On-shell actions with lightlike boundary data

We argue that finite-region observables in quantum gravity are best approached in terms of boundary data on null hypersurfaces. This has far-reaching effects on the basic notions of classical and quantum mechanics, such as Hamiltonians and canonical conjugates. Such radical properties are not unexpected in finite-region quantum gravity. We are thus motivated to reformulate field theory in terms of null boundary data. As a starting point, we consider the on-shell action functional for classical field theory in finite null-bounded regions. Closed-form results are obtained for free scalars and for Maxwell fields. The action of classical gravity is also discussed, to the extent possible without solving the field equations. These action functionals exhibit non-locality and, in special cases, a "holographic" reduction of the degrees of freedom. Also, they cannot be used to define global charges. Whereas for ordinary field theory these are just artifacts of a restrictive formalism, in quantum gravity they are expected to be genuine features. This further supports a connection between quantum gravity and null-boundary observables. In our treatment of the GR action, we identify a universal imaginary term that reproduces the Bekenstein entropy formula.
Comment: 39 pages, 3 figures; v2: fixed minor errors, added refs. to subsequent work on the imaginary part of the GR action