2.6 The measure

One can derive non-trivial conditions on the lattice measure by imposing the Slavnov–Taylor identities in a perturbative lattice calculation. Starting from the general form of the measure for Poincaré gravity,
∏ det(e (n),e (n),e (n),e (n ))N ∕16 ∏ f(e2(n)) ∏ de (n) ∏ dU (n,n + μ ), (5 ) μ ν ρ σ μ μ L nμνρσ nμ n,μ>0 n,μ>0
Menotti and Pelissetto [166] performed a one-loop calculation and found that the Slavnov–Taylor identity can be satisfied for a particular choice of N, f1,f2 (momentum expansion coefficients of f), and of the cosmological constant λ. The solution still depends on a real parameter ξ (related to a residual non-invariance under rotations). This makes it difficult to draw any immediate conclusions on the structure of the full, non-perturbative measure.
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