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The Hamiltonian for the free scalar field, expressed in terms of the creation/annihilation operators, is

[itex] H = \int d^{3}p [\omega_p a^{\dagger}_p a_p + \frac{1}{2}\omega_p \delta^{3}(0)] \hspace{3mm} [/itex]

I thought: [itex] \omega_p [/itex] is a function ofas [itex] \omega^{2}_p = |p|^{2} + m^2 [/itex] and so the dirac delta will sift out the value of [itex] \omega_p [/itex] at [itex] p = 0 [/itex]. Could someone tell me why this statement is incorrect? I think I've made some significant conceptual error. Is the first term divergent for infinite p as well?p

P.S. wherever p appears above, it is to be taken as the 3-momentum

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# Divergence of integral over vacuum energies (Free field)

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