10 The formula we have just arrived at was derived under the assumption that Ω = d ( pdq − Hdt ) . However, it is clear that the expression in the exponent is free of any reference to the Hamiltonian and requires just the clas- sical equations of motion. Therefore it seems reasonable to postulate the probability formula (28) also for non- Lagrangian systems. The approach we shortly presented here does not use the notion of pure states of a non-Lagrangian system. It resembles the Feynman-Vernon approach [10], in which one introduces the influence functional in the presence of dissipative forces. Our probability formula (28) uses the surface functional integral in the extended phase space, while in the Feynman-Vernon approach one just has to calculate a double path integral in the configura- tion space. There is a chance that after a certain dis- cretization we will be able to convert the surface func- tional integral into the double path integral in the con- figuration space (see appendix B), and as a result, we will find the explicit form of the influence functional. Work on this topic is in progress. Σ$_{ext}$ Σ δW ext δ W ( q $_{1}$, t$_{1}$ ) = fi x e d ( q$_{0}$, t$_{0}$ ) = fixe d ˜ γ$_{0}$ ˜ γ$_{1}$ ˜ γ ′ 0 ˜ γ ′ 1 λ$_{1}$ λ ′ 1 λ$_{0}$ λ ′$_{0}$ FIG. 6: Vector field W acting infinitesimally in the extended phase space. As a result, the initial surface Σ$_{ext}$ ∈ U$_{(˜}$ γ$_{1}$, ˜ γ$_{0}$ ) moves to some nearby surface Σ δW ext ∈ U$_{(˜}$ γ ′ $_{1}$, ˜ γ ′ $_{$_{0}$)}$, and its bound- ary ∂ Σ$_{ext}$ = ˜ γ$_{1}$ − λ$_{1}$ − ˜ γ$_{0}$ + λ$_{0}$ is transported to a new boundary ∂ Σ δW ext = ˜ γ ′ 1 − λ ′ 1 − ˜ γ ′ $_{0}$+ λ ′ $_{0}$. Since the variational vector field is assumed to preserve U , W stays tangential to the momentum submanifolds with fixed ( q$_{0}$, t$_{0}$ ) and ( q$_{1}$, t$_{1}$ ) . This research was supported by MŠ SR CERN- Fellowship Program and VEGA Grant 1/1008/09. Special thanks go to Vladimír Balek for his interest and fruitful discussions we have had in the course of the work. where L W stands for the Lie derivative. Let us compute the surface integral on the left hand side: W -fields it holds Acknowledgments Appendix A: Stringy variational principle A . M . D . G . where the the symbol ⸥ denotes the inner product (con- traction) of a vector with a differential form. The first term on the right hand side depends only the values of Ω on the boundary of Σ$_{ext}$ and can be recast into the form In section III we shortly outlined how the transi- tion amplitude (9) can be related to the stringy varia- tional principle, with the action S defined in (11). In what follows we will perform a complete variation of S , with the domain extended from U$_{γ}$$_{cl}$ to a wider class U := ⋃ ˜ γ$_{0}$ ⋃ ˜ $_{γ$_{1}$}$U$_{(˜}$ γ$_{1}$, ˜ γ$_{0}$ $_{)}$. The stringy set U contains all surfaces in the extended phase space trapped between the submanifolds with fixed ( q$_{0}$, t$_{0}$ ) and ( q$_{1}$, t$_{1}$ ) (includ- ing separate histories, regarded as shrunk surfaces). In the course of variation the classical dynamics (5) we have started from will be recovered. 0 = lim δ → 0 S $^{(}$Σ δW ext ) − S $^{(}$Σ$_{ext}$) δ =$^{GLYPH<2>}$ Σ$_{ext}$ L $_{W}$Ω , GLYPH<2> Σ$_{ext}$ L $_{W}$Ω =$^{GLYPH<2>}$ Σ$_{ext}$ d $^{(}$W ⸥ Ω ) + W ⸥ d Ω = = GLYPH<2> ∂ Σ$_{ext}$ W ⸥ Ω +$^{GLYPH<2>}$ Σ$_{ext}$ W ⸥ d Ω , (A1) GLYPH<2> ∂ Σ$_{ext}$ W ⸥ Ω = $_{(}$GLYPH<2> ˜ γ$_{1}$ − GLYPH<2> ˜ γ$_{0}$ + GLYPH<2> λ$_{0}$ − GLYPH<2> λ$_{1}$ $^{)}$W ⸥ Ω . W ∣ $_{λ$_{0}$}$= W$_{a}$ $^{(}$p, q$_{0}$ , t$_{0}$ ) ∂ ∂$_{p}$$_{a}$ , W ∣ $_{λ$_{1}$}$= W$_{a}$ $^{(}$p, q$_{1}$ , t$_{1}$ ) ∂ ∂$_{p}$$_{a}$ . The last two terms here give individually zero contribu- tions, as can be seen from the following quick consider- ation: the vector field W is assumed to preserve U and therefore its restrictions to the λ -boundaries are Suppose we have an extremal surface Σ$_{ext}$ ∈ U$_{(˜}$ γ$_{1}$, ˜ γ$_{0}$ ) ⊂ U and a variational vector field W defined in its neigh- borhood such that the flow of W preserves U . By defini- tion, W moves Σ$_{ext}$ ∈ U$_{(˜}$ γ$_{1}$, ˜ γ$_{0}$ ) to some other worldsheet Σ δW ext ∈ U$_{(˜}$ γ ′ $_{1}$, ˜ γ ′ $_{$_{0}$)}$, where δ is an infinitesimal increment of the parameter of the flow generated by W . The situation is depicted schematically in Figure 6. The extremality of Σ$_{ext}$ means that for all variational