The process $gamma gamma to p bar{p} K^+ K^-$ and its intermediate processes are measured for the first time using a 980~fb$^{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. The production of $p bar{p} K^+ K^-$ and a $Lambda(1520)^0~(bar{Lambda}(1520)^0)$ signal in the $pK^-$~($bar{p} K^+$) invariant mass spectrum are clearly observed. However, no evidence for an exotic baryon near 1540~MeV/$c^2$, denoted as $Theta(1540)^0$~($bar{Theta}~(1540)^0$) or $Theta(1540)^{++}$~($Theta(1540)^{--}$), is seen in the $p K^-$~($bar{p}K^+$) or $pK^+$~($bar{p} K^-$) invariant mass spectra. Cross sections for $gamma gamma to p bar{p} K^+ K^-$, $Lambda(1520)^0 bar{p} K^+ +c.c.$ and the products $sigma(gamma gamma to Theta(1540)^0 bar{p} K^+ +c.c.)BR(Theta(1540)^0 to p K^{-})$ and $sigma(gamma gamma to Theta(1540)^{++} bar{p} K^- +c.c.)BR(Theta(1540)^{++}to p K^{+})$ are measured. We also determine upper limits on the products of the $chi_{c0}$ and $chi_{c2}$ two-photon decay widths and their branching fractions to $p bar{p} K^+ K^-$ at the 90% credibility level.
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