# Is $E_8$ theory any close to reality by any means? [closed]

The E8 theory from Wikipedia:

"An Exceptionally Simple Theory of Everything" is a physics preprint proposing a basis for a unified field theory, often referred to as "E8 Theory", which attempts to describe all known fundamental interactions in physics and to stand as a possible theory of everything. The paper was posted to the physics arXiv by Antony Garrett Lisi on November 6, 2007, and was not submitted to a peer-reviewed scientific journal. The title is a pun on the algebra used, the Lie algebra of the largest "simple", "exceptional" Lie group, E8. The paper's goal is to describe how the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra.

• Did you read the chronology & reactions section of that article? – Allure Jan 30 '19 at 0:02
• Yeah, not satisfying, and looks outdated compared to what I found elsewhere, also wanted to know how people that are studying physics see this theory . – Dhia Hassen Jan 30 '19 at 0:09
• As evidenced by the answers, asking whether a widely-circulated hypothesis & frequent lay idea (which merited a mainstream rebuttal) is consistent with data & theory is entirely on-topic. – alexchandel Sep 30 '19 at 2:29
• @alexchandel yet they put it off-topic ! – Dhia Hassen Dec 3 '19 at 3:49

## 2 Answers

This is a long answer in which I try to convey in detail the ways in which E8 theory is, and is not, close to reality.

Lisi's "E8 theory" is an idiosyncratic variation of gauge field theory (also just called gauge theory), which is how we explain all the forces except gravity.

A gauge theory is defined by a choice of symmetry group, and by a choice of how the fields of the theory transform under the symmetry ("representations" of the symmetry). There's always a gauge field consisting of force bosons, which in a sense implement the symmetry transformations, and then there are usually other fields. The symmetry representation of the other fields, determines how they interact with the gauge bosons. For example, gluons are gauge bosons of an SU(3) symmetry, quarks are in the "fundamental" representation of SU(3), and this determines how quarks and gluons interact.

The standard model (which is everything except gravity, and whatever other physics we don't know about yet) is a gauge theory with symmetry group SU(3)xSU(2)xU(1), the corresponding force bosons, and then an assortment of particles (fermions and the Higgs boson) which inhabit certain representations of the symmetry group. A lot of theoretical effort has gone into constructing "grand unified" gauge theories in which SU(3)xSU(2)xU(1) is just part of a larger symmetry, and in which the known fermions are assembled into representations of that larger symmetry. Such theories imply many new particles, and also new interactions among old particles (none of which has been observed).

E8 is a big symmetry group, part of a sequence of "E" symmetries. Usually one only considers up to E6 in grand unification, because it's the biggest E-group that has "chiral" representations, in which there can be left-handed and right-handed particles that exist separately and behave differently. (This is a feature of the real world.) In a non-chiral theory, left and right always come in pairs, and combine in a way that makes them act the same. An E8 gauge theory should be non-chiral and thus disqualified from corresponding to reality.

E8 gauge symmetries appear in string theory (notably the "heterotic E8xE8 superstring"), but a specifically stringy mechanism allows one to get e.g. an E6 theory after the extra dimensions are compactified.

So that's how the prospects for an E8 theory, from the perspective of ordinary gauge field theory. What does Lisi do differently?

First, he wants to get gravity from E8 as well. He is not alone in wanting to get gravity from a gauge group, this is a known school of thought in quantum gravity. I think it's accurate to say that this idea has problems, but it's a rather technical debate. In any case, Lisi's E8 theory is going to be a "graviGUT" theory that tries to bring gravity into grand unification.

Then - and I think this is far more clearly problematic - he wants to get the fermions - electron, neutrino, quarks, etc - from E8 itself, too. Recall that in the usual gauge theory, there is a gauge field of force bosons which in a sense implement the symmetry transformations, and then there are other fields, distinguished by their "representation" of their symmetry, that the force bosons interact with.

The problem here is precisely that the force bosons are bosons, whereas the matter particles of the standard model are fermions. For bosons and fermions to be in the same group representation, there must be a fermionic element in the symmetry group, that can turn a boson into a fermion or vice versa. In other words, it has to be a supersymmetric group or supergroup. E8 is not a supergroup! There's no way to put bosons and fermions into the same E8 representation.

So why does Lisi even think he can do it? Part of the reason is that there are ways to divide up the elements in E8 so that some are "even" and others are "odd", and this resembles the distinction between boson and fermion. But if you actually make the even elements bosons and the odd elements fermions, you will not have E8 any more, you will have representatives of two separate smaller symmetries.

E8 does have a remarkable property, which is that its "adjoint" representation - the one employed by the force bosons in a gauge theory - is also a "fundamental" representation - the one most often used for the fermions in a gauge theory. So although the gauge bosons and the fermions are not unified in a single E8 representation, they are in identical representations related by a supersymmetry transformation. This is the closest I can come to fulfilling Lisi's intention to place fermions and bosons in the same E8 representation.

Returning to Lisi's actual theory: he is taking a single adjoint representation of E8 symmetry, which ought to give you a gauge theory of E8 force bosons interacting with each other, and he has proposed not just that some of those force bosons are gravitons - which the "graviGUT" people also do - but that some of those E8 particles (presumably the ones which are "odd" under some "Z2 grading") will actually be fermions. This is the step that no-one else tries to take, and the step which no-one knows how to take without disintegrating the E8 symmetry into two lesser symmetries, one for the bosons and one for the fermions.

G. Smith cites a 2009 paper by Distler and Garibaldi as refuting E8 theory. What that paper does is to ignore the problem that I just mentioned, and then says, suppose we go through with such a procedure anyway, what kind of fermions can we get? The fermions of the standard model of the real world are organized into three generations, each with two quarks and two "leptons" (an electron-like particle and a neutrino). The best Distler and Garibaldi can do, is to obtain one generation and one anti-generation from these odd-graded elements of E8. An anti-generation is the chiral partner of a generation, so not only are we two generations short of reality, but even that single generation will have its chirality nullified when it pairs up with the mirror particles in the anti-generation.

With respect to the "one generation not three" problem, Lisi's riposte is that there is a symmetry (outer automorphism) of E8, called triality, which means that there are two other ways of dividing up E8 to get "one generation and one anti-generation". So he's hoping that somehow the three generations of the real world can be obtained by applying triality. But this is his idea that makes the least sense of all. In his latest work, from 2015, he has been doggedly ingenious in trying to make it work, by treating E8 itself as a fundamental 248-dimensional space, and 4-dimensional space as a kind of partial excitation of this E8 space. But although triality will then give you three overlapping ways to get "one generation (and one unwanted anti-generation) of pseudo-fermions in 4-dimensional space", they still don't combine into the "three generations in 4-dimensional space" that you need, but rather some mess with more than 4 dimensions.

Let me try to sum up. Whenever people do grand unification in the usual way, they are trying to squeeze the known forces and particles into the structures implied by some new symmetry group. As I said, this usually implies new particles and new processes, and the usual problem of grand unification is just that you have to tune the theory's parameters to explain why none of that has been seen.

Lisi is doing something in the vein of grand unification, but it requires increasingly problematic steps - gravity in the gauge group, fermions together with bosons, three generations from triality (and anti-generations kept apart from the generations). At this time the two last steps aren't even at the level of a mathematically coherent theory that might be physically right or wrong - there is simply no model there. Possibly you could have a mathematically functioning model that "began with" E8 - but which no longer had E8 symmetry after the odd elements were actually made into fermions - and then you could calculate its properties as in his 2015 work, but there's no reason it would bear any resemblance to reality, it wouldn't even be 4-dimensional.

So Lisi certainly took elements of the real world, and squeezed them into E8, but at the price of breaking the rules that make E8 what it is. His hope is that there's a way to bend the rules that makes sense, that is still mathematically legitimate, and which can define a testable theory, but that's just a hope. So most practitioners of theoretical particle physics have long since moved on. They certainly value creativity, but they believe that Lisi is trying to do things that will never work. There are a handful of physicists who do still take an interest in Lisi's work, but the mainstream would regard most of their theories as fundamentally problematic or ill-defined too.

My guess is that if there is any way of rescuing Lisi's idea, it should start with the naturally supersymmetric version of E8 gauge theory, even though that implies a doubling of particles. At least that is a mathematically coherent and well-defined framework. But, even if the original squeezing of all the particles into a pseudo-E8 looks ingenious, there's no requirement that it has anything to do with reality at all. Theorists have had many attractively ingenious ideas over the years, but they can't all be true, in fact most of them must be false. And that includes many ingenious ideas that do make mathematical sense.

According to Google Scholar, Lisi’s paper had only six citations in 2018, and citations have been on the decline for the last four years. This is objective evidence that the theory is not being taken seriously, presumably because physicists accept Distler and Garibaldi’s 2009 refutation.