Half-level Neveu–Schwarz vector state ($|A;p⟩=A\cdot b_{-1/2}|0;p⟩$)

The lowest fermionic excitation has level $1/2$. Its old-covariant physical conditions are ${\alpha }^{\prime }{p}^2+1/2-a=0$ and $p\cdot A=0$. For nonzero real momentum, timelike $p$ gives a positive polarization space, null $p$ gives a nonnegative space with a longitudinal null direction, and spacelike $p$ permits a negative-norm time polarization. The limiting $a=1/2$ state is a massless gauge vector after quotienting null polarizations. This level-specific test does not prove positivity of every higher NS level.

The usual half-level NS claim that all physical vector polarizations have nonnegative norm for $a\le 1/2$ assumes nonzero particle momentum. At zero momentum and limiting intercept, all positive-mode constraints and transversality allow a timelike polarization with negative norm, and $A\sim A+\kappa p$ removes nothing. Excluding this exceptional momentum restores the ordinary null-vector argument; retaining it makes the unrestricted elementary criterion strict $a<1/2$.