Proposition: Projected constructions and their defects

Suppose for simplicity there is one normalized transverse bound eigenfunction $b(x_{\perp })$ at $-{\kappa }^2$.

1. Removing the entire transverse bound channel gives

   $$G_{\mathrm{cont}}=G_{\mathrm{PV}}-b(x_{\perp })\bar{b}({x^{\prime }}_{\perp })Q_{\kappa }(\Delta ),$$

   where

   $$Q_{\kappa }(\Delta )=\mathrm{PV}\int \frac{{\mathrm{d}}^{2}k}{(2\pi {)}^2}\frac{e^{\mathrm{i}k\cdot \Delta }}{k^2-{\kappa }^2}=-\frac{1}{4}{Y}_0(\kappa |\Delta |),\Delta =(\Delta \tau ,\Delta z).$$

   Since $Q_{\kappa }\sim (2\pi {)}^{-1}\log (1/|\Delta |)$, the subtracted Wick-square limit is $-\infty$ where $b\ne 0$. The kernel also has extra singularities at $\Delta \tau =\Delta z=0$ for distinct transverse points. It is not locally Hadamard for the original field.

2. Restricting only to the full nonnegative spectral subspace keeps the bound-channel modes with $|p_z|>\kappa$. Its Euclidean kernel is

   $$G_{+}=G_{\mathrm{PV}}+b{\bar{b}}^{\prime }D(\Delta \tau ,\Delta z),$$

   where

   $$D(0,z)=0,D(\tau ,0)=\frac{1}{4}{\mathbf{H}}_0(\kappa |\tau |).$$

   The correction tends to zero at the diagonal, so its Wick-square value agrees with the principal-value reference. It is not a smooth regular remainder:

   $$D(\tau ,0)=\frac{\kappa |\tau |}{2\pi }+O(|\tau {|}^3).$$

   Thus the usual covariant stress point-splitting prescription does not apply. The projected covariance is not boost invariant.

Both constructions have projected equal-time CCR and are nonlocal modifications of the original field algebra.

Proof. Spectrally,

Fourier transformation in $(\tau ,z)$ proves (a). The $p^{-2}$ bound-channel tail is precisely the logarithm removed from the full reference. The continuum resolvent difference therefore has the compensating algebraic tail; it must not be described as exponentially decaying.

For (b), the retained bound contribution is

At $\tau =0$ it is $-Y_0(\kappa |z|)/4$. At $z=0$, substituting $u=\sqrt{{p_z}^2-{\kappa }^2}$ gives

Subtract $Q_{\kappa }$. The omitted compact momentum interval gives a bounded Fourier contribution tending to zero with $\tau$, so the coincidence agreement is not restricted to the two axes. The two axis formulas prove failure of Euclidean rotational, hence Lorentzian boost, invariance. Projection replaces the spatial identity in the canonical commutator by the corresponding spectral projection.