Making sense of $1+1$ massless bosonic free field as a "distribution" rather than tempered The question has been motivated by the fact that the $1+1$ massless bosonic free field suffers the infrared problem as a "tempered distribution".
The reason is essentially that $\int_{\mathbb{R}} \frac{dp}{\lvert p \rvert}$ is logarithmically divergent.
Since this is a infrared problem, I am curious whether the issue will be resolved by introducing a infrared cutoff, which is mathematically interpreted as compact supports in the spacetime variable.
In other words, the $1+1$ massless bosonic free field can be defined as a "just distribution" instead of being tempered?
Or more concretely, does the following integral converges for an arbitrary compactly supported smooth function $f(x,y)$ on $\mathbb{R}^2$?:
\begin{equation}
\int_{\mathbb{R}}\frac{dp}{\lvert p \rvert} \int_{\mathbb{R}^2} dxdy 
 f(x,y)e^{i(-\lvert p \rvert x+ py)}
\end{equation}
It seems nontrivial to evaluate the above integral for me. Could anyone please help?
 A: The massless GFF is well-defined as a random tempered distribution modulo constants, i.e. an element of the dual of the space of Schwartz test functions with vanishing integral. If you want it to be defined as a "normal" random tempered distribution, then you have to arbitrarily fix the zero mode somehow. For example, you could enforce that testing against the indicator function of the centred unit ball gives zero.
A: As Martin said, the issue is that one needs to get rid of the possibility of adding a constant to the field. See this older MO answer
Free Boson Correlator $ \langle X(z)X(w) \rangle =- \ln |z - w| $
for a brief discussion of the realization of the massless free field as a probability measure on the dual of the space of test functions with zero integral.
For a much more thorough discussion (starting at page 8), see the review https://arxiv.org/abs/1407.5605 by Duplentier et al.
Finally, note that the same problem occurs also in 1 dimension, with Brownian motion. This is dealt with by imposing that Brownian motion is set equal to zero at time zero.
