I have been doing some reading and calculations in Zwiebach's 2nd edition text of " A First Course in String Theory". After some calculations, I have a conceptual question. Let me ask the question first then motivate it after that.


Is quantum scalar field theory equivalent to the quantum mechanics of a pair of harmonic oscillators? If this is true and with the knowledge that a complex scalar field can be described as two real fields (through a straightforward mathematical mapping), how does this scale the number of oscillators at the quantum level?

Let me lay some foundation and motivate the question

Take in basic form as a starting point the lagrangian density describing a scalar field:

$\mathcal{L} = \frac{1}{2} \partial_0 \phi \partial_0 \phi - \frac{1}{2} \partial_i \phi \partial_i \phi - \frac{1}{2} m^2 \phi^2$

skipping a bit ahead:

$\vec{P} = - \int d^d x (\partial_0 \phi) \nabla \phi$ . . . . (10.46)

After some calculations, one can find

$\vec{P} = \vec{p} (a^* _p a_p - a^*_{-p} a_{-p} )$ . . . . (10.47)

These are two harmonic oscillators.

After some more calculations, it can be shown that at the quantum level, the Hamiltonian becomes an operator

$H = E_p (a^\dagger _p a_p + a^\dagger _{-p}{a_{-p}} )$ . . . . (10.55)

This describes "a pair of simple harmonic oscillators, each with frequency $E_p$ "

The state space can then be constructed.

  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/737996/2451 $\endgroup$
    – Qmechanic
    Nov 24 at 19:54
  • $\begingroup$ You are missing various negative signs. For example, (10.47) as currently written vanishes. $\endgroup$
    – Ghoster
    Nov 24 at 20:03
  • $\begingroup$ That portion of your book is dealing with one only mode of the field, with a particular wavevector. The field has an infinite number of modes. Scalar waves can propagate in any direction with any wavenumber, just like vector waves do in electromagnetism. $\endgroup$
    – Ghoster
    Nov 24 at 20:14
  • $\begingroup$ Quantum field theory would not have mathematical difficulties (such as infinities requiring renormalization) if it reduced to two harmonic oscillators. The difficulties arise because it has an infinite number of arbitrarily high-energy oscillators. $\endgroup$
    – Ghoster
    Nov 24 at 20:20
  • $\begingroup$ @Ghoster I am editing to try to do the _p . Having some difficulties doing the Tex for that, and then I will address the other comments right after $\endgroup$ Nov 24 at 20:22


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