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Chapter IV
Gauge Field
Lecture 3
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
 Advanced Quantum Mechanics by Schwabl
Field Decomposition
As we discussed earlier
--------(1)
Thus, we can write
----(2)
Field Eq
-------(3)
For ν = 0
---------(4)
And for ν = i
----(5)
Which is like mass-less KG Eq.
Sol will be
----(6)
Where
---(7)
Using coulomb Gauge
----(8)
For two polarization direction
-----(9)
We can write following expression for the
decomposition for field
----(10)
where
Exercise:
Using inverse Fourier transform, find
-------(11)
----(12)
Quantization of field
----(13)
Above equations are not consistent with
Transverse character of vector potential
We define following quantization condition
which is consistent with the transverse
condition.
---(14)
Where transverse delta function is defined as
---------(15)
Note that
---(16)
In momentum space
Exercise: Using the expression for fields
in LHS of (14) prove that it gives RHS.
See Schwabl book.
Commutators for annihilation and creation
Operators
---(17)
Normal ordered Hamiltonian and momentum
------(18)
----(19)
We can write
---(20)
---(21)