[Paper Review] Weyl calculus in QED I. The unitary group

Laurent Amour, Richard Lascar+1 more|arXiv (Cornell University)|Oct 18, 2015

Spectral Theory in Mathematical Physics17 references3 citations

TL;DR

This paper establishes that the reduced propagator in quantum electrodynamics (QED) for spin-1/2 particles interacting with the quantized radiation field belongs to the class of infinite-dimensional Weyl pseudodifferential operators within Wiener spaces. Using a semiclassical expansion, it derives estimates for the symbol of the propagator up to any order, providing a rigorous analytical framework for time evolution in QED.

ABSTRACT

In this work, we consider fixed $1/2$ spin particles interacting with the quantized radiation field in the context of quantum electrodynamics (QED). We investigate the time evolution operator in studying the reduced propagator (interaction picture). We first prove that this propagator belongs to the class of infinite dimensional Weyl pseudodifferential operators recently introduced in \cite {A-J-N} on Wiener spaces. We give a semiclassical expansion of the symbol of the reduced propagator up to any order with estimates on the remainder terms. Next, taking into account analyticity properties for the Weyl symbol of the reduced propagator, we derive estimates concerning transition probabilities between coherent states.

Motivation & Objective

• To analyze the time evolution operator in the interaction picture for electrons coupled to the quantized radiation field.
• To establish the mathematical structure of the reduced propagator in quantum electrodynamics (QED).
• To prove that the reduced propagator lies within the class of infinite-dimensional Weyl pseudodifferential operators in Wiener spaces.
• To derive a semiclassical expansion of the symbol of the reduced propagator up to any order with precise error estimates.

Proposed method

• The analysis is conducted within the framework of infinite-dimensional Weyl pseudodifferential operators recently developed in A-J-N on Wiener spaces.
• The reduced propagator is identified as an element of this operator class through structural and analytic properties of the QED Hamiltonian.
• A semiclassical expansion of the symbol of the propagator is constructed using asymptotic series techniques.
• Estimates on the remainder terms of the expansion are derived using bounds in Wiener space norms.
• The method relies on functional analytic techniques and operator-theoretic tools in Fock space.
• The construction is invariant under unitary transformations, ensuring consistency with the interaction picture.

Experimental results

Research questions

• RQ1 Does the reduced propagator in QED for spin-1/2 particles belong to the class of infinite-dimensional Weyl pseudodifferential operators in Wiener spaces?
• RQ2 Can a semiclassical expansion of the symbol of the reduced propagator be rigorously constructed up to any order?
• RQ3 What are the quantitative estimates on the remainder terms in the semiclassical expansion of the symbol?
• RQ4 How does the structure of the propagator relate to the underlying quantum electrodynamics Hamiltonian?
• RQ5 What is the role of the interaction picture in enabling the Weyl calculus framework for the time evolution operator?

Key findings

• The reduced propagator in the interaction picture for QED is rigorously shown to belong to the class of infinite-dimensional Weyl pseudodifferential operators in Wiener spaces.
• A semiclassical expansion of the symbol of the propagator is constructed up to any finite order.
• Uniform estimates on the remainder terms of the expansion are derived, ensuring convergence properties in the operator norm.
• The symbol expansion is valid in the context of the full Fock space and respects the unitary structure of the time evolution.
• The method provides a systematic framework for approximating the time evolution operator in QED with controlled error bounds.
• The results lay a foundation for further analysis of scattering amplitudes and effective dynamics in QED using pseudodifferential operator techniques.