Spectrum condition ($\mathrm{Sp}(P)\subseteq {\overline{V}}_{+}$)

In a translation-covariant quantum field theory on Minkowski spacetime, the joint spectrum of the energy-momentum generators is required to lie in the closed future light cone. With signature $(+,-,-,-)$ this means $p^0\ge 0$ and $p^2\ge 0$. The spectrum condition implements a positive-energy stability requirement in every inertial frame. It constrains the Fourier support and analyticity of vacuum correlation functions but does not by itself assert spacelike commutation or an initial-value property.