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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?

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    $\begingroup$ you are integrating over $p$ but the function $f$ depends on $x$ and $y$; perhaps you want to also integrate over $x$ and $y$? $\endgroup$
    – Carlo Beenakker
    Commented Dec 20, 2022 at 22:22
  • $\begingroup$ Oops, yes, I will edit the question. $\endgroup$
    – Isaac
    Commented Dec 21, 2022 at 3:51

2 Answers 2

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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.

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    $\begingroup$ Thank you for your answer. I would like to make it a "normal" distribution, not necessarily tempered. Then the issue is whether or not the 2D GFF makes sense directly when testing against compactly supported smooth functions only. More concretely, I am concerned about the integral at the bottom of my question. Could you please help me evaluate it? $\endgroup$
    – Isaac
    Commented Dec 21, 2022 at 3:57
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    $\begingroup$ Makes no difference, the problem isn't the behaviour of the test functions at infinity, but whether they average to $0$ or not. $\endgroup$
    – Martin Hairer
    Commented Dec 21, 2022 at 9:51
  • $\begingroup$ I see. I guess the IR cutoff (or behavior at infinity) is somehow related to the average to $0$, which is the crucial property.. Thank you for your answer. $\endgroup$
    – Isaac
    Commented Dec 21, 2022 at 12:09
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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.

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  • $\begingroup$ Thank you for your reference as well. $\endgroup$
    – Isaac
    Commented Dec 21, 2022 at 12:10

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