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To proceed further with the functional integral in ques- tion, introduce a set of regularly distributed nodal points in the parameter space, 

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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 

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The boundary values are required to be consistent with (B3). Explicitly, for all indices τ and σ it must hold 

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The discretization of Σ enables us to approximate the integral of p $^{A}$dq ∧ dt as follows [37]: 

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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 

the reason that will become clear in a moment, we choose the measure as 

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The first step, when dealing with the discretized func- tional integral 

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, 

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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 

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In the exponent there appears the discretized version of the integral 

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: 

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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 

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