How did QED diverge from quantum mechanics mathematically? We have either Heisenberg or Schrodinger picture of quantum mechanics world. So, how did quantum electrodynamics come from mathematical formulations of quantum mechanics? 
Also, QED seems to have Lagrangian formulation, while quantum mechanics does not. Why is it like this?
 A: On good reference available to those curious about the roots of QED is the book Feynman's Thesis.  This book contains paper's by Feynman and one by Dirac which discuss least action formulations of quantum mechanics and the applications to QED.
One paper by Feynman was published in 1948 and is titled, "Space-time Approach to Non-relativistic Quantum Mechanics."
The abstract is revealing:

In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of the sum of complex contributions, one from each alternative way.  The probability that a particle will be found to have a path $x(t)$ lying somewhere within a region of space-time is the square of a sum of contributions, one from each path in the region.  The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of $\hbar$) for the path in question.  The total contribution from all paths reaching $x,t$ from the past is the wave function $\psi (x,t)$. This is shown to satisfy Schroedinger's equation.  The relation to matrix and operator algebra is discussed. Application are indicated, in particular to eliminate the coordinates of the field oscillators from the equations of quantum electrodynamics.

Within the first few paragraphs, Feynman discusses the equivalency of the Schrodinger and Heisenberg picture and then discusses that this paper is about expanding upon a third way that was originally suggested by Dirac.

This paper will describe what is essentially a third formulation of non-relativistic quantum theory.  This formulation was suggested by some of Dirac's remarks concerning the relation of classical action to quantum mechanics.  A probability amplitude is associated with an entire motion of a particle as a function of time, rather than simply with a position of the particle at a particular time.

This gives us an idea of the thought processes that leads one from ordinary quantum mechanics to quantum field theory.  Instead of events, one thinks of paths.
A: The observational prediction should be the same what ever picture you use.
Some things are easier to calculate in certain  pictures.  qft is just qm applied to fields.
