Theorem: Exact free causal BRST recovery

The presentation (eq:freeeq)–(eq:CCR) is a nonzero associative graded polynomial carrier. The derivation $s_0$ preserves its defining relations and induces a nilpotent graded derivation of ${\mathfrak{A}}_0$.

With the displayed normalization, the generator map to the supplied X-G-ST free presentation is the identity, and its inverse is the same generator map.

Proof. The causal Green difference annihilates the free equations. It defines a graded antisymmetric central form on the compact-test quotient. The enveloping algebra of the corresponding central Lie superalgebra, with its central element identified with the unit and the form scaled by $i\hslash$, gives the stated polynomial carrier.

For equation stability, $s_0(Ar+{\mathsf{C}}^{\ast }b)=AKc=0$, and

The antighost equation maps to $\square b$. The metric Bianchi identity and

show that $\square b=0$ follows from the bosonic equations.

The nontrivial initially zero causal relation is the $r,\bar{c}$ relation:

The remaining checks use the zero cross blocks and the zero $cc$ and $bb$ blocks. Nilpotence descends from Lemma lem:s0nil.

The X-G-ST normalization in the supplied passages is exactly (eq:s0), (eq:Ghb), and the same free causal presentation XGST. Identical generators and relations induce mutually inverse maps. This comparison does not verify other results of that manuscript.