Time reversal operation and positivity of the Hamiltonian in the Donaldson-Witten TQFT

A TQFT in terms of general gauge fixing functions is discussed. In a covari ant gauge it yields the Donaldson-Witten TQFT. The theory is formulated on a generalized phase space where a simplectic structure is introduced. The H amiltonian is expressed as the anticommutator of off-shell nilpotent BRST a nd anti-BRST charges. Following the original ideas of Witten a time reversa l operation and the corresponding inner product are defined. We present a n on-covariant gauge fixing that gives rise to a manifestly time reversal inv ariant Lagrangian and a positive definite Hamiltonian, with the previously introduced inner product. As a consequence, the indefiniteness problem of s ome of the kinetic terms of the Witten's action is resolved. The constructi on allows then a consistent interpretation of Floer groups in terms of the cohomology of the BRST charge which is explicitly independent of the backgr ound metric. The relation between the BRST cohomology and the ground states of the Hamiltonian is then completely established. The topological theorie s arising from the covariant, Donaldson-Witten, and non-covariant gauge fix ing are shown to be quantum equivalent by using the operatorial approach.