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Chapter III
Dirac Field
Lecture 3
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
 Advanced Quantum Mechanics by Schwabl
Postulates of special theory of relativity:
(1) All the physical laws are same in all inertial frames of
reference which are moving with constant velocity
relative to each other. There is no absolute or universal
frame of reference.
(2) The speed of light in vacuum is the same in every inertial
frame
 Lorentz Transformations (along x-axis)
Rotation in dimension
---(1)
Where, rotation matrix R (about Z axis) is
-----------(2)
Thus,
----(3)
Rotation matrix R is orthogonal and have unit
determinant
For infinitesimal rotation
----(4)
Anti-symmetric matrix.
Lorentz Transformations
Under Lorentz boost, coordinates transform from one
Inertial frame to other as
Such that
-----(1)
--------(2)
Above eq is written as
In above ω is known as rapidity.
------(3)
In last eq.
------(4)
Such that
As
Thus
Thus, we write the Lorentz transformation as
-------(5)
Where, Lorentz transformation matrix
----(6)
We write
---(7)
Covariant co-ordinate will transform as
Matrix representing Lorentz transformation are
orthogonal
---(8)
And have unit determinant as was the case of rotation
Matrix in 3-D about z-axis.
Infinitesimal Lorentz transformation
-
--(9)
Where
--(10)
Matrix in Eq (10) is anti-symmetric
----(11)
•Lorenz transformations leave the length of vector
Invariant
----(12)
Covariance of Dirac Equation
Consider dynamical Eq
------(13)
Where L is operator.
Above Eq will be covariant if
-------(14)
Consider Dirac Eq
----(15)
Under Lorentz tranfromation
----(16)
If Eq (15) takes form
---(17)
Then it will be form invariant.
Under L.T. Transformed wave function is written
as
----(18)
Where
is a 4 by 4 matrix used to
Represent the effect of LT on wave function.
It must possess inverse
------(19)
From (16) we can write
---(20)
We write
---(21)
Comparing (21) with (15), for invarianve we
should have
-----(22)
Define
--------(23)
We have
----(24)
Consider the infinitesimal Lorentz transformation
--------(25)
Where
Expanding S matrix
----(26)
Where
-----(27)
Also
----(28)
Where
----(29)
From (22), we write
-----(30)
Recall
-----(31)
From (30), identify
------(32)
Thus,
-----(33)
which is RHS of (30).
Thus, for infinitesimal transformation
---(34)
Indentify
as generator of LT