# Why is the Hubbard-Stratonovich transformation also called the “Feynman trick”?

The Hubbard-Stratonovich transformation is a nice way of eliminating squared terms in an exponential. It is heavily used in statistical mechanics. Essentially it boils down to the simple identity:

$$\mathrm{e}^{\lambda a^{2}}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{-x^{2}/2+\sqrt{2\lambda}ax}\mathrm{d}x$$

Wikipedia cites some references to works of Hubbard and Stratonovich justifying its name.

But I have also seen this transformation called "Feynman's trick". Although there are many things named "Feynman trick", for this in particular I could not find any Feynman paper actually exploiting this transformation. Can someone point to some references?

• I believe it's on you to provide some references. Where did you hear someone referring to that transformation as "Feynman's trick"? I could just claim it is also known as the "AFT trick", and ask people to justify that terminology, but that would be pretty off-topic, don't you think? – AccidentalFourierTransform Sep 10 '18 at 16:41
• @AccidentalFourierTransform See the supplementary materials to this paper: 10.1103/PhysRevLett.118.048103. Just look for "Feynman". – becko Sep 10 '18 at 16:55
• Thanks. I'd appreciate it if you included that reference in the post itself (and, if possible, a second reference, by different authors, so as to be sure it wasn't just a lapsus by Tikhonov and Monasson). – AccidentalFourierTransform Sep 10 '18 at 17:15
• AFT trick is called the Cooley-Tukey algorithm. – JEB Sep 10 '18 at 17:28
• @AccidentalFourierTransform Unfortunately I cannot find another right now. I could swear I saw this in a few other papers. But maybe you are right and it was just a mistake by Tikhonov et al. – becko Sep 10 '18 at 18:44