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arXiv:2604.11612

Feynman's linear divergence problem

Alexander Sakhnovich, Lev Sakhnovich

math-phmath.CAquant-phSubmitted Apr 13, 2026• Announced Apr 14, 2026
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Abstract

First, we consider generalized wave and scattering operators and derive modifications of commutation relations (between scattering operators and unperturbed operators) when the corresponding deviation factors behave as $\exp\{i t {\mathcal C}_{\pm}\}$ for $t\to \pm \infty$. Then, we construct so called secondary generalized scattering operators for the related case of linear divergence in QED, which gives a positive answer (in that case) to the well-known problem of J. R. Oppenheimer regarding scattering operators in QED: "Can the procedure be freed of the expansion in $\varepsilon$ and carried out rigorously?"

Comments
This work may be considered as a follow up of the paper arXiv:2511.02847, where the logarithmic divergence was studied

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Consider two agents, Alice and Bob, each of whom takes a quantum input, operates on a shared quantum system KKK, and produces a quantum output. Alice and Bob's operations may commute, in the sense that the joint input-output behaviour is independent of the order in which they access KKK. Here we ask whether this commutation property implies that KKK can be split into two factors on which Alice and Bob act separately. The question can be regarded as a "fully quantum" generalisation of a problem posed by Tsirelson, who considered the case where Alice and Bob's inputs and outputs are classical. In this case, the answer is negative in general, but it is known that a factorisation exists in finite dimensions. Here we show the same holds in the fully quantum case, i.e., commuting operations factorise, provided that all input systems are finite-dimensional.
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