# Duality between Euclidean time and finite temperature, QFT and quantum gravity, and AdS/CFT

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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

∼ High-temperature quantum statistical mechanics in (d+1)-dimensional space

∼ High-temperature 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 gauge-gravity 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 strong-weak 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($N_c$) $\mathcal{N}=4$ super-Yang-Mills 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 thermal-statistical mechanics)

and

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