Physics Asked on May 21, 2021
This relates to page 20 of Peskin and Schroeder.
They state that the Fourier transform of the Klein-Gordon field satisfies the following:
$$left[frac{partial^2}{partial t^2}+(|vec p|^2+m^2)right]phi(vec p,t)=0 tag{2.21},$$
which is the equation of motion of a simple harmonic oscillator with frequency:
$$omega_vec p=sqrt{|vec p|^2+m^2} tag{2.22}.$$
This is fine, however their next equation is the Hamiltonian for the simple harmonic oscillator:
$$H_{SHO}=frac{1}{2}p^2+frac{1}{2}omega^2phi^2,$$
which, confusingly to me, does not have a mass $m$ in the denominator of the kinetic term. I have searched around a bit online and not found any reference to this, have I missed something?
Neither does it have a mass in the numerator for the $phi^2$ term! Peskin & Schroeder just do not bother with a constant $m$ is this context. As you can see, this part introduces you to the ladder operators, in order to apply the formalism to the Klein-Gordon hamiltonian. No need to worry about $m$'s, which are irrelevant to the commutation relations anyway, set it to 1 and work your way through the SHO properties.
Correct answer by gildran on May 21, 2021
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