Example real scalar field

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Example real scalar field

• A relativistic classical field theory described
  by a classical field equation.

• We are able to construct a Lagrangian Density

The Quantum Theory of Fields, Steven Weinberg
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Example real scalar field

• The only dynamical variable is phi, we are able
  to find its canonical conjugate momentum density

• Performing Legendre Transformation, we can
  construct the Hamiltonian Density

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Example real scalar field

• Quantization we promote phi and pi to operators
  and impose suitable commutation relations. Assume
  hbar 1

• By experience, its easier if we perform Fourier
  Transformations

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Example real scalar field

• where

• After that, the commutation relations are
  equivalent to

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Example real scalar field

• We can now express the Hamiltonian in terms of
  the new operators.

• The energy spectrum of the theory can be worked
  out, just like harmonic oscillator. Each creation
  operator creates an excitation particle. The
  particles are bosons.