red boson, blue fermion

I'm not sure I follow. Isn't the point of the four-vector formalism that energy and momentum are not, fundamentally, different things, but projections of the same quantity onto different axes? Our Newtonian perspective on life hides this truth from us, rather as if we (and all our evolutionary forebears) had spent all our lives measuring left-right distances by one unit and front-back distances by another.

In a grand, philosophical perspective, what's the difference between combining scalars and vectors into four-vectors — adding the extra machinery to unify them — and using a supersymmetry to relate bosonic and fermionic sectors of a field theory? Are commutators somehow more pedestrian than anticommutators?

If you've plowed through the discussion over on backreaction or reference frame, a lot of the criticism is the immediate "you can't add bosons or fermions", followed by a "well, not without a super algebra operator or some such".

I was puzzling over this, because it is an ingrained physicsy thing, to not add unlike things.
But, we do, and the familiar example I came up with is the 4-vector, which is precisely ok because under the larger group it is just a projection of a single thing that appears to have these qualitatively different aspects under smaller symmetry groups.
And we have rules - for addition, inner product and outer product with this extended object.

And that is what Lisi seems to be getting at - he can add fermions and bosons, and he doesn't need supersymmetry, because under the large group they are just projections of a larger unified thing which happen to transform differently and in different sectors under the action of smaller symmetry groups.

This makes Lisi's argument more plausible, more worth a closer look.
I don't think he has demonstrated that in fact he has constructed this, but the pieces are there and they may just do something interesting.
It will be a lot of work to see if it works, and then to see if it is interesting for real physics, but to my mind this removes the main "it is obviously wrong" counter-argument.

It is a plausibility argument for me.

YMMV

If you've plowed through the discussion over on backreaction or reference frame, a lot of the criticism is the immediate "you can't add bosons or fermions", followed by a "well, not without a super algebra operator or some such".

It's not the addition or the transformation properties, even. He wants to identify these fields with the adjoint of E_8. The Lie algebra of E8 doesn't contain any Grassman variables, however. Now, I suppose one could just say that this is a representation on a vector space ignoring that some of the variables are Grassman and forget about it being the adjoint. The representation does not respect the grading, then. It's tough for me to imagine getting anything useful out of such a structure given the violation of the grading, but I suppose I can't immediately prove it's impossible. I do know that Lisi did not convince me that he had done it. This is one reason that I'd really like to see the Lagrangian expanded out at once rather than in the piecemeal (and not comprehensible to me) manner in the text.

Yeah, what he said.
Lisi does not convince that he has done it, and the burden of proof is on him.
But with the excess hype in the media, the critique in the semi-pro sites fell into the trap of becoming overly harsh in tone.
However, I'm not going to be the one who goes through it and sees if it can be fit together.
I would not be surprised if someone does not make a more formal argument for why something like this might work, so at that level it can be a successful paper, but right now it is an flawed idea, short of a model and nowhere near a theory.

Hi Steinn,

Thanks for the pointer (it ended up in my junk mail folder though). If you read the comments to my posting you'll find that I never objected on adding 'apples and oranges'. It's just a question of how to properly define the addition. I mean, you can add up forms with different rank, so why not bosons and fermions. That actually is not the problem I have with the paper. My problems are twofold. One is that I don't understand how he gets the properties from the algebra down to the group. The other is that he is avoiding the important step of showing how the E8 structure gives the SM+gravity by just writing down an action. I don't know very much about the BF stuff. If anybody could tell me why one can just choose the B to be proportional to psi? Sorry if that is a stupid question. Best,

B.

While I'm at asking stupid questions, here is another one. The whole discussion about the Coleman Mandula theorem eluded me. Doesn't it say something about the S-matrix? In the full E8 part of the paper, there is no Lagrangian, thus no interaction, thus no S-matrix. In the other part, he has written down a Lagrangian by hand that is not E8 invariant, and it's just made such that it reproduces just the SM. It seems kind of useless to argue it isn't compatible with the SM, as it's made to be. However, as I have mentioned repeatedly, I don't think one can get this Lagrangian by just breaking the symmetry of an YM E8 action, for the simple reason that the fermionic part comes out with the wrong order of derivatives. Just write down (d + A + psi)^2, and you get d psi d psi instead of psi d psi, and there is no psi A psi term that should give the gauge-field fermion vertices? (If you read through the comments to my post, you will find that I have said all that before). Adding vectors in a root diagram doesn't replace having interaction terms in a Lagrangian. I have genuinely no clue how one can get around these problems. I mean, I am generally optimistic, but I think this is the real problem if one is looking for a TOE - not finding an algebra that is large enough to accommodate all quantum numbers of the SM, including spin. The whole media hype really came as a surprise to me.

Best,

B.

let me respond with another stupid question

if the vectors in the root diagram show the interactions, and if the root diagram is complete and closed, then you can write down the lagrangian - either just construct by walking through the root diagram, or write down all possible terms and then strike the forbidden ones - this must be possible if the diagram is a faithful representation of the physics

the Coleman-Mandula theorem, from what I recall and very briefly refreshed myself on from Weinberg III, requires a well defined asymptotic outgoing state

I did read all 284 comments, late the other night, I can not promise I internalised all of them

as I noted below, I am walking through this as I go along

Hi Steinn,

The vectors in the root diagram don't actual show interactions. (I hate to say it, but you also find this in my comments...). What the addition in the diagram shows is which interactions are in principle allowed due to conservation of quantum charges. I.e. if you add the quantum numbers of particle one to particle two, and you end up on a root that corresponds to particle three, this corresponds to an allowed three vertex. It doesn't tell you much about 4-vertices (see the comments...). It also doesn't say which of these interactions actually appear in the Lagrangian, or how many there are. Just imagine, you'd add arbitrary higher oder terms, you can get all kinds of interactions with the same root diagram. I think it might actually be that if you just write down the YM theory with the full symmetry that indeed all of the possible 3-vertices appear, but then that's not what he does anyway. As I said previously, I suspect this would cause a lot of funny looking terms where the fermions appear with wrong dimensionalities.

Reg. CM. Yes, well, isn't that what the S-matrix tells you, how you get the one into the other? Either way, without an Hamiltonian, without an interaction, without a quantization I see no point in arguing about the symmetries of a matrix that wasn't even constructed in the context.

Best,

B.

Being a generous person, I feel all allowed interactions ought to occur.
At best you can suppress them with a small coupling constant...

I do think one of the truths string theory brought us is that there are really no such things a 4-vertices, when they appear in the theory they are just a sign of an incomplete effective theory. So a full theory maybe ought not to have them.

There is something I am missing here: if Lisi's model actually contains the standard model, then all of this is taken care of automatically - the appropriate symmetry groups are in there already as proper subgroups and the standard model Lagrangian comes along for free, what you really need to worry about is that most of the time you get way too many extra terms, new particles, symmetries and interactions that just aren't there in the real world. What Lisi's model ought to be doing is soaking up most of this additional crap into O(1,4) with very little new physics stuck in there as the leftovers.

So, either he did this: he has something that contains the full Standard Model plus some extra bits, or, he is missing pieces of the Standard Model and broke things up wrong.
Then there is a secondary issue of whether what he did is fixable and/or interesting even if it is not right.

I've got no dog in this fight - Lisi is not going to shoot down or confirm any pet theory of mine, I'm just a curious bystander.

Humm, my problem with only 3-vertices is that then he'd need to show how he gets the SM interactions from that. Also, it's not plausible to me if I think about it as a gauge-symmetry since it's non-abelian, so where do these terms go? I am fine with having all kinds of allowed higher order operators, but the lowest order should give the familiar one, and not contain an additional d slash. My problem is that there is no dynamics of the full theory. I am afraid I am somewhat old fashioned in this regard. Even if the symmetry breaking was just done by hand such that it reproduces the standard model, I'd like to know what the full theory looked like to begin with. I just don't see how one can expect the symmetry breaking to account for the dynamical differences between the fermions and bosons, even if the spins work out.

He automatically has the Standard model structure for one generation. His classical lagrangian (which he hasn't written down, but could) will have every interaction consistent with the symmetry. Even if they don't appear at one scale, upon a renormalization group action they will appear at another. Thats highly tweakable, so it shouldn't be a big deal. Or so it seems, except that the fundamental problem with the paper is its still entirely classical -- and really only with one generation unless he enlarges his group to E8 * X (if X is too big, its ugly, if X is too small one worries about horizontal symmetry problems)

Upon standard quantization, he runs into every single other problem people have already mentioned (eg either no Smatrix or a violation of CM, spin-statistics and violation of grading, unitarity issues, etc), so the brunt of the work remains to be done. Being skeptical, I can't help but note that if history is a judge, quantization tends to make things worse, not better, and he will need every ounce of good fortune in this regard.

Well, the three generations, as I understand it, are supposed to come from the "triality", the factorization of E8 that I John Baez pointed out (discovered?).

One thing that could save the S-matrix/C-M conundrum is a careful look at the concept of an S-matrix in de Sitter space. I have not looked into it for years, but from my vague recollection it is hard to define an asymptotic free outgoing state in de Sitter since it is not globally hyperbolic, but looks like RxSn.

I agree a lot of work would have to be done to see if quantization ducks all the other issues.

Yup, there is no Smatrix or boundary correlators defined for DeSitter space (wrong asymptotia). So the conundrum it seems to me is either you settle for that situation and bypass CM, or you insist that there should be an SMatrix (and we do experiments that say that there should be such a concept) and run into CM in the limit lambda --> 0.

Im not sure about the triality he's implemented in his paper, its a little nonstandard afaics (see Distlers blog for discussion). Atm it almost looks like the E8 * X I mentioned is simply three copies of E8, and there is a sort of triality between them. That needs to be flushed out more. Absent his generation hypothesis, its still interesting, but it becomes a lot more standard