Hadamard States in Linearized Quantum Gravity

• Applicant: Dr. Simone Murro
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 431341045

The quantization of general relativity is one of the most debated, hard and treacherous topics in theoretical and mathematical physics. Much has been written on this subject, several models have been proposed and yet no unanimous solution has been found. Recently it has been discovered that the quantization of the linearized gravity can be useful to extract information about the local geometry of the full non-linear phase space of general relativity. In contrast to the case of general relativity, the quantization of linearized gravity seems hard but achievable and the goal of this project is to provide a quantization scheme which should best be seen as a two-step procedure: In the first one assigns to a classical dynamical system a suitable *-algebra of observables A, which encodes the canonical commutation properties, locality, dynamics and causality in particular. In the second one constructs a state, namely a positive linear functional on A from which one recovers the standard probabilistic interpretation via the GNS theorem. While the construction of A can be achieved in a straightforward way, single out a physical meaningful state among the plethora of positive linear functional is notoriously hard. This leads to the notion of Hadamard states, which were characterized by the wavefront set of their two-point functions. Despite being structurally important, their existence for linearized gravity is not guaranteed except for very esclusive background. In this project we propose a systematic way for constructing Hadamard states by using techniques proper of microlocal analysis. This shall be accomplished as follows: First we shall provide an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. Then we obtain Hadamard two-point functions by acting on the initial data.Finally, we shall be restore gauge invariance, by composing the Hadamard two-point function with a suitable spectral projector. Notice that this is the most difficult and delicate part of the project: The action of this projector should not spoil positivity of the two-point function neither its wavefront set. Once that an Hamadard state is assigned to the algebra of observables, the quantization of the linearized gravity is achieved and the standard probabilistic interpretation can be recovered via the GNS theorem.

• DFG Programme: Research Fellowships
• International Connection: France
• Host: Professor Dr. Christian Gérard