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The Quantum Tetrahedron in 3 and 4 Dimensions

Baez, John C.; Barrett, John W.

The Quantum Tetrahedron in 3 and 4 Dimensions Thumbnail


John C. Baez

John W. Barrett


Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the tetrahedron's edges. Repeating the procedure for the tetrahedron in R^4, we obtain a Hilbert space with a basis labelled solely by the areas of the tetrahedron's faces. An analysis of this result yields a geometrical explanation of the otherwise puzzling fact that the quantum tetrahedron has more degrees of freedom in 3 dimensions than in 4 dimensions.


Baez, J. C., & Barrett, J. W. (1999). The Quantum Tetrahedron in 3 and 4 Dimensions

Journal Article Type Article
Publication Date Jan 1, 1999
Deposit Date Jul 30, 2001
Publicly Available Date Oct 9, 2007
Journal Adv.Theor.Math.Phys.
Peer Reviewed Peer Reviewed
Volume 3
Public URL


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