content_image/382480.jpg content_image/382481.jpg content_image/382493.jpg
where 

<img src='content_image/382480.jpg'>

is the metric factor determining the location of the black hole horizon along the radial AdS$_{6}$ direction. The AdS$_{6}$-black hole geometry is illustrated in Fig. 2. The AdS$_{6}$-Schwarzschild metric (18) is an solution of the 6- dimensional Einstein equations with a nonvanishing cos- mological constant. Here θ is taken to be a periodic co- ordinate with period 

<img src='content_image/382481.jpg'>

corresponding to the inverse Hawking temperature of the black hole. Assuming holographic duality, the AdS$_{6}$ met- ric (18) has been used to estimate the low-lying glueball masses, string constant, and other infrared properties of the large- N$_{c}$ Yang-Mills theory with qualitatively good results [43–45]. 

The holographic representation also provides an intu- itive explanation for the rapid quenching of θ -time modes of the strongly coupled boundary gauge theory. Such ex- citations are represented as string-like excitations of the dual gravity theory, which fall into the black hole encod- ing the thermal bath and disappear behind the horizon in a finite time [36, 37]. 

## D. Non-supersymmetric gauge theories 

It may not be necessary to restrict the theory to ei- ther strong coupling or supersymmetry. In fact, in an interacting thermal field theory all time-dependent ex- citations not protected by conserved quantum numbers ultimately decay to the equilibrium ensemble. For ex- ample, in a thermal Yang-Mills theory without matter content all time-dependent gauge field configurations are quenched and only static chromomagnetic fields survive at large times [46]. The time scale for the decay is given by the inverse temperature T$_{5}$ multiplied by a power (and possibly logarithms) of the coupling constant g of the gauge theory. 

This argument suggests that the thermal 5- dimensional Yang-Mills theory in (3 + 2) dimensions reduces to the vacuum Yang-Mills theory in (3 + 1) dimensions on length and physical time scales much larger than the inverse temperature. This statement is known to be true when the physical time dimension t is analytically continued from Minkowski space R 3 , 1 to Euclidean space R $^{4}$. In this case, all static correlation functions in R 4 , 1 are dominated at long distances by the n = 0 Matsubara frequency χ$_{n}$ = 2 πinT$_{5}$ of the thermal unphysical time dimension θ . The effective coupling constant of the 4-dimensional gauge theory is g$_{4}$ = g$_{5}$T$_{5}$ , where g$_{5}$ is the dimensionful coupling of the 5-dimensional gauge theory, which in non-renormalizable and thus requires a physical short-distance cut-off. In 

order to obtain a 4-dimensional coupling of order unity, T$_{5}$ cannot differ very much from this cut-off. 

The physical gauge theory in Minkowski space thus emerges as the long-distance limit of a thermal 5- dimensional gauge theory in a space-time with two time dimensions. Formally, this 5-dimensional theory would be constructed by doubly analytically continuing the 5- dimensional euclidean gauge theory defined on R 4 × S $^{1}$, where the compactified dimension corresponds to the un- physical time dimension, which is hidden from observa- tion by thermal physics. 

## IV. QUANTIZATION BY A SECOND TIME DIMENSION 

Building on the results of the Sect. III.D, we can now argue that a classical dynamics in the ( d +2)-dimensional space-time induces an effective quantum dynamics in the ( d + 1)-dimensional space-time. This concept is not en- tirely new. It is implicit in Beck’s idea of chaotic quanti- zation [47] and explicit in the chaotic quantization model of Bir´ o et al. [48], who argued that quantized non- Abelian gauge theory can be understood as the infrared limit of the corresponding classical gauge theory in a higher space-time dimension. In some respects, our argu- ment can also be viewed as a realization of ’tHooft’s con- jecture [49] that quantum mechanics emerges as a limit of classical dynamics under appropriate conditions of dy- namical information loss. 

The idea that a classical field theory can be equiva- lent to a lower-dimensional quantum field theory is also inherent in the concept of gravitational holography [50], although it is usually explored only in the strong cou- pling limit for the most widely studied models [51]. In these models the additional space-time dimension is usu- ally taken to be spatial and endowed with a nontrivial metric. Here, we consider an additional temporal dimen- sion, not a spatial one. 

In order to explore the quantum mechanics inducing property of an second time dimension it is useful to con- sider a time dimension attached to the euclidean analogue of Minkowski space-time, i. e. by considering the mani- fold defined by the coordinates ( x , t = − iτ, θ ) with the metric 

<img src='content_image/382493.jpg'>

As before, θ denotes the unphysical time coordinate. We denote the conjugate momentum to θ (the θ -“energy”) as p$_{5}$ = H , and the momenta conjugate to ( x , τ ) as ( p , p$_{4}$ ), where p$_{4}$ is related to the Minkowski space energy by E = − ip$_{4}$ . 

Consider a classical dynamical system whose evolu- tion in the unphysical time variable θ is ergodic. We 

## A. Micro-canonical quantization 

