Against Symmetry
Kostant's talk on Lisi's $E8$ physics appears to have renewed interest in the idea of One Big Group to explain everything. But symmetry, on its own, explains nothing at all. The 20th century idea that (standard) model building has sufficient explanatory power in itself is hopelessly inadequate for tackling the problems of quantum gravity. Lots of smart people tried this idea (Lie group based GUTs) and they failed. Did anyone notice? This idea FAILED!
Consider some basic examples of symmetry groups and their representations: say rotations of a sphere. One easy way to shift to a larger group is to increase the dimension of the sphere. But in doing so, observe that nothing in the underlying geometry of the space has been enriched. Whatever the dimension of $\mathbb{R}^{n}$, the symmetry rules for a sphere rely on the basic properties of the real numbers, analysis and the axioms of set theory and topology. In this scheme, what is the symmetry group of a point? You don't know? Shouldn't we actually understand this if we want our spaces to be associated with physical spacetime and matter's internal degrees of freedom?
I am sure Lisi appreciates that his paper is not a final explanation of how to unify the SM with gravity, but rather a new direction to probe effective descriptions of SM particle fields. So where does $E8$ really come from? Maybe $E8$ is pure moonshine...
Consider some basic examples of symmetry groups and their representations: say rotations of a sphere. One easy way to shift to a larger group is to increase the dimension of the sphere. But in doing so, observe that nothing in the underlying geometry of the space has been enriched. Whatever the dimension of $\mathbb{R}^{n}$, the symmetry rules for a sphere rely on the basic properties of the real numbers, analysis and the axioms of set theory and topology. In this scheme, what is the symmetry group of a point? You don't know? Shouldn't we actually understand this if we want our spaces to be associated with physical spacetime and matter's internal degrees of freedom?
I am sure Lisi appreciates that his paper is not a final explanation of how to unify the SM with gravity, but rather a new direction to probe effective descriptions of SM particle fields. So where does $E8$ really come from? Maybe $E8$ is pure moonshine...
posted by Kea | 6:05 PM
7 Comments:
Dear Kea,
Something failed! It was GUTs: the naive and opportunistic idea of extending the group to larger group and get unification as a free lunch.
No one asked what is the fundamental reason why color and Poincare symmetries are exact and why electroweak symmetries are broken. For instance, could electroweak symmetries in higher-D context correspond to holonomies and unbroken symmetries to isometries? And what is really behind family replication- topology perhaps? Could
lepton and baryon number correspond to conserved fermion chiralities in some higher-D theory? And could standard model symmetries have deeper meaning: perhaps related to classical number fields.
Instead of making questions, maximally un-imaginative extension of gauge group was proposed and raised in few years to dogma. Now wonder it did not work. The wonder is that string theorists are still trying to reproduce something like this.
In the case of super-conformal symmetries the extension of symmetries might work. But this not anything naive. Replacing strings with light-like 3-surfaces (metric 2-dimensionality) means a huge extension of these symmetries and explains space-time dimension among other things. But not a single GUT unifier notices!;-)
Concerning Lisi and Kostant. People seem to believe that Kostant's work supports Lisi's model and Distler is wrong. This is not true. Kostant only shows that the product of two copies of SU(5), the smallest of these GUT groups unifying single fermion family, is contained in E(8) and he also considers compact form of E(8). Distler's objections besides a long list of other objections still hold true.
Lisi remains to me one of the media mysteries. E(8) has been proposed as a gauge group long time ago: for instance, Carlos Castro has done this and many people many years earlier.
Kac-Moody type algebra with 8-D Cartan algebra emerges naturally in TGD framework. Indeed, M^4 and 2-D Cartan algebras of ew group and color group indeed give 8-D Cartan algebra. Also representations of E_8 Kac Moody can be constructed by the standard Sugawara construction but they would lead to the same problems as Lisi's approach unless anyonization saves the situation.
Kea, I agree. When I see all the group theoretic / physics complaints about Lisi's E8 I ignore it because to me, the concept that nature is described by a perfect symmetry that is then broken is silly.
A square is a circle with broken symmetry? No, a square is a square. You might be able to approximate it as a circle under some circumstances, but it's still a square and it never was a broken circle.
Nature is perfect, the method of using broken symmetry to describe her is what is broken.
Hear, hear. I thought this point of view was obvious, but this week I have been staggered by the number of comments indicating otherwise.
Thanks for this interesting attack on symmetry groups, Kea. However, I can't really believe that writing like this:
"... symmetry, on its own, explains nothing at all. The 20th century idea that (standard) model building has sufficient explanatory power in itself is hopelessly inadequate for tackling the problems of quantum gravity. Lots of smart people tried this idea (Lie group based GUTs) and they failed. Did anyone notice? This idea FAILED!"
Why cares if "a lot of smart people failed". If they failed, maybe that fact simply implies that they weren't so smart as they (or their professors and their fans) claimed, rather than implying that searching for symmetry is a waste of time.
A lot of smart people can all be wrong, particularly if they all use variants of the same thinking (which happens in today's great educational environment, which manufactures clones).
"Consider some basic examples of symmetry groups and their representations: say rotations of a sphere. One easy way to shift to a larger group is to increase the dimension of the sphere. But in doing so, observe that nothing in the underlying geometry of the space has been enriched."
The key thing from my perspective is that - as Carl Brannen points out - the basic symmetries are broken. Why is the weak isospin force left-handed? Why is electroweak symmetry broken? Why does the SU(3) part of U(1) x SU(2) x SU(3) only apply to quarks, not leptons? In each case, the mainstream answer is broken symmetry, mechanisms for breaking symmetry.
When you have so many exceptions (breaks to symmetry), is it really worth while insisting that the universe is based on symmetry? Clearly, most of the most important things in physics are based on broken symmetry, i.e., asymmetry, which is a different story.
"One easy way to shift to a larger group is to increase the dimension of the sphere. But in doing so, observe that nothing in the underlying geometry of the space has been enriched."
Maybe the point of shifting to SU(5) (or whatever the GUT is supposed to be), is that it looks simpler the groupthink mentality can announce that nature holds a "deep beautiful simplicity", SU(5) or whatever. It doesn't address symmetry breaking, left force handedness, or anything already known for sure.
We know electromagnetic, weak and strong interactions exist, and if U(1), SU(2) and SU(3) describe these interactions, which they're supposed to (although U(1) looks a joke to me because I think there is evidence for a more complex structure behind electromagnetism), then symmetry does at least model stuff.
The description of mesons (quark-antiquark pairs) by SU(2) and the the description of baryon properties by SU(3) does seem to indicate that SU(2) as a description of weak interactions and SU(3) as a description of strong interactions, are sound.
"But symmetry, on its own, explains nothing at all."
Symmetry in particle physics is abstract, it's a mathematical description, not an explanation, so I presume the point you are making about Lisi's use of E8 is that it's not an explanation but just a way to model things abstractly.
You're not going to get anywhere along that road, because the word "explain" is meaningless in physics, which is just about making calculations if you're genuine (or blathering about largely uncheckable speculations if you're not).
"In this scheme, what is the symmetry group of a point? You don't know? Shouldn't we actually understand this if we want our spaces to be associated with physical spacetime and matter's internal degrees of freedom?"
I think the key priority is understanding how to link up a field Lagrangian formulation of say electromagnetism, to a symmetry group. I don't think it is currently being done correctly.
E.g., what is the symmetry group representing quantum gravity?
The Standard Model was constructed from empirical observations of three fundamental forces. Why not just add a fourth?
What's wrong with this idea? Why aren't people doing that? Is it a case can't find the correct Lagrangian of gravity? Most people think gravity is mediated by spin-2 particles so that like gravitational charges (mass, energy) always attract rather than repel as occurs with spin-1 particles mediating forces between similar charges.
I think U(1) is wrong in the S.M. because it is built on a flawed classical model of electromagnetism - the Lagrangian contains Maxwell's equations in tensor form.
This covers up errors in the underlying physics. The chief error in Maxwell's equations is the oversimplification inherent in displacement current, the alleged polarization of the vacuum results in a displacement of virtual charges. We know that this doesn't happen because Schwinger showed that the vacuum can't polarize at low energy (you need strong electric fields, above 10^18 v/m, to get pair production of polarizable charges in the vacuum).
So clearly, there is a more subtle mechanism at work which produces the illusion of displacement current. From evidence in practical electromagnetism, i.e. the propagation of a logic signal guided by conductors occurs at the velocity of light in the insulating medium between the conductors, it seems to be due to exchange radiation consisting of charged, massless bosons. The conventional explanation is some displacement current in the vacuum flowing from one charged conductor to the other (oppositely charged) conductor as the signal passes every point. However, since the field strengths are below Schwinger's threshold for pair production, that can't be the answer. The only way to account for the behaviour of a logic signal flowing the way it does is then to throw away Maxwell's displacement current model, and replace it with a radiation exchange between the conductors as a signal passes. The radiation emulates the mathematical relation that Maxwell gave in some circumstances, but Maxwell's law is only a crude over-simplification in other circumstances. To describe with the exchange radiation between the two conductors which replaces the classical theory of displacement current, quite a lot of new physics - closely linked to quantum field theory - is needed. E.g., the exchange radiation needs described by a lagrangian and evaluated with a path integral.
In order to make it work, it appears to suggest a change is needed to the usual Abelian U(1) formulation of electromagnetism in quantum field theory.
The key problem here is seeing how this relates to the SU(2) lagrangian in the standard model, e.g. see section 8.5 (pages 298-306) in Ryder's Quantum field theory, 2nd ed. Mathematically, Ryder's SU(2) lagrangian description is within my grasp, but trying to actually understand it is hard.
For example, the U(1) symmetry (from mainstream electromagnetism) is supposed to imply a weak hypercharge. I can't understand from reading Ryder (let alone from reading Weinberg's even more mathematically fuzzy writings in his three volumes on QFT), what the experimental evidence for each of the interpretations in the U(1) model is supposed to be: what is weak hypercharge, physicaly? Is there any direct evidence that it physically exists, or is it just a useful mathematical description like conductivity, entropy, frequency, etc. These things don't physically exist like fundamental particles or electric charges do, they are just descriptions of effects. Simply giving a name to something doesn't tell you if there is an underlying mechanism or not, let alone what the underlying mechanism (if one exists) is.
Either physicists should be concerned with mechanism or equations. If they don't believe that mechanisms exist or are loo lazy to investigate them, fine, let's just see the equations and see where the equations come from physically, i.e. what is the experimental evidence for the equations.
If you say this in quantum field theory, you get given lots of equations but the key insights are clearly not directly supported by experimental evidence. The thousands of tests of the Standard Model don't prove that the model used to describe electromagnetism in it is correct, any more than they check anything to do with the Higgs field. It's pretty obvious that the Standard Model, being based on experimental evidence from particle physics, is an accurate quantitative model for such physics. If it wasn't, then it wouldn't have been constructed that way. But is it the correct model, or is it to some extent an epicycle-type model?
As Carl Brannen and Nige emphasize, the basic symmetries are broken. This must be understood without ad hoc constructions. Groups are there - they accompany unavoidably any modern mathematical structure - but they are not necessary symmetry groups or gauge groups.
In TGD framework single hypothesis: H=M^4xCP_2 allows to understand standard model symmetries. What looks strange and weird in standard model framework reduces to a deeper understanding of the notion of symmetry: isometries and holonomies: nonbroken and broken symmetries. Something childishly simple but neglected.
This is not all. At the level of quantum TGD infinite-D dynamical symmetry groups characterizing measurement resolution generated by hermitian elements of subfactor emerge and bring in a new role for the notion of group. Extended conformal symmetries I have already mentioned. p-Adic thermodynamics brings in naturally dynamical symmetry breaking.
Some examples about how more refined conceptualization allows to understand physics.
*The breaking of left-right symmetry results automatically at the level of M^4 chiralities whereas at level of H chiralities are conserved and this corresponds to quark and lepton conservation. Who can count the number of papers published in GUT framework in attempt to make proton long lived enough!
*Quarks and leptons differ because they couple to different multiple of Kahler form: here also a very delicate point is involved. CP_2 does not allow standard spinor structure and one must couple given spinor chirality to an odd multiple of Kahler form of CP_2. Again something highly non-trivial which explains difference between quarks and leptons and gives standard model quantum numbers and couplings.
*Nige mentioned asymmetry of quarks and leptons with respect to color. In TGD the asymmetry between leptons and quarks is that they allow t= 0 and t=1color partial waves and due to differet coupling to Kahler gauge potential.
Here it is good to look what experiments say. In TGD both leptons and quarks can appear in color excited states and there is evidence already from seventies for the existence of bound pion like states of colored excitations of electrons. Last year similar evidence for bound state of colored excitations of muons emerged. The explanation for the gamma rays coming from galactic center regarded as one candidate for dark matter is as decay products of electropions.
To me the most obvious problems of particle physics are incredibily primitive conceptual thinking and complete silencing of experimental facts which do not fit the standard picture. These fellows receiving excellent salaries seem to be waiting that LHC in some miraculous manner does the thinking which they are to lazy to do.
Matti said "... M^4xCP_2 allows to understand standard model symmetries ...".
An easy way to see this is
for gravity, M4 has a conformal group symmetry that can give gravity by the MacDowell-Mansouri mechanism
for SU(3) and SU(2) and U(1) of the Standard Model, CP2 has symmetric space structure
CP2 = SU(3) / SU(2)xU(1)
(as far as I know, Batakis was the first to show how that worked)
Those group structures, seen in the Lagrangian, let you calculate force strengths etc using methods motivated by the work of Armand Wyler.
As Nige said "... physics ... is just about making calculations if you're genuine ..".
(also, Feynman said "... The whole purpose of physics is to find a number, with decimal points, etc! Otherwise you haven't done anything. ...".)
Tony Smith
Dear Tony,
Batakis failed to realize that the holonomy group of CP(2) is identifiable as electroweak weak gauge group and reduces electroweak symmetry breaking and M^4 chiral symmetry breaking to that for CP_2 for fixed imbedding space chirality determining whether quark or lepton is in question.
Instead, he introduced electroweak symmetries as and additional structure.
There are also highly non-trivial details related to color representations and Kaluza-Klein type approach would predict wrong correlation between electroweak and color quantum numbers of fermions. Only when one considers representations of super-conformal algebras one obtains leptons and quarks as massless ground states plus hadron like states from their colored excitations.
Best Regards,
Matti
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