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What is the condition that a test function $f(t^{\prime},t^{\prime\prime})$ has to satisfy in order that:

$$\lim_{t\rightarrow t_{0}}\int_{t_{0}}^{t}dt^{\prime}\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\:f(t^{\prime},t^{\prime\prime})\,\delta^{\prime}(t^{\prime\prime}-t^{\prime})\,=\,0\quad?$$ where

$$\delta^{\prime}(t^{\prime\prime}-t^{\prime})=\frac{d}{dt^{\prime\prime}}\delta(t^{\prime\prime}-t^{\prime})$$ with $\delta(t^{\prime\prime}-t^{\prime})$ is the Dirac delta function.

Yair
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  • Warning, the Dirac delta is not a function and one cannot simply integrate the derivative of the Dirac delta over any interval ... please tell us first your definition of this notation! See e.g. https://math.stackexchange.com/questions/4427947/what-is-the-value-of-the-integral-int-inftya-deltax-a-dx-and-relate/4427952#4427952 for more explanations. – LL 3.14 Jun 01 '23 at 01:22
  • @LL3.14 Yes, I know that I should be careful but I am bot entirely sure about all that. Let me explain how I arrive at this issue and you might be able to understand better than me. All I want is to check that the initial conditions of the Dyson series are satisfied for a theory which satisfies $\hat{H}(t^{\prime})\hat{H}(t^{\prime\prime}),\sim,\lim_{\epsilon\rightarrow0}\frac{f(t^{\prime},t^{\prime\prime})}{(t^{\prime}-t^{\prime\prime}-i\epsilon)^{2}}=f(t^{\prime},t^{\prime\prime})\left[P.v.\frac{1}{(t^{\prime}-t^{\prime\prime})^{2}}-i\pi\delta^{\prime}(t^{\prime}-t^{\prime\prime})\right].$ – Yair Jun 01 '23 at 08:58

1 Answers1

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(Partial answer)

The main "problematic" step with this expression will occur when doing the integration by parts. Indeed, we will get : $$ \left.\int_{t_0}^{t'} f(t',t'')\delta'(t''-t') \,\mathrm{d}t'' = f(t',t'')\delta(t''-t') \right|_{t''=t_0}^{t''=t'} - \int_{t_0}^{t'} f_y(t',t'')\delta(t''-t') \,\mathrm{d}t'', $$ where $f_y$ denotes the derivative of $f$ with respect to its second argument. The first boundary term, namely $f(t',t')\delta(0)$, will be divergent unless you impose $f(t',t') = 0$. I let you finish the derivation, but it seems to me that no more constraint is needed.

Abezhiko
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