This week I have been exploring the length operator in LQG. One of the papers I’ve been reading is this one by Thiemann. Like all of Tiemann’s work it is mathematically quite sophisticated so I’ll precis it in a slightly less mathematical manner.
In this paper the author constructs an operator that measures the length of a curve for four dimensional Lorentzian vacuum quantum gravity. He works in a representation in which a SU(2) connection is diagonal and finds that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization.
In the paper it is shown that the length operator admits self-adjoint extensions. Part of the length operator’s spectrum is computed and found like the volume and area operators to be purely discrete and roughly is quantized in units of the Planck length. It also contains full and direct information about all the components of the metric tensor which aids the construction of a new type of weave states which approximate a given classical 3-geometry.
Spectrum of length operator
The full expression is so complicated that I’ll just give some special cases:
- Trivalent Vertex
For triavalent vertex with J1, J2, J3 we have eigenvalues :
- Divalent Vertex
For divalent vertex with J1 = J2 = Jo we have eigenvalues:
- The Quantum of Length
The quantum of length is achieved for Jo = 1/2 and given by:
The length can change only in packets of:
which approaches zero for large spin so that for high spin the length looks like a continuous operator.
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