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I have a little doubt about the harmonic oscillator hamiltonian written at the beginning of Peskin & Schroeder's "An introduction to quantum field theory"; I enclose the picture of the page.

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It introduces the scalar field and the conjugated momentum field and then as an example there is the hamiltonian of a harmonic oscillator. The harmonic term of the hamiltonian has the scalar field, so why in the kinetic one there is not the conjugated momentum field, rather than the four-momentum?

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I think they are solving the 1D quantum physics harmonic occilator, in which case $p$ is conjugate to $\phi$.

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The variable $p$ in P&S around eq. (2.23) is not the four-momentum, but the conjugate conjugated momentum for a SHO with phase space variables $(q,p)\equiv(\phi,\pi)$, cf. Mikael Fremling's answer. It is unclear why P&S chose to use the confusing mixed notation $(\phi,p)$.

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