Shakthi said:
Why do we study Lorentz and Poincare Groups?
Physicists are interested in theories of motion. To describe motion mathematically, we need to choose a mathematical structure to represent space and time. The obvious choice is the set ℝ
4. But we also need to be able to translate one observer's description of what's going on into another one's. It's natural to try to solve this problem for the "inertial" observers first. They are the ones that never accelerate, and never rotate. Each observer should describe the motion of any of the other inertial observer as a straight line in ℝ
4, right? It seems at least reasonable to try the assumption that they will be able to do that.
This seemingly harmless assumption together with the very natural requirement that the set of functions that represent a coordinate change from one inertial observer's point of view to another's, is a group, is (along with some minor technical assumptions) enough to narrow down our options to two choices: Either the set of coordinate change functions is the Galilei group, or it's the Poincaré group. Non-relativistic classical mechanics and special relativistic classical mechanics are both
defined by a choice of which one of these two groups to use.
If that isn't enough to explain why they're important, you should look into symmetries, in particular Noether's theorem in. Invariance of the laws of physics under translations in time imply that there's a conserved quantity, which we call "energy". Invariance under translations in space imply conservation of "momentum", and so on.
These groups play an even larger role in the corresponding quantum theories, because each particle species is associated with an irreducible representation. The generators of the appropriate one-parameter subgroups (e.g. translations in time) are identified with the measuring devices that measure some quantity (e.g. energy). (These identifications should be considered part of the axioms of quantum mechanics, even though they're never described that way in QM books).
Shakthi said:
Also, is there a way to represent Lorentz transformations, boosts and translations diagrammatically (explicitly - in the sense that one can look at the diagram and understand what's going on)?
As DaleSpam said, spacetime diagrams! A 1+1-dimensional (proper) Lorentz transformation (which is always a boost) can be expressed in the form
\gamma\begin{pmatrix}-v & 1\\ 1 & -v\end{pmatrix}
where
\gamma=\frac{1}{\sqrt{1-v^2}}
As you can verify by having this matrix act on \begin{pmatrix}1\\ 0\end{pmatrix} and \begin{pmatrix}0\\ 1\end{pmatrix}, a boost tilts the "t" axis in one direction, and the "x" axis by the same angle in the opposite direction. This is (the main part of) what ensures that a line with slope 1 represents motion at the invariant speed.