**Significance **

Over the last century a series of mathematical prescriptions have been developed indicating how to alter the laws of classical physics in order to obtain the correct quantum mechanics of a given physical system. This set of axioms, defined by Paul Dirac, goes under the name of canonical quantization. Its effectiveness has been confirmed with astonishing precision in numerous experimental works, despite some ambiguities which make quantization an “art not a science”, as Ludwig Faddeev used to say. Up to now no classes of classical dynamics have been found that fully exhibit the characteristics of quantum dynamics. Furthermore Bell’s inequalities rule out a large class of classical dynamics as possible origin of quantum indeterminacy, but not all.

In a new paper published in the research peer-reviewed journal, Annals of Physics, Prof. Donatello Dolce from the University of Camerino, Italy, has investigated particular classical dynamics in which the time dimension has a cyclic nature as for the hands of an analog clock, finding an exact correspondence to the canonical quantum dynamics down to all the most fundamental details. According to Prof. Dolce’s results, “the mysterious mathematical prescription of canonical quantization is equivalent to prescribing that the classical system must evolve in time loops, without having to modify the laws of classical physics”. The price to pay for reconciling quantum mechanics with classical mechanics is to assume that relativistic time has a cyclical nature in the micro-world. “Particular care is required in putting time into cycles, but if this is properly done it is immediate to realize that the resulting cyclical dynamics are absolutely consistent with all the classical-relativistic principles of physics, in particular causality and locality”. Prof. Dolce says. We know from the wave-particle duality that each elementary system can be in general associated with a wave-function, which is a ‘periodic phenomenon’ as originally named by de Broglie, whose natural recurrence in time is fixed by the energy through the Planck constant. If the system locally exchanges energy due to interactions, then its wave-function has locally varying frequency and therefore the rate of the time recurrence is locally modulated. “We already know that these time recurrences exist and that elementary systems can be described in terms of them. Physicists implicitly use them whenever they apply the wave-function formalism”.

Prof. Dolce has investigated the following legitimate question: “What happens if the natural local time recurrence of the wave-functions typically associated to any generic physical system is imposed as dynamical constraint to the system itself?” This means raising the intrinsic periodicity of elementary systems to a fundamental principle of physics. All in all such kind of intrinsic periodicity was also observed for Time Crystals. The answer is given in Prof. Dolce recent paper in the form of mathematical theorem: “under the very general hypothesis of Hamiltonian systems the result of intrinsic time periodicity is a quantization formally equivalent to the canonical quantization of the system and therefore also potentially violating Bell’s inequalities as much as ordinary quantum mechanics”. The constraint of periodicity is in fact an element of non-locality in Bell’s sense. It transforms ordinary single frequency wave-functions into vibrating strings, similarly to violin strings constrained to vibrate in fixed time cycles rather than in fixed spatial intervals. The quantization is achieved by imposing periodic boundary conditions along the time dimension as a sort of relativistic generalization to the time dimension of the ‘particle in a box’. The local modulations of time recurrence resulting from the interactions are the key point to understand the complexity of Nature, which is emphatically non periodic, in terms of elementary time cycles. “Complex systems can be thought of as musical symphonies, which can be arbitrarily complex depending on the variety of pitch modulations of notes in scores, even though they are generated by ‘periodic phenomena’ such as the standing waves in musical instruments.” Dr. Dolce concludes: “the apparently impenetrable laws of quantum mechanics could in truth hide as fundamental physics the most harmonious laws of nature, that is, the laws of sound, though if fully generalized to relativity”.

Dolce’s paper confirms the equivalences already observed in previous papers for Feynman’s quantization (equivalence with the Feynman path integral) of elementary particles and applied to explain in an original way non-trivial quantum phenomena such as superconductivity and graphene physics. The same idea has also revealed a common origin of gauge and gravitational interactions, and a formal derivation of the Maldacena’s conjecture.

**Reference**

Dolce, D. (2023). **Is time a cyclic dimension? canonical quantization implicit in classical cyclic dynamics**. *Annals of Physics,* *448*, 169182.