The Quantum Tetrahedron in 3 and 4 Dimensions by John C. Baez

John Baez is one of my favourite physicists. He has done a lot of work on quantum gravity especially with regard to loops and knot theory. He used to maintain the ‘ This weeks finds in Mathematical Physics’ webpage.  In this paper Baez states that recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the ‘quantum tetrahedron’.  By starting with a classical phase space whose points correspond to geometries of the tetrahedron in R3, he uses geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labelled by the areas of the faces of the tetrahedron together with another quantum number, such as  the area of one of the parallelograms formed by midpoints of the tetrahedron’s edges.

tetrahedron_parallelogram

Parallelogram formed by midpoints of the tetrahedron’s edges

By repeating the procedure for the tetrahedron in R4, he also  obtains a Hilbert space with a basis labelled solely by the areas of the tetrahedron’s faces.

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