2 This determines the phase of N . stringy approach. In section VII conclusion, discussion and outlook are collected. The section also includes some comments on the relation between the stringy quantization and the Caldeira-Leggett model of quantum Brownian motion [9], as well as the influence functional technique by Feynman and Vernon [10]. A miraculous consequence of the definition of the tran- sition amplitude (1) (not an additional requirement!) is that it satisfies the evolutionary chain rule, or Chapman- Kolmogorov equation, Some rather technical material is left to appendices. Appendix A is devoted to the stringy variational prin- ciple, which plays an important role in the motivation of our approach. In appendix B computational details concerning the surface functional integral for quantum friction force systems are presented. The infinitesimal version of this formula is the celebrated Schrödinger equation. Quantum states of the system are described by the square integrable functions with the standard Hilbert space structure. Physical observables are hermitian op- erators acting on such functions. Given the state Ψ$_{0}$( q ) at the initial moment t$_{0}$ one is able to predict the state Ψ$_{1}$( q ) at any later moment t$_{1}$ according to the formula Historically the first attempt at quantization based on the dynamical equations of motion belongs to Feynman, see [11]. A similar problem was considered by Wigner, Yang & Feldman, Nelson, Okubo and others, see [12]. Among the recent investigations in this field let us men- tion the work of Lyakhovich & Sharapov [13] and Gitman & Kupriyanov [14]. They consider the same problem as us, but their strategy is different. In our opinion, their approach fits much better the context of gauge field dy- namics. In what follows it is assumed that the above concept of states, observables and quantum evolution is valid for the stringy quantization of non-Lagrangian systems as well. The only new element is a modified prescription for the evolutionary integral kernel A ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) . The same kinematical prerequisites can be found also in other phenomenological approaches [2]-[7]. A ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) =$^{GLYPH<2>}$ dq $^{′}$A ( q$_{1}$, t$_{1}$ | q $^{′}$, t ) A ( q $^{′}$, t | q$_{0}$, t$_{0}$ ) . (3) Ψ$_{0}$( q ) ⇝ Ψ$_{1}$( q ) = GLYPH<2> dq $^{′}$A ( q, t$_{1}$ | q $^{′}$, t$_{0}$ )Ψ$_{0}$( q $^{′}$) . (4) A ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = 1 N GLYPH<2> [ D ˜ γ ] exp { i ℏ GLYPH<2> ˜ γ pdq − Hdt $^{}}$. (1) ˙ q a = ∂H ∂p$_{a}$ ≡ p a m , ˙ p$_{a}$ = − ∂H ∂q a ≡ F$_{a}$ . (5) As Ludwig Faddeev noted during the Edward Wit- ten’s talk at the Mathematical Physics Conference: From XX To XXI Century [26], quantization is not a science, quantization is an art . Let us believe that the quantization method proposed here will fit the Ludwig’s dictum and will be meaningful enough to be considered artistic. According to Feynman [15], the probability amplitude of the transition of the system from the space-time con- figuration ( q$_{0}$, t$_{0}$ ) to another space-time configuration ( q$_{1}$, t$_{1}$ ) is A possible step beyond the theory summarized above consists in the elimination of the Hamiltonian function H from formula (1). The price to be paid is the replace- ment of the path integration by the surface functional integration. Our aim is to construct the amplitude for the transi- tion between ( q$_{0}$, t$_{0}$ ) and ( q$_{1}$, t$_{1}$ ) starting from the classi- cal equations of motion (and not from the Hamiltonian function which provides them) III. ONE STEP BEYOND FEYNMAN II. FEYNMAN QUANTUM MECHANICS In the first set of equations, m is the mass of the parti- cle. We restricted ourselves to the simplest case of one particle, although it is trivial to generalize the theory to a system with an arbitrary number of particles. Note also that if the particle is unconstrained and we make use of Cartesian coordinates, the momenta p a defined in (5) reduce to p$_{a}$ . Here the integral is taken over all paths (histories) ˜ γ ( t ) = (˜ q ( t ) , ˜ p ( t ) , t ) in the extended phase space [27], satisfying the conditions ˜ q ( t$_{0}$ ) = q$_{0}$ and ˜ q ( t$_{1}$ ) = q$_{1}$ . The preexponential factor 1 / N in the expression (1) serves just the normalization. To fix it properly we im- pose two physical conditions on the transition amplitude. First we introduce an integral condition that ensures that the total probability is conserved, Suppose that there exists a unique classical trajectory in the extended phase space γ$_{cl}$ ( t ) = ( q$_{cl}$ ( t ) , p$_{cl}$ ( t ) , t ) , connecting the points ( q$_{0}$, t$_{0}$ ) and ( q$_{1}$, t$_{1}$ ) . Then we can assign to any other trajectory ˜ γ ( t ) = (˜ q ( t ) , ˜ p ( t ) , t ) , which enters the path integral in (1), two auxiliary curves This specifies the absolute value of N . Then we add a constraint that expresses the obvious fact that no evolu- tion takes place if the final time t$_{1}$ approaches the initial time t$_{0}$ , λ$_{0}$ ( s )= $^{(}$q$_{0}$ , π ( s ) , t$_{0}$ $^{)}$, where π (0) = p$_{cl}$ ( t$_{0}$ ) , π (1) = ˜ p ( t$_{0}$ ); λ$_{1}$ ( s )= $^{(}$q$_{1}$ , φ ( s ) , t$_{1}$ $^{)}$, where φ (0) = p$_{cl}$ ( t$_{1}$ ) , φ (1) = ˜ p ( t$_{1}$ ) . GLYPH<2> dq$_{1}$ A ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) A ( q$_{1}$, t$_{1}$ | q ′ $_{0}$, t$_{0}$ ) = δ ( q$_{0}$ − q ′ $_{0}$) . (2a) lim t$_{1}$ → $_{t$_{0}$}$A ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = δ ( q$_{1}$ − q$_{0}$ ) . (2b)