Spacetime background of Quantum mechanics Why is it said that the Schrodinger equation suggests a fixed, non-dynamical background spacetime, with time as an external parameter? How does this interpretation come about from the Schrodinger equation?
I guess what I am asking in general is, how does one go about to infer anything about the spacetime background QM is formulated upon?
 A: The spacetime geometry may be indirectly read from the differential operators in the equations controlling other objects – particles and (matter) fields. Schrödinger's equation contains the differential operator 
$$\Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}  $$
which is the operator of the general form 
$$\sum_{ij} g^{ij} \frac{\partial}{\partial x_i}\frac{\partial}{\partial x_i}$$
with $g^{ij}=\delta^{ij}$ i.e. $1$ for $i=j$ and $0$ for $i\neq j$.
This special form of the metric tensor (the coefficients $g^{ij}$) is known as the geometry of the flat 3-dimensional space. Because the usual Schrödinger's equation never contains any difficult coefficients in front of the second spatial derivatives, it means that the wave function "propagates" in the flat 3-dimensional space.
Because the flat 3-dimensional space has a geometry that isn't affected by the objects, matter, and fields that are present in the geometry – indeed, it's completely constant as a function of time $t$ here – we say that it is "non-dynamical". The word "dynamical" means "dependent on time", actually just "dependent on time through the dependence on other objects".
Similarly, there is no time-dependent coefficient in Schrödinger's equation in front of the time derivative $\partial / \partial t$, so the time is non-dynamical as well.
To make spacetime dynamical means to introduce new fields $g_{\mu\nu}(x,y,z,t)$ – either classical fields or quantum fields (or something that is effectively equivalent) – whose dependence on $x,y,z,t$ is affected by the matter around; and to use this $g^{\mu\nu}$ as coefficients in front of all differential operators, and other terms, that appear in the equations for fields and wave functions etc.
When we introduce this dynamical $g_{\mu\nu}$, the spacetime becomes curved in general, and the curvature depends on the matter and energy distributed in the space. This curvature will manifest itself as the gravitational force. That's how Einstein's general theory of relativity explains the gravitational force – it's due to the curved space.
