6 Since both actions are quadratic functions of the end- points q$_{0}$ and q$_{1}$ , and since we require that the total probability is conserved, the preexponential factors for both transition amplitudes can be computed from the Van Vleck formula Using this formula we arrive at the normalized transition amplitudes GLYPH<Minus> 1 1 2 3 4 GLYPH<Minus> 5 5 10 15 t$_{1}$ CK CK Stringy Stringy 1 $_{N}$=$^{√}$ i 2 π$_{ℏ}$ ∂ $^{2}$S ∂q$_{1}$ ∂q$_{0}$ . A$_{1}$$_{str}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ )= √ κ 4 πi$_{ℏ}$ tanh κ $_{2}$( t$_{1}$ − t$_{0}$ ) e i $^{$_{ℏ}$S$_{1}$$_{cl}$}$, (17) A$_{CK}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ )= √ κ 2 πi$_{ℏ}$ (e − κt$_{0}$ − e − $^{κt$_{1}$}$) e i $^{$_{ℏ}$S$_{CK}$}$, (18) FIG. 3: Expectation values of position (solid color lines) and momentum (dashed color lines) as functions of time for the Gaussian wave packet (the best fit for the moving classical particle), drawn for the stringy and CK propagators (17) and (18), respectively. Initial wave packet characteristics are 〈 q 〉 ( t$_{0}$ ) = 0 m and 〈 p 〉 ( t$_{0}$ ) = 5 m s − 1 , and the friction constant κ is set to 0 . 6 s − 1 . One realizes that at the time t$_{1}$ − t$_{0}$ = ln 3 κ − 1 the stringy mean momentum becomes zero. Classically, for t$_{1}$ − t$_{0}$ > κ − 1 the physical relevancy of the solution breaks down. This explains the time bound on the applicability of stringy propagator (17). which trivially satisfy (2). Moreover, one immediately verifies that in the frictionless limit κ → 0 both transition amplitudes coincide with the free particle propagator. A short inspection shows that the stringy propagator de- pends on t$_{0}$ and t$_{1}$ only through t$_{1}$ − t$_{0}$ , but this is not the case for the CK propagator. The same observation holds for S$_{1}$$_{cl}$ and S$_{CK}$ . This is a typical feature for all auxiliary actions defined in terms of integrators of Ω , see [21]. We pay attention to this point because the classical equation of motion ¨ q = − κ ˙ q which we have started with is invariant with respect to time translations, and one would naturally expect the same invariance on the quan- tum level. Since only A$_{1}$$_{str}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) has this feature, we obtain an efficient argument for the stringy functional integral (9) when compared to the CK one. The evolu- tion of Gaussian wave packets rather than damped; thus, they evolve according to the time reversed dynamical equations. In our case we have: These equations of motion can be derived from the least action principle with the quadratic and time independent Bateman(-Morse-Feshbach) Lagrangian: whose dynamics is governed by (17) and (18), is visual- ized in Figure 3. For more detail, see [22]. The canonical as well as path integral quantization of the theory encounters various difficulties because non- normalizable states of the amplified system must be em- ployed [23]. This problem will be discussed later, when the treatment of the auxiliary Q -degrees of freedom in this approach will be described. The path integral eval- uation of the transition probability amplitude based on L$_{B}$ is straightforward [24]. The result is: From the figure we can see that the stringy averaged momentum 〈 p 〉 becomes negative (i.e. meaningless) for t$_{1}$ − t$_{0}$ > ln 3 κ − $^{1}$. Let us try to explain this peculiarity. It is well known that the relevancy of the classical solu- tion q$_{cl}$ ( t ) of ¨ q = − κ ˙ q breaks down when the time t$_{1}$ − t$_{0}$ exceeds the relaxation time κ − $^{1}$. Since q$_{cl}$ is used in the derivation of the formula (17) for the stringy propagator, its applicability is automatically restricted as well. After this restriction is taken into account, the stringy evolu- tion of 〈 p 〉 becomes more acceptable from the physical point of view then the evolution in the CK approach. ¨ q = − κ ˙ q (damped) AND ¨ Q = + κ ˙ Q (amplified) . Ψ$_{0}$( q ) ∝ e − q $^{2}$+ i $_{ℏ}$p$_{0}$ q ⇝ Ψ$_{1}$( q ) ∝$^{GLYPH<2>}$ dq $^{′}$A ( q, t$_{1}$ | q $^{′}$, t$_{0}$ )Ψ$_{0}$( q $^{′}$) , L$_{B}$ ( q, ˙ q, Q, ˙ Q ) = ˙ q ˙ Q + κ 2 ( q ˙ Q − Q ˙ q ) . (19) A$_{B}$ ( q$_{1}$, Q$_{1}$, t$_{1}$ | q$_{0}$, Q$_{0}$, t$_{0}$ ) = κ 4 π$_{ℏ}$ sinh κT 2 × × exp { i ℏ α $^{[}$Q$_{1}$ ( q$_{1}$ − q$_{0}$β$_{−}$ ) + Q$_{0}$ ( q$_{0}$ − q$_{1}$β$_{+}$ ) ] $^{}}$, (20) α = κ 2 1 tanh κT 2 , β$_{±}$ = e ± κT 2 cosh κT 2 , T = t$_{1}$ − t$_{0}$ . where Our aim, however, is to find the transition amplitude for the damped (sub)system only. In order to obtain it we The key element of the Bateman(-Morse-Feshbach) ap- proach [3] are new subsidiary degrees of freedom intro- duced in addition to the initial ones, which are amplified B. Stringy versus Bateman approach