One requirement of the Hilbert space is that it is normed. Furthermore, in the case of special relativity, the norm has the following problematic requirements:
$$ ||u||\geq 0 \\ ||u||= 0 \implies u=0 $$
This requirements are respected in 3d space with metric $ds^2=dx^2+dy^2+dz^2$.
However, in special relativity, the interval is given by
$$ ds^2=dx^2+dy^2+dz^2-dt^2 $$
In the case of a photon traveling at the speed of light, the interval is always zero, yet u could be (3,4,0,5) such that $3^2+4^2+0^2-5^2=0$. In this case $u\neq 0$, but $||u||=0$. In some cases, the norm is negative such as $u=(0,0,0,5)$.
Consequently, the space of functions whose inner product is the interval of special relativity does not form a Hilbert space.
How does one reconcile Hilbert space with special relativity? Or are Hilbert spaces only relevant for non-relativistic quantum mechanics?