In his famous paper "There are no Goldstone Bosons in Two Dimensions", this link http://projecteuclid.org/euclid.cmp/1103859034, first he states that the $\delta(k^2)$ is not a distribution in two dimension, and the Lemma (after Eq(10)) does not apply to $\delta(k^2)$ (in the bracket at the end of the Lemma). And in Eq. (11a), the definition of $F$ is just the Fourier transform of (6), which means $F(k)= A \delta(k^2)\theta(k^0)$, where $A$ is an irrelavent constant. Thus $F(k)$ is not a well-defined distribution and the Lemma can not be directly used on it. But before Eq. (18), he states " By the lemma and the assumed support properties of $g$, the first integral (of Eq. (15)) vanishes as $\lambda$ goes to infinity. " Eq.(15) is \begin{eqnarray} [\int d^2k F(k) |\tilde h(k)|^2][\int d^2k F_{00}(k) |\tilde h(k)|^2]\ge|[\int d^2k F_0(k) |\tilde h(k)|^2]|^2 \end{eqnarray} and $\tilde h$ is defined in (16) \begin{eqnarray} \tilde h(k)= f(\lambda k_-)g(k_+)+f(\lambda k_+)g(k_-) \end{eqnarray} and the support of $g(x)$ does not include zero, $f(k)$ being a peaked test function defined around Eq(8). As $\lambda\to \infty$, the support of the two terms in $\tilde h(k)$ does not overlap and the first integral in (15) becomes \begin{eqnarray} \int d^2k F(k) (|f(\lambda k_-)g(k_+)|^2+|f(\lambda k_+)g(k_-)|^2) \end{eqnarray} It seems that Coleman's statement means that from the Lemma, $\lim_{\lambda\to \infty}\int dk_- F(k)|f(\lambda k_-)|^2\sim \delta(k_+)$ and since $g(k_+=0)=0$ and similar to the second term, the integral vanishes. Since as I stated above $F(k)\sim \delta (k^2)$ and the Lemma can not be used on it, how can he reach this result? And then he stated that since "$F_{00}$ is a positive distribution, and the second integral is monotonic decreasing". It seems to me that since $F_{00}$ is a well-defined distribution, the Lemma could be used on it, and thus the second integral on the left of Eq.(15) is vanishing. It seems that Coleman's statement about $F_{00}$ integral just applies to the $F$ integral, and the one for $F$ can be used on $F_{00}$.

Is this a typo of Coleman or a misunderstanding of mine?


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