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Interchange of stochastic averaging with perturbation series

Ascertain whether stochastic averages of observables in the nonlocal Dirac dynamics with stochastic potentials can be taken term-by-term in the perturbative expansion, i.e., justify the interchange of stochastic averaging with summation in the retarded perturbation series.

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Background

Solutions are constructed by a retarded perturbation series. Even when convergence can be established under smallness assumptions, it remains unclear whether stochastic averages may be interchanged with the perturbation expansion.

Clarifying this would rigorously ground the use of stochastic fields within the perturbative framework employed to connect causal fermion systems to quantum field theoretic dynamics.

References

Even if this result applies, the subtle issue remains that it is not clear whether the stochastic averages may be taken order by order in perturbation theory.

— Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems (2410.18045 - Dappiaggi et al., 23 Oct 2024) in Section 3.1 (Strategy for Obtaining Bosonic Field Operators)