Gauge-invariant subsystems and entanglement

One of the most basic notions in physics is the partitioning of a system into subsystems, and the study of correlations among its parts. In standard quantum mechanical systems, this partitioning is encoded in a tensor product structure of the Hilbert space of the system. However, for quantum field theories, this tensor product structure fails due to the universal divergence of the entanglement entropy, and so subsystems must be identified with commuting subalgebras of observables that encode a notion of locality. In canonical approaches to quantum gravity, this algebraic construction is complicated by the fact that gauge invariant Dirac observables are inherently non-local and can be difficult to construct in practice. For a quantum theory with gauge constraints, I will show that while the unconstrained (kinematical) Hilbert space may admit a tensor product structure, the physical (gauge-invariant) Hilbert space need not inherit this tensor product structure upon implementation of the constraints. A theorem will be proven providing necessary and sufficient conditions for the physical Hilbert space to inherit the kinematical subsystem structure. I will also leverage the framework of quantum reference frames to show that different reference frames induce generically different subsystem algebras of observables. The main thesis of this work is the identification of a gauge-invariant but frame-dependent notion of subsystems and entanglement in constrained systems. Since gravitation is an example of such a system, I will briefly comment on the issue of constructing physical gravitational subsystems.