Functional integral for non-Lagrangian systems

A functional integral formulation of quantum mechanics for non-Lagrangian systems is presented. The approach, which we call “stringy quantization,” is based solely on classical equations of motion and is free of any ambiguity arising from Lagrangian and/or Hamiltonian formulation of the theory. The functionality of the proposed method is demonstrated on several examples. Special attention is paid to the stringy quantization of systems with a general $A$-power friction force $-\kappa {\mathrm{q̇}}^A$. Results for $A=1$ are compared with those obtained in the approaches by Caldirola-Kanai, Bateman, and Kostin. Relations to the Caldeira-Leggett model and to the Feynman-Vernon approach are discussed as well.

Figure 1

Schematic picture of two auxiliary curves ${\lambda }_0(s)$ and ${\lambda }_1(s)$ that connect the classical history ${\gamma }_{\mathrm{cl}}(t)$ with the given history $\mathrm{\gamma ̃}(t)$ in the extended phase space. $\lambda$-curves are located in the $n$-dimensional subspaces of the extended phase space in which the momenta are varying while the coordinates and time are kept fixed. The contour $\partial \Sigma =\mathrm{\gamma ̃}-{\lambda }_1-{\gamma }_{\mathrm{cl}}+{\lambda }_0$ forms a boundary for plenty of extended phase space surfaces. One of them, denoted as $\Sigma$, is drawn in the figure.

Figure 2

Four elements (two open, one closed, and one shrunk to ${\gamma }_{\mathrm{cl}}$) of ${\mathcal{U}}_{{\gamma }_{\mathrm{cl}}}$, anchored to the given classical history ${\gamma }_{\mathrm{cl}}$ in the extended phase space. The two open surfaces belong to two different stringy classes ${\mathcal{U}}_{(\mathrm{\gamma ̃},{\gamma }_{\mathrm{cl}})}$, whereas the closed and the shrunk surface are both elements of the same stringy class ${\mathcal{U}}_{({\gamma }_{\mathrm{cl}},{\gamma }_{\mathrm{cl}})}$. As the worldsheet $\Sigma$ varies, the front and rear boundary curves, denoted in the text ${\lambda }_0$ and ${\lambda }_1$, vary as well.

Figure 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 [Eqs. (17) and (18)], respectively. Initial wave packet characteristics are $\langle q\rangle (t_0)=0\hspace{0.5em}\mathrm{m}$ and $\langle p\rangle (t_0)=5\hspace{0.5em}\mathrm{m}\mathrm{\hspace{0.5em}}{\mathrm{s}}^{-1}$, and the friction constant $\kappa$ is set to $0.6\hspace{0.5em}{\mathrm{s}}^{-1}$. One realizes that at the time $t_1-t_0=\ln 3\hspace{0.5em}{\kappa }^{-1}$ the stringy mean momentum becomes zero. Classically, for $t_1-t_0>{\kappa }^{-1}$ the physical relevancy of the solution breaks down. This explains the time bound on the applicability of stringy propagator Eq. (17).

Figure 4

Expectation values of the position and momentum as functions of time for the Gaussian wave packet, drawn for the stringy and effective Bateman propagators [Eqs. (17) and (22)], respectively. The initial wave packet characteristics and the meaning of solid and dashed color lines are the same as in Fig. 3.

Figure 5

Expectation value of the momentum as a function of time for an eigenstate of initial momentum ${\Psi }_{p_0}(q,t_0)\propto \exp \{\frac{i}{\hslash }{\mathit{qp}}_0\}$, drawn for the stringy propagator Eq. (17) (blue dashed lines) and the propagator given by the Kostin-Schrödinger equation [Eq. (23)] (red dashed lines), respectively. The values of the friction constant $\kappa$ are indicated next to the curves. The initial momentum is $p_0=5\hspace{0.5em}\mathrm{m}\mathrm{\hspace{0.5em}}{\mathrm{s}}^{-1}$.

Figure 6

Vector field $W$ acting infinitesimally in the extended phase space. As a result, the initial surface ${\Sigma }_{\mathrm{ext}}\in {\mathcal{U}}_{({\mathrm{\gamma ̃}}_1,{\mathrm{\gamma ̃}}_0)}$ moves to some nearby surface ${\Sigma }_{\mathrm{ext}}^{\delta W}\in {\mathcal{U}}_{({\mathrm{\gamma ̃}}_1^{{}_{{}^{\prime }}},{\mathrm{\gamma ̃}}_0^{{}_{{}^{\prime }}})}$, and its boundary $\partial {\Sigma }_{\mathrm{ext}}={\mathrm{\gamma ̃}}_1-{\lambda }_1-{\mathrm{\gamma ̃}}_0+{\lambda }_0$ is transported to a new boundary $\partial {\Sigma }_{\mathrm{ext}}^{\delta W}={\mathrm{\gamma ̃}}_1^{{}_{{}^{\prime }}}-{\lambda }_1^{{}_{{}^{\prime }}}-{\mathrm{\gamma ̃}}_0^{{}_{{}^{\prime }}}+{\lambda }_0^{{}_{{}^{\prime }}}$. Since the variational vector field is assumed to preserve $\mathcal{U}$, $W$ stays tangential to the momentum submanifolds with fixed $(q_0,t_0)$ and $(q_1,t_1)$.

Figure 7

Rectangular nodal web in the parametric space $[t_0,t_1]\times [0,1]$ of a surface $\Sigma \in {\mathcal{U}}_{(\mathrm{\gamma ̃},{\gamma }_{\mathit{cl}})}$. Each elementary tile encloses the area $\Delta ·\varepsilon =(t_1-t_0)/(\mathit{KL})$ (at the end, the numbers $K$ and $L$ will be sent to infinity). The points marked by crosses are constrained by the conditions [Eq. (B3)].