Feynman Path Integral Formulation

274 8 Numerical Studiesare not affected by uncontrollable errors, such as for example the 2 + ε expansionof Sect. 3.5. Since the lattice cutoff and the method of dimensional regularizationcut the theory off in the ultraviolet in rather different ways, one needs to compareuniversal quantities which are cutoff-independent. One example is the critical exponentν, as well as any other non-trivial scaling dimension that might arise. Withinthe 2 + ε expansion only one such exponent appears, to all orders in the loop expansion,as ν −1 = −β ′ (G c ). Therefore one central issue in the lattice regularizedtheory is the value of the universal exponent ν.Knowledge of ν would allow one to be more specific about the running of thegravitational coupling. One purpose of the discussion in Sect. 3.3 was to convincethe reader that the exponent ν determines the renormalization group running ofG(μ 2 ) in the vicinity of the fixed point, as in Eq. (3.22) for the non-linear σ-model,and more appropriately in Eq. (3.117) for quantized gravity. From a practical pointof view, on the lattice it is difficult to determine the running of G(μ 2 ) directly fromcorrelation functions , since the effects from the running of G are generally small.Instead one would like to make use of the analog of Eqs. (3.29), (3.59) and (3.60)for the non-linear σ-model, and, again, more appropriately of Eqs. (3.121) and possibly(3.127) for gravity to determine ν, and from there the running of G. Butthecorrelation length ξ = m −1 is also difficult to compute, since it enters the curvaturecorrelations at fixed geodesic distance, which are hard to compute for (genuinelygeometric) reasons to be discussed later. Furthermore, these generally decay exponentiallyin the distance at strong G, and can therefore be difficult to compute dueto the signal to noise problem of numerical simulations.Fortunately the exponent ν can be determined instead, and with good accuracy,from singularities of the derivatives of the path integral Z, whose singular part isexpected, on the basis of very general arguments, to behave in the vicinity of thefixed point as F ≡− 1 V lnZ ∼ ξ −d where ξ is the gravitational correlation length.From Eq. (3.121) relating ξ (G) to G − G c and ν one can then determine ν, aswellas the critical coupling G c .8.2 Observables, Phase Structure and Critical ExponentsThe starting point is once again the lattice regularized path integral with action asin Eq. (6.43) and measure as in Eq. (6.76). Then the lattice action for pure fourdimensionalEuclidean gravity contains a cosmological constant and Regge scalarcurvature term as in Eq. (6.90)I latt = λ 0 ∑hV h (l 2 ) − k∑δ h (l 2 )A h (l 2 ) , (8.1)hwith k = 1/(8πG), and leads to the regularized lattice functional integral∫Z latt = [dl 2 ] e −λ 0 ∑ h V h +k ∑ h δ h A h, (8.2)