 Solution

For translations represented by $U(a)=e^{i{a}_{\mu }{P}^{\mu }}$, the energy-momentum generators have a joint spectral measure. The spectrum condition isusing signature $(+,-,-,-)$. It excludes negative-energy and spacelike momentum states in the vacuum representation, apart from the allowed zero momentum of the vacuum. It is stronger than bounding the Hamiltonian below in just one chosen frame: energy is nonnegative in every future-directed inertial frame. Thermal infinite-system representations need not satisfy the same vacuum spectrum condition, so the representation matters.

The condition supports stability and controls analyticity. For a translation-invariant vacuum, insertion of the spectral measure into a two-point function gives a formReplacing $x$ by $x-iy$, with $y$ in the future timelike cone, inserts the decaying factor $e^{-p\cdot y}$ and gives analytic continuation into the appropriate tube, subject to the usual distributional bounds. This is a consequence of spectral support, not an assertion of compact spatial support of $W$.

Positive energy is not a commutator condition or a guarantee of finite-speed particle-position propagation. It supplies none of the region-algebra inclusions used in primitive causality by itself. Its tension with sharp localization is precisely why the particle-localization discussion needs separate hypotheses. Spectrum, spacelike commutation and operational no-signalling constrain different structures and cannot simply substitute for one another.