Quantum Field Theory in Curved Space-Time Backgrounds

  • Dieter LüstEmail author
  • Ward Vleeshouwers
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)


The scalar field action is given by \(S\left[ \phi \right] = - \frac{1}{2} \int d^4x \left[ \eta ^{\mu \nu } \partial _{\mu } \phi \partial _{\nu } \phi + m^2 \phi ^2 \right] \). We promote the field \(\phi ( t, \vec {x})\) to an operator \(\hat{\phi } ( t, \vec {x})\) with associated creation and annihilation operators, which we can then make time-dependent as
$$ \left. \begin{array}{cc} {\hat{a}}_{\vec {k}}^{+} &{} \rightarrow \hat{a}_{\vec {k}}^{+}(t) := e^{i \omega _k t} \hat{a}_{\vec {k}}^{+} \\ \hat{a}_{\vec {k}}^{-} &{} \rightarrow \hat{a}_{\vec {k}}^{-}(t) := e^{- i \omega _k t} \hat{a}_{\vec {k}}^{-} \end{array}\right\} \text {Dispersion relation:} ~ \omega _{\vec {k}} = c |\vec {k} |~~,~~~ m=0~. $$

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Arnold-Sommerfeld-CenterLudwig-Maximilians-UniversitaetMunichGermany
  2. 2.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands

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