differential equations is well posed. Let us, for simplicity, consider the scalar n = p + q dimensional wave equation χ ( n $^{)}$, the dispersion relation in real physical time: 3 only yields real Minkowski space energies ω . The theory thus does not contain tachyonic modes. Finally, ghost- like modes are eliminated if the modes of vector fields polarized in the θ -time direction are screened. Differential equations of this type with q > 1 are called ultra-hyperbolic . The Cauchy problem for linear ultra- hyperbolic differential equations of the type (5) has been studied by various authors [23, 24] and explicit solutions were given by Owens [25]. The minimal conformal rep- resentations of solutions of the wave equation were con- structed by Kobayashi and Ørsted [26] for even values of n . The possibility of the cosmological emergence of ad- ditional time dimensions was considered by Sakharov [16]. Aref’eva and Volovich [17] showed that the prob- lem of ghost modes in Kaluza-Klein theories with addi- tional time dimensions can be avoided, if the additional time dimensions reside in compact manifolds that have no Killing vector field at all. Examples of such spaces are Kummer’s K 3 manifold or, more generally, compact hyperbolic manifolds. Aref’eva and Volovich showed further that Kaluza- Klein type theories with additional time dimensions would generally compactify on hyperbolic manifolds. Geodesics in such spaces have stochastic behavior; as a matter of fact, geodesic motion on Riemannian manifolds with constant negative curvature serves as a tractable mathematical model for dynamical systems with strongly chaotic properties [18]. As we argued above, this prop- erty may also ameliorate or even eliminate the problem of macroscopic causality violations. If there is more than one temporal dimension, the ini- tial data must be given on a ( n − 1)-dimensional hypersur- face of mixed metric signature, i. e. one which contains temporal directions. Without loss of generality, we as- sume that the initial data are given on the hypersurface t$_{1}$ = 0 and denote the vector of the remaining temporal coordinates by t ′ = ( t$_{2}$, . . . , t$_{q}$ ). Building on a general- ized mean value theorem by Asgeirsson [27], John showed that, if the initial value problem has a solution, this so- lution is unique [23]. Courant [28] pointed out that Asgeirsson’s theorem also implies that the initial value problem is generally not well-posed, because a global solution of the wave equation (5) does not exist for arbitrary initial conditions. This can be seen as follows [28] (see also Tegmark [29]). First, Asgeirsson’s theorem implies that knowledge of the ini- tial value data in an infinitesimal region in the temporal coordinates, Bars and collaborators have extensively considered theories in spaces with two time dimensions, which pos- sess an additional local symplectic gauge symmetry con- necting position and momentum variables [19–21]. The imposition of a gauge constraint then allows one to reduce the dynamics to a single time dimension. This avoids all undesirable aspects of theories with two time dimensions, but at the cost that the quantum field cannot, even in principle, dynamically explore the second time dimen- sion. Nevertheless, the possible progeny from a higher dimensional space-time may help resolve certain unex- plained properties of the Standard Model [22]. on the hypersurface. In other words, while data on the whole spatial domain contained somewhere in the past light-cone of a space-time point is needed to uniquely de- termine the value of the function u ( x , t ), the data need to be known only on an infinitesimal strip in the temporal directions t$_{2}$, . . . , t$_{q}$ (see Fig. 1). Because the infinitesimal disk (6) is sufficient to determine the Cauchy data for a solution of eq. (5) over the whole cone (7), it is not possi- ble to prescribe arbitrary data in the cone, but only those which are compatible with the data in the infinitesimal disk. Since a solution of the ultra-hyperbolic equation is unique if it exists, this implies that no solution exists for arbitrary Cauchy data on the cone-like region. (The same argument applies to the disk-like region itself, be- cause it is always possible to find a narrower disk which already fixes the Cauchy data on the larger disk.) The generalization of Minkowski space with p spatial and q temporal dimensions is denoted as R $^{p,q}$, and we specify a vector in R p,q as ( x$_{1}$, . . . , x$_{p}$ ; t$_{1}$, . . . , t$_{q}$ ). We will refer to t$_{1}$ as the “physical” time coordinate. Later we will be interested in the special case p = 3 , q = 2. In this case, we denote the five-dimensional vector as ( x ; t, θ ), where t represents the physical time coordinate. The prerequisite for a causal dynamics in any space- time manifold is that the Cauchy problem for the relevant A. The Cauchy problem III. ULTRA-HYPERBOLIC WAVE EQUATIONS is sufficient to fix the data in the finite cone-shaped region t$_{1}$ = 0; | t ′ − t ′ $_{0}$|  ≤ ε ; | x − x$_{0}$ | ≤ a (6) t$_{1}$ = 0; | t ′ − t ′ $_{0}$| + | x − x$_{0}$ | ≤ a (7) p ∑ i =1 ∂ $^{2}$u ∂x 2 i − q ∑ k =1 ∂ $^{2}$u ∂t 2 k = 0 . (5) ω ( k , χ ) = k 2 − χ 2 n + m 2 = k 2 + (2 πnT$_{5}$ ) 2 + m 2 (4) C. Other studies of multiple-time theories