, respectively.
These variables will be used in the quantum theory; see Chapter 6.