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Quantising a scalar field

  1. Feb 26, 2013 #1
    Hi I am trying to derive this fairly common cosmological scalar field equation in a FR metric, it is not a nomework question and is fairly advanced so i posted here. ([itex]dt^2-a^2(t)d\underline{X}^2[/itex]), where $\chi$ is quantised as [itex]\chi=\int \frac{d^{3}K}{(2\pi)^{\frac{3}{2}}}[\chi \hat{a_k}e^{-ikx}+\chi^{*}\hat{a_k}^{\dagger}e^{+ikx}][/itex], and the scalar field in curved space-time is [itex]g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi + 3Hg^{\mu \nu} \nabla_{\nu}\phi + g^{2}\phi^{2}\chi = 0[/itex]. H is the hubble paramater equal to $\dot{a}/a$ and a is the scale factor.

    [itex]\ddot{\chi} + 3H(t)\dot{\chi}+(\frac{k^2}{a^2(t)}+g^{2}\phi^{2})\chi=0 [/itex] with the potential [itex]V=\frac{1}{2}m^{2}\phi^{2} + \frac{1}{2}g^{2}\phi^{2}\chi^{2} [/itex].

    I can happily derive all these terms in the scalar field equation but I get an additional [itex]ik\chi\hat{a}e^{ikx} - ik\chi^{*}\hat{a}^{\dagger}e^{+ikx} [/itex] term, the only way I can see to make it cancel is by using that $\chi $ is hermitian so [itex]\chi^{*}=\chi [/itex], but that wouldn't explain the conjugate exponential and creation/annhilation operator or would it? guidance?
  2. jcsd
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