We are indebted to Andrey Mironov and Niels Obers for useful discussions. Introducing we rewrite the measure of [6] for the integration over reparametrizations as The integration over v$_{i}$ ’s in Eq. (A2) can be represented through the integration over a scalar field as follows. Writing we have and This represents the continuous measure as where t is a certain parametrization of the contour (e.g. through the proper time) and D ψ is the usual measure The scalar field ψ , that appears in Eqs. (A1), (A3), is in fact a discretization of the boundary value of the Liouville field ϕ ( τ, s ) ∣ boundary = ψ ( s ) , (A8) 13 ∫ D ψ = s$_{f}$ ∏ s = s 0 ∫ $^{+}$∞ −∞ d ψ ( s ) . (A7) ∫ s$_{f}$ s 0 D $_{diff}$s = ∫ D ψ δ (1) ( s$_{f}$ − s$_{0}$ − ∫ s$_{f}$ s 0 d t e − ψ ( t ) / 2 ) , (A6) ∫ s$_{f}$ s 0 D $_{diff}$s = lim $^{N}$→∞ 1 2 N N ∏ i =1 ∫ $^{+}$∞ −∞ d ψ$_{i}$ δ (1) ( s$_{f}$ − s$_{0}$ − N ∑ j =1 e − ψ$_{j}$/ 2 ) . (A5) ∫ ∞ 0 d v$_{i}$ v$_{i}$ · · · = 1 2 ∫ $^{+}$∞ −∞ d ψ$_{i}$ (A4) v$_{i}$ = e ψ$_{i}$/ 2 , (A3) ∫ s$_{f}$ s 0 D $_{diff}$s ≡ lim $^{N}$→∞$^{N}$ − 1 ∏ i =2 ∫ s$_{i}$$_{+1}$ s 0 d s$_{i}$ ( s$_{i}$$_{+1}$ − s$_{i}$ ) ∫ s 2 s 0 d s$_{1}$ ( s$_{2}$ − s$_{1}$ )( s$_{1}$ − s$_{0}$ ) = lim $^{N}$→∞ N ∏ i =1 ∫ ∞ 0 d v$_{i}$ v$_{i}$ δ (1) ( s$_{f}$ − s$_{0}$ − N ∑ j =1 v$_{j}$ ) . (A2) v$_{i}$ = s$_{i}$ − s$_{i}$ − 1 s$_{N}$ = s$_{f}$ , (A1) Appendix A: Measure for integrating over reparametrization Acknowledgments