4 we write the Fourier representation of the solution u ( x , t ) of eq. (5) as with ω ( k , χ ) = $^{√}$k 2 − χ $^{2}$, the constraint (10) ensures that only real values of ω ( k , χ ) appear in the representa- tion, and thus u ( x , t ) remains bounded. The problem with the result of Craig and Weinstein is that the constraint (10) only works for a free field, but fails to protect the global evolution of u if the field is interacting, i. e. if the wave equation contains nonlin- ear terms in u . For illustration, let us consider a small quadratic term in the wave equation: u ( x , t ) = ∫ d $^{p}$k dq − $^{1}$χ (2 π ) p + q − 1 ( ˜ u$_{0}$ ( k , χ ) cos[ ω ( k , χ ) t$_{1}$ ] + ˜ u$_{1}$ ( k , χ ) ω ( k , χ ) sin[ ω ( k , χ ) t$_{1}$ ]$^{)}$ e i k · x − iχ · t ′ (11) FIG. 1: Initial data domain for the Cauchy problem for the scalar wave equation in an space-time with two time dimen- sions. The spatial extent of the domain on the co-dimension one hypersurface t$_{1}$ = 0 can be chosen arbitrarily large (here as a sphere of radius a ), while an infinitesimal extent in the second temporal dimension t$_{2}$ is sufficient. The data given in the infinitesimal disk completely fix the data in the entire cone-shaped region indicated in the figure. This result implies that the Cauchy problem on any finite volume on the t$_{1}$ = 0 hypersurface has generally no solution. If a solution exists, however, it is unique. If αu 2 ≪ u , we can consider the nonlinear term as a perturbation, at least for a very short time interval 0 ≤ t$_{1}$ ≤ τ . The time derivative of the solution of (12) at time t$_{1}$ = τ is given by The Fourier transforms of the first and last term only have support in the domain χ 2 ≤ k $^{2}$, if this is true for the initial data at t$_{1}$ = 0. The second term, however, has the Fourier transform These results, however, leave the question unanswered which Cauchy data are admissible. Recently, Craig and Weinstein [30, 31] showed that the existence of a unique causal solution can be ensured by imposing a nonlocal constraint on the initial values. They proved that a unique global solution of the ultra-hyperbolic wave equa- tion (5) exists if the ( p + q − 1)-dimensional Fourier rep- resentation which generally is nonzero for k = 0 even if χ = 0. This consideration implies that the constraint (10) is gener- ally not preserved by the physical time evolution of the field in the presence of interactions. The physical inter- pretation of our result is that two space-like momentum modes on the hyperplane t$_{1}$ = 0 can fuse into a time-like momentum mode, if their space-like momenta cancel or nearly cancel. of the initial data satisfies the constraint The physical interpretation of the constraint (10) is that the initial conditions of the field only have support in the region of space-like momenta on the ( n − 1)-dimensional hyperplane t$_{1}$ = 0. This result is easy to understand. If The consideration at the end of the previous subsec- tion shows that the initial value problem for interacting fields in space-times with more than a single time dimen- sion can be well-posed only if time-like momentum modes B. Fields with quenched time-like modes ˜ u$_{0}$ ( k , χ ) = ˜ u$_{1}$ ( k , χ ) = 0 for χ 2 > k $^{2}$. (10) u$_{0}$ ( x , t $^{′}$) = u ( x , 0 , t $^{′}$) u$_{1}$ ( x , t $^{′}$) = ∂u ∂t$_{1}$ ( x , 0 , t $^{′}$) (9) u$_{0}$ ( x , t $^{′}$) = ∫ d $^{p}$k dq − $^{1}$χ (2 π ) p + q − 1 ˜ u$_{0}$ ( k , χ ) e i k · x − iχ · t $^{′}$, u$_{1}$ ( x , t $^{′}$) = ∫ d $^{p}$k dq − $^{1}$χ (2 π ) p + q − 1 ˜ u$_{1}$ ( k , χ ) e i k · x − iχ · t ′ (8) τ α ∫ d $^{p}$k ′ d q − $^{1}$χ ′ (2 π ) p + q − 1 ˜ u$_{0}$ ( k − k $^{′}$, χ − χ $^{′}$)˜ u$_{0}$ ( k $^{′}$, χ$^{′}$) , (14) ∂u ∂t$_{1}$ ( x , τ, t $^{′}$) ≈ ∂u ∂t$_{1}$ ( x , 0 , t $^{′}$) + ∫ τ 0 dt$_{1}$ ∂ $^{2}$u ∂t 2 1 ( x , 0 , t $^{′}$) = u$_{1}$ ( x , t $^{′}$) + ∫ τ 0 dt$_{1}$ ( αu 2 0 + ∂ $^{2}$u$_{0}$ ∂ x 2 − ∂ $^{2}$u$_{0}$ ∂ t ′ 2 ) = u$_{1}$ ( x , t $^{′}$) + τ α u$_{0}$ ( x , t $^{′}$) 2 − τ ∫ d $^{p}$k dq − $^{1}$χ (2 π ) p + q − 1 ω ( k , χ ) $^{2}$˜ u$_{0}$ ( k , χ ) e i k · x − iχ · t $^{′}$. (13) p ∑ i =1 ∂ $^{2}$u ∂x 2 i − q ∑ k =1 ∂ $^{2}$u ∂t 2 k + αu 2 = 0 . (12)