content_image/436591.jpg content_image/436604.jpg content_image/436603.jpg content_image/436602.jpg content_image/436590.jpg content_image/436607.jpg content_image/436606.jpg content_image/436605.jpg
<img src='content_image/436591.jpg'>

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 

<img src='content_image/436604.jpg'>

of the initial data 

<img src='content_image/436603.jpg'>

satisfies the constraint 

<img src='content_image/436602.jpg'>

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 

we write the Fourier representation of the solution u ( x , t ) of eq. (5) as 

<img src='content_image/436590.jpg'>

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: 

<img src='content_image/436607.jpg'>

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 

<img src='content_image/436606.jpg'>

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 

<img src='content_image/436605.jpg'>

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. 

## B. Fields with quenched time-like modes 

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 

