Path-integral approach to ’t Hooft’s derivation of quantum physics from classical physics

Abstract

We present a path-integral formulation of 't Hooft's derivation of quantum from classical physics. The crucial ingredient of this formulation is Gozzi et al. supersymmetric path integral of classical mechanics. We quantize explicitly two simple classical systems: the planar mathematical pendulum and the Rössler dynamical system.

't Hooft's quantum determinism: path integral viewpoint

“…Consider the class of systems described by Hamiltonians of the type (8), and let us try to quantize them using path integrals [15]. Because of the absence of a leading kinetic term quadratic in the momenta p a , the system can be viewed as singular and the ensuing quantization can be achieved through some standard technique for quantization of constrained systems.…”

“…The information loss together with certain accompanying non-trivial geometric phases may explain the observed non-locality in quantum mechanics. This idea has been further developed by several authors [4,5,[8][9][10][11][12], and it forms the basis also of this paper.…”

“…The constrained dynamics enters into 't Hooft's scheme twice: first, in the classical starting Hamiltonian which is of first order in the momenta and thus singular in the Dirac-Bergmann sense [14]. Second, in the information loss condition that we impose to achieve quantization [8]. It is thus clear that a better understanding of 't Hooft's quantization scheme is closely related to a proper treatment of the involved constrained dynamics.…”

“…Our aim is to study 't Hooft's quantization procedure by means of path integrals, as done in our previous work [8]. However, in contrast to Ref.…”

“…It is thus clear that a better understanding of 't Hooft's quantization scheme is closely related to a proper treatment of the involved constrained dynamics. In our previous paper [8] this has been done by means of the customary Dirac-Bergmann technique, which is often cumbersome. Here we want to point out the simplifications arising from the alternative Faddeev-Jackiw method, which allows a clearer exposition of the basic concepts.…”