Non-integrable Dynamics and Physics Beyond the Standard Model

The evolution of integrable classical systems leads to conserved quantities and vanishing Poisson brackets. In contrast, such invariants do not exist in the dynamics of non-integrable systems, which include (but are not limited to) deterministic models with long-term chaotic behavior. The object of this review is to briefly survey the mathematical background of nonintegrability and its role in the physics unfolding well-above the Standard Model (SM) scale.


Introduction
A cornerstone postulate of perturbative Quantum Field Theory (QFT) is the cluster decomposition principle (CDP), according to which local processes are insensitive to distant environments [1]. In a nutshell, CDP allows one to isolate a subsystem from all others that are widely separated in either spacetime or energy scale. In doing so, CDP ultimately secures consistency of both Renormalization Group and the perturbative formulation of QFT. One may reasonably ask if CDP survives or is ruled out somewhere above the SM scale.
There are well motivated reasons to suspect that the deep Terascale sector prevents thermalization of quantum fluctuations and creates an environment favoring the onset of complex dynamics [2]. In this setting, the underlying principles of classical statistical physics and perturbative QFT are likely to break down. The ergodic theorem, the fluctuation-dissipation theorem, analyticity, unitarity, locality, finiteness in all orders of perturbation theory and renormalizability are either violated or lose their conventional meaning.
It is known that, while the dynamics of integrable classical or quantum systems leads to conserved quantities, such invariants do not generally exist for non-integrable systems.
The goal of this report is to sketch the plausible connection between non-integrability of phenomena lying well above the SM scale and the breakdown of CDP.

Final remarks
Historically, Prigogine and his Brussels-Austin school of non-equilibrium thermodynamics were the first to point out that, under certain conditions, Poincaré resonances can render Quantum Mechanics inconsistent and deny its celebrated predictivity [7][8]15]. Few observations are required to further clarify this point: 3.1) Singular denominators are also present when computing scattering amplitudes in perturbative QFT. One must bear in mind that, while the so-called i prescription of QFT evades singularities by isolating the propagator poles [1,9], Poincaré resonances describe persistent scattering, a genuinely non-perturbative process falling outside the boundaries of conventional QFT. By coupling the creation and annihilation operators, Poincaré resonances generate time irreversibility and nonlocality at the quantum level of observation [15].

3.2)
A tacit assumption of all covariant field theories is that coordinate and gauge transformations, along with their inverse, are well-behaved functions that can be differentiated arbitrarily many times. But this assumption fails to be true in general due to the broad spectrum of existing non-differentiable functions and trajectories, as repeatedly discovered since the introduction of fractal geometry in 1983. Many authors have emphasized that fractional calculus and fractional dynamics provide the adequate framework for the analysis of such objects [10][11][12][13][14].

3.3)
Unlike conventional dynamics on smooth manifolds, fractional dynamics implies nonlocality, dissipation, and power-law correlations [10][11][12][13][14]. These attributes are inherently linked to the mixing of ultraviolet and infrared sectors above the SM scale and lead to the violation of the CDP postulate. Indeed, the formulation of CDP stems from the assumption of Gaussian statistics and Central Limit Theorem, which likely break down in non-equilibrium dynamical settings [2,10].

3.4)
One recalls that the rigorous construction of interacting QFT is confronted by specific challenges brought up by the Haag's theorem [1]. This theorem asserts that the quantum interaction picture breaks down when the free-field Hamiltonian ( 0 H ) and its interacting part (V ) differ from each other by the integral of some local operator density. The prevailing view is that the Haag theorem can be circumvented by full regularization of QFT, in which both a spatial infrared cutoff (that is, a finite spatial volume) and an ultraviolet cutoff (that is, a finite lattice spacing) are introduced. It is unclear, however, if evading the Haag theorem by regularization survives a dynamical regime where non-integrability sets in and CDP no longer holds.