Proposition: Spectral splitting and charged commutator

At the operator-kernel level,

For the charged spacetime kernels, the second term in (eq:odd) is the reversed opposite-charge ordering:

Thus the state has the extension-selected canonical commutator. The pointwise real and imaginary parts of the scalar expression for $W_{\beta }$ in a parallel frame are not, in general, the two operators in (eq:even)–(eq:odd).

Proof. The first two identities are the elementary spectral cosine and sine identities for (eq:W). Complex conjugation of the spatial bundle character, followed by endpoint reversal, identifies

This can be checked directly by reindexing the angular modes. The antiparticle Fock construction gives precisely this reversed operator ordering, proving (eq:chargedCCR).

In a twisted parallel representation, the angular factors $e^{\mathrm{i}(n+\beta )\delta }$ can make the time-even spatial kernel complex. Therefore scalar complex conjugation at the same ordered pair is not the operation $T\mapsto -T$ with spatial endpoints retained.