If I recall correctly, Sidney Coleman wrote a few papers on "No go theorems" many years ago, where he showed that any renormalizable quantum field theory that included a non-compact gauge group (e.g. the Lorentz group required for gravity) must have supersymmetry. My understanding is that if local quantum field theory is to include gravity, then the No Go Theorems require supersymmetry. On the other hand, if quantum field theory turns out to be non-local, like string theory, then this may not be true. But all of this is my understanding from 20 years ago and may no longer be true. And the underlying issue is singularities, which we have all been worrying about forever, and once again relates to infinite precision calculations in a universe where space-time should be quantized. At least that is my knuckle-head kid from Texas interpretation... On Tue, Oct 8, 2013 at 3:36 PM, Warren D Smith <warren.wds@gmail.com> wrote:
Here's some notes by me attempting to figure this out...
Supersymmetry (SUSY) is an additional hypothetical symmetry (or more than one such) of space-time which converts bosons into fermions (and vice versa). For example it would convert the photon, a boson, into a "photino" (hypothetical fermion) and the electron (a fermion) into a "selectron" (hypothetical boson). It is probably the biggest open question in fundamental physics whether SUSY exists, and its existence would be a prerequisite for supergravity (SUGRA) and superstring theory, as well as a possible explanation for "dark matter" (which would be composed of one or more of the superpartner particles which must exist if SUSY is true).
So far there is zero convincing evidence for SUSY -- and indeed it seems clear from evidence that "unbroken SUSY" cannot be correct in our universe, only "broken SUSY"s could be correct, which are much more mysterious. But there are some highly unconvincing but tantalizing clues which perhaps constitute some kind of pro-SUSY "evidence" if you are an incredible optimist. Far as I can tell physicists like it just because it seems amazing and cool, not since there is any significant evidence. SUSY solves or partly solves some fundamental problems in physics provided you are enough of an optimist (i.e. are willing to accept highly speculative heuristic-aided reasoning in place of proofs, to get the "solves").
It was initially proved to be impossible for such a symmetry to exist -- before it was proved to be possible. Fortunately the inventors had not known about the impossibility proof.
The reason is that the construction required setting everything not in conventional space-time at all (that cannot work), but instead in a higher-dimensional space in which the extra dimensions were not real numbers, but rather "grassman numbers."
The simplest superspace R^(1|1) has 1 real coordinate and one non-real coordinate, got by adjoining an additional magic symbol eps to the reals, obeying eps*eps=0 and otherwise acting normally, i.e. commutative, associative, distributive. (Analogous to getting the complex numbers by adjoining i with i*i=-1.)
[Do not confuse this superspace with the word "superspace" which has also been used in a better way by J.Wheeler to denote something entirely unrelated -- the space of metrics. Wheeler as usual makes good name choices while the other physicists make bad ones... that's just an inexplicable observed fact about Wheeler versus most other physicists.]
"Grassman numbers," also called "fermionic dimensions," act like real numbers except multiplication within them anticommutes, causing all squares of pure-grassman quantities to be zero. The whole notion is plainly logically self-consistent because real matrices can implement it. For example, the general element of R^(1|1) is a+b*eps with a,b, real, which can be regarded as the 2x2 matrix a b 0 a then just matrix multiply and add as usual.
Grassman numbers are simpler than real numbers, e.g. polynomials (and power series!) with grassman argument always have degree=1.
Trying to do integral and differential calculus, delta functions, partial differential equations, etc, inside superspaces is a bit more challenging or anyhow head-hurting.
Trying to do serious physics work in spacetime manifolds with grassman dimensions, appears to be hellish. Tensor notation was fairly nasty, but spinor notation is nastier, and then throw in grassman dimensions on top of spinor notation and voila -- welcome to utter notation hell; plus say goodbye to whatever phsyical intuitions you used to have to help you. My personal threshold is tensors. Due to my mental limitations I've never been able to achieve any sort of happiness dealing with (the numerous) spinor notations and their even-hairier descendants. I might be able to push further were I motivated, but the trouble is to motivate me you'd probably have to convince me that SUSY were true (or some equivalent motivation) which means I'm basically useless for the present-day's task of determining that.
When you set up a superspace that includes our usual spacetime as an all-real subspace... it includes within it the usual Poincare-Lorentz relativistic group which all physical laws are invariant under... as a subgroup.
Now within superspace, the magic boson<-->fermion exchanging symmetry operation is: to interchange the grassman and real coordinates. Well, no. Actually, it's hairier than that, an operator involving differentials acting on quantum fields. Anyhow, it is definable.
Do I want SUSY to be true? No. I want it to be false, for the purely greedy reason that dealing with the hellish complexities of it is something I'd prefer humans not to have to do. (SUGRA? String theory? Those are even harder.)
Despite the messiness, quite a lot has been actually worked out in the MSSM ("minimal supersymmetric extension of the standard model" of particle physics) that actually is reasonably close to stuff that experimentalists can try to observe. This contrasts with SUGRA and string theory where essentially nothing has ever been worked out that is anywhere near to being experimentally verifiable. The main obstacle is the whole business of "broken" supersymmetry, which must happen if SUSY happens, but is basically total guesswork and fantasized hopes, and gives those guys ENORMOUS leeway for dodging inconvenient experimental results and still trying to assert SUSY exists no matter what the experiments say. Far as I can see, absolutely nothing stops that from continuing forever. On the bright side, if they experimentally do see supersymmetry, then wow, big progress. So SUSY is kind of 1-sided: You can prove it, but you cannot disprove it. At least that's the way I perceive it. As such, it seems somewhat problematic whether we should consider it "science."
Here's two "easy introductions" to this subject: http://arxiv.org/abs/hep-th/0108200 http://www.kitasato-u.ac.jp/sci/resea/buturi/hisenkei/sasaki/Kiryuu.pdf which I find very unsatisfactory. I have not yet found any introduction I like.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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