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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
Solution for Dirac Equation
Plane wave solution
------(1)
Using this, Dirac Eq
-----(2)
where
We can write
-----(3)
We use following two component form
for 4-component spinor (also known as bispinor)
For upper two components
--------(4)
For lower two components
From Eq (22), we can write now
---------(5)
Above eq lead to following coupled equations
-----(6)
From 2nd relation in Eq (26), we have
Using above Eq. in 1st relation of (26), we get
Which is relativistic energy momentum relationship.
Note that
-----(7)
Now from Dirac Eqs., (28), we have
--------(8)
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 .
Now we use
------(11)
and this give
----(12)
which is for E<0.
Thus, we write
-----(13)
Exercise: Discuss the non-relativistic limit of above
Solutions.
Normalisation method
Defining
----(1)
We write solution as
----------(2)
Where, α and β are normalization constants.
are normalized as
----(3)
which is for same spin components. For different
spin components it vanish.
We now calculate
-----(4)
Negative energy solutions
---------(5)
Also
-----(6)
Wave function (adjoint spinor)
---(7)
e.g.
----(8)
----(9)
Using (8)
-----(10)
Similarly, using (9)
-------(11)
For relativistic normalization, we will not have
normalization condition
-----(12)
Probability density transform like time component
of a four vector
For relativistic covariant normalization, we need
---(13)
In rest frame
Independent free particle wave function
With above normalization condition (eq 13),
----------(14)
Using (4), (5) and (13)
--------(15)
--------(16)
Normalized +Ve and –Ve energy solutions are
----(17)
Also
---(18)
Which is Lorentz scalar.
Positive and negative energy solutions are orthogonal
= 0.
----------(19)
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)
From this, we have
-----(22)
--------(23)
Also
---(24)
Which is again scalar.
More on Solutions and orthogonality relations
Positive energy sol of Dirac Eq satisfy
----(25)
where
----(26)
Negative energy sol satisfy
---(27)
We write positive and negative energy sol as
----(28)
Using above from (25),
----(29)
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)
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.
 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)
Compare last Eq of (34) with Eq (20). Is there
anything wrong?
From (34) we can write
---(35)
Also
---(36)
Projection operator and Completeness
Conditions:
We define the operators
----(37)
----(38)
Consider the operation of above operators on
solutions
---(39)
---(40)
---(41)
---(42)
Note
--(43)
---(44)
Also
----(45)
Also
---(46)
We now consider the outer product of the
solutions. Consider the elements of P matrix
----(47)
Acting matrix P on positive spinor give the
positive spinor
--(48)
----(49)
Also
-----(50)
Thus, we can write
---(51)
For negative sol, we define outer product
---(52)
Operating on spinors, we get
----(53)
----(54)
Also
---(55)
Matrix Q project on to space of –Ve energy sol
---(56)
Completeness condition
---(57)
Or in Matrix form
---(58)