A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of coordinates ('fields') have two superpartners ('antifields'). The quantization on such a triplectic manifold requires introducing several specific differentialgeometric objects, whose properties we study. These objects are then used to impose a set of generalized master-equations that ensure gauge-independence of the path integral. The theory thus quantized is shown to extend to a level-1 theory formulated on a manifold that includes antifields to the Lagrange multipliers. We also observe intriguing relations between triplectic and ordinary symplectic geometry.

Recent results on finite open group transformations are reviewed.

It is shown that the BRST charge $Q$ for any gauge model with a Lie algebra symmetry may be decomposed as $$Q=\del+\del^{\dag}, \del^2=\del^{\dag 2}=0, [\del, \del^{\dag}]_+=0$$ provided dynamical Lagrange multipliers are used but without introducing other matter variables in $\del$ than the gauge generators in $Q$. Furthermore, $\del$ is shown to have the form $\del=c^{\dag a}\phi_a$ (or $\phi'_ac^{\dag a}$) where $c^a$ are anticommuting expressions in the ghosts and Lagrange multipliers, and where the non-hermitian operators $\phi_a$ satisfy the same Lie algebra as the original gauge generators. By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del^{\dag}|ph\hb=0$ which is naturally solved by $c^a|ph\hb=\phi_a|ph\hb=0$ (or $c^{\dag a}|ph\hb={\phi'_a}^{\dag}|ph\hb=0$). The general solutions are shown to have a very simple form.Comment: 18 pages, Late

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form |ph = e [Q,ψ] |ph 0 where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge fixing operator, and |ph 0 BRST invariant states determined by a hermitian set of BRST doublets in involution. |ph 0 does not belong to an inner product space although |ph does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for |ph 0 and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that |ph 0 are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set |ph 0 are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.

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