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Pyramid scheme

My ex-adviser posted a written version of his talk dedicated to Murray Gell-Mann for his 80th birthday,

Holographic space-time and its phenomenological implications.


Tom Banks is presenting several cute yet profoundly controversial pet ideas he has been nurturing for a decade. I would see four different ideas:
  1. Formulation of a full theory of quantum gravity from a set of holographic axioms about the Hilbert spaces
  2. The importance of a black hole fluid, with the "p = +rho" equation of state
  3. A solution to the cosmological constant problem based on SUSY breaking induced by virtual black-hole states near the cosmic horizon
  4. A phenomenological model of particle phenomenology, extending "trinification" unified theories to a tetrahedron: the Pyramid Scheme
Tom usually presents these four ideas as a coherent whole. Frankly speaking, I don't really understand the full logical relationship between any pair among the four entries above. The main unifying theme that connects them seems to be Tom himself. But I admit that there can exist some links that I simply haven't appreciated yet. And the individual ideas are arguably appealing - and conceivably viable - in isolation, at least some of them.




Holographic axiomatics

The first idea is a set of axioms that are believed by Banks to be rooted in holography. In his approach, Hilbert spaces are associated with causal diamonds - rhomboid-shaped regions in spacetime with light-like boundaries. Tom attributes Hilbert spaces to such diamonds and their arbitrary intersections and believes various things about them.

Well, I surely do share the belief that regions with light-like boundaries - event horizons of various kinds - are very important for a proper understanding of quantum gravity, especially its information aspects. They also behave in a special way when you analytically continue things into the Euclidean spacetime (for example, the black hole interior completely disappears). However, I seem to disagree with pretty much every other "detailed" statement that Tom is making.

First, I don't believe that there are any "single pixel Hilbert spaces", namely that all Hilbert spaces are naturally written as tensor powers of this minimal "single pixel Hilbert space". Well-understood vacua in string theory show that this idea - similar to various "quantization of area" and related discrete approaches to physics - is just way too constraining and naive. The spaces of states are much more complex than simple powers of simple low-dimensional spaces.

But even more generally, I don't believe that in quantum gravity, Hilbert spaces may be precisely constructed as tensor products of Hilbert spaces coming from regions: the independence of several regions of spacetime is what quantum gravity violates, in my understanding of holography. In fact, I am confident that even weaker assumptions of Tom's sort are incorrect. I don't believe that well-defined Hilbert spaces with integral dimensions can be "exactly" associated with finite causal diamonds.

Also, I don't believe that the Hilbert space of quantum gravity should depend on the diamond, on the slicing, and other things. Tom claims that this dependence exists. The space must be completely universal - and quantum gravity must explain how the new degrees of freedom may "emerge" when the space gets bigger. In fact, I even believe that morally speaking, even the degrees of freedom behind the cosmic horizons must be already included in your local ones, by a generalized "black hole complementarity". An empty region of space (e.g. in the black hole interior) must also be explained as an averaging over very many microstates associated with the event horizon or another light-like boundary and these are the constructions that are still waiting to be fully hacked.

Another fact is that I don't believe that the "intersections of causal diamonds" are natural regions of space that should have nicer Hilbert spaces associated with them than generic regions with light-like boundaries of an arbitrary shape. Note that the intersection of two diamonds is not a diamond itself (if the spacetime dimension exceeds two) but you cannot get an arbitrary shape with light-like boundaries by intersections, either.

Also, I don't think that the unitarity equivalence of two density matrices from page 3 is deep or staggeringly complex. It looks naive to me. To summarize, there are just way too many assumptions here that i consider to be either completely incomprehensible or manifestly incorrect. So I don't know what to do with this pillar of the theory. I don't know where to start if we want to follow it.

In Spring 1999, when Tom came to give lectures at TASI in Boulder, Colorado, I was first asked to complete Tom's ingenious ideas about the holographic axioms into a full-fledged theory of everything. Needless to say, I failed and I am still failing in this task. ;-)

Black hole gas

At any rate, Tom claims that his complicated conditions imposing the "generalized unitarity equivalence" between density matrices from different causal diamonds can be solved but the only solution is the "black hole fluid".

Recall that if the equation of state is "p = w rho" for some constant "w", the speed of sound, "sqrt(d |p| / d rho)", is still forbidden to exceed the speed of light which implies that "w" must be between "-1" and "+1". The minimum value "-1" is achieved by the cosmological constant which locally preserves the Lorentz symmetry. Tom is interested in the opposite extreme value, "+1".

One can derive this "extreme" relationship between the positive pressure and the positive energy density, "p = +rho", from the energy-entropy power law that follows from a "gas of black holes" - a hypothetical continuum of black holes whose separation is not too different from the Schwarzschild radii.

It's an interesting heuristic representation of the extreme equation of state. But otherwise I don't understand anything about it. Is that actually realized somewhere in Nature? Or everywhere in Nature? If it is, how is it compatible with the manifestly different equations of state that we observe in the known forms of matter? Isn't the observation of the "more mundane" equations of state a falsification of Tom's equations that don't admit other solutions than the black hole fluid?

If you add this extreme, and Lorentz-breaking, "p = +rho" equation of state, what do you exactly explain? It's necessary to be honest here: after a decade, I am just not still getting it.

Solution to the C.C. puzzle near the cosmic horizon

Tom argues that this black hole fluid is related to another cool idea of his: a solution to the cosmological constant problem. The normal way of looking at the problem is to assume that the Planck scale is "natural" and the vacuum energy density is "unnaturally tiny", by 123 orders of magnitude (or 60+ of them if you assume a low-energy broken supersymmetry).

Banks' perspective is the opposite one. He adopts the units linked to the cosmological constant whose value is therefore "natural". So he has to explain why the mass scale equal to the fourth root of the vacuum energy density is so gigantic in these new "natural" units. That's surely a legitimate reversal of the perspective. To explain the paradox in the new language, Tom assumes that supersymmetry exists but it is broken by some effects where the virtual particles pretty non-locally probe the vicinity of the cosmic horizon, and return back to you.

These virtual effects induce huge mass splittings between the superpartner masses - relatively to the new "natural" mass scale - but they don't affect the vacuum energy density. If it were true, it would be just super-neat. Except that I don't understand why it should be true, why the two effects of the virtual particles should be so different. Lots of nice exotic concepts such as the "black hole fluid" are being used here but I don't understand how these concepts can do the job that is expected from them.

The Pyramid Scheme trinification of particle physics

Finally, there is a fourth building block that Tom seems to link to the previous ideas: the pyramid scheme of particle physics. He's been working on it with various collaborators including Jean-Francois Fortin whose name can also be spelled as "Fourtin" in Canada, at least by Paul Frampton. :-)

By the requirement of the absence of Landau poles; preservation of minimal SUSY GUT's one-loop gauge coupling unification; and the absence of the undesired new pseudo-Nambu-Goldstone bosons, Tom argues that the previous principles single out a pretty specific model for particle physics beyond the Standard Model. That's how the hugely speculative ideas above are supposed to converge to a very specific technical conclusion about particle physics.

I don't understand the logic in its entirety but it is conceivable that the logical relationship between this "Pyramid Scheme" of particle physics and the "cosmological breaking of supersymmetry" announced in the previous points is tighter than I am able to realize.

Trinification and friends

It is useful to discuss an older model that led to the "Pyramid Scheme". It was called "trinification" and studied by many physicists including Sheldon Glashow. It is a mutated version of "grand unification".

In ordinary grand unification, the Standard Model's gauge group, "U(1) x SU(2) x SU(3)", is embedded into a simple group such as "SU(5)" or "SO(10)" or "E_6". Such a simple group is "one", just like one God, and it has therefore one coupling constant only. This constant is still undetermined, but because it runs, there is effectively no free gauge-coupling-like dimensionless parameter associated with the GUT theories.

In trinification, one replaces "SU(5)" by something like "SU(3) x SU(3) x SU(3)". One of the factors is colorful; another one contains the weak isospin while the hypercharge is a combination of generators from several factors. All the elementary quarks and leptons come in bi-fundamental representations such as "(3,3bar,1)" under various pairs of the "SU(3)" groups.

Despite the largely Jewish physicists who gave it life, trinification is a Christian version of unification. ;-) In unification, there is only one God with one coupling constant. However, in trinification, there are three simple factors. However, because this Holy Trinity is composed out of three players that are isomorphic to each other, namely "SU(3)" - recall that God the Son is actually isomorphic to His father and the Holy Spirit as well -, one can impose a "Z_3" symmetry that exchanges these three factors.

Requiring the three coupling constants to be invariant under this "Z_3" (at high energies) effectively means that there's only one coupling constant, just like in ordinary unification. The lesson is that there's no moral difference between one God and three Gods: as you have surely known, Christianity works just like any other religion. ;-)

I will mention some aspects of the trinification momentarily. But Banks claims that something better than the trinification is needed. It's the "Pyramid Scheme" although one could call it "tetranification", too. You need four "SU(3)" groups instead of three and they can be arranged to vertices of a tetrahedron. There are three families of "(3,3bar)" representations associated with any pairs of the vertices of the tetrahedron.

Let me spend some time with terminology here. It is useful to check the Latin and Greek numerical prefixes to follow the following superficial discussion.

Tom uses the term "Pyramid Scheme" because the last time he visited Egypt, the bases of the pyramids were triangles rather than squares which they became after the modern reconstruction. ;-) So the pyramids used to be tetrahedrons and the word "pyramid" is therefore appropriate for a tetrahedron.

More seriously, the prefix "uni-" is a Latin cardinal prefix. The corresponding prefix for the number 3 is "tri-" - which happens to be identical to the Greek cardinal prefix for 3. However, one might wonder whether the right word shouldn't be just "trification" rather than "trinification".

At any rate, the extrapolation to the number 4 should be "quadrification" or "quadrufication". However, that sounds bad because this Latin prefix creates a wrong impression that the four elements are ordered along the vertices of a square rather than a tetrahedron. That's why it's superior to commit the linguistic sin and switch to the Greek prefixes and call it "tetrafication" instead. :-)

That's been enough linguistic fun for this text.

Back to some maths. You might think that the trinification is a completely different construction than unification and that trinification can't be "embedded" into unification. However, you would be wrong.

In fact, the "SU(3) x SU(3) x SU(3)" trinification group is a subgroup of "E_6" which is a viable grand unified group. The complex "27" representation transforms as "(3,3bar,1)" plus the two cyclic permutations of it while the adjoint decomposes as
78 = (8,1,1) + (1,8,1) + (1,1,8) +
+ (3,3bar,1) + (1,3,3bar) + (3bar,1,3) +
+ (3bar,3,1) + (1,3bar,3) + (3,1,3bar).
Note that 78 = 24 + 2x 27. So the gauge group of tetranification naturally decomposes under the trinification subgroup. However, the "(3,3bar,1)" terms in the decomposition above can't be interpreted as the lepton and quark chiral families - for example because we only got one of them, not three of them.

Nevertheless, it's useful to remember that the groups can be embedded in this way. Consequently, one may try to construct trinification models in heterotic string theory, too. However, because of the bi-fundamental matter fields, the obvious preferred approach to embed these systems into string theory are intersecting braneworlds.

However, Tom needs four copies of an "SU(3)". Can you embed it into an exceptional group? You bet. Because "E_8" contains an "E_6 x SU(3)" subgroup and I have just explained that "E_6" contains an "SU(3)^3" subgroup, it is clear that "E_8" contains an "SU(3)^4" subgroup. The four "SU(3)" subgroups can indeed be arranged at vertices of a tetrahedron and the fundamental representation of "E_8" decomposes as
248 = 4 terms like (8,1,1,1) +
+ 6 terms like (3,3bar,1,1)
+ 6 terms like (3,3bar,3,1).
The treatment of the four "SU(3)" subgroups in this decomposition is not symmetric with respect to the "S_4" permutations of the four "SU(3)" factors.

By the way, you may have been impressed that four "SU(3)"s whose total rank equals eight can be squeezed into this single "E_8" group whose rank is also eight. But "E_8" is pretty big for a Lie group of rank eight (its dimension divided by squared rank equals 31/8, nearly 4, much more than 1 for large "SU(n)" groups or 2 for large "SO(2n)" or "Sp(2n)" groups) and such embedding is pretty common.

You can also embed "A_4 x A_4" i.e. "SU(5) x SU(5)" as a subgroup into "E_8", a trivial fact that eluded a "top E_8 expert", the surfer dude Garrett Lisi. ;-) The decomposition is
248 = (1,24) + (24,1) +
+ (5,10) + (10bar,5) +
+ (5bar,10bar) + (10,5bar).
Also, one can embed "(A_1)^8" i.e. "SU(2)^8" into the "E_8" group simply because "SU(2) x SU(2)" is locally isomorphic to an "SO(4)", and "SO(4)^4" can be embedded into "SO(16)" which is clearly a subgroup of "E_8". Under the "SU(2)^8" subgroup, the fundamental representation decomposes as
248 = 8 terms like (3,1,1,1,1,1,1,1) +
14 terms like (2,2,2,2,1,1,1,1).
Note that 248 = 8x 3 + 14x 16. Those 8 terms are obvious - the triplet can be in any factor. The 14 terms that are doublets under four "SU(2)" groups are more non-trivial. If you represent the four "SU(2)" factors under which you have a doublet as "a,b,c,d" written as four binary numbers between "000" and "111" each, then the "EXOR" of all these four numbers "a,b,c,d" must be zero.

It's easy to see that we can distribute the four doublets among 8 places - given the condition above - in 14 ways. Three of the digits "2" can be placed arbitrarily, and "(8 choose 3)" is equal to "8 x 7 x 6 / (3 x 2 x 1) = 56". The fourth factor that will see the representation as a doublet is already determined by the "EXOR" condition. But in this method, we count each quadruplet four times (because the last, completed factor may be any one among the four), so we only find "56 / 4 = 14" distinct quadruplets of this kind.

This decomposition can be easily seen if you decompose "248" as "120+128" under "SO(16)" and then embed "SO(4)^4" into "SO(16)", and so on.

So the trinification and the "Pyramid Scheme" allow you more freedom about the choice of the matter (non-adjoint) representation - and string theory may also reshuffle, suppress, or proliferate the matter spectra if a gauge group is broken, either by field-theoretical mechanisms or the stringy ones - but the basic structure of the trinified or pyramid gauge fields can still be embedded into the unified field theories.

Summary

But again, I don't see how this simple and relatively conventional model of particle physics can lead to a solution of the cosmological constant problem via the virtual liquid black holes near the cosmic horizon. Tom clearly wants to modify the rules to calculate observables in quantum field theories dramatically, so that the effect on the mass splittings is huge while the effect on the vacuum energy is negligible. I have tried to do something similar many times in many ways but I don't understand what the modified rules actually are in Tom's setup and why they don't contradict the standard tests of the quantum field theory methods that have been empirically established.

However, despite my misunderstanding of the logic of these constructions in most of their aspects, I still haven't gotten rid of the feeling that Tom may know something really deep that is still waiting for others who may understand it and follow it in the future. This feeling is partially given by my experience showing that Tom is not only very bright but he is actually a conservative physicist (in the apolitical sense only, of course) so he can't possibly be saying completely crazy things without any reason.

We may see the right answer sometime in the future.

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reader Mitchell said...

I think these lectures he gave a few months later are the best discussion I've seen from him about his philosophy of quantum gravity. They're just marvellously profound, full of insights to think about, and also full of useful technical leads. Very high quality.

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