The spectral theory of Laplace(-Beltrami) and Dirac operators is analyzed in a global Lorentzian geometric setting. The two types of Lorentzian manifolds to be considered are 1) Time dependent, globally hyperbolic manifolds and 2) Static or stationary, non-globally hyperbolic manifolds. Self adjoint extensions of the above operators are studied. In addition to proving the existence of self adjoint extensions of those operators, we study the uniqueness of those extensions by analyzing essential self-adjointness. Applications to General Relativity and quantum field theory in curved spacetimes are worked out.
The main method for the wave equation is to apply recent results by Shubin showing that on a complete Riemannian manifold, the weighted Laplace-Beltrami operator plus a locally square integrable potential is essentially self-adjoint on the space of smooth functions of compact support. Preliminary works show that a sufficient condition for essential self-adjointness is that, after a suitable conformal transformation, the induced Riemannian metric is geodesically complete on each leaf of the foliation. Consequently, the second part of this project is to classify the time-dependent globally hyperbolic manifolds for which this condition can be satisfied. The third part of the project is to extend these methods to the study of essential self-adjointness of the Laplace and Dirac operators in static or stationary, non globally hyperbolic manifolds. In the last part we use these results to construct complex structures on the solution spaces as needed for the quantization.
Publications
The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.
Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.
| Journal | Math. Phys. Anal. Geom. , 7 (2025) 42pp |
| Publisher | Springer Nature |
| Volume | 28 |
| Pages | 7, 42pp |
| Link to preprint version | |
| Link to published version |
Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
The fermionic Rényi entanglement entropy is studied for causal diamonds in two-dimensional Minkowski spacetime. Choosing the quasi-free state describing the Minkowski vacuum with an ultraviolet regularization, a logarithmically enhanced area law is derived.
Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.
| Journal | Ann. Henri Poincaré (2023) |
| Publisher | Springer Nature |
| Volume | 24 |
| Pages | 1185-1209 |
| Link to preprint version | |
| Link to published version |
Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
This chapter is an up-to-date account of results on globally hyperbolic spacetimes and serves as a multitool; we start the exposition of results from a foundational level, where the main tools are order-theory and general topology, we continue with results of a more geometric nature, and we finally reach results that are connected to the most recent advances in theoretical physics. In each case, we list a number of open questions and we finally introduce a conjecture, on sliced spaces.
| Journal | In: Parasidis, I.N., Providas, E., Rassias, T.M. (eds) Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications, vol 179. Springer, Cham. |
| Publisher | Springer, Cham |
| Book | Springer Optimization and Its Applications |
| Volume | 179 |
| Pages | 281–295 |
| Link to preprint version | |
| Link to published version |
Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
Team Members
Prof. Dr. Felix Finster
Project leader
Universität Regensburg
Dr. Albert Much
Researcher
Universität Leipzig