Hadamard renormalized scalar field theory on anti–de Sitter spacetime

We consider a real massive free quantum scalar field with arbitrary curvature coupling on $n$-dimensional anti–de Sitter spacetime. We use Hadamard renormalization to find the vacuum expectation values of the quadratic field fluctuations and the stress-energy tensor, presenting explicit results for $n=2$ to $n=11$ inclusive.

Figure 1

$⟨{\mathrm{\Phi }}^2{⟩}_{\text{ren}}$ for varying $n$ with $a=1$ and $M=\frac{e^{\mathit{\gamma }}}{\sqrt{2}}$.

Figure 2

$⟨{\mathrm{\Phi }}^2{⟩}_{\text{ren}}$ for (a) $n=2$, (b) $n=4$, (c) $n=6$, (d) $n=8$ and (e) $n=10$. In each case we have set $a=1$ and $M=\frac{e^{\mathit{\gamma }}}{\sqrt{2}}10^{j-6}$.

Figure 3

$\frac{1}{n}⟨{T_{\mu }}^{\mu }{⟩}_{\text{ren}}$ with $\xi =0$, $a=1$ and $M=\frac{e^{\mathit{\gamma }}}{\sqrt{2}}$.

Figure 4

$\frac{1}{4}⟨{T_{\mu }}^{\mu }{⟩}_{\text{ren}}$ for $n=4$, $\xi =\frac{1}{6}$, $a=1$, $M=\frac{e^{\mathit{\gamma }}}{\sqrt{2}}10^{j-6}$.

Figure 5

$\frac{1}{n}⟨{T_{\mu }}^{\mu }{⟩}_{\text{ren}}$ as a function of both $\eta$ and $\xi$, with $a=1$ and $M=\frac{e^{\mathit{\gamma }}}{\sqrt{2}}$, for (a) $n=2$, (b) $n=3$, (c) $n=4$, (d) $n=5$, (e) $n=6$, (f) $n=7$, (g) $n=8$, (h) $n=9$, (i) $n=10$ and (j) $n=11$.