Relativistic Quantum Mechanics Lecture 1 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

Why relativistic quantum mechanics ? Quantum Mechanics + Special theory of relativity  High energy phenomenon's require relativistic wave equations  Relativistic wave equations should be invariant Under Lorentz transformations  Space and time co-ordinates have to consider on equal footing

 Relativistic theory must be able to explain pair annihilation and creation processes.  Schrodinger equation is given by (1) For spinless non-relativistic particle (2)

In position basis (3) In order to get relativistic version, we proceed as Follows: ) With above H, S.E. will be ---(5)

Problems with Eq (5):  Space and time derivative are on different footing. (First order time derivative and second order space derivative. Thus Eq (5) is not Lorentz invariant  Also, expanding square root in power of, we get infinitely higher order space derivatives acting on wave function and equation will not be local in space.

Notations/Conventions: In three dimension Euclidean space Scalar product: Position Angular momentum Some arbitrary vector

Length of vector In four dimensional space (where space and time are considered on equal footing) any point is represented by four coordinates and also a vector in this space will have four components.

We have two kind of vectors in 4-dim space-time Above vectors are related through metric tensor Contavariant vector Covariant vector

Metric tensors are defined as And are related to each other Metric tensors are symmetric

Defining two arbitrary vectors: The scalar product is defined as Above scalar product is invariant under Lorentz transformation and is called Lorentz scalar. (Prove!)

Length of vector in Minkowski space Length of a vector need not always to be positive as was the case for 3-dim Euclidean space. Note that which is the invariant length of any point from origin (Prove!).

The length between two point infinitesimal close to each other is given by is proper time. Prove above statement!

Space-time region is time-like if (using c =1, natural unit) Space-like if Light-like region if

All future processes takes place in future light cone or forward light cone defined by Contragradient and cogradient vectors are defined as (using c = 1) respectively.

 We define Lorentz invariant quadratic operator Known as D’ Alembertian operator as (c =1)  Energy and momentum are defined in terms of energy-momentum four vector Using above we define the Lorentz invariant scalar Here we used c = 1, otherwise, we have to use E/c

We know the Einstein relationship Where m is the mass of particle. Last two equations define the mass as Lorentz invariant scalar quantity

We know the operator forms Co-ordinate representation of energy momentum four-vector will be

Four dimension Levi-civita tensor which is ant -symmetric four dimensional tensor and

 Natural System of Units Here we consider ћ = 1 and c =1 This lead to