Quantum entanglement and thermodynamics of bosonic fields in noncommutative curved spacetime

Abstract

Within the quantum field theory approach and using the technique of Bogoliubov transformations, the von Neumann boson-antiboson pair creation quantum entanglement entropy is studied in the context of the noncommutative Bianchi I universe. The investigations have shown that the behavior of entanglement entropy is strongly influenced by the choice of the noncommutativity $\theta$ parameter, $k_{\perp }$-frequency modes and the structure of the curved spacetime. Moreover, the relationship between thermodynamics, Bose–Einstein condensation and quantum entanglement is also discussed.

A N.C. mathematical formalism

We propose the following action:

$$\begin{aligned} S=\frac{1}{2{k}^{2}}\int d^{4}x\left( {\mathcal {L}}_{G}+ {\mathcal {L}}_{SC}\right) \end{aligned}$$

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where ${\mathcal {L}}_{G}$ and ${\mathcal {L}}_{SC}$ stand for the pure gravity and matter scalar densities corresponding to the charged scalar particle.

We define

$$\begin{aligned} {\mathcal {L}}_{G}=\hat{e}*\hat{R} \end{aligned}$$

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and

$$\begin{aligned} {\mathcal {L}}_{SC}=\hat{e}*\left( \hat{g}^{\mu \nu }*\left( \hat{D}_{\mu }\hat{\varphi }\right) ^{\dagger }*\hat{D}_{\nu }\hat{\varphi }\right) \end{aligned}$$

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with:

$$\begin{aligned} \hat{R}=\hat{e}^{\mu }_{*a}*\hat{e}^{\nu }_{*b}*\hat{R}^{ab}_{\mu \nu } \end{aligned}$$

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Applying the principle of least action, it is easy to show that the modified field equations are given bssy:

$$\begin{aligned}{} & {} \frac{\partial {\mathcal {L}}}{\partial \hat{\varphi }}-\partial _{\mu }\frac{\partial {\mathcal {L}}}{\partial \left( \partial _{\mu }\hat{\varphi }\right) }+\partial _{\mu }\partial _{\nu }\frac{\partial {\mathcal {L}}}{\partial \left( \partial _{\mu }\partial _{\nu }\hat{\varphi }\right) }-\partial _{\mu }\partial _{\nu }\partial _{\sigma }\frac{\partial {\mathcal {L}}}{\partial \left( \partial _{\mu }\partial _{\nu }\partial _{\sigma }\hat{\varphi }\right) }\nonumber \\{} & {} \quad +O\left( \theta ^{3}\right) =0 \end{aligned}$$

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