Spacetime and quantum mechanics In special relativity, the particle has a fixed world line in spacetime. So its whole trajectory is determined. But how can we represent the world line of the particle in spacetime when we take quantum mechanics into account? Because if we represent it by a line, then its motion would be deterministic.
 A: This is precisely what Richard Feynman achieved. This is known as the path integral approach to quantum mechanics. In classical mechanics, the world line is considered to be the solution to the Euler-Lagrange equations. So if you had an action, all you have to do is find the solution to the Euler-Lagrange EOMs and there you go - you get a parametrized world line. 
However, in quantum mechanics, the particle's evolution from point $A$ to point $B$ is given by the weighted average over all the possible paths from $A$ to $B$. The "weighted average" is in fact derived from a probabilistic measure and therefore, the notion of the world line in quantum mechanics is still probabilistic. So the line that you are referring to in the last line of your question is basically the classical limit which is the probabilistic average of an infinite number of world lines.
As Sanya points out in his/her answer, the notion of Schrödinger quantum mechanics is not compatible with SR because of ghost modes, i.e. propagating states of negative norm. This is fully and consistently merged with SR only using the Dirac formulation. So you need to be a bit careful as to how you introduce QM and SR - the Schrödinger equation does not treat space and time equally and you can see this in the fact that it is a second order derivative in space while it is a first order derivative in time.
A: Schrödinger Quantum Mechanics is built in a world with an absolute time and absolute space - thus combining special relativity and simple quantum mechanics is not a good idea. From a Kuhnian perspective I would even go as far as to call the theories incompatible. First approaches to generalise quantum mechanics towards a relativistic theory led to the Dirac theory and - because there were particles with negative energies (or something like that) which seemed strange - in the end towards todays quantum field theories, which present a fully relativistic theory of quantum effects.
A: You are partially right. In Special Relativity, the trajectory of a mass is a "worldline" in four dimensional Minkowski-spacetime.  
But in Quantum Mechanics, the development of quantum systems are not described as trajectories in Spacetime, rather they are described by the Schrödinger-equation, which is a three-dimensional partial differential equation.
There have been unifications of these two concepts by the Klein-Gordon-equation or the Dirac-equation, which are the foundation of relativistic Quantum Mechanics.
