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[26] It took place at EPFL Laussane 16.-17.3.2009. 

[27] For the record, the extended phase space is locally de- scribed by the coordinates in the configuration space q a , the canonically conjugated momenta p$_{a}$ and the time co- ordinate t (the index a runs from 1 to the number of degrees of freedom n ). One assumes the standard Pois- son brackets { p$_{a}$, q $^{b}$} = − δ b a . In a more elevated language, the extended phase space is the space T $^{∗}$M ×$_{R}$ , where M stands for the configuration space and R for time. 

[28] The integrals over λ$_{0}$ and λ$_{1}$ give trivially zero contribu- tions to the contour integral, since on both λ ’s dq = 0 and dt = 0 . 

[29] The precise definition is: suppose there are two histories ˜ γ$_{0}$ ( t ) and ˜ γ$_{1}$ ( t ) in the extended phase space whose pro- jections Proj (˜ γ$_{0}$ ) and Proj (˜ γ$_{1}$ ) on the extended configu- ration space connect ( q$_{0}$, t$_{0}$ ) with ( q$_{1}$, t$_{1}$ ) . Then U$_{(˜}$ γ$_{1}$, ˜ γ$_{0}$ ) contains all continuous and oriented surfaces Σ in the extended phase space such that 

<img src='content_image/402590.jpg'>

If ˜ γ$_{0}$ ̸ = ˜ γ$_{1}$ , the orientation of Σ is supposed to be such that ˜ γ$_{1}$ − ˜ γ$_{0}$ is a part of the boundary of Σ . The Proj ection is simply the momentum forgetting map, or to put it in a more elevated way, the canonical projection from the cotangent bundle to the base manifold. A special case is the set U$_{(˜}$ γ, ˜ γ ) , comprised of all closed surfaces in the extended phase space containing ˜ γ . In what follows, we 

consider any history ˜ γ as a degenerated (shrunk) closed surface, hence ˜ γ ∈ U$_{(˜}$ γ, ˜ γ ) by definition. 

[30] Strictly speaking, the expression for the friction force can be used only for the motion with increasing q . The uni- versally applicable expression is − κ sgn ( ˙ q ) | ˙ q | A . 

[31] The prefix A in the subscript Acl , as well as in the sub- script Astr introduced later, refers to the power in the expression for the friction force. 

[32] The external source in the functional integral (14) can be viewed as a sort of additional time-dependent po- tential, therefore one would expect that the Chapman- Kolmogorov equation (3) will be satisfied. However, this argument is not applicable in the present situation. The point is that if we merge two paths KL and LM into a new path KM , then the time-dependent potentials − κqp A cl − $_{KL}$( t ) and − κqp A cl − $_{LM}$( t ) differ, for a general po- sition of L , from the potential − κqp A cl − $_{KM}$( t ) in the cor- responding time (sub)intervals, 

<img src='content_image/402589.jpg'>

Since this differs substantially from the standard case, the argument about the validity of the Chapman- Kolmogorov equation cannot be used. 

[33] An exhaustive list of various quantization techniques ap- plicable in the presence of linear dissipation, including those mentioned in the introduction, can be found in [19] and [20]. 

[34] The ordering problem and its consequences are not dis- cussed here. 

[35] Since the traveling wave | p$_{0}$ 〉 is not a square integrable function, we evolved by A$_{1}$$_{str}$ ( q, t$_{1}$ | q $^{′}$, t$_{0}$ ) a modified (nor- malized) state | Ψ$_{ϵ}$ 〉 = GLYPH<1> dp e − ϵ ( p − p$_{0}$ ) $^{2}$| p 〉 ( ϵ > 0 ). After the stringy evolution of | Ψ$_{ϵ}$ 〉 , the expectation value of the momentum was computed and the regularization pa- rameter ϵ was set to zero. 

[36] The situation here strongly resembles that discussed in section III. However, there is one substantial difference between the two cases. In the present case, we allow the λ - curves to vary in both coordinates q and p , only the time t must stay unchanged along them. In the case considered in section III we were more stringent: the λ -curves were allowed to vary only with respect to p and had to stay unchanged with respect to both q and t . GLYPH<31> 

[37] From now on we are using a new symbol for all dis- cretized integrals. In the continuum limit K and L ap- proach independently infinity and GLYPH<31> → $^{GLYPH<1>}$. However, we must keep in mind that we must perform simultaneously the limits Δ → 0 and ε → 0 so that the quantities KΔ = t$_{1}$ − t$_{0}$ and Lε = 1 stay finite. 

