1. Relativistic Quantum Mechanics
Lecture 2
Books Recommended:
Lectures on Quantum Field Theory by Ashok Das
Advanced Quantum Mechanics by Schwabl
Relativistic Quantum Mechanics by Greiner
Quantum Field Theory by Mark Srednicki

2. Lorentz Transformations (along x-axis)
Postulates of special theory of relativity:
All the physical laws are same in all inertial frames of reference which are moving with constant velocity relative to each other. There is no absolute or universal frame of reference.
(2) The speed of light in vacuum is the same in every inertial frame
Lorentz Transformations (along x-axis)

3. Lorentz Transformations
Under Lorentz boost, coordinates transform from one
Inertial frame to other as
Such that (1)
(2)

4. Above eq is written as
------(3)
In above ω is known as rapidity.

5. In last eq.
------(4)
Such that As
Thus

6. Thus, we write the Lorentz transformation as
(5)
Where, Lorentz transformation matrix
----(6)

7. We write
---(7)
Covariant co-ordinate will transform as

8. Matrix representing Lorentz transformation are
orthogonal

9. Lorenz transformations leave the length of vector
invariant