What question did this study set out to answer?

To rigorously prove the preservation of the Slavnov–Taylor identity under the Reynolds Projector in SU(3) Yang–Mills theory.

April 21, 2026Open Access

Resolution of Open Problem O3: Proof of the Slavnov–taylor Identity Under the Reynolds Projector

Key Points

To rigorously prove the preservation of the Slavnov–Taylor identity under the Reynolds Projector in SU(3) Yang–Mills theory.
Defined BRST symmetry axiomatically.
Constructed the Reynolds Projector using the Haar measure on SU(3).
Established the null commutator via color-singlet analysis and Haar measure invariance.
Proved the Slavnov–Taylor Ward identity is preserved under the Reynolds Projector.
Confirmed that restricting gluon phase-space does not break gauge invariance.
Established the independence of topological projection and quantum gauge structure.

Abstract

We provide a rigorous proof that the Slavnov–Taylor Ward identity of pure SU(3) Yang–Mills theory is preserved under the action of the Reynolds Projector . The central result is the Null Commutator Theorem: This theorem is established in five steps: Step 1: Axiomatic definition of BRST symmetry; Step 2: Integral construction of the Reynolds Projector via the Haar measure on SU(3); Step 3: Proof of the null commutator via the color-singlet nature of the BRST charge and the invariance of the Haar measure; Step 4: Derivation of the transversality of the projected vacuum polarization tensor; and Step 5: Connection to Theorem I of the T-DFT framework via the exactness of the vacuum differential form. The proof resolves the structural concern that restricting the gluon phase-space to the color-singlet fraction f = 1/64 might anomalously break gauge invariance. It establishes that the topological projection and the quantum gauge structure are geometrically independent and mutually transparent.

Resolution of Open Problem O3: Proof of the Slavnov–taylor Identity Under the Reynolds Projector

Key Points

Abstract

Cite This Study