1. Chapter III
Dirac Field
Lecture 2
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
A First Book of QFT by A Lahiri and P B Pal

2. Solution for Dirac Equation
Plane wave solution
------(1)
Using this, Dirac Eq
-----(2)
where

3. We use following two component form
We can write
-----(3)
We use following two component form
for 4-component spinor (also known as bispinor)
(4)
For upper two components
For lower two components

4. From Eq (22), we can write now
(5)
Above eq lead to following coupled equations
-----(6)

5. From 2nd relation in Eq (26), we have
Using above Eq. in 1st relation of (26), we get
Which is relativistic energy momentum relationship.

6. Note that
-----(7)
Now from Dirac Eqs., (28), we have
(8)

7. We consider first solution
(9)
Using (28) and (29) in (24), we can write
------(10)
With p = 0, above Eqns. reduces to free particle
Solution with E> 0 .

8. Now we use
------(11)
and this give
----(12)
which is for E<0.

9. Thus, we write
-----(13)
Exercise: Discuss the non-relativistic limit of above
Solutions.

10. Normalisation method
Defining
----(1)
We write solution as
(2)
Where, α and β are normalization constants.

11. are normalized as
----(3)
which is for same spin components. For different
spin components it vanish.

12. We now calculate
-----(4)

13. Negative energy solutions
(5)
Also
-----(6)

14. Wave function (adjoint spinor)
---(7)
e.g.
----(8)
----(9)

15. Using (8)
-----(10)

16. Similarly, using (9)
(11)
For relativistic normalization, we will not have
normalization condition
-----(12)
Probability density transform like time component
of a four vector

17. For relativistic covariant normalization, we need
---(13)
In rest frame
Independent free particle wave function
With above normalization condition (eq 13),
(14)

18. Using (4), (5) and (13)
(15)
(16)

19. Normalized +Ve and –Ve energy solutions are
----(17)
Also
---(18)
Which is Lorentz scalar.

20. Positive and negative energy solutions are orthogonal
= 0.
(19)

21. Note that
(20)
Normalization discussed above is for massive particle
Only.
Alternative, normalization condition which work
Well for massive and mass-less particles is
------(21)

22. From this, we have
-----(22)
(23)

23. Also
---(24)
Which is again scalar.

24. More on Solutions and orthogonality relations
Positive energy sol of Dirac Eq satisfy
----(25)
where
----(26)
Negative energy sol satisfy
---(27)

25. We write positive and negative energy sol as
----(28)
Using above from (25),
----(29)

26. And for Eq (27), we have
----(30)
Which is for negative energy sol.
Adjoint Eq corresponding to (29) (take hermitia
-n conjugate and multiply by on right) :
------(31)

27. Adjoint Eq corresponding to (30) is written as
----(32)
Two +Ve and two –Ve energy solutions can be
Denoted a
----(33)
r actually represent spin projection.

28. Each sol. is a component spinor. For spinor index we use α
Each sol. is a component spinor. For spinor index we use α. Thus, α = 1, 2, 3, 4.
We can write the Lorentz invariant conditions
studied earlier in Eq. (18) using above notations as
--(34)

29. Compare last Eq of (34) with Eq (20). Is there
anything wrong?
From (34) we can write
---(35)
Also
---(36)

30. Projection operator and Completeness
Conditions:
We define the operators
----(37)
----(38)

31. Consider the operation of above operators on
solutions
---(39)
---(40)
---(41)
---(42)

32. Note
--(43)
---(44)

33. Also
----(45)
---(46)

34. We now consider the outer product of the
solutions. Consider the elements of P matrix
----(47)

35. Acting matrix P on positive spinor give the
--(48)
----(49)

36. Also
-----(50)
Thus, we can write
---(51)

37. For negative sol, we define outer product
---(52)
Operating on spinors, we get
----(53)
----(54)

38. Also
---(55)
Matrix Q project on to space of –Ve energy sol
---(56)

39. Completeness condition
---(57)
Or in Matrix form
---(58)