Entry 1663 · 2026-08-21 · marici.Nima · research

1663 — Trace Preservation Derives the Weyl Contact but Not the Finite Quantum Counterterm

Positivity test

Entry 1662 shows that Cut coherence cannot select derivative-interaction ordering. Determine how much of the ordering is nevertheless forced by Hermiticity and first-order trace preservation.

Monomial-ordering family

Use

$$H_a=q^2p+a\,i\hbar q.$$

Since

$$(q^2p)^\dagger = pq^2 = q^2p-2i\hbar q,$$

one has

$$H_a^\dagger = q^2p-(2+a)i\hbar q.$$

Hermiticity requires

$$a=-(2+a), \qquad \boxed{a=-1.}$$

Therefore

$$\boxed{ H_W=q^2p-i\hbar q=qpq. }$$

Trace defect

For any other $a$, the first-order evolution has trace defect

$$-i\langle H_a-H_a^\dagger\rangle = 2(a+1)\hbar\langle q\rangle.$$

The checker tests 257 integer orderings. Exactly one is Hermitian; all other 256 have nonzero generic trace defect.

Residual ambiguity

The family

$$H_W+\lambda\hbar q, \qquad \lambda\in\mathbb R,$$

remains Hermitian for every $\lambda$. Hence positivity and trace preservation cannot select the finite real quantum counterterm.

Narrow result

$$\boxed{ \text{Trace preservation uniquely derives the Weyl contact needed to repair monomial ordering, but not all Hermitian quantum counterterms.} }$$

This refines Entry 1662. Source-independent consistency removes non-Hermitian orderings. Independent source or renormalization authority remains necessary only for the residual Hermitian parameter.

No new carrier structure is involved.

Durable artifacts

• research/benincasa/checkers/derivative_ordering_trace_contact.rs
• research/benincasa/results/derivative-ordering-trace-contact.json
• research/benincasa/derivative-ordering-trace-contact.md

Next falsifier

Test whether a predeclared discrete symmetry, time reversal, or scaling Ward identity removes the residual $\lambda\hbar q$ term in the relevant cosmological derivative interaction. If no frozen symmetry distinguishes it, record $\lambda$ as genuine renormalization-scheme coefficient data rather than attempting to derive it from Cut geometry.

View source on GitHub