Yesterday there were some sessions on Lattice techniques aimed at non-specialists attending the ICHEP conference. Apparently the attendance was disappointing. That is not very surprising given the competition from other parallel sessions where new physics could be announced. Lattice theory has been around for a long time and mostly looks at QCD which is far from new.

As an ex-lattice gauge theorist myself I think there are some aspects of it that people working on more sexy subjects such as quantum gravity would benefit from understanding better. In particular they should understand how the phase diagram of QCD at high temperature and density is being charted using these non-perturbative methods. The reason they need to know this is that a similar phase structure should exist in quantum gravity and there is likely to be a strong (but approximate) correspondence through AdS/CFT duality that relates quantum gravity to a QCD-like theory.

In the QCD theory of the strong interactions there is believed to be a temperature known as the Hadgdorn temperature above which nuclear matter breaks down into a quark gluon plasma. This happens at around 10 billion degrees Kelvin. In quantum gravity according to string theory (if you don’t like string theory dont switch off, this is just a short diversion) there is another Hagdorn temperature at around the Planck scale. That’s about 10^{32} degrees Kelvin. What happens there?

According to string theory the length of strings becomes very large and effectively the concept of the string breaks down. Sometimes string theorists call this the topological phase of string theory because they think that spacetime loses its geometry in the hotter phase. The truth is that not much is known about what really happens because most of string theory is based on perturbative calculations and phase transitions are very non-perturbative. What might happen is that not only geometry of space-time is lost but topology too. In that case it should be called the non-topological phase, or pregeometric phase. To put it another way, spacetime evaporates. Even if you don’t believe in string theory you might still consider this possibility. Some non-string theorists talk about geometrogenesis which is the process of cooling from the high temperature pregeometric phase to the more familiar geometric phase at the start of the big bang.

For now we can get some feel for the phase structure of quantum gravity by looking at the phase structure of QCD which brings me to one of the ICHEP talks from yesterday. However I’ll do that in a separate post in case people get confused and think it was about quantum gravity.

If the Planck scale is there because of the rise in temp. (=decoherence) reach a treshold value, then what brings in the Pauli exclusion? Because the Pauli exclusion is also the startpoint for gravity, and it can be the startpoint for inflation in Big Bang. Seen in that light it is hard to see how it can be ‘not topological’.

I think Pauli Exclusion can exist without topology, it is just about states :)

The Pauli exclusion principle is topological. The operator b or Q = 1/sqrt{2m}b is such that Q^2 = 0. This is the same thing as d^2 = 0. The intrusion of more topology enters in for Ψ \in kerQ^*/imQ^*, so the state is not determined by a boundary operator (here Q or Q^*) on some physical state. In this case the physical states are entirely topological in the BRST quantization.

http://lanl.arxiv.org/abs/0901.3816 Richard M. Weiner 2010

It is conjectured that all known fermions are topological solitons. This could explain the non-observation of bosonic leptons and baryons and provide a physical mechanism for the Pauli exclusion principle.

The link suggested by the Skyrme mechanism between quantum numbers of fermions and the Pauli principle might imply that a possible violation the conservation of fermion quantum numbers is associated with the violation of the Pauli principle and vice versa.

That’s the interesting part. You can construct simple operators with D^2 = 0 using matrices. The challenge is to get the topology back from the algebra of these operators. That’s the kind of thing the non-commutative geometry aims to do.

The skyrmion is a case of a non-linear sigma model. This is sort of close to what I have been banging away on for some time, where there is an equation of motion of the form

dg_{ij}/dt – nabla_iN_j – nabla_jN_i – Nk^2/2sqrt{g}G_{ijkl}δW/δg_{kl} = 0

where N and N_j are the lapse and shift functions. This is a form of the renormalization group equation The target or fixed point solution are the Einstein field equation for

W = 1/k^2∫d^3x(R – 2Λ).

The Skyrmion as a fermi field enters the picture when this potential energy portion has Bogoliubov coefficients, which both define quantum field in curved spacetime and superconductivity.

I have become very fond of the skyrmion myself:) It can be seen as a QCD model, and suits the standard model, and condensed matter physics. It’s topology is interesting. I have seen the same picture for the electron somewhere. But this is off-topic.

Is the topology ever lost in algebra? No need to get it back? The chirality is already in the vacuum field.

Two standard facts probably known by all. Finkelstein published decades ago a nice paper about a possible reduction of fermionic statistics to topology by demonstrating that 2pi rotation and a topological homotopy representing exchange of particles are homotopic. Dirac demonstrated that the so called orientation entanglement relation demonstrates explicitly that 2pi rotation is not equivalent with no rotation at all but 4pi rotation is.

Matti,

Can you tell me if the skyrmion fits TGD? With easy text for a simpleminded biologist :) I like the magnetic pic.

Where is 2 pi rotation and 4pi rotation used? Remember I have seen 4pi in CKM? But topology is saved also in CKM?

Spin and isospin are both magnetic, one weak gauge force, and one strong (QCD). They must have a relation. Then also EM-force from photons have magnetic field wave. I have tried to understand the difference, but not sure it is right.

Here an interesting link, with the two coupled to quantum Hall effect.

http://macbeth.if.usp.br/~gusev/ferromagnet.pdf