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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)