Spin network
From Free net encyclopedia
A spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. It was invented by Roger Penrose in 1971. Spin networks were applied to the physics problem of quantum gravity by Carlo Rovelli, Lee Smolin, Fotini Markopoulou-Kalamara, and others to reformulate loop quantum gravity in the canonical approach. There, they chop off the Lorentz gauge group Spin(3,1), which is noncompact to SU(2), which is compact. Later, it was generalized to gauge theories with connections in general.
In the context of loop quantum gravity
One of the key results of loop quantum gravity is quantization of areas: according to several related derivations based on loop quantum gravity, the operator of the area <math>A</math> of a two-dimensional surface <math>\Sigma</math> should have a discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals
- <math>A_{\Sigma} = 8\pi G_{\mathrm{Newton}} \gamma
\sum_i \sqrt{j_i(j_i+1)}</math>
where the sum goes over all intersections <math>i</math> of <math>\Sigma</math> with the spin network. In this formula,
- <math>G_{\mathrm{Newton}}</math> is the gravitational constant,
- <math>\gamma</math> is the Immirzi parameter and
- <math>j_i=0,1/2,1,3/2,\dots</math> is the spin associated with the link <math>i</math> of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.
Similar quantization applies to the volume operators but mathematics behind these derivations is less convincing.
More general gauge theories
(Outside the context of LQG, the name spin networks is a bit of a misnomer...)
As mentioned, it was noticed that analogous constructions can be made for general gauge theories with a compact Lie group G and a connection form. This is actually an exact duality over a lattice. Over a manifold however, assumptions like diffeomorphism invariance are needed to make the duality exact (smearing Wilson loops is tricky). Later, it was generalized by Robert Oeckl to representations of quantum groups in 2 and 3 dimensions using the Tannaka-Krein duality. Michael Levin and Xiao-Gang Wen have also defined another generalization of spin networks which they call string-nets using tensor categories. String-net condensation produces topologically ordered states in condensed matter.
Publications
Some random early papers (none of them actually called them spin networks; that is Penrose's name for them):
- Hamiltonian formulation of Wilson's lattice gauge theories, John Kogut and Leonard Susskind, Phys. Rev. D 11, 395–408 (1975)
- The lattice gauge theory approach to quantum chromodynamics, John B. Kogut, Rev. Mod. Phys. 55, 775–836 (1983) (see the Euclidean high temperature (strong coupling) section)
- Duality in field theory and statistical systems, Robert Savit, Rev. Mod. Phys. 52, 453–487 (1980) (see the sections on Abelian gauge theories)
Modern papers:
- The dual of non-Abelian lattice gauge theory, Hendryk Pfeiffer and Robert Oeckl, hep-lat/0110034.
- Exact duality transformations for sigma models and gauge theories, Hendryk Pfeiffer, hep-lat/0205013.
- Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants, Robert Oeckl, hep-th/0110259.
- Spin Networks in Gauge Theory, John C. Baez, Advances in Mathematics, Volume 117, Number 2, February 1996, pp. 253–272.
- Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions, Xiao-Gang Wen, [1]. (Dubbed string-nets here.)
