The little I know about LQG is that it is 3+1-D theory with 3-geometries as basic objects expressed in terms of loop variables. Witten considers 3-D theory with Chern Simons action: in this case 2-geometries would be the basic dynamical objects. Witten himself made clear that he has no idea about how to generalize the theory to 4-D context.
A connection to 4-D theory in TGD framework is obtained if one brings in holography, replaces 3-metrics with light-like 3-surfaces (light-likeness constraint is possible by 4-D general coordinate invariance), and accepts the new view about S-matrix implied by the zero energy ontology.
- Light-like 3-surfaces can be regarded as solutions vacuum Einstein equations with vanishing cosmological constant (Witten considers solutions with non-vanishing cosmological constant). The effective 2-D character of the induced metric is what makes this possible.
- Zero energy ontology is also an essential element: quantum states of 3-D theory in zero energy ontology correspond to generalized S-matrices: Matrix or M-matrix might be a proper term. Matrix is a "complex square root" of density matrix -matrix valued generalization of Schrödinger amplitude - defining time like entanglement coefficients. Its "phase" is unitary matrix and might be rather universal. Matrix is a functor from the category of Feyman cobordisms and matrices have groupoid like structure.
- Theory becomes genuinely 4-D because S-matrix is not universal anymore but characterizes zero energy states.
- 4-D holography is obtained via the Kähler metric of the world of classical worlds assigning to light-like 3-surface a preferred extremal of Kähler action as the analog of Bohr orbit containing 3-D lightlike surfaces as submanifolds (analogs of blackhole horizons and lightlike boundaries). Interiors of 4-D space-time sheets corresponds to zero modes of the metric and to the classical variables of quantum measurement theory (quantum classical correspondence). The conjecture is that Dirac determinant for the modified Dirac action associated with partonic 3-surfaces defines the vacuum functional as the exponent of Kähler function with Kähler coupling strength fixed completely as the analog of critical temperature so that everything reduces to almost topological QFT.
- The counterpart of ordinary S-matrix is between zero energy states. I call it U-matrix. It has nothing to do with particle reactions. It is crucial for understanding consciousness via moment of consciousness as quantum jump identification.
2 comments:
"The little I know about LQG is that it is 3+1-D theory with 3-geometries as basic objects expressed in terms of loop variables. Witten considers 3-D theory with Chern Simons action: in this case 2-geometries would be the basic dynamical objects. Witten himself made clear that he has no idea about how to generalize the theory to 4-D context."
Peter Woit does give a discussion of the basic principle of LQG in his book:
‘In loop quantum gravity, the basic idea is to use the standard methods of quantum theory, but to change the choice of fundamental variables that one is working with. It is well known among mathematicians that an alternative to thinking about geometry in terms of curvature fields at each point in a space is to instead think about the holonomy [whole rule] around loops in the space. The idea is that in a curved space, for any path that starts out somewhere and comes back to the same point (a loop), one can imagine moving along the path while carrying a set of vectors, and always keeping the new vectors parallel to older ones as one moves along. When one gets back to where one started and compares the vectors one has been carrying with the ones at the starting point, they will in general be related by a rotational transformation. This rotational transformation is called the holonomy of the loop. It can be calculated for any loop, so the holonomy of a curved space is an assignment of rotations to all loops in the space.’ - P. Woit, Not Even Wrong, Jonathan Cape, London, 2006, p189.
I watched Lee Smolin's Perimeter Institute lectures, "Introduction to Quantum Gravity", and he explains that loop quantum gravity is the idea of applying the path integrals of quantum field theory to quantize gravity by summing over interaction history graphs in a network (such as a Penrose spin network) which represents the quantum mechanical vacuum through which vector bosons such as gravitons are supposed to travel in a standard model-type, Yang-Mills, theory of gravitation. This summing of interaction graphs successfully allows a basic framework for general relativity to be obtained from quantum gravity.
It's pretty evident that that "loops" are the closed exchange cycles of gravitons going between masses (or other gravity field generators like energy fields), back and forward, in an endless cycle of exchange. That's the loop mechanism, the closed cycle of Yang-Mills exchange radiation being exchanged from one mass to another, and back again, continually.
According to this view, the graviton interaction nodes are associated with the 'Higgs field quanta' which generates mass.
Hence, in a Penrose spin network, the nodes represent the points where quantized masses exist.
I think the mainstream is being misled by spin-2 graviton ideas, and the U(1) component of the Standard Model is wrong and SU(2) describes electromagnetism (as well as isospin). The SU(2) symmetry models two types of charges, hence negative and positive charges without the wrong method U(1) uses where it specifies there are only negative charges and positive ones are negative ones going backwards in time. In addition, SU(2) gives 3 massless gauge bosons, two charged ones (which mediate the charge in electric fields) and one neutral one (which is the spin-1 graviton, that causes gravity by pushing masses together).
In addition, SU(2) describes doublets, charge-anticharge pairs. We know that electrons are not produced individidually, only in lepton-antilepton pairs. The reason why electrons can be separated a long distance from their antiparticle (unlike quarks) is simply the nature of the binding force, which is long range electromagnetism instead of a short-range force.
The problem is that string people reflexively stamp the label "crackpot" immediately on to any alternative ideas for unifying the Standard Model and gravity.
Concerning LQG the description of Woit more or less summarized my basic understanding.
I am not very enthusiastic about functional integral approach nor about replacement of space-time with a discrete structure. I see discretization as something related to the finite accuracy of cognition and to finite measurement accuracy: this leads to number theoretic vision based on number theoretic braids involving only algebraic points.
The notion of finite measurement accuracy can be described using inclusions of hyper-finite factors of type II_1 and quantum groups emerge naturally.
Post a Comment