On the completeness of physics, see my response to metaphysicist here.
As for the determinism of physical law, I’m afraid that’s not looking too good these days either. Initial value problems for the gravitational field equations satisfy existence and uniqueness conditions only within the domain of dependence of the initial data surface, and this need not extend across all of spacetime. In particular, if the spacetime is not globally hyperbolic, the initial value problem will not (in general, although there are specific exceptions) have a unique solution. Global hyperbolicity fails if, for instance, there are naked singularities (singularities without event horizons). It is for this sort of reason that Penrose came up with the cosmic censorship conjecture, outlawing naked singularities.
Unfortunately, the conjecture seems to be in tension with what we know about quantum gravity. Consider Hawking radiation, the process by which black holes gradually evaporate away. If (and this is a pretty big if) the evaporation can be described by a classical general relativistic spacetime, then it must eventually result in a momentarily naked singularity, violating cosmic censorship (and global hyperbolicity).
There are also problems involving quantum mechanics, even if you don’t adopt a collapse interpretation. There are configuration spaces that possess singularities. In the classical case, these singularities are usually protected by a potential barrier, preventing the system from falling into them. But in the quantum case, it is possible for the system to tunnel through the barrier, leading to non-unitary evolution. Mathematically, this corresponds to a Hamiltonian operator that is not essentially self-adjoint (its closure is not self-adjoint). This problem would arise if we were doing quantum mechanics on a spacetime with a timelike or naked singularity.
There are also interesting issues related to the failure of narratability in relativistic quantum theory, discussed in this cool paper.
On the completeness of physics, see my response to metaphysicist here.
As for the determinism of physical law, I’m afraid that’s not looking too good these days either. Initial value problems for the gravitational field equations satisfy existence and uniqueness conditions only within the domain of dependence of the initial data surface, and this need not extend across all of spacetime. In particular, if the spacetime is not globally hyperbolic, the initial value problem will not (in general, although there are specific exceptions) have a unique solution. Global hyperbolicity fails if, for instance, there are naked singularities (singularities without event horizons). It is for this sort of reason that Penrose came up with the cosmic censorship conjecture, outlawing naked singularities.
Unfortunately, the conjecture seems to be in tension with what we know about quantum gravity. Consider Hawking radiation, the process by which black holes gradually evaporate away. If (and this is a pretty big if) the evaporation can be described by a classical general relativistic spacetime, then it must eventually result in a momentarily naked singularity, violating cosmic censorship (and global hyperbolicity).
There are also problems involving quantum mechanics, even if you don’t adopt a collapse interpretation. There are configuration spaces that possess singularities. In the classical case, these singularities are usually protected by a potential barrier, preventing the system from falling into them. But in the quantum case, it is possible for the system to tunnel through the barrier, leading to non-unitary evolution. Mathematically, this corresponds to a Hamiltonian operator that is not essentially self-adjoint (its closure is not self-adjoint). This problem would arise if we were doing quantum mechanics on a spacetime with a timelike or naked singularity.
There are also interesting issues related to the failure of narratability in relativistic quantum theory, discussed in this cool paper.