Functional Schroedinger and BRST Quantization of (1+1)--Dimensional Gravity

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We discuss the quantization of pure string--inspired dilaton--gravity in (1+1)--dimensions, and of the same theory coupled to scalar matter. We perform the quantization using the functional Schroedinger and BRST formalisms. We find, both for pure gravity and the matter--coupled theory, that the two quantization procedures give inequivalent ``physical'' results.