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## Summary:

- Renormalized fields give the wrong values for the free Hamiltonian.

## Main Question or Discussion Point

When introducing renormalization of fields, we define the "free Lagrangian" to be the kinetic and mass terms, using the renormalized fields. The remaining kinetic term is treated as an "interaction" counterterm. If we write down the Hamiltonian, the split between "free" and "interaction" terms, used in defining the interaction picture, will presumably be the same. So apparently field renormalization involves multiplying the free Hamiltonian by a factor ##Z^{-1}##.

How does this make sense? The free Hamiltonian has a well-defined scale - it is just the sum of the particle number operators for given momenta, with the appropriate energies (plus an infinite constant). And this only works with a particular normalization of the field operators: the derivation uses the canonical commutation/anticommutation relations, implying canonical normalization.

How does this make sense? The free Hamiltonian has a well-defined scale - it is just the sum of the particle number operators for given momenta, with the appropriate energies (plus an infinite constant). And this only works with a particular normalization of the field operators: the derivation uses the canonical commutation/anticommutation relations, implying canonical normalization.