Spin network

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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.

A spin network, immersed into a manifold, can be used to define a functional on the space of connections on this manifold. One simply computes holonomies of the connection along every link of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way. A remarkable feature of the resulting functional is that it is invariant under local gauge transformations.

[edit] In the context of loop quantum gravity

In loop quantum gravity, a spin network represents a "quantum state" of the gravitational field on a 3-dimensional hypersurface. The set of all possible spin networks (or, more accurately, "s-knots" - that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG Hilbert space.

One of the key results of loop quantum gravity is quantization of areas: the operator of the area A of a two-dimensional surface Σ should have a discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals

A_{\Sigma} = 8\pi G_{\mathrm{Newton}} \gamma \sum_i \sqrt{j_i(j_i+1)}

where the sum goes over all intersections i of Σ with the spin network. In this formula,

  • GNewton is the gravitational constant,
  • γ is the Immirzi parameter and
  • j_i=0,1/2,1,3/2,\dots is the spin associated with the link i of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.

According to this formula, the lowest possible non-zero eigenvalue of the area operator corresponds to a link that carries spin 1/2 representation. Assuming Immirzi parameter on the order of 1, this gives the smallest possible measurable area of ~10-66 cm2.

The formula for area eigenvalues becomes somewhat more complicated if the surface is allowed to pass through the nodes ( it is not yet clear if these situations are physically meaningful. )

Similar quantization applies to the volume operator. The volume of 3-d submanifold that contains part of spin network is given by a sum of contributions from each node inside it. One can think that every node in a spin network is an elementary "quantum of volume" and every link is a "quantum of area" surrounding this volume.

[edit] 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 A. 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.

[edit] 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.)
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