Chapter II Klein Gordan Field Lecture 1 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Quantum Field Theory by Michio Kaku

Review of Simple Harmonic Oscillator Hamitonian for Harmonic oscillator ------(1) Introducing annihilation and creation operators -------(2)

We can write ----(3) Hamiltonian will be ----(4)

Commutation of Hamiltonian with creation And annihilation operators ----(5) Considering -----(6) We can write -----(7)

Ground state energy -----(8) For excited states -----(9)

Klein Gordon Field Lagrangian density for real scalar field ------(1) Momentum field density -----(2)

Hamiltonian density ----(3)

Total Hamiltonian ----------(4) Dimensions : (write (1) using and c)

Canonical quantization for scalar fields -------(5)

Field Decomposition We use plane wave solutions as complete set of basis and expand field operator as -------(6)

Satisfy Klein Gordon Eq -----(7) Using (6) in (7) -----(8)

Fourier transform has non-vanishing component only on the mass shell -----(9) We can write ---------(10)

Using (10) in (6), -------(11) Argument of Dirac delta vanished when ----(12)

We write Dirac delta function as -------(13)

Using (13) in (11) -------(14)

In 2nd term of Eq (14) use We have ---(15)

Real field will be hermitian ---------(16)

We have ---(17) And we write ------(18)

Note that ---(19) And using ----(20) Using above in (18), -----(21)

Positive and negative energy part ----(21) And thus --------(22)