Language shapes thought, and in no field is this more true than in physics. For over three centuries, the basic grammar of theoretical physics has remained that of Newton, consisting of two basic components: A `state vector’ representing the system at any given time, and an evolution equation for the state vector. Implicit in those two is also the way in which our actions in a `tabletop experiment’ are modeled: By choosing initial conditions for the state vector, and/or choosing parameters for the evolution equation, such as applying an external force or field.

Newtonian grammar1 (NG) even survived the 20th century turmoil. The quantum revolution only redefined the space of the state vector—n-dimensional configuration space ↦ infinite dimensional Hilbert space. More remarkably, even the relativistic revolution, which did away with the very notion of absolute time, did not seem to require a novel grammar.

Its robustness to paradigm shifts has erroneously promoted NG to an attribute of nature rather than that of our descriptive language, and physical models not expressible in NG are generally met with instinctive skepticism. However, the picture emerging from previous papers by the author suggests that, NG’s robustness to paradigm shifts might itself have been a product of wrongly shaped (Newtonian) thought. Maxwell-Lorentz Classical electrodynamics (CED) of point charges—an alleged success of NG—is ill-defined, and a century of attempts, expressed in NG, to cure its pathologies all failed. Currently, the only well-defined CED of interacting point-like charges (to the author’s best knowledge), dubbed ECD [1,2], is not expressible in NG. Precisely for this reason ECD can serve as a (classical) ontology underlying QM statistics without conflicting with various no-go theorems, all implicitly assuming an ontological description using NG [3]. That QM is an NG theory can be traced to local constraints, noatably energy-momentum conservation, satisfied by that ECD ontology; see caption of Figure 1. Finally, there are indications that advanced solutions of Maxwell’s equations, mandated by ECD, are at play not only in microscopic physics, where they create the illusion of `photons’ (among else) but also in astronomy’s missing mass problem [4]. And although advanced solutions per se are not in conflict with NG, their incorporation into a consitent mathematical formalism, involving both matter and radiation, certainly is—time-symmetric action-at-a-distance electrodynamics [5] being yet another, better known example.

The appearance of NG in macroscopic (classical) physics is likewise explained by local constraints satisfied by the underlying (ECD) ontology—see, e.g., appendix D of [1] for how the Lorentz force equation is obtained as a local approximation for the center of an extended charge. The question of whether the underlying microscopic ontology is likewise described by NG, might therefore seem irrelevant to macroscopic physics. We argue to the contrary. Microscopic local constraints are just what they are: constraints; long-time integration of their manifestation in macroscopic, coarse grained quantities is not only often unreliable due to the so-called butterfly effect, but moreover meaningless in a non NG universe, as are most statistics derived therefrom. This realization, in conjunction with the above indications of a non NG ontology, point to new experiments which might appear far-fetched, but only because of a NG bias. If the universe is indeed non mechanistic, the implications for physics and science in general would go far beyond quantum weirdness.

The settings of what follows is the block-universe (BU): 4D spacetime hosting a locally conserved (symmetric) energy-momentum tensor, constructed from the basic building blocks (fields in the case of ECD) of an ECD-like theory (Figure 1).

The BU is an infinitely redundant representation of a NG ontology as its full content, statistical aspects included, can be faithfully reproduced from any of its space-like slices. Philosophy aside, it is therefore an unnecessary complication even in relativistic theories. However, for a non-NG ontology, the BU is arguably the only faithful representation. Consider a non-machine—for lack of a better name: A system which, unlike a machine, does not admit an NG representation, viz., its contribution to the BU is not the result of propagating some initial conditions. To illustrate the basic idea with as little technical complications as possible, consider the following (formal) non relativistic, toy non-machine action for $q:(-\infty ,\infty )\to {\mathbb{R}}^n$ where K is any diagonal matrix of non compactly supported, once integrable symmetric functions, and $\ddot{(·)}$ denotes double differentiation. It is the non compactness of K on ${\mathbb{R}}_{-}$ which prevents translating the associated Euler-Lagrange equations, into NG language2. Note that, in and of itself, K’s non singular support does not necesserily lead to temporal paradoxes (although, importantly, unlike ECD, does violate scale covarince); More accurately: non tachyonic extrema of a relativistic version of (1), e.g. the self-term in [13], do not do so. It is violation of this no-tachyons condition which can create paradoxes3 rather than a non-tachyonic $q\left(s\right)$ (s a Lorentz scalar) functionally depending on $q\left(s^{\prime }\right)$ for $s^{\prime }>s$—hence also on $V(q\left(s^{\prime }\right),s^{\prime })$—which, at `worst’, can be labeled retro-causal, and does not entail a paradox, as explained below.

Nonetheless, twice integrating the first term in (2) by parts, we get the classical e.o.m. with a residual term vanishing in the `delta function limit’: $K(·)\mapsto {lim}_{\lambda \to \infty }\lambda K(\lambda ·)$ (and is $O\left({\lambda }^{-2}\right)$ for a finite $\lambda$). A different way of seeing a machine in (2) involves the Noether currents associated with the symmetries of action (1), which are exactly conserved even at finite $\lambda$. For sufficiently slowly varying V on the scale set by K’s extent, translation invariance, e.g., lends itself to a good mechanistic description of the coarse grained momenta and positions viz., $\dot{Q}=m^{-1}P,\dot{P}\simeq -\nabla V\left(Q\right)$. Without such quasi locality it would be impossible to explain the reproducibility of many experiments notwithstanding (spacetime-) translation covariance. It would further allow to directly infer with certainty future attributes of V based on current behavior of q, which would lead to a paradox whenever those attributes could still be altered.

Nonetheless, the local machine approximation, however good, is still only an approximation, whose validity strongly depends on the context in which it is used. For chaotic potentials (more generally: chaotic systems) the time-scale, $T_M$, over which a non-machine’s solution and that of its machine approximation remain close (t-wise less than some small constant) could grow very slowly with increasing $\lambda$ especially if K has a long, i.e., algebraic tail (as is the case in ECD). One way of seeing this lies in the residue, R, playing the role of an external force in (3) which, when acting on q in unstable directions, leads to its subsequent exponential separation from the unperturbed path. The dimensionality, n, of the system plays a crucial role in tempering the growth of $T_M$ with increasing $\lambda$, as larger n implies (statistically) higher maximal Lyapunov exponent—and it only takes one such ultra unstable direction for even a meager R to rapidly drive an entire chaotic system `off course’. Now, realistically speaking, there is no “unperturbed path”. And indeed, for a sufficiently large $\lambda$ most `coarse grained’ statistics associated with (2), e.g. attractor manifold and power spectrum, would most likely be experimentally indistinguishable from those of a noisy machine, viz. (3) with R a random noise. However, as R is far from being random, such noisy machine approximation becomes moot with regard to the global spacetime structure of (at least some) trajectories. In other words, a noise history reproducing a global path of (2) would need to be too `structured’, or non random, for any realistic noise source (e.g. Gaussian White).

However, the experiments proposed here do not attempt to `implicate’ individual spacetime structures as belonging to non-machines. Instead, a non mechanistic statistical signature is sought in ensembles of suspected structures, and there is a crucial difference in this regard between machines and non-machines. In the former, e.g. (3) with $R=0$, its solution set can be 1-to-1 mapped to a subset of ${\mathbb{R}}^{2n}$ equipped with a natural, i.e. Liouville measure. In contrast, the solution set of (2) is some infinite dimensional function space having no obvious counterpart measure. Single-system equations of non-machines, then, cannot be a complete description of the experiment, necessitating a compatible statistical description of ensembles of solutions, not deriviable from the single-system theory alone, hence being equally fundamental.

The significance of this last point is illustrated clearly in a scattering experiment ($n=1$) of monoenergetic particles off a chaotic potential, e.g. a crystal lattice. In the non-machine case, uniformity over the impact parameter does not define an ensemble since two incoming particles can have identical (freely moving, asymptotic-) solutions yet different outgoing ones; As the two particles approach the target, their distinct future paths, gradually render their R’s non negligible and distinct. Now, suppose that the past asymptotic motion takes place in some time-independent, chaotic potential V. At some fixed plane orthogonal to the average propagation direction of the particles, V transitions into either $V_1$ or $V_2$. The BU statistical view of this scenario now involves two ensembles of word-lines, q, shifted in time such that $q\left(0\right)$ lies on the $V-V_{(·)}$ interface plane. Define the past ensemble as that collection of partial world-lines $q\left(t\right)$ for $t<0$. On time-scales shorter than $T_M$, any such partial world-line whose form is not excluded by (3) with $R=0$ could appear in either past ensemble. But does it? More accurately: Must its statistical weight (frequency of appearance) be the same in both? An NG physicist would answer in the affirmative, so long as the two ensembles originate from a common distribution of past initial conditions, but in a non mechanistic BU an ensemble can’t even be $1-1$ mapped to such a distribution; past and future potentials seen by ensemble’s members are both relevant. A non mechanistic statistical signature should therefore be present in ensembles of non-machines even when individual members are examined on time-scales shorter than $T_M$. Obviously, one can do better by examining them on time-scales longer than $T_M$, hence the choice of a chaotic V (almost any non chaotic $V\neg \equiv 0$ could do as well but the much longer $T_M$ introduces much more noise which, in turn, requires greater statistical power to distinguish between the two ensembles).

Zooming out now from our toy model, insofar as a physicist obeys the physics of the systems he is studying—and there is no evidence to the contrary—he is represented by some (extended) world-line in the BU, and his free will is technically an illusion. This tension with one’s subjective feeling is present also in Newtonian-grammar physics, and this never disocuraged physicists from doing physics. But unlike its Newtonian counterpart, our physicist cannot meaningfully model this illusary free will by chosing the initial conditions of a system (or distribution thereof) from which the system (ensemble thereof resp.) then evolves. Instead, his `freely chosen’ actions constrain the global, spacetime structure of systems he is studying (in a way which depends on both the system and the actions) and in general, infinitely many such systems are compatible with a given constraint. His actions therefore only define an ensemble, with the relative frequency of each ensemble-member being a statistical property of the BU (revealed in a single lab only as a means of saving the hassle of sampling the whole BU for similarly constrained, spontaneously occurring systems [3]). Note that the Newtonian resolution can be viewed as a private case of ours.

In the case of closed systems, according to [3], QM provides a rich statistical description of the ensemble, encoded in the wave-function, which is therefore an attribute of the ensemble rather than of any single system (see caption of Figure 1). By “closed” it is meant that the system’s full energy-momentum balance is known and exactly incorporated into its Hamiltonian. In contrast, the statistical description of open systems [12] is currently not nearly as detailed and conceptually problematic. It boils down to treating an open system as a small subsystem of a large closed system, tracing out the extra degrees of freedom, not before making simplifying assumptions about their interaction with the subsystem and with one another. However, this attempt is no more reliable than similar attempts to model irreversibility within the framework of (classical) Hamiltonian dynamics, if only because it ignores an essential source of dissipation and thermodynamic irreversibly: the radiation arrow-of-time (manifested in ECD systems which are out of equilibrium with the zero-point-field [4]). When used in the context of macroscopic irreversible systems it furthermore pushes the mysterious `collapse postulate’ of QM far beyond its empirically validated domain, making it even harder to defend: Is the `observer’—and what is meant by that—part of the environment? Moreover, when that irreversible system is chaotic, this approach, at best, would prove consistent with the classical description, which, in turn, comes with its own conceptual and practical difficulties. Ergodic theory [11], for example, seeks a flow-invariant measure on phase/configuration space, and is indeed a valid starting point for predicting the steady-state distribution of those (exceptional) systems for which such flow exists (e.g. type-1 circuits in Figure 3). However, as it typically yields a fractal set which includes infinitely many (unstable) periodic orbits, neither ergodic theory nor its noisy versions, e.g. the associated Fokker-Planck equation, are sufficient for that. More relevant to out point, though (and as already pointed out) if this flow only locally approximates the 4D structure of chaotic systems, then ergodic theory is mute with regard to more complex statistics, e.g., the measure on the past ensemble of chaotic solutions from the above example. A similar objection applies to ensemble propagation when used to predict the long-time behavior of chaotic systems, or to inter-system correlators of previously coupled chaotic systems (see sect.3.3).

Summarizing, when leaving the domain of closed quantum systems, QM becomes an unreliable tool. When then stepping into the realm of chaotic irreversible systems, one is already in largely uncharted territory. There, presumably, lies new physics which is nevertheless consistent with well established theories.

To distinguish machines from non-machines, we first propose testing whether a system can `remember its future’, diving deeper into the experiment described in sect. 2 and its consequences, but without committing to the toy model used there. Machines can obviously remember their past, meaning that a perturbation, p, to a machine in its past can be inferred from its present state m (`memory’). For simplicity, a binary type perturbations shall be used, labeled `L’(eft) and `R’(ight). In contrast, machines cannot `remember’ their future. Inferring a future perturbation from a machine’s present state entails, among else, the following: The state, m, of a machine is measured and projected onto the set $\{L,R\}$ at some initial time. The machine then propagates to a later time when its world-line intersects that of a random bit generator (RBG) applying a random $p\in \{L,R\}$ to it, and miraculously $p=m$. This must happen everywhere throughout the BU, to all copies of the machine, which is clearly not our BU.

This evident truth can be extended to non-deterministic machines in which the rest of the universe is (realistically) treated as the source of randomness and possibly dissipation. In this case, the machine’s stochastic evolution leads to a certain probability distribution over its future states, parametrically depending on the nature of the initial perturbation (L or R in our case, marginalized over possible additional `hidden variables’). If the two distributions are distinguishable, i.e., if a random perturbation, p, can be deduced from m with probability $>0.5$, then the machine is said to fuzzily remember a bit. As in the deterministic case, machines can fuzzily remember a past bit but not a future one.

Like machines, non-machines can remember their past. A sufficiently strong perturbation to a fully developed turbulence, for example, leaves an obvious signature on the streamlines for a short time thereafter. It is even conceivable that this `short term memory’ would extend much further into the past had only a different, more suitable signature been used—as in the case of seasonal weather forecasts which are based on large statistical tables rather than propagation of differential equations. But why should non-machines not remember their future, at least fuzzily?

In previous work by the author a detailed proposal was laid out for a classical ontology (ECD), of which QM is a very partial statistical description, applicable to ensembles of closed systems where the effects of the radiation arrow-of-time—an inescapable attribute of an ECD universe—are negligible. This view implies that current use of QM is grossly overreaching (and accordingly, with minimal empirical success to back it up). However, quantum weirdness allegedly lies in individual members of an ensemble, which are spacetime `structures’ associated with non-machines, whether closed or open. It is only QM’s statistics of ensembles of such structures which is applicable exclusively to simple closed systems. Quantum weirdness is therefore not expected to disappear altogether from macroscopic irreversible physics but, instead, take a new statistcal form, perhaps even weirder. The illusion of `dark matter’ is one example [4] with immediate, testable predictions. Non mechanistic correlations, proposed in the current paper, is another. The use of machine learning to this end is by no means essential, reflecting the author’s failure to hitherto come up with XM—that rich statistical description of irreversible systems. Nonetheless, from epicycles to GR and QM, breakthroughs in physics were always preceded by advances in modeling tools, and it is conceivable that new, fully analytic laws of physics are a thing of the past.