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The question is as in the title. In QFT literature, $\delta^4(0)$ is said to stand for the volume of entire $\mathbb{R}^4$, where $\delta^4(x)$ is the $4-$dimensional delta function.

Or when defining normal-ordered product of free scalar field operator $\phi(x)$ at a coinciding point, they use the limit operation \begin{equation} :\phi^2:(x):=\lim\limits_{x \to y} \Bigl( \phi(x)\phi(y)-iD^+(x-y) \Bigr) \end{equation} where $iD^+(x-y)$ is a causally supported two-point function made from commutator of $\phi$. $iD^+(x-y)$ is a tempered distribution and it is written in standard literature that $iD^+(x-y) \simeq \delta^4(x-y)$ as $x \to y$.

I have been always curious how to make sense of these expressions rigorously. The free scalar field $\phi(x)$ is in fact an operator-valued tempered distribution as well. So, every object written above is in fact a distribution. How should we understand their limits written in the form of ordinary functions?

In summary,

  1. What does it mathematically mean by "$\delta^4(0)=$ the volume of entire $\mathbb{R}^4$"?
  2. How should I make rigorous sense of expression like $\lim\limits_{x \to y} \delta^4(x-y)$?

Could anyone please clarify? I think I am essentially curious about defining a multivariate tempered distribution $T(x,y)$ on the thin diagonal $x=y$.

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  • $\begingroup$ The book by Streater and Wightman, briefly talks about this on pages 103, 104. I don't have it in front of me, but the green book by Bogoliubov et al should explain that too. $\endgroup$ Commented May 3, 2023 at 14:53
  • $\begingroup$ You mean the "Introduction to the Theory of Quantized Fields"? $\endgroup$
    – Isaac
    Commented May 3, 2023 at 15:06
  • $\begingroup$ @AbdelmalekAbdesselam I forgot to tag you in order for you to get a notice.. I believe the book by Bogoliubov is "Introduction to the Theory of Quantized Fields"? $\endgroup$
    – Isaac
    Commented Dec 8, 2023 at 10:57
  • $\begingroup$ No it's not that one. Bogoliubov and Shirkov is more of a theoretical physics book. The one I am talking about is "Introduction to axiomatic quantum field theory" by Bogoliubov Logunov and Todorov which is a mathematical physics book. Note that there is also a kind of follow up "General principles of quantum field theory" by Bogoliubov, Logunov, Oksak and Todorov. $\endgroup$ Commented Dec 8, 2023 at 15:54

1 Answer 1

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Q1 refers to the identity $$(2\pi)^{-4}\int_{\mathbb{R}^4}e^{i\mathbf{k\cdot x}}d^4 x=\delta^4(\mathbf{k}).$$ So "$\delta^4(0)=(2\pi)^{-4}$ times the volume of $\mathbb{R}^4$".

My answer to Q2 would be that you want to start from a finite volume $V=L^4$ in $\mathbb{R}^4$. With periodic boundary conditions the wave vector components are discrete, $\mathbf{k}\mapsto(2\pi/L)(n_1,n_2,n_3,n_4)$ with $n_\alpha\in\mathbb{Z}$. The delta function $\delta^4(\mathbf{k}-\mathbf{k}')$ becomes a Kronecker delta, $(2\pi)^{-4} V\delta_{n_1-n'_1}\delta_{n_2-n'_2}\delta_{n_3-n'_3}\delta_{n_4-n'_4}$, and you can take the limit $\mathbf{k}\rightarrow\mathbf{k}'$.

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  • $\begingroup$ How does the finite volume $V$ and Fourier transform of $\delta^4$ on it have to do with the limit in my Q2? $\endgroup$
    – Isaac
    Commented May 3, 2023 at 15:05
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    $\begingroup$ if you insist on an infinite volume there is no way to make sense of the limit in Q2, you need to regularize the limit by taking a finite volume. $\endgroup$ Commented May 3, 2023 at 15:24
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    $\begingroup$ Yes, that is my question. How exactly do I regularize by introducing a finite volume? There is no explicit mention of $\mathbb{R}^4$ in the limit in Q2. $\endgroup$
    – Isaac
    Commented May 3, 2023 at 15:26
  • $\begingroup$ I have filled in the details. $\endgroup$ Commented May 3, 2023 at 15:33

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