(Alpha) Unification of gravity, gauge fields, and Higgs bosons

[Farsight]

I thought this was an interesting paper: Unification of gravity, gauge fields, and Higgs bosons by A Garrett Lisi, Lee Smolin, and Simone Speziale:

Abstract:
We consider a diffeomorphism invariant theory of a gauge field valued in a Lie algebra that breaks spontaneously to the direct sum of the spacetime Lorentz algebra, a Yang-Mills algebra, and their complement. Beginning with a fully gauge invariant action – an extension of the Plebanski action for general relativity – we recover the action for gravity, Yang-Mills, and Higgs fields. The low-energy coupling constants, obtained after symmetry breaking, are all functions of the single parameter present in the initial action and the vacuum expectation value of the Higgs.

Here's the last paragraph of the conclusion:

Much remains to be done to investigate this theory. The gravitational sector needs to be better understood [16, 17]. Since the action and equations of motion are low-order polynomials, we believe that progress can be made on the quantization of the unified theory, but this remains to be investigated. In addition, one can consider more general versions of an extended Plebanski action in which Φ³, in (1), is replaced by a scalar function, U(Φ), as in [11]. While much remains to be done, it is now clear that the line of thought that began with the work of Plebanski and Ashtekar yields a natural and simple proposal for the unification of all known interactions.

What does everybody else think?

 

[BenTheMan]

Equation (8) looks pretty dubious, and since all of his conclusions seem to follow from that, I'm not very inspired.

Also it looks like there is trouble with actually getting matter (i.e. fermions) out of the whole treatment.

Finally, equation (27) seems to predict a relationship between the higgs VEV and the cosmological constant which looks pretty hard to satisfy phenomenologically. There's also a relationship between g (which is, presumably, the coupling constant of the gauge theory), the higgs vev, and the Newton constant, G_N, which looks difficult to understand. We know that g ~ 1/2 from low energy data, so barring some large threshold corrections (which they don't calculate, or even estimate), I would think it would be hard to realize this relationship

 

[BenTheMan]

Also they write the higgs potential down, they don't explain it. This is a step backwards from SUSY, for example, which explains (a term you're familiar with, surely) the ``mexican hat'' shape of the higgs potential.

 

[Farsight]

Thanks Ben. I've ploughed through the "symmetry breaking" section 2.1 a couple of times now, and still can't see what they're getting at with "Here we show a spontaneous symmetry breaking of our new action that produces the dynamics of gravity and a Higgs field as well as Yang-Mills". When I follow the reference to http://arxiv.org/abs/0712.0977, I see "This is in fact a version of a proposal of Misner and Wheeler from 1957 that matter might be nothing but the mouths of Planck scale wormholes", which has me splurting my tea. Then in section 3 on fermions I'm seeing the torsion and chirality but nothing like a bosonic action trapped in its own curvature. And ouch, equation 27. I didn't see that it predicts a relationship between the Higgs VEV and the cosmological constant. But phenomenologically speaking, maybe that's to do with the vacuum catastrophe and a hidden strong force that's always in the space. I'm thinking low-energy proton-antiproton annihilation to neutral pions thence gamma photons and where does the strong force go?

Maybe they avoid explaining the Higgs potential because it's "the source of all our troubles". I'm getting mixed messages from this paper. In the intro they said the metric is relegated to an auxiliary role, analogous to that of the Higgs field which rather downplays it to space itself. But on page 2 they say the complete spectrum of the theory also contains Higgs bosons. We've got the gμv, geometric formulation, classical solution, chiral spinors, a perturbation to a topological background field, and overall something that feels coherent and new, and yet they seem to be trying to toe "rather arbitrary" Higgs sector line too. I'm not sure what to make of it, so all input gratefully received.

 

[Guest254]

What does everybody else think?

I think the paper is complicated, at least from a mathematical viewpoint. Could you break down some of the results for us - say to graduate level at least!

 

[BenTheMan]

But phenomenologically speaking, maybe that's to do with the vacuum catastrophe and a hidden strong force that's always in the space.

Well, their point is to show that a single theory can combine the standard model and gravit. If they can't show why their theory is finite, then what's the point? That is, they're trying to propose something that unifies gravity and the standard model. So things like ``vacuum catastrophes'' are pretty important to understand.

I'm thinking low-energy proton-antiproton annihilation to neutral pions thence gamma photons and where does the strong force go?

Well, the pions are strongly coupled quark-anti-quark pairs. Not that the strong force has to ``go anywhere''.

Maybe they avoid explaining the Higgs potential because it's "the source of all our troubles".

Again, then what's the point? I already have a well motivated theory which explains the higgs potential---it's called the minimal supersymmetric standard model. Why should I give that up?

Also notice that there are no predictions of this theory other than the ones that the standard model already gives us.

 

[Farsight]

Guest: no, sorry.

Ben: I'm not saying you should give up anything, thanks for the feedback.

 

[BenTheMan]

Ben: I'm not saying you should give up anything, thanks for the feedback.

I mean, the essential point of the paper is that you can do some mathematical magic and get gravity coupled to a gauge field and a scalar field. But that sounds just like Kaluza-Klein theory, which was figured out 100 years ago---you take gravity in five dimensions, compactify on a circle, and you get (gasp) gravity + gauge theory + scalar. You can add a potential by hand (which is what Smolin/Lisi do), and give mass to the gauge boson, but you still haven't explained anything other than you can come up with a potential for the scalar field.

Morally, I don't see how the two are different. It amazes me that people take Smolin seriously, when I read papers like this.

 

[Guest254]

Guest: no, sorry.

You don't have to go into any great detail or anything. Obviously you understand the paper - could you just give us a quick run down?

Might make the thread more interesting, since more people can get involved with the discussion.

 

[Farsight]

Noted Ben.

Guest254: No. I found it pretty inpenetrable, I'm not sure I do understand it. Hence any "quick" run down I tried to give would take me hours, and might be very badly misleading. It's just that some aspects sounded interesting, so I thought I'd seek some input.

 

[Guest254]

Guest254: No. I found it pretty inpenetrable, I'm not sure I do understand it. Hence any "quick" run down I tried to give would take me hours, and might be very badly misleading. It's just that some aspects sounded interesting, so I thought I'd seek some input.

Ah, right. I figured you understood the paper (and found it interesting?!?). No worries.

 

[Farsight]

Guest: I should explain that on my initial skim there seemed to be an undercurrent that I felt I recognised as the "right" approach, involving action, geometry, and torsion. I found myself saying looks like they're getting warm. But then I went back to plod through it and found myself scratching my head over the mathematics, realising that I didn't understand what they were really saying, and asking myself if Lisi Smolin and Speziale were missing some simple relationships because they'd gotten themselves too bogged down in the rigor. I found myself dwelling on action for gravity, wondering if Smolin was still thinking he can quantize gravity, then thinking do these guys really have much of a clue? From what Ben was saying, it sounds as if the answer is no. And from my own "take" on gravity which I won't go into here, I rather think I agree with him.

 

[BenTheMan]

I found myself dwelling on action for gravity, wondering if Smolin was still thinking he can quantize gravity,

Well, you can write the Einstein-Hilbert action for gravity down, which doesn't mean that you've quantized anything.

 

[Guest254]

Guest: I should explain that on my initial skim there seemed to be an undercurrent that I felt I recognised as the "right" approach, involving action, geometry, and torsion. I found myself saying looks like they're getting warm. But then I went back to plod through it and found myself scratching my head over the mathematics, realising that I didn't understand what they were really saying, and asking myself if Lisi Smolin and Speziale were missing some simple relationships because they'd gotten themselves too bogged down in the rigor. I found myself dwelling on action for gravity, wondering if Smolin was still thinking he can quantize gravity, then thinking do these guys really have much of a clue? From what Ben was saying, it sounds as if the answer is no. And from my own "take" on gravity which I won't go into here, I rather think I agree with him.

I don't really know what you're saying to be honest! Speaking from the point of view of someone who knows a modest amount of mathematical physics, I think the paper is mathematically demanding. It's essentially an outline of a new mathematical model, from which the authors extrapolate in regards to the physics.

I'm not sure how you tell if they're "getting warm" if the mathematics of the model isn't transparent! But perhaps you understand the mathematics better than I do - hence the request for a quick run down.

 

[Farsight]

Thanks Ben. On a related point, I was kicking something around with some people a few weeks back, I wonder if you can assist? In The Foundation of the General Theory of Relativity on page 185 of Doc 30 Einstein says "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy". The people I was talking to said that this energy is not included in the Einstein Field Equations. They seemed to know what they were talking about, but I thought surely not? I've asked around in a couple of places but can't seem to get a clear answer. It's quite an important point, so if you could clear it up I'd be grateful.

Guest: I don't understand the mathematics better than you do. I see you're a maths teacher. My formal maths education extends only as far as A-level, and whilst I've given paid maths tuition to schoolchildren and am comfortable with some further mathematics, I'm often having to stop at every expression and do laborious lookups to work out what it means. Even then I don't always get the picture. For example if you say "action" I'm immediately thinking of a cubic lattice which has been subjected to some kind of "kick" distorting the lattice in a pressure-pulse way, wherein gravity is the result of the surrounding density gradient which can be described in terms of the curvature this imparts on the path of another action. I didn't get this picture here.

 

[Guest254]

Guest: I don't understand the mathematics better than you do. I see you're a maths teacher. My formal maths education extends only as far as A-level, and whilst I've given paid maths tuition to schoolchildren and am comfortable with some further mathematics, I'm often having to stop at every expression and do laborious lookups to work out what it means. Even then I don't always get the picture. For example if you say "action" I'm immediately thinking of a cubic lattice which has been subjected to some kind of "kick" distorting the lattice in a pressure-pulse way, wherein gravity is the result of the surrounding density gradient which can be described in terms of the curvature this imparts on the path of another action. I didn't get this picture here.

Mathematics teacher might be slightly downplaying it! I'm a mathematician with a keen interest in mathematical physics.

I have absolutely no idea how you can extract any meaningful information from the paper if your mathematical background is A-Level. In fact, I don't think it's possible! I have a PhD in mathematics and I think it's tough going. For me, the interesting part of a paper with a "big conclusion", in this case unification, is not the actual conclusion or starting the assumptions - but the meat and bones of how the authors got from A to B.

But everyone's different - obviously you're more interested in the conclusion.

 

[Farsight]

Sorry Guest254, I got the "maths teacher" from http://www.sciforums.com/showthread.php?t=84340. It certainly isn't possible for me to simply read a paper like this and extract the meaning, but one can often "slog through" with laborious lookups to squeeze something out. However on this particular paper it would take me a year or two, so I was looking for shortcuts and assistance to try to appreciate the concepts and the approach. I share your sentiment that a conclusion isn't particularly valuable if somebody has jumped to it.

 

[Guest254]

Sorry Guest254, I got the "maths teacher" from http://www.sciforums.com/showthread.php?t=84340.

No problem - I wasn't really offended!

It certainly isn't possible for me to simply read a paper like this and extract the meaning, but one can often "slog through" with laborious lookups to squeeze something out. However on this particular paper it would take me a year or two, so I was looking for shortcuts and assistance to try to appreciate the concepts and the approach. I share your sentiment that a conclusion isn't particularly valuable if somebody has jumped to it.

I don't know what "slog through" really means in this context. Sure, every time you see a word you don't know, you can look up its meaning. But this doesn't offer any sort of understanding - it's how the words interplay and the context in which they are used. For instance, this paper is all about setting up a field theory on a particular fibre bundle. If you look up the definition of a fiber bundle, it won't mean a thing if you're not intimately familiar with graduate+ level differential geometry. Even my eternal optimism doesn't think its possible to understand physics/maths/anything with that sort of approach. Indeed, if it was, then I would read (and hope to understand) papers on areas I'm not familiar with, by simply googling words I didn't understand.

To put the level of mathematics in the paper into some sort of perspective: it would require undergraduate and graduate degrees in mathematics, and lots of graduate level mathematical physics knowledge to fully comprehend all the material. And that's assuming the student has the mathematical ability to get through all the prerequisites to get to that level. If your current knowledge doesn't stem beyond A-Level maths, the 2 year comment might be a bit of an underestimate!

If you'd like, I'm more than happy to recommend some undergraduate text books on maths/physics.

 

[CptBork]

Yeah math is kinda crazy like that, it gets incredibly intricate and everything builds on previous concepts. I think there are things in the field that no one on Earth could hope to learn in a short span like 2 years- that would be like climbing Mount Everest with 100 foot leaps. But once you have that background, the technical understanding, it's like having a hammer and a screwdriver whereas before all you had were your bare hands. Too bad you can sit down and set up all kinds of clever contraptions that work based on detailed mathematical knowledge, but you don't get any cool Jedi powers that you could command on the spot.

 

[Guest254]

You don't get Jedi powers after learning a sufficient amount of mathematics? Well that's put a dampener on my day.