I don't have Wald handy, so here's a guess. Localizing a particle to a point in 1-particle quantum theory results in a singular wavefunction (a delta function), and, similarly, Localizing a quantum field to a point in spacetime results in a singular field.
In non-relativistic quantum theory, consider a particle on a 1-dimensional ring of circumference L. This is like a particle in a 1-dimensional box, where the box has been bent around until the ends join. Because x and x + nL represent the same location on the ring for any integer n, the wavefunction satisfies the periodic boundary condition \psi \left( x + L \right) = \psi \left( x \right) for all x.
A particle localized to the point x = a should have a wavefunction that looks something like \psi \left( x \right) = \delta \left( x - a \right). However, this doesn't satisfy the periodic boundary condition. A delta function is needed at a + nL for each n, so
<br />
\psi \left( x \right) = \sum^{\infty}_{n = -\infty} \delta \left( x - \left(a + nL \right) \right).<br />
For simplicity take a = 0, and try and write \psi as the Fourier expansion
<br />
\psi \left( x \right) = \sum^{\infty}_{n = -\infty} c_{n} e^{i2\pi nx/L},<br />
and solve formally for the Fourier coefficients c_{n}.
<br />
\begin{equation*}<br />
\begin{split}<br />
c_{n} &= \frac{1}{2\pi} \int_{-L/2}^{L/2} \psi \left( x \right) e^{-i2\pi nx/L} dx\\<br />
&= \frac{1}{2\pi} \int_{-L/2}^{L/2} \delta \left( x \right) e^{-i2\pi nx/L} dx\\<br />
&= \frac{1}{2\pi},<br />
\end{split}<br />
\end{equation*}<br />
since only one of the delta functions is within the range of integration. Thus, the wavefunction for a particle localized at x = 0 is
<br />
\psi \left( x \right) = \sum^{\infty}_{n = -\infty} \delta \left( x - nL \right) = \frac{1}{2\pi} \sum^{\infty}_{n = -\infty} e^{i2\pi nx/L}.<br />
The Fourier expansion for \psi doesn't converge pointwise because \psi is a (singular) distribution, not a function. The expansion does converge weakly, i.e., in the sense of tempered distributions.
In quantum field theory, at point x in spacetime, the field is represented by an "operator" \Psi \left( x \right) that is really an operator-valued distribution with, for states f and g, <f|\Psi \left( x \right)|g> a distribution.
For my money, Chapter 9, Quantum Field Theory in Curved Spacetime, from Sean Carroll's new GR book is the best introduction in print to Hawking radiation. Carroll doesn't concern himself with mathematical niceties like operator-valued distributions but does give a nice physics-style intro to things like Bogolubov transformations. If you do want to see the math done carefully, stick with Wald or Wald.
Regards,
George
) talks about operators, so I answered in terms Fourier expansions of quantum field operators, which don't even converge in norm.
