History of Canonical Quantum Gravity

- 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)

- Use
*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

- Diffeomorphism invariant

- Non-perturbative (in contrast to string/M-theory)

- 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)