A BRST-closed operator is annihilated by the differential defining a BRST transformation: . In operator language, is a graded commutator with the BRST charge, so closure does not generally mean the ordinary product vanishes.
Physical local observables are sought in the ghost-number-zero BRST cohomology: a closed operator is identified with another differing by a BRST-exact operator. Since , every exact operator is closed but represents the zero class. Thus closure by itself does not guarantee an independent physical observable; the ghost number, quotient by exact operators and the appropriate quantum consistency conditions also matter. A gauge-invariant operator provides the basic motivating example in an anomaly-free gauge theory.
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