This is a question about how the amplitudes of QM might be generalized. I'm not sure this post belongs here or perhaps some other forum. But I believe I am talking about extending QM beyond the standard. Correct me if I'm wrong, but what we have in flat space is states, [tex]\[ |\psi> \][/tex], projected onto eigenvectors of the position operator [tex]\[|x > \][/tex] to give us a wavefunction, [tex]\[ < x|\psi> = \psi(x)\] [/tex], whose modulus squared is a probability density, [tex]\[ < \psi|x > < x|\psi> = |\psi(x)|^2 = p(x)\][/tex]. These position eigenvectors form an orthonormal basis, [tex]\[ < x_i |x_j > = \delta (x_i - x_j )\][/tex], so that the inner product is preserved, [tex]\[ < x_i |x_j > = < x_k |x_l > = 1\][/tex]. This give a unitary transformation from one place to another and allows a probability to be calculated to go from some initial state to a final state. But it occurs to me that a probability is just one type of measure, and the orthonormal inner product, [tex]\[ < x_i |x_j > = \delta (x_i - x_j )\][/tex], is just one type of metric. So quantum mechanics establishes a relationship between the metric and the measure. My question is can this relationship between the metric and the measure be generalized to curved spacetime in such a way that as the metric becomes flat, the measure becomes a probability? I suppose that if the metric were to change like this, then the evolution of states would cease to be unitary, it would be more difficult to describe states in a basis of position eigenstates, and the measure would cease to be the probabilities we are familiar with. New interpretations would have to be employed to describe such a situation, no doubt. I believe that the study of QFT in curved spacetime might have something to say about this. But I am hoping to find something more fundamental. Is there anything in the study of metric spaces, measure theory, of functional analysis that would justify this kind of general relationship between the metric and the measure? Thanks.