# mp.mathematical physics – Propagation of Klein-Gordon solutions in extra dimensions

In his paper “Von Neumann Algebras of Local Observables for Free Scalar Field” Araki used the solutions of the equation

$$frac{partial ^{2}h}{partial x^2}-frac{partial ^{2}h}{partial t^2}-m^2h=0$$

where he put the suport of the initial data $$h(t, 0)$$ and $$frac{partial h}{partial x}(t, 0)$$ at $$x=0$$.

Then He used this solution (noted as $$h_{0}$$) to pose another Klein-Gordon Problem in $$mathbb{R}^{4}$$:
$$(Box+m^2)G=0$$
with initial conditions

$$G(0,vec{x})=h_{0}(0,x_{1})h_{1}(vec{x})$$ and $$frac{partial G}{partial x^0}(0,vec{x})=frac{partial h_{0}}{partial t}(0,x_{1})h_{1}(vec{x})$$

where $$h_{1}$$ is a function wich is equal to 1 in a ball (of radius $$t_{2}$$) which contains the suports of the initial conditions for $$h_{0}$$ and then goes to zero smoothly.

Araki asserts that $$G(x^0,x^1,0,0)=h_{0}(x^0,x^1)$$ for $$|x^0|+|x^1|, where $$t_2$$ satisfies $$h_{1}(vec{x})=1$$ for $$|vec{x} |.

My question is why the extra dimensions doesn’t affect the equality, at least for near points. I know the initial contitions for $$G$$ has translation simmetry in $$x^3$$ and $$x^4$$ and I think this would be important.

Some parts of Araki’s paper

And the problematic part for my

Many thanks in advance.