11 Second, the wider class U contains plenty of degenerate (shrunk) surfaces, the histories ˜ γ . These are obviously stationary surfaces of S when being varied within U , since The one-form we are integrating is W ⸥ Ω = W$_{a}$ $^{(}$dq a − ( p $^{a}$/m ) dt $^{)}$, but since both q and t stay unchanged on λ , the integral is zero. As a result, to annihilate the bound- ary term for all variational fields W we are forced to chose the curves ˜ γ$_{0}$ = (˜ q$_{0}$ ( t ) , ˜ p$_{0}$ ( t ) , t ) and ˜ γ$_{1}$ = (˜ q$_{1}$ ( t ) , ˜ p$_{1}$ ( t ) , t ) in the extended phase space in such a way that their instant tangent vectors annihilate Ω at any moment t ∈ [ t$_{0}$, t$_{1}$ ] . Thus, it must hold for any variational vector field W . However, only one of these histories, namely γ$_{cl}$ , satisfies both equations (A2) and (A3) at the same time. This fictitious problem is avoided when one works from the beginning with the stringy subclass U$_{γ}$$_{cl}$ ⊂ U . or, equivalently, Let us explain here in some detail how the surface functional integral is computed. To be as tangible as possible consider a one-dimensional system only, so that the extended phase space will be the three dimensional space R $^{3}$[ q, p, t ] (more dimensions represent only a tech- nical problem). The particle is supposed to move under the combined action of the potential U ( q ) and the fric- tion force with a general A -power law. The dynamical equations are for both ˜ γ$_{0}$ and ˜ γ$_{1}$ . Here one recognizes the dynamical equations (5) as desired. From the boundary term analysis and the assumption about the uniqueness of the history γ$_{cl}$ which we adopted from the very beginning we can conclude that Σ$_{ext}$ ∈ U$_{(}$$_{γ}$$_{cl}$$_{,γ}$$_{cl}$$_{)}$ ⊂ U$_{γ}$$_{cl}$ . Hence Σ$_{ext}$ has the topology of a closed string attached to the classical trajectory γ$_{cl}$ . To specify its shape the second term in (A1) must be employed. The resulting variational equation, written in a coordinate- free notation, is S (˜ γ $^{δW}$) − S (˜ γ ) = GLYPH<2> ˜ γ δW Ω − GLYPH<2> ˜ γ Ω = 0 − 0 = 0 0 = d ˜ γ dt ⸥ Ω , (A2a) 0 = ( F$_{a}$ − ˙ p$_{a}$ ) dq a +$^{(}$ ˙ q a − p a m$^{)}$ dp$_{a}$ + + $^{(}$˙ q $^{a}$F$_{a}$ − ˙ p$_{a}$ p$_{a}$ m$^{)}$ dt , (A2b) Appendix B: Surface functional integral – computational details and in addition to them, we require that the particle satisfies the boundary conditions q ( t$_{0}$ ) = q$_{0}$ and q ( t$_{1}$ ) = q$_{1}$ . According to the definition (8), the two-form Ω is Our aim is to compute the surface functional integral (9) over the stringy set U$_{γ}$$_{cl}$ = ⋃ ˜ $_{γ}$U$_{(˜}$ γ,γ$_{cl}$ $_{)}$. The direct application of (10) and (B1) together with the Stokes theorem yields This is equivalent to the following system of partial dif- ferential equations for the unknown functions q $^{a}$( t, s ) and p$_{a}$ ( t, s ) (indices a and k run from 1 , . . . , n ): One solution of these equations, satisfying all boundary conditions, is trivial. It is the shrunk surface Σ$_{ext}$( t, s ) = γ$_{cl}$ ( t ) . After recalling the assumption of uniqueness of γ$_{cl}$ once more, we can see that this is the only solution of (A3). If there existed a closed unshrunk extremal surface Σ$_{ext}$ , the initial set of equations of motion (5) would have at least one one-parametrical family of classical solutions between the given pair of endpoints. ˙ q = p , ˙ p = − d dq U ( q ) − κ ˙ p A , Ω = d $^{(}$pdq − 1 2 p $^{2}$dt − U ( q ) dt $^{)}$− κ p $^{A}$dq ∧ dt . (B1) 0 = d Ω $^{(}$∂$_{t}$ Σ$_{ext}$ , ∂$_{s}$ Σ$_{ext}$ , . $^{)}$. (A3a) 0 = ∂q k ∂s ∂F$_{k}$ ∂p$_{a}$ , 0 = ∂q k ∂s ( ∂F$_{a}$ ∂q k − ∂F$_{k}$ ∂q a $^{)}$+ ∂p$_{k}$ ∂s ∂F$_{a}$ ∂p$_{k}$ . (A3b) GLYPH<2> U$_{γ}$$_{cl}$ [ D Σ] exp { i ℏ GLYPH<2> Σ Ω } = (B2) = GLYPH<2> [ D ˜ γ ] exp { i ℏ $_{(}$GLYPH<2> ˜ γ − GLYPH<2> γ$_{cl}$ ) $^{(}$pdq − 1 2 p $^{2}$dt − U ( q ) dt ) $^{}}$× × GLYPH<2> U$_{(˜}$ γ,γ cl ) [ D Σ] exp $^{{}$− i ℏ κ GLYPH<2> Σ p $^{A}$dq ∧ dt $^{}}$. Σ( t, 0) = γ$_{cl}$ ( t ) Σ( t, 1) = ˜ γ ( t ) and Σ( t$_{0}$, s ) ∈ R $^{3}$[ q$_{0}$, p, t$_{0}$ ] Σ( t$_{1}$, s ) R $^{3}$[ q$_{1}$, p, t$_{1}$ ] . (B3) Σ ∈ U$_{(˜}$ γ,γ$_{cl}$ ) ⊂ U$_{γ}$$_{cl}$ ↦→ S (Σ) = $_{(}$GLYPH<2> ˜ γ − GLYPH<2> γ$_{cl}$ $^{)}$( pdq − Hdt ) . Σ : [ t$_{0}$, t$_{1}$ ] × [0 , 1] → R $^{3}$[ q, p, t ] , ( t, s ) ↦→ Σ( t, s ) , The nontrivial part of this is the functional integral over the stringy subset U$_{(˜}$ γ,γ$_{cl}$ $_{)}$. As mentioned earlier, Σ ∈ U$_{(˜}$ γ,γ$_{cl}$ ) is a map such that for ∀ t ∈ [ t$_{0}$, t$_{1}$ ] and ∀ s ∈ [0 , 1] there holds The variational principle we described above operates on a wider stringy class U than it is in fact necessary. The surface integral formula (9) requires just the restricted subset U$_{γ}$$_{cl}$ ⊂ U . The transition to U$_{γ}$$_{cl}$ is advisable for two reasons. First, in the Lagrangian case with Ω = d ( pdq − Hdt ) we obtain an equivalence (modulo additive constant) between the least action principle using S and the standard Hamilton least action principle. Explicitly,