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?



Making sense of $1+1$ massless bosonic free field as a "distribution" rather than tempered
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.


