‘Alien calculus’ could save particle physics from infinities
So one question I have in the introduction section is that it seems the article misses the difference between the number of Feynman diagrams to calculate a_n and the value of a_n.  They point out that the the number of Feyman diagrams grows ~n!, which is much larger than the rate x^n shrinks (given 0&lt;x&lt;1).  If the number a_n calculated by Feyman diagrams doesn&#x27;t grow proportional to the number of Feyman diagrams, then it is still entirely possible for x^n to shrink faster than any change in a_n.<p>Based on my limited knowledge of particle physics, physicists are currently able to calculate using Feynman diagrams because a_n does grow less than x^n shrinks.  There are some equations (I think dealing with specific forces&#x2F;fields) where the constant it larger than others which makes calculates much harder.  x ~=.7 shrinks much slower than x~=.007.  Yet even then the general trend does hold and it does allow for making calculations which can then be tested against experimental data.<p>What we find is that our calculations do match the experimental data.  It isn&#x27;t a perfect match, there is room for error and confidence intervals and such.  The important point is that what this article suggest doesn&#x27;t seem to happen.  If at some point a_n grew much faster than x^n shrunk, then the real world solution would diverge and our answer from calculating n out to 5 wouldn&#x27;t closely match the data.  It almost sounds like the article is suggesting things will diverge only once we calculate out for n&gt;100 or so, but reality doesn&#x27;t await for those calculations.  If this problem really existed, it would happen because reality is calculating out n all the way to infinity even while the physicists can not.<p>So I&#x27;m left with two possible conclusions.<p>1.  The article is misunderstanding the relationship between the number of Feynman diagrams needed to calculate a_n and a_n itself.<p>2.  The real critique is that the current model is wrong because the model diverges, not that reality itself diverges.  Thus while this model is approximate for what we currently calculate, it is inherently wrong.<p>The second issue is an interesting idea.  A model that looks correct and is correct for all calculations done so far, but which may no be correct for more detailed calculations but which we do not and will not have the computation power to test at that level.