Third party funded individual grant
Start date : 16.08.2026
End date : 15.08.2028
In recent years, the Tomita-Takesaki theory of modular flows of von Neumann algebras has been extensively used both in high energy and condensed matter physics, as a tool for studying energy, entropy and entanglement in quantum field theory (QFT) and semiclassical gravity. The central object in modular theory, the modular or entanglement Hamiltonian associated to a given quantum state and spacetime region, is however only known in a handful of examples. These mainly include theories and regions with high symmetry, such as conformal field theories in double cone regions. The only modular Hamiltonian known for a general QFT in Minkowski spacetime is the Bisognano-Wichmann one, which applies to theories restricted to a Rindler wedge. However, for non-conformal theories in finite regions, even as simple as a free massive scalar field in a double cone, the explicit form of the modular Hamiltonian is unknown, and its determination is a major long-standing open problem.
It is the goal of this proposal to use mathematically rigorous methods developed for QFT in curved spacetime to develop a general framework for computing the modular Hamiltonian for arbitrary interacting field theories. The main idea is to exploit the KMS condition satisfied by the modular flow to derive a relation between the integral kernel of the modular Hamiltonian and the correlation functions of the interacting theory, generalizing a similar relation that exists for free bosons and fermions. This framework will be applied to a number of instructive examples, including the λφ⁴ scalar theory and the Thirring model in a double cone and the Rindler wedge. For the wedge, we will verify the consistency with the Bisognano-Wichmann result, while the results for the double cone will be new and will clarify the relation with the trace anomaly that arises from the breaking of conformal invariance in the quantum theory.
We will furthermore resolve an open conjecture regarding the inverse of this problem, namely the construction of a quantum state for a prescribed modular Hamiltonian. This conjecture is an important ingredient in the construction of the algebra of observables associated to a local observer in semiclassical gravity, and for its resolution we will leverage methods developed for perturbations of KMS states in von Neumann algebras. Another goal is the determination of the modular Hamiltonian for a free massive scalar field in a double cone, which has resisted all attempts at a solution so far, and for which we propose a novel indirect way: to first show the convergence of the perturbative expansion in the mass for massive fermions, and then use supersymmetry to map the result to the bosonic theory.
As a final application of the developed framework, we will compute the entropy generated by the expansion of our universe, by employing the Araki formula for the relative (entanglement) entropy between the Bunch-Davies and the Minkowski vacuum state.