9 B. Surface functional integral versus Feynman-Vernon Formulas are mostly written down in one (local) chart. From this, however, one can ascend to a global, coor- dinate free description employing bundles and jet pro- longations. In order that we did not cloud up the main idea of the paper, we did not follow such fluffy approach. However, in the future it could be useful to analyze ob- structions to the implementation of our method which arise from the global properties of the underlying space- time geometry. The goal of our paper was to write down a functional integral formula for the quantum transition amplitude in terms of the underlying classical equation of motion. However, we did not eliminate the artificial notions of pure states and classical action for non-Lagrangian sys- tems. In the surface functional integral (9), these notions were implicitly present. A possible alternative to our approach consists in ex- pressing the transition probability (not the probability amplitude!) using the functional integral. The prob- ability that the system evolves from the mixed state ρ$_{0}$ ( q$_{0}$, q ′ $_{0}$, t$_{0}$ ) at the time t$_{0}$ to the mixed state ρ$_{1}$ ( q$_{1}$, q ′ $_{1}$, t$_{1}$ ) at some later time t$_{1}$ is GLYPH<2> Special attention was paid to the stringy quantiza- tion of dissipative systems and of the Douglas system. The latter cannot be quantized by any known quantiza- tion technique while the former can. In fact, dissipative (friction force) systems were studied extensively in the past and several approaches to their quantization were proposed. The stringy quantization was compared with three of them, Caldirola-Kanai, Bateman and Kostin, and it was shown that it either gives better results (in the first two cases) or is simpler to apply (in the third case). Here ˜ γ and ˜ γ ′ are curves in the extended phase space whose q -projections connect q$_{0}$ with q$_{1}$ and q ′ 0 with q ′ 1 respectively. Using the Stokes theorem one is able to convert the difference of the line integrals of the one- form pdq − Hdt into the surface integral of the two-form Ω = d ( pdq Hdt ) . Explicitly, As mentioned in the introduction, the quantization proposed here shares a common property with the well established approaches of Caldirola-Kanai, Bateman, Kostin and others: they all fail to cover decoherence phe- nomena. On the other hand, as we have seen, the stringy quantization produces a quantum propagator that does not respect the memoryless Chapman-Kolmogorov equa- tion. We believe that this essential feature of the theory is measuring/reflecting the non-Lagrangeanity of the sys- tem on the quantum level. where Σ represents again a map from the parametric space ( t, s ) ∈ [ t$_{0}$, t$_{1}$ ] × [0 , 1] to the extended phase space, such that ∂ Σ = ˜ γ $^{′}$( t ) − λ$_{1}$ ( s ) − ˜ γ ( t )+ λ$_{0}$ ( s ) . The sideways λ -boundary curves[36] of Σ P$_{ρ}$$_{0}$$_{→}$$_{ρ}$$_{1}$ ∝ dq$_{0}$dq ′ $_{0}$dq$_{1}$ dq ′ $_{1}$ρ$_{0}$ ( q$_{0}$, q ′ $_{0}$, t$_{0}$ ) ρ$_{1}$ ( q$_{1}$, q ′ $_{1}$, t$_{1}$ ) × × GLYPH<2> [ D ˜ γ ][ D ˜ γ $^{′}$] exp { i ℏ $_{(}$GLYPH<2> ˜ γ − GLYPH<2> ˜ γ ′ $^{)}$( pdq − Hdt ) $^{}}$. $_{(}$GLYPH<2> ˜ γ − GLYPH<2> ˜ γ ′ $^{)}$pdq − Hdt = − GLYPH<2> Σ Ω −$^{$_{(}$GLYPH<2>}$ λ$_{1}$ − GLYPH<2> λ$_{0}$ $^{)}$pdq , (27) A. Stringy versus Caldeira-Leggett model Σ : ( t, s ) ↦→ Σ( t, s ) ≡ $^{(}$q$_{Σ}$ ( t, s ) , p$_{Σ}$ ( t, s ) , t$_{Σ}$ ( t, s ) = t $^{)}$, λ$_{0}$ ( s ) ≡ Σ( t$_{0}$, s ) and λ$_{1}$ ( s ) ≡ Σ( t$_{1}$, s ) From the author’s point of view one can rise a concep- tual objection also against the particle-plus-environment model by Caldeira and Leggett [9]. Their approach is microscopical except the way in which the spectral den- sity ρ$_{D}$ ( ω ) for the reservoir degrees of freedom is intro- duced. This density is not obtained from any kind of microscopical theory and the form it assumes is moti- vated solely by the necessity to obtain the dissipative term − κ ˙ q in the effective theory. On the top of it, the interaction Hamiltonian is such that it produces, after integrating out the reservoir degrees of freedom, an ad- ditional Langevin stochastic force F ( t ) . Thus, the effec- tive classical motion of the particle is governed by the Brownian equation of motion ˙ p = − κp + F ( t ) , which is conceptually different from the physical situation consid- ered in section V. It is also worth to point out that the Caldeira-Leggett model cannot describe satisfactorily the friction force proportional to − p A with A = 1 . There is no doubt that for A = 1 the model gives better results than all effective theories mentioned before including the stringy one, but it is fair to say that it describes a differ- ent physical phenomenon, namely the quantum Brownian motion of a particle which is in thermal equilibrium with a reservoir. live in the instant phase spaces of R 2 n $^{+1}$[ q $^{a}$, p$_{a}$ , t ] , i.e. in the submanifolds R 2 $^{n}$[ q $^{a}$, p$_{a}$ , t$_{0}$ ] and R 2 $^{n}$[ q $^{a}$, p$_{a}$ , t$_{1}$ ] . Moreover, as the intrinsic parameter s varies from 0 to 1 , the q$_{Σ}$ -components of λ$_{0}$ ( s ) and λ$_{1}$ ( s ) vary from q$_{0}$ to q ′ 0 and from q$_{1}$ to q ′ 1 respectively. Let us denote by U the space of all Σ -maps for which Since the right hand side of (27) depends only on Σ and its two boundaries, the double path integral GLYPH<1> [ D ˜ γ ][ D ˜ γ $^{′}$] · · · entering P$_{ρ}$$_{0}$$_{→}$$_{ρ}$$_{1}$ can be rewritten as a surface functional integral GLYPH<1> [ D Σ] · · · , q$_{Σ}$ ( t$_{0}$, 0) = q$_{0}$, q$_{Σ}$ ( t$_{0}$, 1) = q ′ $_{0}$, q$_{Σ}$ ( t$_{1}$, 0) = q$_{1}$, q$_{Σ}$ ( t$_{1}$, 1) = q ′ $_{1}$, λ$_{0}$ ( s ) ⊂ R 2 $^{n}$[ q, p, t$_{0}$ ] , λ$_{1}$ ( s ) R 2 $^{n}$[ q, p, t$_{1}$ ] . P$_{ρ}$$_{0}$$_{→}$$_{ρ}$$_{1}$ ∝$^{GLYPH<2>}$ dq$_{0}$dq ′ $_{0}$dq$_{1}$ dq ′ $_{1}$ρ$_{0}$ ( q$_{0}$, q ′ $_{0}$, t$_{0}$ ) ρ$_{1}$ ( q$_{1}$, q ′ $_{1}$, t$_{1}$ ) × ×$^{GLYPH<2>}$ U [ D Σ] exp $^{{}$− i ℏ $_{[}$GLYPH<2> Σ Ω +$^{$_{(}$GLYPH<2>}$ λ$_{1}$ −$^{GLYPH<2>}$ λ$_{0}$ $^{)}$pdq$^{]}}$. (28)