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.
Lisi's homepage (currently severly slashdotted) and wiki
There is extensive discussion at Backreaction, here and some followup here
Woit also has something to say, as, of course, does Lubos
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
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I thought he had an actual expectation value for the cosmological constant.
"with a cosmological constant related to the Higgs vacuum expectation value, "
One may argue that we don't know the Higgs vacuum expectation value, and no one has calculated that from this, um, model, but we have idea of what it should be, right?
you can not add an apple and a bolt of lightning
Where do you think applesauce comes from?
It is lateral thinking like that which wins Nobel Prizes!
See. Here I was thinking, one apple + one orange = two fruit.
Easy.
But, no, someone has to find way to add lightning.
Actually I though that was how you got cider, and then you just have to add some grain, hops, yeast and blackcurrants, and I distinctly recall finding unification.
As for the VEV of the Higgs.
Yes, the fourth root of the cosmological constant will be a small number times the Higgs VEV.
First we need to calculate the Higgs field, then renormalize it or something, and then find a natural reason to divide it by a few trillion.
Fortunately E8 has a lot of large numbers that can be squared or exponentiated, which could get us to a few trillion easy.
Which few trillion though...
I had a spirograph when I was young.
What happens when you add two apples and one lightning bolt? Thrre apples? I think that operation is more akin to multiplication.
Chancellor Brian May
I liked my spirograph...
I think the phrase we need is "direct product"
Oo. Chancellor May, what a career track!
About the compatibility of gravity and QFT/SM, you wrote:
"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."
That statement is wrong! Quantized general relativity is a perfectly fine quantum theory. It is however an effective field theory which is doomed to fail at latest at the Planck scale. Its quantum predictions are well defined and calculable, but very very tiny. See Donoghue's papers! So as long as we are concerned with phenomena at scales below the Planck scale and on backgrounds which are sufficiently smooth (i.e. where classical GR itself works), quantized GR is a valid QFT with a spin-2 boson.
Well, no, I am not wrong.
GR and SM meet at the Planck scale, and are irreconcilable there as formally formulated currently.
That is why there IS a Planck scale, because that is where those two theories meet.
This shows up at low energies in many places, the simplest of which is the issue of scattering off a black hole - an astrophysical black hole is low energy, but the propagation of any quantum mechanical particle into it and subsequent asymptotic outgoing states is unresolved.
And the solutions involve unpalatable options, like giving up unitarity, or Lorentz invariance.
I think we both agree there is a problem, I think it is a problem that is pervasive.
Wasn't Hitler named Chancellor at some point in his career?
Just maybe he is onto something, but it is way overhyped
I thought it was very interesting to compare the initial response, when this hit Backreaction and Not Even Wrong, to the later response, when the Telegraph wrote an article and it hit slashdot. In the early response all the bloggers were sometimes hopeful but very very skeptical-- even Lisi was very skeptical, Garrett Lisi showed up in some of those blog discussions and was falling over himself to emphasize how speculative everything was and highlight places where he thought the theory needed more work. In the Telegraph-period response though all that skepticism seemed to disappear-- the Telegraph article somehow or other wound up not attributing any skepticism to Lisi himself, and the "counterpoint" quotes all said nothing more negative than "this is a very long shot", something even Lisi seems to more or less acknowledge. It seems like the media only has two modes when it comes to science, "HYPE" and "IGNORE"...
Turns out that an assumption of the [Coleman-Mandula] 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 ... So the relevant group is SO(4,1) not SO(3,1) and Coleman-Mandula does not strictly apply... 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.
I have heard this last, bolded part a couple of times, and I was wondering if I could ask you to clarify it a little.
As I understand it, Garrett's specific claim (well, one of them) is not so much that the universe is de sitter, but the somewhat simpler claim that his decomposition of E8 does not include the Poincare group as a subgroup. I interpret this to mean that Lisi's E8 theory does not, in fact, predict Poincare invariance; and the whole dark energy/de sitter thing is just his excuse for why he should be allowed to get away with not predicting Poincare invariance.
To me-- with my only-math background and limited physics intuition-- this sounds entirely straightforward: If Coleman-Mandula is a theorem, then it holds only when its prerequisites are met. Talking about finding a "loophole" in a theorem sounds to me like a strange attitude, either the theorem holds or it doesn't. Where the theorem doesn't hold you don't need a "loophole". Now, maybe a more-general form of Coleman-Mandula holds but just hasn't been proven yet, and probably one could have a reasonable expectation that the more general form is true; but until the general form is proven you don't know that. Still, a lot of people who know way more about physics than I do seem to think of the Coleman-Mandula thing as settled proof or no proof, so I feel like I ought to try to understand that viewpoint.
In particular I'm curious about the claim, which you make here but I've seen in more than one place, that because poincare invariance is "apparently" true (i.e. its experimental predictions bear out in practice), then Coleman-Mandula should be expected to hold in any real-world theory even if you have a model which claims there is a more fundamental level where it doesn't hold. (A commenter on NEW named "amused" worded this as "The remark in the paper about C-M not being relevant because the spacetime is deSitter is nonsense, since Poincare symmetry can be assumed for all practical purposes when applying the theory to particle scatterings in labs"). I don't yet understand Coleman-Mandula itself, but can you try to help me understand why this follows? It seems like what you and amused are really saying can be reduced to an argument that poincare invariance is a very close working approximation to whatever symmetry spacetime "really" has. But if that's all, is this really close enough to trigger Coleman-Mandula? It doesn't seem like it should be. Is there something else I am missing?
Or is the idea that you expect that Lisi's claims that poincare invariance doesn't hold at the e8 scale can't possibly be right, since poincare invariance is observed to hold at the macro/particle scales and there is no known mechanism whereby poincare invariance could be false at the e8 scale but suddenly emerge at larger scales?
Ok, tough one.
This partly is the issue of "effective theory" vs "true theory".
Locally spacetime is very close to Poincare symmetric, but one of the "outs" for a lot of physics problems is to relax Poincare invariance (Lorentz invariance, specifically), at a price of getting some new problems of course.
Now, with a cosmological constant, the global spacetime is really de Sitter, it is not asymptotically flat, there is no background Minkowski space to propagate into.
Really truly.
This trashes theorems, particularly any that make statements about asymptotic states, because they are incorrectly defined from the outset.
With mathematical physics theorems an incorrect assumption is like being a little bit pregnant, it voids the theorem. It is almost as good as mathematics that way...
And then SO(1,3) is not the true space-time symmetry, it is only approximate and O(1,4) is the true symmetry group. So Poincare symmetry is irrelevant.
But, on the other hand, you don't expect an electron scattering in a collider and ending up in a detector to "know" about the asymptotic state of the universe at t -> positive infinity, locally everything looks flat (except of course the cosmological constant can look like a pervasive scalar field which the electron is propagating in...).
But people can't have it both ways - if it is all effective theory and minor violations don't matter, then you can do funky stuff.
But if people want to be all proper and mathematical, well then we have to seriously contemplate that SO(1,3) is not it - it really is O(1,4) because that is what spacetime is actually like. It is not that O(1,4) appears at high energies, it is that it really is our spacetime symmetry group and SO(1,3) symmetry is an approximate local illusion.
This makes a lot of stuff wrong. Including Coleman-Mandula (well, it is not wrong, just bypassed).
Now an interesting question is whether for O(1,4) spacetime symmetries is there something analogous to the Coleman-Mandula theorem which makes any mixing of gauge symmetries and spacetimes symmetries "no-go". It is not clear to me whether it does and I expect other people will be pounding Weinberg Volume III to see if it works or not.
The Higgs VEV is well-known to be 247 GeV. It's the Higgs mass that is unknown.
In THIS theory, the Higgs VEV is not known because it has not been calculated.
Ideally it ought to come out to the "right" value.
Clearly the Higgs VEV is related to the cosmological constant in that the root of the cosmological constant is some multiple of the Higgs VEV - the trick is figuring out that li'l number, and why it has the value it does. If indeed there is a reason.
Then you need to get a mass as well.
I am very pleased by your knowledge of this kind of math and math. physics - I thought that you were only counting planets and maybe Martians producing CO2 on Mars using integers. ;-) Best, LM
I am so glad my years at Caltech doing a physics PhD were not entirely wasted...
Preskill seared some aspects of axiomatic QFT into my brain forever, and looking back over my notes, I learned a lot more from Feynman's class than I appreciated at the time.
Not that you really need anything more than integers, of course.
"Die ganze Zahl schuf der liebe Gott, alles Ubrige ist Menschenwerk"
"God created the integers; all else is the work of man"
Leopold Kronecker [7 Dec 1823 in Liegnitz, Prussia (now Legnica, Poland) - 29 Dec 1891 in Berlin, Germany]
So who made E8?
And is that connected to Feynman asking, of the muon, "who ordered that?"
57 is an integer, 248 is an integer, but symmetries of 248 points in 57-dimensional space, well, that seems more that Kronecker would have liked.
I have no dog in the Lisi fight either, but last night at a Thankgiving dinner in Beverly Hills, I was asked about E8 and Lisi by a student who wanted to go to Veterinary school, the world's leading expert on Afro-Cuban Salsa dance, and the Concertmaster of the L.A. Opera. So this story is spread wider than I'd thought.
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.
Does it show up experimentally, anywhere?
The GR vs QFT discrepancy does not show up experimentally anywhere, as of now.
There were great hopes for the GZK cutoff, but that turned out to be experimental error.
IF there were an experimental discrepancy the situation would go from interesting niggle to major crisis of science; right now it allows people to speculated (possibly to excess, or not) and to argue about which theory has to give.
An interesting analogy is the situation at the end of the 19th century where there were known anomalies not consistent with theory, but which were thought to be minor niggles, and turned out to be seeds of major conceptual revolutions.
It is of course possible that the universe is simply inconsistent, logically.
That is slightly bothersome if you then figure that the universe is otherwise consistent with mathematics, since then anything goes... sort of.
Or the universe might just be incomplete.
Steinn Sigurdsson: "It is of course possible that the universe is simply inconsistent, logically." This could happen in many ways. It could be locally inconsistent (with patches of mutually incommeasurable characteristics in some way). It could be temporally inconsistent, with some oscillation between different characteristics. It could be paraconsistent.
Steinn Sigurdsson: "That is slightly bothersome if you then figure that the universe is otherwise consistent with mathematics, since then anything goes... sort of." Depends on WHICH mathematics. This gets us back to the problem with Davies' op ed piece, as we don't know how to define the space of all possible consistent physical laws, let alone the inconsistent ones.
Steinn Sigurdsson: "Or the universe might just be incomplete." There's the question of whether only Godel-consistent universes "exist" in the multiverse. And in what way the universe might be logically incomplete.
Oh, remember "Luminous" by Greg Egan? The universe had regions in which integer arithmetic worked differently, and the border between our and the next one over began to move...
Hi Steinn,
I'm the "amused" guy that Coin mentioned above. The answer you gave to him is at odds with how I see the Coleman-Mandula situation, so I'd like to give my argument and get your feedback. (If I'm wrong it's a good chance to learn something.)
I'm no expert on C-M but am under the impression that the theorem can be (loosely) formulated as follows: If there is a symmetry group of the theory which contains the Poincare group in a nontrivial way (i.e. not as a direct product with something else), and if certain reasonable technical assumptions hold, then the scattering matrix of the theory is trivial.
Now consider theoretically the scattering of particles in a collider experiment described by a theory with a symmetry group which contains the Poincare group in a nontrivial way. C-M tells us that the scattering is trivial, i.e. the particles don't interact. Now, theoretically, consider what happens if we "turn on" a positive cosmological constant Lambda. Globally spacetime becomes deSitter but we can surely still consider the scattering of particles in the lab experiments. (In fact, apparently Lambda really is nonzero in our universe, but that hasn't stopped earthlings from scattering particles in labs.) OK, Poincare symmetry is no longer exact but only holds as an approximation in the lab experiments (albeit a very good one if Lambda is small). So we can no longer invoke C-M to conclude that the scattering must be trivial. But surely the scattering matrix depends continuously on Lambda! For if Lambda is very small then the scattered particles will only "feel" Lambda to a very small extent. So the scattering matrix for very small nonzero Lambda must be very close to the scattering matrix for Lambda = 0. Since the latter is trivial by C-M, we conclude that the scattering matrix for very small nonzero Lambda must be very close to trivial. But in the real world the scattering matrices for particle scatterings are very far from trivial. Hence we conclude that a theory with a symmetry group which contains the Poincare group in a nontrivial way cannot describe quantum physics in our world, since in our world Lambda is very small yet scattering matrices are far from trivial.
Do you agree with this argument? If it is right it seems to contradict your claim that Coleman-Mandula is "bypassed", since C-M at Lambda=0 was still enough to rule out the theory at very small nonzero Lambda provided one makes the eminently plausible assumption that the scattering matrix for particle scattering in labs depends continuously on Lambda.
Actually, in Lisi's theory C-M gets bypassed in another way, namely due to the fact that the ad hoc action he uses is not invariant under the full E8 gauge group. But for there to be any reason to consider his theory it should surely have a high-energy completion where E8 gauge symmetry is restored, and then C-M comes back to bite him for this high energy theory. The only way out would be if spacetime gets messed up in the high energy theory so that Poincare symmetry no longer holds (for Lambda=0), which was a possibility that a commenter over at N.E.W. suggested.
A final thing: Pardon my ignorance, but where does the O(1,4) that you mentioned come from? (It must be acting on some 5-dimensional vectorspaces but I have no idea where they would come from since the spacetime is 4-dimensional.)
amused wrote:
I thought the C-M theorem said that if there was a non-trivial scattering matrix then there couldn't be a scattering of particles by gravity (or rather, by the part of the particle theory that's meant to represent gravity), if it had a Poincare symmetry.
The suggestion here is that if there almost a Poincare symmetry, that is compatible with there being almost no scattering by gravity.
Hi Steinn, it is my understanding that the Superstringers were also looking at using the E8 structure for something within their theory too. Do you have an idea what it was going to be used for?
Without PHD courses in physics, my conceptual understanding of the two theories' seems remarkably similar. In SST, you have objects (stings) floating around in 6D Calabi-Yau spaces interacting with our standard 4D spacetime. In Lisi's vision, it sounds like he wants to replace 6D Calabi-Yau spaces with 8D polytopes floating around in 4D spacetime. To me, as a layman, it makes almost no difference whether we have 6D or 8D objects floating around out there, both are equally outside our perception. So I guess my question was do you think SST was looking to replace (evolve?) 6D extra-dimensional Calabi-Yau-shaped spaces with 8D extra-dimensional E8-shaped spaces?
Another question, I've heard of E8 described as 8D (and it is the definition I've been using for the moment), but I've also heard it described as 57D. Which is right?
Heterotic superstrings have E8xE8 as a natural symmetry group.
One of the problems recovering the standard model from superstrings is that the symmetry groups are just too large, they imply a lot of physics we just don't see.
On C-M: I'll have to go back and read Weinberg III, again, but as I recall the triviality follows from the absence of a mass-gap. ie if SO(3,1) is a symmetry group then the lowest mass state is zero, and this drives all the interactions (not just gravity) to be trivial. This is from memory.
So, let me make two contradictory claims - I can do that, since I was at Cambridge, where I learned it is always good to claim both sides of an argument in advance -
1) it is not clear the these things go smoothly to zero as Lambda goes to zero, because going from an exactly zero cosmological constant, to any non-zero global cosmological constant changes the spacetime topology discontinuously. So... it is not implausible to imagine that there is a jump condition associated with this. Which is a potential out.
2) This is numerology, but MHiggs ~ √ ( 4√ Λ MPlanck )
Which is kinda cute, if I did the conversion from real (G=1) to fake (hbar=1) units correctly.
So ramp Λ to zero, and the Higgs particle goes away smoothly and therefore presumably all the induced mass terms and hence most of the couplings. I don't know that this drives you to a trivial effective theory, but it is not implausible that it might.
I kinda rambled on this in the "almost massless world" post I did over thanksgiving, but that was last week.
Re. (1): It would be amazing (to me at any rate) if particles scattering in labs were sensitive to (a change in) the global spacetime topology.
Re (2): If this senario holds, it is just as well that astronomical measurements led to nonzero Lambda, since if they found Lambda=0 the Standard Model would be dead! (Although maybe I misunderstood something here.)
yup, it's be amazing, but not inconceivable.
Arguably if anything like Feynman's sum-over-histories picture is correct, then in fact all scattering amplitudes are sensitive to the global topology
the "prediction" of the Higgs mass really is just numerology, although I could imaging something analogous to a see-saw mechanism driving such a relation in some theory.
If I were really cleverbold I would have predicted that the Standard Model required a non-zero Lambda ;-)
But, yeah, smoothly running the Higgs mass with Lambda uses the cosmological constant to solve the hierarchy problem and postdicts the cosmology...
Other way of looking at it is that it reduces antrhopism by one implausible step.
I really did re-read large chunks of Weinberg III last night and noticed something amusing, since adS is generally unstable to vacuum tunnelling, and a zero Lambda vacuum is stable against tunnelling to adS states, a general vacuum has only one way to go and that is to a de Sitter space, so we should on probabilistic grounds predict we live in a de Sitter space, and hence a non-zero Lambda.
Although I may be overinterpreting the argument.
As Dr. Philip Vos Fellman commented to me by email:
There's also a kind of Godelian sense in which consistency may not be decidable. At least one of my limited takes on Godel is that he demonstrates that the law of the excluded middle isn't really a law.
I'm not always sure what people mean by "consistency" any more except in some layman's sense.
Isn't there a contradiction with locality though? E.g., if for idle amusement god decided to change the topology by doing something to a faraway region of spacetime, while leaving the geometry of our current region in spacetime unchanged, surely that shouldn't affect our particle scattering experiments here on Earth??
Is Lisi's paper the first to suggest this connection between Higgs and Lambda? If so, I guess that's one interesting physics idea to come out of it.
yeah, but...
if you change the topology you change the boundary conditions
and at least in the path integral formulation that is potentially bad.
There has been work on this, since spactimes with fluctuating topology are a major issue.
I think this shows up in S-matrix formalisms since formally you take the asymptote, so you don't get out of it by saying it is local.
I couldn't tell you what all has been done on it, but I don't think Lisi is the first to link Higgs and Lambda, in fact I'm pretty sure I remember an attempt from someone at Harvard back about 20 years ago.
It seems like there ought to be something there, them both looking like scalar fields etc.
I don't like string theory, I have no particular scientific logic behind it but it is really boring. Besides who likes 11 dimensions.
Ps. Ãslendingar hér, nú er ég svo aldeilis hlessa.