# How do we give a rigorous mathematical meaning to expressions like $\delta^4(0)$ or $\lim\limits_{x \to y} \delta^4(x-y)$?

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 $$$$:\phi^2:(x):=\lim\limits_{x \to y} \Bigl( \phi(x)\phi(y)-iD^+(x-y) \Bigr)$$$$ where $$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 $$x=y$$.

• 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. Commented May 3, 2023 at 14:53
• You mean the "Introduction to the Theory of Quantized Fields"? Commented May 3, 2023 at 15:06
• @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"? Commented Dec 8, 2023 at 10:57
• 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. Commented Dec 8, 2023 at 15:54

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 $$\mathbf{k}\rightarrow\mathbf{k}'$$.
• How does the finite volume $V$ and Fourier transform of $\delta^4$ on it have to do with the limit in my Q2? Commented May 3, 2023 at 15:05
• 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. Commented May 3, 2023 at 15:26