1. Relativistic Quantum Mechanics
Lecture 4
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. Dirac Equation
Klein Gordon lead to negative energy solutions.
Probability density can be negative. This is
because it has first derivative of wave function w.r.t. t.
We need to have first derivative of ϕ w.r.t time in
wave equation. But then, spatial derivative should.
Also be of first order because of Lorentz invariance.
We can do this if energy and momentum are
linearly dependent.

3. Dirac proposed equation of the form
----(1)
Where, Hamiltonian =
-----(2) Or
Or (In natural units)

4. -----(3)
(in natural units)
Coefficients αi cannot be simply numbers because
if is so then equation will not be invariant
αi and β should be hermitian matrices so that H is
hermitian say N by N matrix
N component column vector

5. Following conditions will be imposed
Klien Gordon Eq and hence Relativistic energy-
moment relationship should be satisfied
E2 = p2c2 + m2c4
Conserved four-current density with positive zero
component probability density should exist
Should be Lorentz invariant

6. Differentiating eq n (3) w.r.t. t
(4)

7. For eq (4) to satisfy KG eqn, we get
-----(5)
------(6)
(7)
Thus, we have four anticommuting Hermitian
Matrices.

8. Now we will show that above matrices should by
4 by 4 dimensions
Matrices β and αi will be traceless
From (6), we write
Taking trace and using cyclic property, we have

9. As α and β are hermitian, they can be diagonalzed
to get eigen values. As the square of these
matrices is unity, eigen values should be ±1
As the trace of matrices is zero and hence matrices
should be of even dimension
Matrices cannot be of 2 by 2 dimensions, as
there will not be 4 traceless ant commutating
hermitian matrices. Only 3 e.g. Pauli matrices.
Thus they will be of at least 4 by 4 dimension

10. Following representation is used for Dirac matrices
(8)
Where Pauli matrices are
Exercise: Show that matrices in Eq (8) satisfy
eq. (5) to (7)

11. Dirac Equation in the Covariant form
Dirac Eq is
------(9)
Multiplying by β/c, we get
----(10)
Now we use representation
(11)

12. Properties of Gamma matrices:
Proof:
------(12)

13. Also
-----(13)
Eq (12) and (13) gives
(14)

14. In terms of Gamma matrices, Dirac Eq (9) can be
written as
-----(15)
Which is Covariant form of Dirac eq.
Feynman slash notation
(16)
Where v is some arbitrary vector

15. In terms of Feynman slash notation, Dirac Eq will be
------(17)
In terms of
Dirac eq (15) can be written as
(in natural units)
-----(18)
-----(19)

16. Pauli’s fundamental theorem: If satisfy
Clifford algebra, then they must be related through
the similarity transformations.
i.e. if
-----(20)
------(21)
Then there exist non-singular matrix S such that
------(22)

17. Starting from
We have
Where,