and respectively two — are field independent), it follows that the BRST differential s reduces to a sum of two fermionic derivations

\[ s = \delta + \gamma , \eqno ( 2 2 ) \]

with \( \delta \) the Koszul–Tate differential, graded in terms of agh \( ( \mathrm { a g h } ( \delta ) = - 1 \)),and \( \gamma \) the longitudinal exterior derivative (in this case a true differential), graded by pgh \( ( \mathrm { p g h } ( \gamma ) \) = 1). These two degrees do not interfere \( ( \mathrm { a g h } ( \gamma ) \) = 0,\( \mathrm { p g h } ( \delta ) \) = 0), such that the total degree of the BRST differential (and of each of its components), namely gh, becomes equal to 1:\( \operatorname { g h } ( s ) = \operatorname { g h } ( \delta ) = \operatorname { g h } ( \gamma ) = \) 1. One of the major requirements of the BRST setting, namely the second-order nilpotency of s, becomes equivalent to three separate equations

\[ s ^ { 2 } = 0 \Leftrightarrow ( \delta ^ { 2 } = 0 , \, \delta \gamma + \gamma \delta = 0 , \, \gamma ^ { 2 } = 0 ) , \eqno ( 2 3 ) \]

which confirms that \( \gamma \) can indeed be constructed as a true differential in the case of the free model under study. The actions of \( \delta \) and \( \gamma \) on the BRST generators that enforce (23) as well as the fundamental cohomological requirements of the antifield BRST theory [27, 28, 29, 30, 31, 32] are given by

\[ \delta \chi ^ { \Delta } = 0 , \qquad \qquad \qquad \delta \varphi ^ { * } = \partial _ { \lambda } H ^ { \lambda } , \qquad \qquad \delta A ^ { * \mu } = \partial _ { \lambda } B ^ { \lambda \mu } , \qquad \qquad ( 2 4 ) \]

\[ \begin{array} { r } { \delta H _ { \mu } ^ { * } = - \partial _ { \mu } \varphi , \qquad \qquad \delta B _ { \mu \nu } ^ { * } = - \frac 1 2 \partial _ { [ \mu } A _ { \nu ] } , \qquad \quad \delta \phi _ { A } ^ { * } = k _ { A B } \sqcap \phi ^ { B } , \qquad ( 2 5 ) } \end{array} \]

\[ \delta \eta ^ { * } = - \partial _ { \lambda } A ^ { * \lambda } , \qquad \quad \delta C _ { \mu \nu } ^ { * } = \partial _ { [ \mu } H _ { \nu ] } ^ { * } , \qquad \qquad \delta \eta _ { \mu \nu \rho } ^ { * } = \partial _ { [ \mu } B _ { \nu \rho ] } ^ { * } , \qquad ( 2 6 ) \]

\[ \delta C _ { \mu \nu \rho } ^ { * } = - \partial _ { [ \mu } C _ { \nu \rho ] } ^ { * } , \qquad \delta \eta _ { \lambda \mu \nu \rho } ^ { * } = - \partial _ { [ \lambda } \eta _ { \mu \nu \rho ] } ^ { * } , \qquad \delta C _ { \lambda \mu \nu \rho } ^ { * } = \partial _ { [ \lambda } C _ { \mu \nu \rho ] } ^ { * } , \qquad ( 2 7 ) \]

\[ \gamma \chi _ { \Delta } ^ { * } = 0 , \qquad \qquad \qquad \gamma \varphi = 0 , \qquad \qquad \qquad \gamma A _ { \mu } = \partial _ { \mu } \eta , \qquad \qquad ( 2 8 ) \]

\[ \gamma H ^ { \mu } = - 2 \partial _ { \lambda } C ^ { \lambda \mu } , \qquad \gamma B ^ { \mu \nu } = - 3 \partial _ { \lambda } \eta ^ { \lambda \mu \nu } , \qquad \gamma \phi ^ { A } = 0 , \qquad \qquad ( 2 9 ) \]

\[ \gamma \eta = 0 , \qquad \qquad \qquad \gamma C ^ { \mu \nu } = - 3 \partial _ { \lambda } C ^ { \lambda \mu \nu } , \: \: \: \: \: \gamma \eta ^ { \mu \nu \rho } = - 4 \partial _ { \lambda } \eta ^ { \lambda \mu \nu \rho } , \: \: \: \: ( 3 0 ) \]

\[ \gamma C ^ { \mu \nu \rho } = - 4 \partial _ { \lambda } C ^ { \lambda \mu \nu \rho } , \quad \gamma \eta ^ { \lambda \mu \nu \rho } = 0 , \qquad \qquad \gamma C ^ { \lambda \mu \nu \rho } = 0 , \qquad \qquad ( 3 1 ) \]

where both operators are assumed to act like right derivations and \( \sqsupset \equiv \)\( \partial _ { \mu } \partial ^ { \mu } \) symbolizes the d’Alembertian. We notice that the actions of \( \gamma \) on all fields/ghosts can be obtained in this particular situation simply by replacing all gauge or reducibility parameters from the right-hand sides of relations (3)–(5), (7), (10), and (13) with the corresponding ghosts introduced in (15)–(17).

A striking feature of the antifield approach, in spite of its essentially La-grangian origins, resides in the (anti)canonical action of the BRST differen-tial [23, 24, 25, 26, 27, 28, 29, 30, 31, 32], \( s \cdot = ( \cdot , S ) \), where its (anti)canonical