# Malament theorem in curved spacetime?

Malament's theorem roughly assert that given a very general theory of a point particle, characterized by some operator $P_D$ such that for a region of space $D$ at a given time $t$, $P_D | \Psi \rangle$ corresponds to the certainty that the particle is localized within $D$, which obeys the following rules

• Translation covariance: For a translation $a$, we have a unitary operator $U(a)$ such that $U(a) P_D U(-a) = P_{D+a}$.

• Localizability: For two disjoint subsets of the same spacelike hyperplane $D_1$, $D_2$, $P_{D_1} P_{D_2} = P_{D_2} P_{D_1} = 0$.

• Lower bound on energy : For a timelike vector $\xi$, $U(\lambda \xi)$ has a spectrum bounded from below.

• Microcausality: For any two spatial regions in two spacelike hypersurfaces $D_1$, $D_2$, we have $P_{D_1} P_{D_2} = P_{D_2} P_{D_1}$.

The Localizability just says that a particle can't be at two places at once (fairly important for a point particle), and microcausality is just the usual relativistic causality.

With this, we have that any quantum theory obeying those rules has $P_D = 0$ for all $D$ : the only theory is the theory of $0$ particles.

Does this theorem generalizes well to curved spacetime? Quantum theories on curved spacetime lack unitary operators, both in space and time, which seems like the biggest issue (localizability can probably be adapted well enough using achronal slices and microcausality the lack of causal curves from $D_1$ to $D_2$), as we know that asking for unitary transformations for spacetimes without Killing vectors is not reasonable.