5 the system is a part of a larger system and hence it should be affected by the memory effect. It is not difficult to compute A$_{Astr}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) for the general power A and the potentials U ( q ) = 0 and U ( q ) = 1 $_{2}$ω$^{2}$q$^{2}$. In both cases we can carry out the path integration in (14) explicitly to obtain In this example the surface functional integral can be cal- culated explicitly (for more detail see [17], or Appendix B). In the course of calculation, the surface functional integral in the extended phase space reduces to path in- tegral in the configuration space, where the non-Lagrangian action S$_{Acl}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) is given by the expression (13). An open problem is to de- termine the preexponential factor 1 / N$_{Astr}$ ( t$_{1}$ − t$_{0}$, q$_{1}$, q$_{0}$ ) for all powers A except for A = 1 (at least the author is incapable to do that). If A = 1 , the preexponential factor is apparently dependent not only on the time difference t$_{1}$ − t$_{0}$ , but also on the endpoint positions q$_{0}$ and q$_{1}$ . This hypothesis is supported by the results obtained in [18], where the path integral for a non-conservative force quadratic in velocity is calculated. The normalization prefactor which appears there is explicitly dependent on the endpoints positions as well as on t$_{1}$ − t$_{0}$ . In what follows we will restrict ourself to the case when the power A is equal to 1. This simplified setup enables us to compare the stringy approach with the Caldirola- Kanai, Bateman and Kostin approaches [33]. This combined with the formula (9) suggests that it will be convenient to define the classical action as [31] When doing so, we obtain a controlled cancelation of exp { i $_{ℏ}$S$_{Acl}$ } with the preexponential factor arising from (12). The final probability amplitude then assumes a compact and reasonable form: A$_{Astr}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = = 1 $_{N$_{Astr}$}$exp$^{{}$ i ℏ S$_{Acl}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ )$^{}}$ , (15) GLYPH<2> U$_{γ}$$_{cl}$ [ D Σ] exp { i ℏ GLYPH<2> Σ Ω $^{}}$∝ ∝ exp $^{{}$− i ℏ t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ q 2 cl − U ( q$_{cl}$ ) − κq$_{cl}$p A cl $^{)}$dt $^{}}$× × GLYPH<2> [ D q ] exp { i ℏ t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ q 2 − U ( q ) − κqp A cl $^{)}$dt$^{}}$. (12) S$_{Acl}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ q 2 cl − U ( q$_{cl}$ ) − κq$_{cl}$p A cl $^{)}$dt . (13) A$_{Astr}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = = 1 N$_{Astr}$ GLYPH<2> [ D q ] exp { i ℏ t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ q 2 − U ( q ) − κqp A cl $^{)}$dt$^{}}$. (14) L$_{CK}$ ( q, t ) = e κt [ 1 2 ˙ q 2 − U ( q ) $^{]}$. (16) A$_{CK}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) = 1 N$_{CK}$ GLYPH<2> [ D q ] exp { i ℏ t$_{1}$ GLYPH<2> t$_{0}$ L$_{CK}$ ( q, t ) dt } = 1 $_{N$_{CK}$}$exp { i $_{ℏ}$S$_{CK}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) } . ¨ q = − dU dq − κ ˙ q A cl , A. Stringy versus Caldirola-Kanai approach The dissipative system under consideration is weakly non-Lagrangian. This means that d Ω = 0 , but Ω pos- sesses a local integrator f = 0 . Let us define the auxiliary Lagrangian function (one of plenty) as The preexponential factor 1 / N$_{Astr}$ can in principle be obtained by subjecting A$_{Astr}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) to the condi- tions (2). Let us explain why one should consider the propaga- tor formula (14) reasonable. In the path integral above there appears an unconventional “external source” term − κqp A $_{cl}$. Its appearance guarantees that the quantum dy- namics governed by A$_{Astr}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) transforms into classical mechanics in the limit ℏ → 0 . This follows from the simple fact that the unique solution of the saddle point equation for the path integral (14), This Lagrangian is usually called CK-Lagrangian after its inventors P. Caldirola and E. Kanai [2]. Henceforward a damped free particle is considered only, i.e. the potential energy U ( q ) is supposed to be zero. It is immediately clear that the transition amplitude computed from (15) differs from the CK-amplitude which satisfies the given initial and final conditions q ( t$_{0}$ ) = q$_{0}$ and q ( t$_{1}$ ) = q$_{1}$ , is the classical trajectory q$_{cl}$ ( t ) . The external source term in (14) breaks the valid- ity of the Chapman-Kolmogorov (memoryless) equation [32]. From the physical point of view it is a desired phenomenon. The microscopic origin of friction is some environmental interaction. This, however, was not ac- counted for here explicitly. What has been considered is some effective (macroscopic, phenomenological) interac- tion emerging on the classical level only. Microscopically Indeed, the stringy and CK actions which enter the cor- responding transition amplitudes are different, S$_{1}$$_{cl}$ = κ 4 ( q$_{1}$ − q$_{0}$ ) ( q$_{0}$ + 3 q$_{1}$ )e − κt$_{1}$ − ( q$_{1}$ + 3 q$_{0}$ )e − κt$_{0}$ e − κt$_{0}$ e − κt$_{1}$ , S$_{CK}$ = κ 2 ( q$_{1}$ − q$_{0}$ ) 2 e − κt$_{0}$ e − κt$_{1}$ .