The thoughts below have occurred to me, several years ago (since 200x), again and again, since I learn quantum field theory(QFT) and statistical mechanics, and later AdS/CFT. It is about the duality and the
correspondence between Euclidean time and finite temperature, QFT and quantum gravity, and AdS/CFT.
So let me summarize the duality between them below as (1)~(5). This can be read from, for example A Zee's book on QFT and for AdS/CFT review, read from any arXiv review of it.
(1). Euclidean QFT in (d+1) dimensional spacetime
∼ Classical statistical mechanics in (d+1) dimensional space
∼ Classical statistical mechanics in (d+2) dimensional spacetime (where time does not play much the role)

(2). Euclidean QFT in (d + 1)dimensional spacetime, $0 \leq \tau \leq \beta$
∼ Quantum statistical mechanics in (d)dimensional space
∼ Quantum statistical mechanics in (d+1)dimensional spacetime

(3). Euclidean QFT in (d+1)dimensional spacetime
∼ Hightemperature quantum statistical mechanics in (d+1)dimensional space
∼ Hightemperature quantum statistical mechanics in (d+2)dimensional spacetime

The above (1)~(3) relations can be more precise from considering the partition function of both sides of the duality:
$$
Z=\text{tr}[e^{\beta H}]=\int_{\text{periodic boundary}} D\phi e^{\int^\beta_0 d\tau_E \int d^\text{d} d x L(\phi)}
$$
here $\tau_E$ is the Euclidean time with periodic boundary condition.

In AdS/CFT correspondence, or gaugegravity duality, we learn that
(4). QFT in (d+1)dimensional spacetime
∼ quantum graivty in (d+2)dimensional spacetime
where the bulk radius plays the rule of the renormalization group (RG) energy scale. Such a correspondence has a strongweak coupling duality, e.g.
(5). QFT in (d+1)dimensional spacetime at strong coupling
∼ classical graivty in (d+2)dimensional spacetime at weak coupling
The duality can be made more precise between:
SU($N_c$) $\mathcal{N}=4$ superYangMills and AdS$_5 \times S_5$
$\frac{R^2}{\alpha'} \sim \sqrt{g_s N_c} \sim \sqrt{\lambda},\;\;\; g_s \sim g_{YM}^2 \sim \frac{\lambda}{N_c},\;\;\; \frac{R^4}{\ell_p^4} \sim \frac{R^4}{\sqrt{G}} \sim N_c$
the large $N_c$, number of colors indicates the small gravitational constant $G$.
My Question:
Given the relation between the AdS bulk radius as a energy $E$ scale, which is basically related to a time $t$ scale and the temperature via the dimensional analysis
$$[E]\sim 1/[t] \sim [T]$$
and given the hinting relations between gravity and thermodynamics via S Hawking and T Jacobstein works, and perhaps E Verlind, etc.
and Given the suggestive relations between Quantum in (d+1) dim and classical thermo/gravity
in (d+2) dim of (1)~(5).
How much have we known and have been explored in the literatures about the relations between (1)~(3) and (4)~(5)? e.g. the relations between:
the duality between Euclidean time and finite temperature
(e.g. QFT and thermalstatistical mechanics)
and
AdS/CFT duality?
This post imported from StackExchange Physics at 20140625 20:52 (UCT), posted by SEuser Idear