12 the reason that will become clear in a moment, we choose the measure as The first step, when dealing with the discretized func- tional integral U$_{(˜}$ γ,γ cl ) [ D Σ] :=$^{+}$$^{∞}$ GLYPH<2> −∞ K − 1 ∏ τ =1 L − 1 ∏ σ =1 ( AκΔ 2 πi$_{ℏ}$ p A − 1 ( τ,σ ) $^{)}$dq$_{(}$$_{τ,σ}$$_{)}$ dp$_{(}$$_{τ,σ}$$_{)}$. (B5) Int ≡ U$_{(˜}$ γ,γ cl ) [ D Σ] exp $^{{}$− i ℏ κ Σ p $^{A}$dq ∧ dt $^{}}$, xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx .... .... + + + + + + + + + + + + + + + + + + + + + + + + Δ ε t s ( τ, σ ) t$_{0}$ + τ Δ 0 + σ ε τ = 0 τ = 1 τ = K σ = 0 σ = 1 σ = L FIG. 7: Rectangular nodal web in the parametric space [ t$_{0}$, t$_{1}$ ] × [0 , 1] of a surface Σ ∈ U$_{(˜}$ γ,γ$_{cl}$ ) . Each elementary tile encloses the area Δ · ε = ( t$_{1}$ − t$_{0}$ ) / ( KL ) (at the end, the numbers K and L will be sent to infinity). The points marked by crosses are constrained by the conditions (B3). consists in the integration over the internal stringy posi- tions dq$_{(}$$_{τ,σ}$$_{)}$ . After this integration, the integrand trans- forms into a chain of delta functions, To proceed further with the functional integral in ques- tion, introduce a set of regularly distributed nodal points in the parameter space, The next step is the integration over the internal stringy momenta dp$_{(}$$_{τ,σ}$$_{)}$ . After performing this trivial integra- tion we arrive at the expression The points of the set are labeled by two discrete indices, the time index τ = 0 , . . . , K and the space index σ = 0 , . . . L , see Figure 7. In this way we obtain for any Σ - map from U$_{(˜}$ γ,γ$_{cl}$ ) a discretized ( K + 1) × ( L + 1) -tuple In the exponent there appears the discretized version of the integral Int = + ∞ GLYPH<2> −∞ K − 1 ∏ τ =1 L − 1 ∏ σ =1 $^{(}$A p A − 1 ( τ,σ ) $^{)}$dp$_{(}$$_{τ,σ}$$_{)}$ δ $^{(}$p A ( τ,σ − 1) − p A ( τ,σ ) $^{)}$× × exp $^{{}$− i ℏ κ$^{K}$$^{−}$$^{1}$ ∑ τ =1 Δ $^{[}$q$_{(}$$_{τ,L}$$_{)}$ p A ( τ,L − 1) − q$_{(}$$_{τ,}$$_{0)}$ p A ( τ, 0) ] $^{}}$. $^{{}$( t$_{0}$ + τ Δ, σ ε ) ∈ $^{[}$t$_{0}$ , t$_{1}$ ] × [0 , 1] } . Int = exp $^{{}$− i ℏ κ$^{K}$$^{−}$$^{1}$ ∑ τ =1 Δ $^{[}$q$_{(}$$_{τ,L}$$_{)}$ p A ( τ, 0) − q$_{(}$$_{τ,}$$_{0)}$ p A ( τ, 0) ] $^{}}$. Σ ↭ $^{{}$Σ( t$_{0}$ + τΔ, σε ) ≡ $^{(}$q$_{(}$$_{τ,σ}$$_{)}$ , p$_{(}$$_{τ,σ}$$_{)}$, t$_{0}$ + τΔ )} ( K,L ) ( τ,σ )=(0 , 0) . t$_{1}$ GLYPH<2> t$_{0}$ dt $^{{}$˜ q ( t ) p A $_{cl}$( t ) − q$_{cl}$ ( t ) p A $_{cl}$( t ) $^{}}$, q$_{(}$$_{τ,}$$_{0)}$ = q$_{cl}$ ( t$_{0}$ + τΔ ) , p$_{(}$$_{τ,}$$_{0)}$ = p$_{cl}$ ( t$_{0}$ + τΔ ) , q$_{(}$$_{τ,L}$$_{)}$ = ˜ q ( t$_{0}$ + τΔ ) , p$_{(}$$_{τ,L}$$_{)}$ = ˜ p ( t$_{0}$ + τΔ ) , q$_{(0}$$_{,σ}$$_{)}$ = q$_{0}$, q$_{(}$$_{K,σ}$$_{)}$ = q$_{1}$. GLYPH<2> U$_{(˜}$ γ,γ cl ) [ D Σ] exp $^{{}$− i ℏ κ GLYPH<2> Σ p $^{A}$dq ∧ dt } = = exp $^{{}$− i ℏ κ t$_{1}$ GLYPH<2> t$_{0}$ dt $^{[}$˜ q ( t ) p A $_{cl}$( t ) − q$_{cl}$ ( t ) p A $_{cl}$( t ) ] $^{}}$. (B6) where ˜ q ( t ) stands for the q -projection of ˜ γ ( t ) . Conse- quently, after returning back to the continuum limit we obtain the following important result: The boundary values are required to be consistent with (B3). Explicitly, for all indices τ and σ it must hold The discretization of Σ enables us to approximate the integral of p $^{A}$dq ∧ dt as follows [37]: After substituting (B6) into the initial formula (B2) we recover a path integral in the extended phase space. The integral is quadratic in momenta, with the discretized (standard) Liouville measure Formally, the discretized functional integral over all stringy configurations from U$_{(˜}$ γ,γ$_{cl}$ ) is a multiple integral over all unconstrained variables x$_{(}$$_{τ,σ}$$_{)}$, p$_{(}$$_{τ,σ}$$_{)}$ which are needed to specify Σ . The only problematic part, as usual, is the choice of an appropriate integration measure. For [ D ˜ γ ] · · · = + ∞ GLYPH<2> −∞ d ˜ p$_{K}$ 2 π$_{ℏ}$ K − 1 ∏ τ =1 d ˜ p$_{τ}$ d ˜ q$_{τ}$ 2 π$_{ℏ}$ · · · , Σ p $^{A}$dq ∧ dt := K − 1 ∑ τ =0 L − 1 ∑ σ =0 Δε $^{{}$p A ( τ,σ ) q$_{(}$$_{τ,σ}$$_{+1)}$ − q$_{(}$$_{τ,σ}$$_{)}$ ε } = = K − 1 ∑ τ =1 L − 1 ∑ σ =1 Δ $^{{}$q$_{(}$$_{τ,σ}$$_{)}$$^{(}$p A ( τ,σ − 1) − p A ( τ,σ ) $^{)}}$+ + K − 1 ∑ τ =1 Δ $^{{}$q$_{(}$$_{τ,L}$$_{)}$ p A ( τ,L − 1) − q$_{(}$$_{τ,}$$_{0)}$ p A ( τ, 0) $^{}}$. (B4)