In Welton(1983), Memories of Feynman, Welton mentions two formulas which he denotes as Feynman's Famous Formula (FFF) and FFF #2.

Which famous formulas is he talking about? Is he maybe talking about the Bethe-Feynman efficiency formula? Are these formulas still classified or does somebody know how these formulas look like?


There is something famously called "Feynman's famous formula", which comes up in QFT calculations, which I imagine must be the second FFF referred to in Welton's account. It reads:

$$\frac1{a_1 a_2 \ldots a_n} = \int_{x \in \Delta^{n-1}} \frac1{(\sum_{i=1}^n a_i x_i)^n} d\sigma$$

where $\Delta^{n-1}$ denotes the simplex $\{x = (x_1, \ldots, x_n) \in \mathbb{R}^n: x_i \geq 0, x_1 + \ldots + x_n = 1\}$ of dimension $n-1$, and $d\sigma$ denotes Lebesgue measure on this simplex, normalized so that its total volume is $1$. For example, in the case $n=2$, it may be written as

$$\frac1{ab} = \int_0^1 dx \frac1{(ax + b(1-x))^2}.$$

In any case, many people do refer to this as "Feynman's famous formula", for example Witten in the book Quantum Fields and Strings (Proposition 1.3, page 427), or in the article here (from Particle Physics: Càrgese 1989). Apparently Feynman first wrote about this in a letter to Bethe:

"I am the possessor of a swanky new scheme to do each problem with one less energy denominator. It is based on the great identity [the previously displayed formula] so 2 energy denominators may be combined to one -- reserving the parametric $x$ integration to the indefinite future (there's the rub of course)."

(Source: QED And The Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga, by Silvan Schweber, page 453.)

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