  The generalized second law is proven for semiclassical quantum fields falling
across a causal horizon, minimally coupled to general relativity. The proof is
much more general than previous proofs in that it permits the quantum fields to
be rapidly changing with time, and shows that entropy increases when comparing
any slice of the horizon to any earlier slice. The proof requires the existence
of an algebra of observables restricted to the horizon, satisfying certain
axioms (Determinism, Ultralocality, Local Lorentz Invariance, and Stability).
These axioms are explicitly verified in the case of free fields of various
spins, as well as 1+1 conformal field theories. The validity of the axioms for
other interacting theories is discussed.
