The blogosphere is a-twitter over surfer dude paper modestly titled An Exceptionally Simple Theory of Everything by Garrett Lisi
Just maybe he is onto something, but it is way overhyped
And so it is.
Sean, uncharacteristically, declines to comment, sensible chap.
So, like a fool, I tread there.
I was asked to nicely. It is late, and I may be doing some re-edits of this over the week, it is not a trivial issue.
I will also try to speak astrophysics-speak rather than true physics-speak, which may lose or loosen some jargon in translation. I’ll try to find time to tighten this up and add pointers over the week. Have not had time to read through the mainstream media spin on this, heard it was hyper.
So… all True Physicists Quest for the One True Theory of Everything.
It seems a bit silly to be looking for The ToE all the time, but honestly, that is what we do. Even those playing in astrophysics, we’re just looking in a different place.
Looking out, rather than introspecting, as it were.
Now, as you know Bob, physics has an extraordinarily successful pair of theories under its belt: Ye Olde General Theory of Relativity, which is a classical geometric theory of gravity, extraordinarily successful, very useful, very robust; on the sinister hand, is the Standard Model, a quantum field theory, which unifies electromagnetism, the strong nuclear force and the weak force in a compact theory that has totally predicts just about everything calculable in quantum.
And, as you know Bob, the two don’t mix. At all.
The Standard Model does not contain gravity, and Gravity is not quantized.
What is more, we know that the two are not just irreconcilable internally, they meet in a contradiction. This shows up in a lot of places, and it is bad.
So… people try to fix it, and there are many approaches – the best known is superstring theory, which incorporated gravity in a natural way, and we think incorporates the Standard Model somewhere.
Non-stringy quantum field theorists also work on this, really, some do; as do quantum gravity folks, notably the dreaded quantum loop gravity theorists.
Lisi has a model, which has a heritage, which he claims incorporates the Standard Model in a quite minimalistic way, and adds in gravity.
Now, as you know Bob, the Standard Model incorporates its field into a gauge invariant theory, with symmetry groups SU(3)xSU(2)xU(1)
U(1) is electromagnetic gauge invariance, think phase invariance.
SU(n) is glorified n-dimensional complex rotation, so weak (SU(2)) and strong forces (SU(3)) have bigger symmetries, they are invariant under more local rotations of the gauge fields. You also get a free, Lie, algebra with your group.
Now, that is one ugly symmetry, but people immediately realised that if you pick a larger symmetry, it can be “broken” into smaller symmetry groups – ie the ugly Standard Model symmetry is the pieced together shards of a bigger, more beautiful symmetry group.
The smallest such is SU(5), which was a serious candiate for grand unification of the Standard Model. Simple SU(5) predicts proton decay, which is not observed at the rate predicted. Bad theory.
People played with other Lie groups, bigger, but their heart wasn’t in it.
It is possible to fix the proton decay by suppressing the responsible coupling, but the bigger groups imply additional particles – you add more parameters to the theory and it all gets very silly.
In the meantime supersymmetry came along – this posits a symmetry between the integer spin, bosonic gauge particles, and the half spin, fermionic to which the gauge fields couple. Very exciting, only doubles the number of particles, and is still an option, not ruled out by experiment and may be observed Real Soon Now.
Then came strings, which helped a lot.
Superstrings brough gravity on board, at the expense of adding six extra dimensions (and then seven).
Strings also bring in, quite naturally, some particular symmetry groups, like SO(10), rotations in ten dimensions, and E8 – the largest of the five exceptional Lie groups.
E8 is large, certainly large enough to contain the Standard Model, but that leaves a lot of extra bits, which need to be accounted for.
E8 is very pretty, it is very big, and Lisi has had some fun with it.
Basically he noticed that E8 can be decomposed into a piece that contains the Standard Model, and a piece that looks like spacetime.
He then posits that this IS the Standard Model plus gravity.
That is it.
There’s only a few extra pieces, notably a trio of coloured scalar fields that might be observable.
So, there are two problems, that have stick-in-the-mud ye olde particle physicists trying to squelch all the fun.
One is that Lisi cheerfully adds fermions and bosons as needed it.
I mean he sums them – which is generally frowned upon, since they are different.
The whole apples and lightning thing (because you can add apples and oranges, but you can not add an apple and a bolt of lightning).
Now, there is an intriguing issue, which is that this is what you do in super theories, where you introduce an additional operator to generate a symmetry between the fermions and bosons.
Lisi asserts that the BRST quantization scheme enables us to naturally add fermionic and bosonic fields under E8. Essentially they live in different subgroups of E8 and he chooses a representation where you can add them piecewise (what happen if change representation and mix these is not addressed and is a serious issue).
Secondly, there is a famous theorem in mathematical physics, the Coleman-Mandula theorem, which forbids this whole process.
It is a “no-go” theorem, which says you cannot do a non-trivial mixing of gauge symmetries and spacetime symmetries and get a non-trivial theory (ie any theory which does this is basically empty, it has no interesting interactions).
This kinda sucks, since it is very tempting to think of gauge symmetries as real geometric symmetries on some compact geometry.
So… naturally clever people will try to get around this.
Lisi adds his fermions as “ghosts” – ghosts are a mathematical trick used in some quantization schemes, usually they are considered unphysical, they carry negative probability amplitudes, but can be propagate as long as the total probabilities sum to positive (think Konus densities in classical mechanics), and as long as they don’t propagate asymptotically, they should not be free particles in any scattering.
Lisi circumvents this, near as I can tell, by fiat. The fermions are ghosts, but he forces them to be (hopefully) real by gauge fixing the theory and letting them propagate. It is, shall we say, not clear that this is possible or will work, but it is at least thinking outside the box.
It is the sort of technicality that might be worked around.
Its been done before.
If it doesn’t work, unitarity is broken – ie probabilities are not conserved.
That is Bad.
The other, more serious problem, is the Coleman-Mandula restriction.
Turns out that an assumption of the theorem is that spacetime is Poincare symmetric.
Which it is, to very high precision, locally.
Lisi points out, that observationally the universe is not Poincare, it is de Sitter (this is because of the cosmological constant – the universe is accelerating exponentially).
So the relevant group is SO(4,1) not SO(3,1) and Coleman-Mandula does not strictly apply.
Formally, there are no outgoing states in S-matrix theory, in de Sitter space, because there is no asymptotic flat Minkowski space to propagate into. Bummer.
That could be a loophole.
On the other hand, the local universe is very, very close to being Poincare symmetric, and it really shouldn’t matter that it is de Sitter globally. Plus there are generalizations of Coleman-Mandula (which I have never looked at, and just heard about). But, effective quantum field theories ought to look like they satisfy Coleman-Mandula, and Lisi’s model does not.
So… what do we have.
Well, using E8 as the umbrella group has been tried, both in field theory and in string theory.
Taking most of the excess structure and identifying it with gravity is novel and cute.
It may not actually work, but there is a reason that a lot of people started talking about it, other than to annoy rigorous algebraists.
Lisi does not actually have a theory yet, he doesn’t really have a model even.
He has a couple of insights that may go somewhere, he has a beautiful framework for exploring the structure he is looking at, and he has a possibility of being onto something.
If the rather dubious bits he pulls off tighten into something half-rigorous, and if there is another loophole in Coleman-Mandula, and if the theory is calculable (like masses of the new scalar fields, or actual expectation value for cosmological constant) then it may be very interesting.
That is a lot of ifs.
Sure are pretty pictures though.
E8 in all its glory