Natural integral (as we will define it) is a distinguished antiderivative of a function that can be understood as interpolation of the sequence of consecutive derivatives to the $-1$. It has a naturally defined integration constant. While it is possible to define natural integral in various ways, it all boils down to the following property:

$$f^{(-1)}(x)=\int_0^x f(t) \, dt+\frac{1}{2} \left(\int_{-\infty }^0 f(t) \, dt-\int_0^{\infty } f(t) \, dt\right),$$

where the integrals in the brackets should be understood in the sense of regularization, if they diverge. The $0$ is not important here and can be replaced by any point, this will not affect the answer.

That said, I wonder, if such natural antiderivative ever appears in physical applications?


As @Qmechanic pointed out, it appears in many areas of physics. I prefer examples from classical mechanics or elementary quantum theory.


1 Answer 1


The natural antiderivative $$f^{(-1)}(x)~:=~\frac{1}{2}\int_{\mathbb{R}}\!\mathrm{d}x^{\prime} ~{\rm sgn}(x\!-\!x^{\prime})~f(x^{\prime})$$ is the most symmetric choice of integration constant. The kernel ${\rm sgn}(x\!-\!x^{\prime})$ appears all over physics. It is the Fourier transform of ${\rm PV}\frac{1}{k}$ up to a multiplicative constant. Examples:

  1. The symplectic potential $\frac{1}{2}z^I\omega_{IJ}\dot{z}^J$ in the Lagrangian leads to the Feynman propagator/Greens function $$G^{IJ}_F(t\!-\!t^{\prime})~=~\frac{1}{2}\omega^{IJ} {\rm sgn}(t\!-\!t^{\prime}).$$ See also my related Phys.SE answer here.

  2. The Poisson commutation relations $\{\phi(x),\phi(y)\}=\frac{1}{2}{\rm sgn}(x\!-\!x^{\prime})$ for a self-dual boson field becomes $$\{\Phi[f],\Phi[g]\}=\frac{1}{2}\int_{\mathbb{R}}\!\mathrm{d}x \int_{\mathbb{R}}\!\mathrm{d}x^{\prime}~{\rm sgn}(x\!-\!x^{\prime})f(x) g(x^{\prime})$$ in terms of test functions $f,g$.

  • $\begingroup$ I wonder why the mathematicians are not aware of this. when it comes to differintegral definitions or generalizing the polygamma function to the negative orders, all kinds of definitions pop out. $\endgroup$
    – Anixx
    May 17, 2020 at 12:39

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