Backreaction has reminded me of the time I worked at Parkes telescope back in 1989, just before the Voyager Neptune encounter. Marissa Shaw and I were running the night shift (radio telescopes can run 24/7). NASA was there laying extra cables for the Voyager encounter. Many readers would be amazed at the banks of computers that lined the control room: their processing power wouldn’t beat a modern laptop, and tracking data was backed up with handwritten logbooks.

But despite such primitive technology, Voyager was out there, exploring the solar system. It was possible because the physics was simple, allowing a focus on the essential effort of strategy and engineering. Are we really supposed to believe that, in a mere 20 years, the laws of physics have suddenly become ridiculously complicated? Physics is the science of building simple tools for the computation of useful physical quantities, such as particle masses.

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## nige said,

September 16, 2007 @ 1:22 pm

I agree with what you say about building simple laws and tackling particle masses. I know Carl Brannen is working on this.

Maybe sometime you could write about Lie algebras and symmetry groups, and whether different symmetry groups can be related by category theory. E.g., if leptons and quarks are unified at very high energy, how do you get the SU(3) symmetry group emerge, or more to the point (seeing that the universe in its earliest and simplest stages was at the highest energy!), if the basic symmetry group is something like SU(4), can you use category theory to deal with the combinations of subgroups it can produce? At present I think that symmetry groups are extremely difficult to understand by the existing methods, so maybe category theory could be applied to them to make it simpler to see what the facts are concerning all possible symmetries and where broken symmetries are needed, etc.

Woit has some introductory lectures starting with this one located near the end of this page. He teaches that stuff and has also recently linked to new English notes on it by ‘t Hooft here.

Quite a lot of stuff there I know in a non-technical way because I was interested in the 8 fold way and how all the hadrons are explained in terms of quarks using the SU(2) and SU(3) symmetries (expressed as geometric drawings with the different particles plotted at the vertices). Also, I did quantum mechanics and feel familiar with the Schroedinger equation, etc.

What I don’t understand are most existing (modern) textbooks about Lie groups which immediately get into technical details of the mathematical tools and steer clear of solid physics. It’s just a drone of trivia, relying on the ability of the reader to memorise it.

Recently however I decided to buy a quantity of books on Lie algebra from the time (or just before the time) that the standard model was being developed, and “Lie Groups for Pedestrians” by Harry Lipkin (2nd ed 1966, reprinted 2002 by Dover) is a relatively painless introduction to the key maths.

From experimental data back in the 1960s, SU(3) was shown to be the correct (predictive, experimentally confirmed) symmetry for strong interactions between hadrons, and to imply quarks.

From the 1970s to the present, we know SU(2) models the weak interaction because the neutral currents of exchanged Z_0 massive, chargeless bosons were observed in interactions in the 1970s and evidence for the massive charged W’s was obtained at CERN in 1983.

So there is plenty of evidence that SU(2) with 3 massive gauge bosons models the weak force which left-handed spin fermions experience, as well as quark-antiquark pairing in mesons, and that SU(3) with its 8 gluons describes the triplets of quarks and their binding in baryons.

Where I think the standard model U(1)xSU(2)xSU(3) falls down is in the simplest group U(1) which is used for electromagnetism, and also in lacking gravity.

There are various issues with electromagnetism being modelled by U(1) – it has only a single charge, and only a single electrically neutral gauge boson. You can’t get a ‘charged’ field mediated by such gauge boson exchange without invoking the fact that the gauge bosons of U(1) are special and have 4 polarizations, the 2 additional polarizations obviously are manifested as electric field. But then you effectively have charged massless gauge bosons, and since they have two charges (positive, negative), you really have 2 distinct gauge bosons. Feynman argues that you can treat positive charge as negative charge going back in time to make U(1) work, but this isn’t going anywhere useful. In addition, you then have to introduce a Higgs field to break the U(1)xSU(2) electroweak symmetry, and that can’t make falsifiable exact predictions of the Higgs mass because there are various possibilities available.

It would be far more convenient to have SU(2) account for both electromagnetism and the weak force, so that you have two charges but you also have the 3 gauge bosons exist in both massive and massless versions. The two massless charged gauge bosons mediate positive and negative electric fields. The neutral massless gauge boson gives gravity.

The main problem then is introducing correctly the mechanism for the weak force (mediated by 3 massive SU(2) gauge bosons) to only act on left-handed spinors, and working out all the mathematical structure and predictions from this model that differ from the standard model.

Lipkin’s book at page 110 discusses the SU(4) group, which can contain SU(2)xSU(2) and SU(3) as subgroups. So maybe something like SU(4) will contain the entire set of symmetries for all forces, when the physical conections are properly understood in detail.

Perhaps category theory could help to untangle the problem of what is the correct symmetry group of the universe?

I think that energy conservation might be helpful for fundamental forces. It seems as if Louise’s equation arises if the energy needed to cause the big bang E=mc^2 is equal to the potential energy of the gravitational field which would be released if the matter all collapsed, E = mMG/R = mMG/(ct).

Then you get E = mc^2 = mMG/(ct) which gives tc^3 = MG (Louise’s equation). There are various other ways of deriving it.

If you think about it, it’s pretty logical that the energy used to blast matter apart against gravity should be equal to the gravitational potential energy!

By analogy, for a Saturn V, the energy needed to make it get to the moon is basically the gravitational potential energy difference. (OK, air drag plays a factor near the ground, but such effects aren’t important for the Big Bang I’m considering.)

If an explosion is open and doesn’t collapse due to gravity, then it must have contained enough energy E = mc^2 for it to have overcome the gravity force tending to collapse the fireball.

The gravitational potential energy of mass m with respect to mass of universe M located at at average radius R is E = mMG/R. This is very simple physics. I fail to understand why people ignore it.

More generally, energy conservation is vital for understanding the force unification problems. When particles are bought together in new ways and binding is done by weak or strong or electromagnetic forces, energy is tied up in that process and if you know the energy density of the field (which is well known for electromagnetism as a function of field strength, but is more controversial for the other forces), you can easily integrate that field strength over space to get total energy.

I wonder whether category theory can help to simplify the complex table of particle charges by showing physically (with energy conservation principles) how weak forces emerge when particles approach one another? Can a very high energy (as yet unobserved) transformation of leptons into quarks occur?

## nige said,

September 16, 2007 @ 1:23 pm

Sorry Kea – comment above is far too long.

## Kea said,

September 16, 2007 @ 10:02 pm

Hi Nigel. Occasional long comments are allowed! One of my motivations for working with category theory is to get away from Lie theory and ordinary manifolds. Eg. the eightfold way is actually about Lattices. Since the N of SU(N) is associated with categorical dimension, this suggests leaving the recovery of Lie symmetries until after the whole ‘Koszul heirarchy’ is sorted out. My view of ordinary SU(N) unifications is that SU(5) is interesting because it relates to Calabi-Yau spaces (see Baez’ stuff) but these are then better viewed as 3D analogues of elliptic curves for Number Theory. In the end, the Lie theory can only be an effective description for commutative spaces, which cannot be more than part of the story. Carl’s method is promising because it also takes a qubit point of view of spacetime events. My thesis is on qubit logic in toposes, also with the hope of better understanding causality outside of the realm of Lie theory.

## L. Riofrio said,

September 17, 2007 @ 4:58 pm

That must have been cool working at “The Dish.” The Parkes telescope made some big contributions to Space missions.

Since humans went to the Moon and Space probes to the outer solar system without them, one wonders whether all these computers have done us any good. Today’s students have far less arithmetic ability. The online archives are full of mediocre papers, and original papers still have difficulty getting published. Then there is Spam.

“Physics is the science of building simple tools for the computation of useful physical quantities…” Your conclusion is a valuable one. The availability of computers has led to over-complicated theories of the Universe. If computers had been available in the Rennaissance, we might still be using epicycles.

## Kea said,

September 17, 2007 @ 9:51 pm

Today’s students have far less arithmetic ability.Yes, I must admit to being stunned at how hopeless my students are at algebra. Even the ones with obvious talent in physics can barely cancel common factors from an abstract fraction without breaking out in a sweat. I was taught arithmetic by my mother as a toddler, via the stern methods she learnt in the 1930s from some Irish nuns in the North Island countryside, so I find the inability of young people to manipulate fractions in their head frightening.