- #106
physics8553
DataBase said:(1) How come there are no equations in that paper?
(2) Where are the lie groups? The Teichmuller spaces or what ever?
(3) I don't see how Einstein's equations can be derived without even doing any calculations to begin with.
About 1 and 3: There is an intermediate step that is described in my 6th volume. Einstein's field equations follow from the fact that c^4/4G is an invariant quantity that is also a limit quantity. This happens in the same way that special relativity follows from the fact that c is an invariant quantity that is also a limit quantity. For this reason, to derive general relativity, it is sufficient to find a model that reproduces the invariant limit c^4/4G. The strand model does so by design; this limit is part of the basic posulate that a crossing switch defined the Planck units. As a result, Einstein's field equations follow from the strand model. The way this is done uses the old 1995 argument about the thermodynamics of space-time. The 6th volume gives all the details (it is about 1 page in total).
The same is valid for hbar. If a model reproduces the observer-invariance of hbar, and also spin 1/2 behaviour, then it contains Dirac's equation. This has been shown in 1980 already. I explain it in the 6th volume in more modern language. In particular, the least action principle also follows, and thus the existence of Lagrangains.
Any unified model for general relativity and quantum theory thus only has to reproduce the observer-invariance of hbar and of c^4/4G. Any model that does so contains the two theories. The riddle then is to find the simplest such model. Since we need extension to get black hole entropy, the strand model comes up as the simplest such model. A crossing change is a crossing change for any observer (with some subtleties); thus the basic postulate already includes general relativity and relativistic quantum theory.
About 2: In the strand model, the Lie algebra and Lie group structure follow from the definition of wave function and wave function phase, and from the three ways that tangle cores can be deformed.
In more detail, the wave function is the short-term average crossing density (produced by the short-time fluctuations of a tangle). The phase is seen very naively, as the short-time averaged orientation of tangle crossings.
Given this definition of the phase, the three possible ways to deform tangle cores (which in turn define wave functions) yield three possible ways to change phases. These three possible ways are the three interactions. Each deformation can be generalized to a Lie albegra and then to a Lie group, and it turns out that the three Lie groups U(1), SU(2) and SU(3) follow from the three Reidememister moves.
Given that strands contain the DIrac Lagrangian, the appearance of these Lie groups yields the QED, QCD and then (with more details) the broken electroweak SU(2) Lagrangian. In short, the Lie groups are seen as reults of deformations of tangle cores, and all this is happening in a 3d background. There are no other complex abstract spaces involved. This is a simple summary of the ideas leading to the strand model, and of the way the Lie groups appear.