[Paper Review] On microlocalisation and the construction of Feynman Propagators for normally hyperbolic operators

Onirban Islam, Alexander Strohmaier|arXiv (Cornell University)|Dec 17, 2020

Advanced Differential Geometry Research4 citations

TL;DR

This paper develops a global microlocal construction of Feynman propagators for normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes, generalizing the Duistermaat-Hörmander theory to non-scalar geometric operators. It establishes that Feynman propagators can be constructed with a positivity property reflecting Hadamard states in quantum field theory, even for Dirac-type operators despite non-positive-definite bundle metrics, using microlocal analysis of Lagrangian distributions and wavefront sets on the cotangent bundle.

ABSTRACT

This article gives global microlocalisation constructions for normally hyperbolic operators on a vector bundle over a globally hyperbolic spacetime in geometric terms. As an application, this is used to generalise the Duistermaat-Hörmander construction of Feynman propagators, therefore incorporating the most important non-scalar geometric operators. It is shown that for normally hyperbolic operators that are selfadjoint with respect to a hermitian bundle metric, the Feynman propagators can be constructed to satisfy a positivity property that reflects the existence of Hadamard states in quantum field theory on curved spacetimes. We also give a more direct construction of the Feynman propagators for Dirac-type operators on a globally hyperbolic spacetime. Even though the natural bundle metric on spinors is not positive-definite, in this case, we can give a direct microlocal construction of a Feynman propagator that satisfies positivity.

Motivation & Objective

• To extend the Duistermaat-Hörmander theory of Feynman parametrices to vector bundle-valued normally hyperbolic operators on globally hyperbolic spacetimes.
• To establish a geometric, microlocal characterization of Feynman propagators using wavefront sets and Lagrangian distributions.
• To prove the existence of Feynman propagators satisfying a positivity condition equivalent to the existence of Hadamard states in quantum field theory on curved spacetimes.
• To provide a direct microlocal construction of Feynman propagators for Dirac-type operators, even when the natural spinor metric is not positive-definite.
• To unify the construction of Feynman propagators across scalar and non-scalar geometric operators via microlocal analysis on the cotangent bundle.

Proposed method

• The authors use global microlocal analysis on the cotangent bundle to define Lagrangian distributions associated with the positive and negative energy Lagrangian cones $C^\pm$.
• They construct parametrices via iterative solutions to transport equations on the wavefront set, solving $(-\mathrm{i}X_\square + \sigma^\mathrm{sub}_\square)a_0 = b$ to determine principal symbols.
• The construction relies on the Egorov-type theorem and propagation of singularities to ensure that the wavefront set of the parametrix lies within the Lagrangian $C^\pm$.
• For Dirac-type operators, a direct microlocal construction is achieved by solving the transport equation on the spinor bundle with a non-positive-definite metric, ensuring positivity of the resulting propagator.
• The method uses the square root construction for pseudodifferential operators (Proposition F.2) to build a selfadjoint parametrix whose wavefront set satisfies the required microlocal positivity condition.
• The theory is applied to show that the resulting Feynman parametrices satisfy the Hadamard condition and thus define Hadamard states.

Experimental results

Research questions

• RQ1 Can Feynman propagators be constructed for normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes using only geometric and microlocal data?
• RQ2 Does the Duistermaat-Hörmander uniqueness result for Feynman parametrices extend to non-scalar operators, such as those on vector bundles?
• RQ3 Can a positivity condition on the Feynman propagator be established microlocally, reflecting the existence of Hadamard states in quantum field theory?
• RQ4 Is it possible to construct a Feynman propagator for Dirac-type operators without relying on a positive-definite bundle metric?
• RQ5 How can the wavefront set of a parametrix be controlled to ensure it corresponds to the correct energy sign (positive/negative) in the absence of a global timelike Killing vector?

Key findings

• The paper constructs Feynman propagators for normally hyperbolic operators on vector bundles over globally hyperbolic spacetimes using global microlocal analysis, generalizing the Duistermaat-Hörmander framework.
• The constructed Feynman propagators satisfy a positivity property that corresponds to the existence of Hadamard states in quantum field theory on curved spacetimes.
• For Dirac-type operators, a direct microlocal construction of a Feynman propagator is achieved even when the natural bundle metric is not positive-definite.
• The wavefront set of the constructed propagator lies within the positive and negative energy Lagrangian cones $C^\pm$, ensuring correct causal and energy-sign behavior.
• The construction is invariant under equivalence modulo smoothing operators, confirming the uniqueness of the Feynman parametrix in the microlocal sense.
• The method provides a systematic way to build Feynman propagators via iterative solution of transport equations on the wavefront set, ensuring microlocal positivity.