content_image/442121.jpg content_image/442122.jpg content_image/442123.jpg
SU(2) Yang-Mills theory in (3+1) dimensions [71]. 

We have argued that additional time dimensions are not ruled out, if the dynamics governing the evolution of elementary fields in these dimensions is thermal or quasi- thermal. There are several distinct scenarios for this: 

Any scenario of a thermal, additional time dimension raises a number of questions. What is the origin of the high temperature T$_{5}$ associated with the additional time dimension and why is T$_{5}$ so high? While the thermal cosmic background radiation is naturally understood as remnant of the thermalized energy density contained in the excited vacuum state that drove cosmic inflation, the temperature associated with an additional time dimen- sion must be of a more primordial nature. It also needs to remain constant as a function of physical time; oth- erwise fundamental laws of nature would be observed to change over cosmic time scales. 

A partial answer to this question emerges from the ob- servation that, in the presence of two time dimensions, temperature is a two-component vector. If the observable universe is homogeneous, the physical time direction will simply be the time direction that is orthogonal to the pri- mordial temperature vector. The ekpyrotic cosmological model [72], which posits that the Big Bang was initiated by a violent process, e. g. the collision between two en- ergetic branes, could then be re-interpreted to provide an explanation for the high temperature associated with 

## V. SUMMARY AND OUTLOOK 

1. If the universe is fundamentally quantum mechan- ical, an additional, thermalized time dimension is equivalent to a compactified spatial dimension on time scales long enough for all non-equilibrium pro- cesses to be dampened out. Because fermions and bosons obey different periodicity conditions, this scenario implies the breaking of any fundamental supersymmetry and thus could explain why super- symmetry is not realized at low energies. At dis- tances and times much larger than $_{ℏ}$/T$_{5}$ , the pres- ence of the additional time dimension is essentially invisible, except for the supersymmetry breaking and its influence on various low-energy constants. 

2. If the universe is fundamentally classical, an addi- tional, highly excited and quasi-thermal time di- mension will generate quantum dynamics in the four-dimensional space-time. Quasi-thermal means that the dynamics is micro-canonical globally, and the local ergodic ensemble can be approximated by a thermal ensemble. Planck’s constant ℏ is given by the product T$_{5}$a , where a is the ultraviolet cut- off length of the classical field theory. In this sce- nario, the additional time dimension is reflected in the quantum mechanical nature of Minkowski space physics. 

the additional time dimension. It would be interesting to construct a model that explains why the temperature of the universe associated with the physical time dimension decreased over time, but the temperature associated with the second time dimension remained high. 

We hasten to emphasize that the physical laws will differ severely from those customarily assumed in those domains of space-time, e. g. during or before the cos- mic Big Bang, where the full dynamics in one of the two time dimensions is not effectively screened by thermal or quasi-thermal dynamics. For example, the law of causal- ity will not apply in such regions, at least not the usual notions of causality. How such physical laws will break down in domains where two time dimensions are dynam- ically active may depend quite strongly on the details of the cosmological model. 

Another question is whether and how the presence of an additional temporal, thermal dimension could be de- tected experimentally. With respect to θ -time averages, the second temporal, but thermally excited dimension behaves just like a compactified spatial dimension with compactification length L = 1 /T$_{5}$ . Experimental probes of average properties thus will require energies E of or- der of the temperature, or precision of the order of some power of E/T$_{5}$ . However, it may be possible to probe for the existence of real θ -time dynamical phenomena, which may have a much longer duration if the dynamics of rele- vant fields is weakly coupled. For most field theories, the longest lived excitations are of hydrodynamic nature, be- cause their decay is protected by local conservation laws. It may thus be possible to probe for violations of causal- ity or unitarity induced by such dynamical processes in θ -time. It would be interesting to derive a general theory of such microscopic causality violations. 

Acknowledgments: This work was supported in part by DOE grant DE-FG02-05ER41367. We thank T. S. Bir´ o and I. Ya. Aref’eva for valuable comments on an earlier version of this manuscript. 

## Appendix: The Five-Dimensional Propagator 

Let us consider a massless scalar field with the La- grangian 

<img src='content_image/442121.jpg'>

and its Feynman propagator 

<img src='content_image/442122.jpg'>

Here k denotes the conjugate momenta to the spacetime coordinates, and we can break k up into k = ( k$_{x}$, k$_{y}$, k$_{z}$ ), ω , the momentum conjugate to t , and χ , the momentum conjugate to the second time direction θ . After a Fourier transformation, the five-dimensional wave equation with a source term 

<img src='content_image/442123.jpg'>

