To D.Nordstrom, editor, Phys Rev D. prd@ridge.aps.org ``Zero point energies, the cosmical constant, and supersymmetry'' (df8058) by Warren D. Smith (me). It was rejected by a referee for Phys Rev D. On reconsideration.. I've decided there IS something wrong with my paper (which you had rejected anyhow). The following describes what. So: I'm quitting complaining. --Warren D. Smith. ----------------------------- REFUTATION OF MY "ZEROPOINT" REPORT ====Warren D. Smith 30 Nov 2001======= Well, I hate to admit this, but... it appears that my report "Zeropoint energies, cosmical constant, and supersymmetry" is WRONG. This is a pity, since it appeared to be a beautiful argument, and it appeared to solve (or at least offer the potential to solve) one of the biggest problems in theoretical physics, the "cosmical constant crisis." Unfortunately, the grandest of theories can be laid low by some simple facts. I'm still leaving the "Zeropoint" report on my web page even though I currently believe it is wrong (but it now is accompanied by this here warning), since it seems to me to contain interesting and useful ideas which perhaps can be used for another purpose - or which perhaps can be resuscitated for its originally-intended purpose (see bottom for possible ways that might be done?). Incidentally, this has something to do with the referee's rejection of this paper. The referee was definitely wrong in the sense that, apparently, he did not even look at over half of my paper - the part with my new ideas that overcame the problems with old ideas! He simply saw my paper had some old ideas in the half he looked at, those ideas were "known" not to work, for reasons the ref listed (but reasons I had already considered in my paper & shown how to overcome - but the ref ignored that utterly) hence: reject. So the ref did an incompetent job. BUT, just because the ref was wrong, does not mean I was right! It now appears we BOTH did a lousy job... I will now explain what my paper did and the problem with it. Essentially, the "cosmical constant problem" is the fact that space should be filled with "zeropoint modes" of various fields (e.g. electromagnetic) at all frequencies up to some very high frequency (perhaps the Planck frequency, but anyway very high). The existence of these modes has experiemtnal consequences - which have been seen. The total mass-energies of these modes should be enormous. The vacuum therefore should weigh (i.e. gravitate) a lot. But in fact, experimentally, it does not. Experimentally, the "cosmical constant" (vacuum energy) is tiny, or zero. The disagreement between theory and experiment here is enormous - that is the problem that needs to be solved. My proposed solution of this problem had 9 steps: 1. According to a proposal popular among theoretical physicists called "supersymmetry", (SUSY for short) there must be 1 fermionic mode for each bosonic mode. 2. Fermionic modes have NEGATIVE zeropoint energies. 3. I can write an asymptotic expansion, in decreasing powers of the (assumed high) frequency-cutoff, of the vacuum energy. 4. The coefficient of the first term in this expansion is EXACTLY ZERO due to (2) - the negatives exactly cancel the positives at first order. 5. The coefficient of the second term in this expansion is ZERO due to a mathematical identity (called by various names, one of them is the "supertrace theorem") discovered in the 1970s by Ferrara. This identity is supposed to hold "at tree level" in all supersymmetric theories including broken SUSY. ("Tree level" means at the level of the first term in a different asymptotic expansion in decreasing powers of some different large numbers - e.g. the supersymmetric breaking energy.) 6. Great. We have now reduced the cosmical constant problem (assuming SUSY) by 2 terms down in that expansion. But what about the 3rd, 4th, etc terms? 7. Well, it turns out that my asymptotic expansion was based on a certain approximation of a sum by an integral. If you really do it right, i.e. use that sum, then a function which (in the integral-based version) is nice and smooth, turns out (in the better, sum-based version) to be "rough." It behaves irregularly when you look at it closely. This ultimately is traceable to irregularities in the distribution of prime numbers, etc. It can be analysed. There are theorems about it (which I used) by Tsang. 8. So now, when this "roughness" is put into the picture, it turns out to be BIGGER than the 3rd, 4th, etc terms. In fact, in the limit as the high-freq cutoff ---> infinity, it is bigger even than the 2nd term would have been (if it had had some other coefficient than 0) - and bigger by a factor which goes to infinity. 9. This roughness causes the energy of the vacuum to be extremely sensitive to tiny perturbations in various quantities (such as the precise value of the cutoff, or the precise size of the universe). I then made an argument that tremendous forces should automatically arise, which will have a "feedback" effect causing the size of the universe to self-adjust, tremendously quickly, by a tiny amount, to get to a location on the "rough" curve, at which the zeropoint energy should be zero almost exactly. One can crudely estimate the amount by which it still might be nonzero, and that amount is consistent with experimental evidence. This solves the cosmical constant problem wonderfully! But now to explain the bug: (A) In step (8), the "factor which goes to infinity" only does so logarithmically. Furthermore, the Ferrara identity, despite being a "theorem" (physicists seem to have a different notion of the word "theorem") is currently believed by physicists who support SUSY, to be wrong ("softly broken SUSY") and thus my second coefficient is NOT zero. (If the Ferrara identity held reasonably exactly, supposedly this would force light particles to exist, which have never been observed experimentally.) (B) So now the crux question is: we must compare the estimated now-nonzero value of that 2nd coefficient, with the value of log(infinity)! Is log(infinity) greater? Well, of course it is. BUT... if the high frequency mode-cutoff is NOT infinite, but in fact is the Planck frequency (which is large, but finite) then according to some numerical estimates I just made (which in retrospect, I should have made long ago...) in fact, it is NOT greater! It is many orders of magnitude smaller. Oops. This seems to blow my solution out of the water! Can it be resuscitated? I do not know. The two most obvious ways to try are (I) Resuscitate the Ferrara identity - at least for the present purposes. (II) Argue that, for some reason, it is legitimate to take the limit as the cutoff frequency tends to infinity - and I mean infinity, not merely the Planck frequency.