Emergence of Planck’s Constant from Iterated Maps

Iterations of continuous maps are the simplest models of generic dynamical systems. In particular, circle maps display several key properties of complex dynamics, such as phaselocking and the quasi-periodicity route to chaos. Our work points out that Planck’s constant may be derived from the scaling behavior of circle maps in the asymptotic limit.


Introduction
In is known that the probability amplitude for a quantum particle to start at the spacetime point ( , ) The classical approximation of (2) corresponds to the case S  , with the phase S being a large angle. Shifting a given path by a small amount x  leads to a negligible change of the action on the classical scale, yet it produces significant changes at a scale commensurate with . At this scale, the exponential function in (2) undergoes fast fluctuations in sign and the overall contribution to (1) cancels out to zero. Neighboring paths in the proximity of the classical path () xt reinforce each other by constructive interference. This is the path that minimizes the action functional ( 0 S   ), a condition consistent with 0  .
In the standard formulation of Quantum Mechanics, there are no additional fluctuations of the exponential besides the intrinsic oscillations discussed above. It is not unreasonable to assume that, on sufficiently short time scales, background fluctuations start to develop and act upon S as periodic or non-periodic perturbations. The "noise-like" nature of these perturbations follow from the inherent instability brought about by probing phenomena on short time scales. In light of these considerations, it makes sense to model the dynamics of the action S using the general theory of coupled oscillators. This is the object of the next section.

Circle maps and the perturbed dynamics of classical action
We assume that the background fluctuations are the "driver" oscillator with period () TB and . Likewise, let the intrinsic fluctuations of S arising from (2) which can be alternatively presented as Here, nN is the iteration index, K denotes the coupling strength and and characterizes the average number of 2 rotations per iteration [2]. It is important to note that (6) is different from (10) when 0 K  . By contrast, ( 0) WK   when there is no coupling between background and action fluctuations, in which case (0) W denotes the angular increment at each iteration step n .
Two situations of interest exist concerning the behavior of (10) when 01 K  , namely [1-2]: a) When (10) is a rational number, the motion corresponds to stable cycles in a finite range of bare winding numbers 01    , b) Stable quasi-periodic orbits develop when (10) is an irrational number and the motion occurs on a Cantor set. Moreover, if the quasi-periodic orbits have winding numbers (10) that reside on the Cantor set at 1 K  , these orbits necessarily transition to chaos for 1 K  .
Analysis shows that the transition to chaos in either (7) Note that the condition 1 K  is consistent with the separation of the high and low energy scales in effective Quantum Field Theory [3]. b) According to (14) and (15), Planck's constant emerges as a fixed number related to the golden-mean. The precise numerical value of this constant is irrelevant as Planck's constant can be set to 1  in natural units.