[Paper Review] Global and microlocal aspects of Dirac operators: propagators and Hadamard states

Matteo Capoferri, Simone Murro|arXiv (Cornell University)|Jan 28, 2022

Geometric and Algebraic Topology4 citations

TL;DR

This paper presents a geometric, invariant construction of global Cauchy evolution operators and Feynman propagators for the Lorentzian Dirac operator on globally hyperbolic 4-manifolds using oscillatory integrals with complex-valued geometric phase functions. The key contribution is a direct, coordinate-invariant method to construct quasifree Hadamard states and the Feynman propagator without relying on resolvents or complex analysis, establishing a rigorous microlocal framework for quantum field theory on curved spacetimes.

ABSTRACT

We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realise the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals -- the positive and negative Dirac propagators -- global in space and in time, with distinguished complex-valued geometric phase functions. As applications, we relate the Cauchy evolution operators with the Feynman propagator and construct Cauchy surfaces covariances of quasifree Hadamard states.

Motivation & Objective

• To develop a geometric, invariant method for constructing Cauchy evolution operators for the Lorentzian Dirac operator on globally hyperbolic 4-manifolds.
• To provide a direct construction of the Feynman propagator without using resolvents or complex analysis.
• To realize quasifree Hadamard states via pseudodifferential projections and oscillatory integral techniques.
• To establish a microlocal framework for quantum field theory on curved spacetimes using global, invariant Fourier integral operators.
• To extend Riemannian microlocal techniques to the Lorentzian setting for systems, particularly the Dirac equation.

Proposed method

• Construct the Cauchy evolution operator as the sum of two invariantly defined oscillatory integrals: the positive and negative Dirac propagators.
• Use Levi-Civita phase functions as distinguished complex-valued geometric phase functions to ensure globality in space and time.
• Apply an algorithmic construction of pseudodifferential projections via Riesz projections to achieve exact satisfaction of spectral conditions.
• Modify pseudodifferential projections by adding smoothing operators to satisfy orthogonality and spectral conditions exactly.
• Realize basis projections on the solution space using the involution Θ and the adjunction map Υ to construct Hadamard states.
• Derive the Feynman propagator as a combination of the positive/negative propagators and the Cauchy evolution operator, using the time-ordered kernel.

Experimental results

Research questions

• RQ1 How can the Cauchy evolution operator for the Lorentzian Dirac operator be constructed globally in space and time using geometric, invariant methods?
• RQ2 What is the role of complex-valued geometric phase functions in constructing global oscillatory integral representations of propagators?
• RQ3 How can quasifree Hadamard states be constructed directly from pseudodifferential projections without relying on resolvent techniques?
• RQ4 In what way do the wavefront sets of the propagators reflect the Hadamard condition in the microlocal sense?
• RQ5 How is the Feynman propagator related to the positive and negative Dirac propagators and the Cauchy evolution operator?

Key findings

• The Cauchy evolution operator is realized as the sum of two globally defined oscillatory integrals with complex-valued geometric phase functions, ensuring invariance and globality.
• The positive and negative Dirac propagators are constructed via an algorithmic method based on pseudodifferential projections and Riesz projections, achieving exact spectral conditions.
• Quasifree Hadamard states are constructed via basis projections on the solution space, explicitly linked to the modified pseudodifferential projections.
• The Feynman propagator is derived as $\mathsf{G}_{F}(t,s) = U^{(+)}(t,s) - \theta(s-t)U(t,s)$, with wavefront set of Hadamard form.
• The wavefront set of the propagators satisfies the microlocal spectrum condition, confirming their physical relevance in quantum field theory.
• The construction avoids resolvent techniques and complex analysis, offering a direct, geometric alternative to existing methods.