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)