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I have to prove the following statement:

$f\mapsto \phi(f)\vert\Omega\rangle$ is continuous, where $\phi$ is a scalar Wightman quantum field, $\Omega$ the vacuum state of the theory and $f$ a Schwartz-class function.

First of all, continuity means that $\Vert\phi(f_{n})\vert\Omega\rangle\Vert\to 0$ for $n\to\infty$, if $\Vert f_{n}\Vert\to 0$ for $n\to\infty$.

Per definition of the Wightman quantum field as a operator-valued distribution, I know that the map $f\mapsto \langle\Omega\vert\phi(f)\vert\Omega\rangle$ is a tempered distribution. Therefore I can write

$$\Vert\phi(f_{n})\Omega\Vert = \langle\Omega\vert\phi(f_{n})\phi(f_{n})\vert\Omega\rangle=\int\mathrm{d}^{4}x\mathrm{d}^{4}y\langle\Omega\vert\phi(x)\phi(y)\vert\Omega \rangle f_{n}(x)f_{n}(y)$$

My next step is to use the Källen-Lehmann representation: $$\Vert\phi(f_{n})\Omega\Vert=\int\mathrm{d}^{4}x\mathrm{d}^{4}y f_{n}(x)f_{n}(y)\int\mathrm{d}\rho(m^{2})\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}2\omega_{p}}e^{ip(x-y)}$$

Is this right up till know? Now I don`t know how to continue...According to a hint on the exercise, I am allowed to use, that $\rho(m^{2})$ is polynomially bounded measure and the Fourier transformation is linear and continuous....

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The statement in the box normally is an axiom (part of Wightman's). You are not supposed to prove it. Rather, I suspect, you are supposed to use it in order to prove something else, like the Källen-Lehman representation.

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  • $\begingroup$ According to the exercise I shoud proove the statement claimed in the box.... $\endgroup$
    – B.Hueber
    Feb 7, 2020 at 21:55
  • $\begingroup$ Then you have to explain: prove this continuity from what? Can't prove a theorem without hypotheses. $\endgroup$ May 12, 2020 at 19:55

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