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A spin network is a graph whose edges are associated with representations of a 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 and gauge theory.
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 of a two-dimensional surface
should have discrete spectrum. Every spin network is an eigenstate of each such operator, and the areaeigenvalue equals
where the sum goes over all intersections of
with the spin network. In this formula, is the gravitational constantAccording to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is called , the gr, is the Immirzi parameterThe Immirzi parameter (also known as the Barbero-Immirzi parameter is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. and is the spinSpin has several meanings, including those primarily discussed as spinning For physical rotation or an analogous phenomenon in sub-atomic physics, see spin (physics) For the periodical, see Spin Magazine Computer: For unproductive repetition in a computer associated with thelink 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.