1. Chapter III
Dirac Field
Lecture 1
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
A First Book of QFT by A Lahiri and P B Pal

2. Dirac Equation (For details on Dirac Eq and its Sol
see lectures of QM-II, III Semester )
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. In terms of Gamma matrices, Dirac Eq is given by
-----(1)
Which is Covariant form of Dirac eq.
Feynman slash notation
(2)
Where v is some arbitrary vector

4. In terms of Feynman slash notation, Dirac Eq will be
------(3)
In terms of
Dirac eq (1) can be written as
(in natural units)
-----(4)
-----(5)

5. Properties of Gamma matrices:
Gamma matrices are even dimensional
Matrices (Pauli-Dirac representation)
---(6)
Where sigma Pauli matrices are
---(7)

6. Gamma matrices satisfy the Clifford algebra
----(8)
From above we have
-----(9)

7. -----(10)
--(11)

8. ---(12)
---(13)
We define six antisymmetric matrices
----(14)

9. (15)
----(16)

10. Matrices represent spin operator for
Dirac particle.
We can identify using
-----(17)
as spin operator.

11. We constructed following 16 matrices
----(18)

12. 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)

13. Starting from
We have
Where,

14. Other representations for Dirac matrices
Majorana Representation: In this are
Purely imaginary. We have for
Useful for study of Majorana fermions which
Are charge neutral particle.

15. Dirac matrices in Dirac Pauli representation and
Majorana representation are related through
the unitary transformation

16. Weyl Representation: Used for study of
massless fermions
where
Dirac Pauli representation and Weyl representation are related through