Ashtekar 1986: Departure of metric-
(background-) dependent formulation of GR, i.e. the affine connection as fundamental:
new variables: connection and conjugated momentum (this idea
goes back to discussions held by Einstein and Schrödinger)
Since 1995 Rovelli, Smolin and
Baez formulated a
new representation of canonical quantum gravity: loop quantum gravity
(LQG): quantum mechanical representations in which the wave functions are
functionals of the loop, which carry the gauge invariant information of the
theory (holonomies along families of closed curves)
Useof spin networks
(invented by Penrose in 1971, generalization of knot theory): representations
of a gauge group used to label the loops (spin networks determine
the gauge invariant functional of the connection):
Edges are assigned half-integer
labelings of irreducible representations of SU(2)
Vertexes are assigned tensor
products of representations, e.g. Clebsh-Gordon coefficients
States of LQG are represented
by spin networks (Dirac
quantization of constraint of canonical formalism of general relativity)
Baez: Spin foam models,
dynamically evolving spin networks, as path integral formalism: "forces
as edges and matter as loose ends of spin networks"
Thiemann (student of Ashtekar):
Complete formulation of LQG (gave rigorous definition of the
Hamiltonian constraint)
Some
Features
Geometrical operators for length,
area and volume are discrete; discreteness used for:
calculation of black
hole entropy up to a constant
resolving big bang
singularities
proof of the finiteness
of the path integral formalism
Manifestly background independent
(in contrast to string/M-theory)
Manifestly UV and IR convergent
without renormalization: volume operator as natural regulator, serving as
cut-off
No extra dimensions, no supersymmetry
- but also no constraint on number of dimensions and particle families (should/can
LQG be merged with string/M-theory,
i.e. are strings and loops variations
of the same theme?)
New notion of perturbative quantum
field theory very different from Fock space idea (no Minkowski metric, no
Poincaré group)
Possible experimental verification
(gamma-ray burst effect, Amelino-Camelia)
Possible defect of string/M-theory:
AdS/CFT and Rehren's duality: either string theory on a background
is not a theory to which the usual Haag-Kastler framework applies (would be
surprising) or there is no Lagrangian origin for M-theory - not even
in its low energy limit
LQG can reproduce
Einstein's field equations in 2 and 3 dimensions, however, the full 4-dimensionall
case remains a challenge
Ref.: Thiemann, "Introduction
to Modern Canonical Quantum General Relativity", gr-qc/0110034;
Carlip, "Quantum Gravity: a Progress Report", gr-qc/0108040
Some General
Issues/Problems
Epistemological (i.e. problem
due to lack of knowledge of reality):
"What is the structure of
space-time?" New space-time structure, or none at all? Qunatize the topology?
"What is time?" Conflict between
the notion of time in general relativity (no preferred time slicing) and
quantum mechanics (fixed times), or xxx.lanl.gov/abs/quant-ph/0207029,
what about more than one time dimension (mathematically)?
"What is matter?" What is
mass, what are particles, and what about the Higgs mechanism?
Mathematical models:
Symmetry: what is the ultimate
symmetry/invariance priciple?
"Wave function of universe"
Quantum cosmology brings up all the conceptual problems that were swept
under the carpet
Quantum field theory and scattering
states: in general relativity is the free field theory a good approximation
even asymptotically?
Axioms: What can be derived,
what is required (Standard Model operators, symmetry groups, scalars, spinors,
tensor-particles,...)?
Implications of a discrete
nature of reality: time evolution, 0-point energy, cosmological constant problem,
holographic principle
How does complexity emerge
from fundamental simplicity? Wolfram
and the continuation of Mandelbrot's work (chaos theory, fractal geometry):
"A New Kind of Science"
stating a new paradigm (the
algorithmic or computational foundation of reality and formal thought systems)