A 4d Lorentzian Spin Foam Model With Timelike Surfaces
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We construct a 4d Lorentzian spin foam model capable of describing both spacelike and timelike surfaces. To do so we use a coherent state approach inspired by the Riemannian FK model. Using the coherent state method we reproduce the results of the EPRL model for Euclidean tetrahedra and extend the model to include Lorentzian tetrahedra. The coherent states of spacelike/timelike triangles are found to correspond to elements of the discrete/continuous series of SU(1,1). It is found that the area spectrum of both spacelike and timelike surfaces is quantized. A path integral for the quantum theory is defined as a product of vertex amplitudes. The states corresponding to timelike triangles are constructed in a basis diagonalised with respect to a noncompact generator. A derivation of the matrix elements of the generators of SL(2,C) in this basis is provided.