Functional integral for non-Lagrangian systems Denis Kochan ∗ Department of Theoretical Physics, Comenius University, Bratislava, Slovakia & Theory Group, CERN, Genève, Switzerland A novel functional integral formulation of quantum mechanics for non-Lagrangian systems is pre- sented. The new 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 − κ ˙ q A . Results for A = 1 are compared with those obtained in the approaches by Caldirola-Kanai, Bate- man and Kostin. Relations to the Caldeira-Leggett model and to the Feynman-Vernon approach are discussed as well. Dedicated to my father on the occasion of his 60th birthdays. PACS numbers: 03.65.Ca, 11.10.Ef, 31.15.xk Quantization is a phenomenon that changes our bright classical perspective into a bit uncertain and at first sight rather nonintuitive picture. This picture, however, is more rigorous than the classical one, possesses many fas- cinating features and has produced a lot of successful predictions. A and B. We also make sure that whenever the system under consideration becomes Lagrangian, the stringy de- scription reduces to the standard one with the Feynman path integral. The subtle problem of transition from classical to quantal attracts attention from the early days of quan- tum mechanics. Over the years various techniques and methods for solving this puzzle have been invented. Our aim is not to trace back the complete (hi)story of the milestone ideas in this field (for the review we refer to [1]). What we want to do is to give a concise exposi- tion of the method we have developed. However, since we have generalized the original Feynman’s path integral approach, we will recapitulate this approach shortly in section II. The main goal of our paper is to obtain a func- tional integral formula for the quantum propagator which would not refer to Lagrangian and/or Hamiltonian func- tion. Quantum propagator is the probability amplitude A ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) for the transition of the system from the initial configuration ( q$_{0}$, t$_{0}$ ) to the final configuration ( q$_{1}$, t$_{1}$ ) . We will derive a closed expression for this quan- tity starting from the given set of classical dynamical equations of motion. The proposed method uses functional integration in the extended phase space, but instead of integration over path histories we introduce integration over stringy sur- faces. This crucial element of our approach is explained in full detail in sections III and IV and in appendices Section VI contains the analysis of a curious two- dimensional Douglas system [8] which is strongly non- Lagrangian, i.e. not derivable from any sort of La- grangian or Hamiltonian. This causes a fundamental problem for all conventional quantization methods, but can be dealt with in a rather transparent way in the One can argue against the stringy quantization of dis- sipative systems that it is unable to describe decoher- ence phenomena. This is true, but the same objection can be raised against the generally accepted heuristic approaches by the authors cited above, as well as those by Dekker [5], Razavy [6], Geicke [7] and others, sim- ply because their kinematical and dynamical prerequi- sites are different from the prerequisites of the particle- plus-environment quantum models. However, a possible argument for the stringy quantization is that the particle- plus-environment models are not able to handle satisfac- torily the case with the friction force − κ ˙ q A for the gen- eral power A . Moreover, they describe a rather different phenomenon, namely the quantum Brownian motion for which the total force equals − κ ˙ q + stochastic term . In sections V and VI some simple examples are scruti- nized. It is well known that there are classical dynamical systems that cannot be described within the traditional Lagrangian or Hamiltonian framework. The stringy ap- proach enables us to quantize them straightforwardly. In section V the quantization of a weakly non-Lagrangian system is performed. The transition amplitude is com- puted for a particle in a conservative field, whose motion is damped by a general A -power friction force F = − κ ˙ q $^{A}$. The stringy results for A = 1 are compared with the re- sults obtained in the approaches by Caldirola-Kanai [2], Bateman [3] and Kostin [4]. arXiv:1001.1863v2 [hep-th] 22 Feb 2010 I. INTRODUCTION $^{∗}$Electronic address: kochan@fmph.uniba.sk