I saw a neat talk at Perimeter a couple weeks ago on new integration techniques:
Speaker: Achim Kempf from University of Waterloo.
Title: “How to integrate by differentiating: new methods for QFTs and gravity”.
Abstract: I present a simple new all-purpose integration technique. It is quick to use, applies to functions as well as distributions and it is often easier than contour integration. (And it is not Feynman’s method). It also yields new quick ways to evaluate Fourier and Laplace transforms. The new methods express integration in terms of differentiation. Applied to QFT, the new methods can be used to express functional integration, i.e., path integrals, in terms of functional differentiation. This naturally yields the weak and strong coupling expansions as well as a host of other expansions that may be of use in quantum field theory, e.g., in the context of heat traces.
(Many talks hosted on PIRSA have a link to the mp4 file so you can directly download it. This talk does not, but you can right-click here and select “save as” to get the f4v file.a )
The technique is based on the familiar trick of extracting a functional derivate inside a path integral and using integration by parts. The basic idea is to promote this from one-off trick to systematic tool.
It would be surprising to me if this idea wasn’t described elsewhere before, but I haven’t seen it and these guys do a nice job expressing it. If this technique really is novel, it is the sort of thing I expect to never enter into the physics canon taught to students, regardless of how useful it is for understanding key integration steps in the foundational topics themselves (like quantum field theory).b
This talk had a few nice tricks I hadn’t seen before but that are obvious in retrospect (and I’m sure familiar to people who work with this stuff). For instance: 
, considered as a operator on functions of 
, just translates a function to the left by 
.
Kempf was very clear and a bit apologetic about the fact that they haven’t actually found an example integral that can be computed with this method that no one had done before. But this actually doesn’t bother me. People have had decades to throw the kitchen sink at strange integrals, so I am not so worried that this method doesn’t have strictly greater “integrating power” than the entire sink. I am much more interested in the idea that you can “condense” the grab bag of tricks and heuristics used by experienced integrators (whether human or computer) into concepts/methods easily applied by relative laymen. This is actually a lot more valuable than finding yet another obscure ansatz integration trick that no one will actually remember.
I still am interested in how to understand the “power” of this technique. Is it always as useful as contour integrals, in the sense that it allows you to evaluate an integral if and only if it can be done with a complex contour? This is somewhat plausible based on the examples from the talk and the fact that the whole technique is very much dependent on analyticity. Kempf thought it was possible this was so, but he didn’t know.
I’d also like to know how this is related to the crude technique of just Taylor expanding an integral and integrating term-by-term.
The related paper appears to be “New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory” (arXiv:1404.0747v3) by A. Kempf, D.M. Jackson, and A.H. Morales. Added: Peter Morgan points out that Kempf et al have just released a more directly related article, “How to (Path-) Integrate by Differentiating” (arXiv:1507.04348).
Footnotes
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- This file format can be watched with VLC player. You can find it for any talk hosted by PIRSA by viewing the page source and searching the text for “.f4v”. There are many nice things about learning physics from videos, one of which is the ability to easily speed up the playback speed and skip around. In VLC player, playback speed can be incremented in 10% steps by pressing the left and right square brackets, ‘[‘ and ‘]’.↵
- The reason is that it isn’t strictly necessary to perform these integral, and essentially zero refactoring is ever done of the physics canon. Historically the integrals were done using opaque tricks, and if it was good enough for Feynman it ought be good enough for you….↵
Yesterday new on math-ph, http://arxiv.org/abs/1507.04348, is presumably much closer to the talk at PIRSA.
Thanks much, Peter.
I feel ambivalent about making much of this as a method in QFT. Much of QFT can be grounded in the free field CCR’s, which are just to say that the free field acts as a derivation, together with the free field vacuum state, which, to put it over-briefly, can be used to place us in the context of Hermite polynomials and of Gaussian integrals, where integration and differentiation are closely linked. Needless to say, there are delicate issues because of the various infinities that we introduce in QFT; those are manageable for free fields but for now there appears to be no suggestion of any new way to think about interacting fields.
At the level of ordinary integration, it seems pertinent that the Fourier transform already maps convolution to multiplication by an arbitrary function and differentiation to multiplication by a power, and that for the method as given to be much usable would need us to know the function spaces for which it works rigorously.
(CCR = Cutkosky cutting rules?) Fair enough. I’d be plenty happy for an improvement in free-field techniques and how they are taught. The world would be a better place if more things were as elegant as the relationships between Fourier transforms and convolutions.
Sorry, no, I meant Canonical Commutation Relations. My bad. For an operator-valued distribution hatphi(x) of a free quantum field we have [hatphi(x),hatphi(y)]=multiple of the identity operator (with the multiple being a distribution in x-y to ensure translation invariance), which is similar to what we have for an elementary derivative operator, [partial_x,x]=1 (though this, being positive, is /more/ similar to the commutation relation that applies for annihilation and creation operators). Free field can be done very cleanly, but there has to be a lot of other preparation because what we’re really interested in are interacting fields.
Hi Jess,
I didn’t attend the talk but I skimmed over the paper. My impression is that this type of technique had been used in formal study of field theories before. Ken Wilson, Franz Wegner, Christof Wetterich and many others used this technique when they derived exact functional differential equations governing the RG flow of various generating functionals. In some sense, all beta functions are packed in one equation. If you need the beta function of one specific interaction vertex, you just unpack the equation and end up with a hierarchy of ordinary differential equations. Pretty neat stuff.
Any chance you have a reference? I’m not too surprised the idea has appeared before, but if it languishes in the obscurity of some high-energy journal article it doesn’t do the world any good.
Hi Jess,
I learned this subject by reading Tilman Enss’ PhD thesis:
http://arxiv.org/abs/cond-mat/0504703
The relevant part is section 2.1.4. It came across to me as a nice trick back then. I think this trick (introducing a source conjugate to the physical field and replacing physical field by derivative w.r.t. the source ) has been around for a while. It seems that it has caught attention from mathematicians. The difference seems to be that this trick is a means of deriving some functional identities in field theory whereas here it is used for explicit evaluation of integrals.