The origin of spatiotemporal order in physical and biological systems is a key scientific question of our time. How does microscopic matter self-organize to create living and non-living macroscopic structures? Do systems capable of generating spatiotemporal complexity obey certain universal principles? We propose that progress along these questions may be made by searching for fundamental properties of non-linear field models which are common to several areas of physics, from elementary particle physics to condensed matter and biological physics. In particular, we've begun exploring what models that support localized coherent (soliton-like) solutions – both time-dependent and time-independent – can tell us about the emergence of spatiotemporal order. Of interest to us is the non-equilibrium dynamics of such systems and how it differs when they are allowed to interact with external environments. It is argued that the emergence of spatiotemporal order delays energy equipartition and that growing complexity correlates with growing departure from equipartition. We further argue that the emergence of complexity is related to the existence of attractors in field configuration space and propose a new entropic measure to quantify the degree of ordering of localized energy configurations.

SOLITONS AND SELF-ORGANIZATION

A key question across the natural sciences is how simple material entities self-organize to create coherent structures capable of complex behavior. As an example, phenomena as diverse as water waves and symmetry-breaking during phase transitions can give rise to solitons, topologically or non-topologically stable spatially-localized structures (“energy lumps”) that keep their profiles as they move across space. They beautifully illustrate cooperative behavior in Nature, that is, how interacting discrete entities work in tandem to generate complex structures that minimize energy and other physical quantities (Infeld & Rowlands, 2000; Walgraef, 1997; Cross & Hohenberg, 1993; Rajamaran, 1987; Lee & Pang, 1992).

This book examines the foundational consistency of quantum mechanics incorporated within relativistic frameworks. Quantum physics remains a perplexing formalism that, although very successful in explaining physical phenomena, poses many philosophical and interpretational questions. Several of the subtleties of quantum physics become more manifest when quantum processes are described using relativistic dynamics. For instance, the successful connection of spin to quantum statistics is a consequence of the consistent incorporation of special relativity into the quantum formalism. There should be similar profound explanations awaiting discovery as gravitating phenomena are successfully incorporated into quantum formulations.

The common theme of this manuscript is the examination of the incorporation of relativistic behaviors upon the foundations of quantum physics. The approach is to keep all formulations as close to observed phenomena as possible, rather than to present a set of speculative models whose primary motivations are internal aesthetics. In the search for the most elegant models of physical phenomena, one must recognize that at its core, physics is an experimental science. The dimensional analysis of fundamental units, taught at the very beginning of introductory physics classes, demonstrates that phenomenology lies at the foundations of physics. Fundamental ideas such as correspondence, the principle of relativity, and complementarity provide direct contact with the physics used to guide this exploration. This manuscript is an elaboration and expansion on previously published work, but also contains some new material.

A synopsis of Big Bang cosmology

One of the fundamental principles guiding the expected behaviors of the cosmology is a generalization of ideas of Copernicus, known as the cosmological principle. This principle presumes that no non-rotating observer at rest to the CMB radiation is more special than any other. Since the observed universe has large-scale uniformity, the cosmological principle imposes an overall homogeneity and isotropy to the universe. In addition, the dynamics of most of the aggregate features in the universe can be described assuming that the energy-momentum content of the cosmology is consistent with being an ideal fluid.

The Friedmann-Lemaitre equations 8.6, which describe the dynamics of an ideal fluid cosmology, are spatially scale invariant (if the cosmological constant is negligible), but not temporally scale invariant. The form of those equations that govern a spatially flat (k = 0) expansion satisfies spatial scale invariance (at least to a very good approximation), due to the fact that the energy densities that drive the dynamics are intensive thermodynamic variables. However, there is apparently a beginning time t0 ≈ 13.7(±0.2) billion years ago, which represents the earliest backwards-looking extrapolation of the standard model expansion called the Big Bang. The physics during these earliest moments is an active field of research. Thus, the cosmological principle does not refer to the temporal evolution of the universe.

Boolean algebras

In this appendix, X is always a (nonempty) set and S is a nonempty set of subsets of X.For any A⊆X, the symbol A⊥ denotes the set theoretical complement of A in X;thatis, A⊥ = X\A.

Definition A.1S is a (Boolean) ring if for every A, B ϵ S we have (A∪B) ϵ S and (A\B) ϵ S.

A ring is a set of sets that is closed with respect to set theoretical union and difference. If S is a ring, then ∅ ϵ S (because ∅ = A\A); however, X does not necessarily belong to S. If it does, then the (Boolean) ring is called Boolean algebra:

Definition A.2 A Boolean ring S is called Booelan algebra if X ϵ S.

A Boolean algebra S is thus closed with respect to the complement: if S is a Boolean algebra and AϵS, then A⊥ ϵ S.

One can define the notion of Boolean algebra directly: S is a Boolean algebra with respect to the set theoretical operations ∪, ∩, ⊥ if X ϵ S and if it holds that if A, B ϵ S, then (A∩B), (A∪B), and A⊥ are all in S.

The history of philosophy teaches us that metaphysical claims of sweeping generality are neither verifiable nor conclusively falsifiable. One can only aim at assessing their plausibility on the basis of the best available evidence provided by the sciences – both formal and empirical sciences. This is what has been done in this book in connection with the Principle of the Common Cause.

In Chapter 2, Reichenbach's notion of common cause and the related Common Cause Principle was formulated explicitly in terms of classical, Kolmogorovian probability measure spaces. The Definition 2.4 of common cause followed Reichenbach's original definition closely, and insisting on the quite obvious methodological principle that probabilistic concepts and statements (in particular claims about random events being correlated or probabilistically independent) are only meaningful within the context of a fixed probability measure space in terms of which some segment of reality is modeled, we specified the notions of common cause incomplete and common cause complete probability theories: a theory was defined to be common cause complete if it contains a proper common cause of every correlation it predicts, common cause incomplete otherwise. (There is a strong version of common cause completeness as well: a probability space was defined to be strongly common cause closed if it contains common causes of all admissible types – Definition 4.4.)

These explicitly defined notions form the basis on which one can start assessing the status of the Common Cause Principle in the spirit of empirical philosophy. Suppose we have an empirically confirmed scientific theory T describing some segment of reality using (possibly among other mathematical structures) a probability measure space (X, S, p).

Common cause partitions

Confronted with a common cause incomplete probability space (X, S, p)in which a direct causal influence between the correlated events is excluded, one can have in principle two strategies aiming at saving the Common Cause Principle: one may try to argue that S is not “rich enough” to contain a common cause, but there might exist a larger probability space (X′, S′, p′) that already contains a common cause of the correlation. As we have seen in Chapter 3 this strategy always works in the sense that it is always possible to enlarge (X, S, p) in such a way that the enlarged probability space already contains an event C that satisfies the Reichenbachian conditions.

Another natural idea is to suspect that the correlation between A and B is not due to a single factor but may be the cumulative result of a (possibly large) number of different “partial common causes,” none of which can in and by itself yield a complete common-cause-type explanation of the correlation, all of which, taken together, can however account for the entire correlation. In this chapter we elaborate this idea by formulating precisely a notion of the Reichenbachian Common Cause System (RCCS) and prove propositions on the existence and features of such systems.

As we have seen in Chapter 2, if the events A, B, C satisfy the Reichenbachian conditions (2.5)–(2.8) then there is a positive correlation between A and B (Proposition 2.5).

Common causes and common common causes

Proposition 3.9 tells us that every common cause incomplete classical probability space can be strongly common cause extended with respect to any (finite) set of correlations; Proposition 4.19 states that every classical probability space is even common cause completable. Note that what these propositions say is not that for a set of correlations between (Ai, Bi)(i = 1, 2 …n) there exists a single, common common cause C in the extension (or completion) for the whole set of correlations; in fact, the common causes Ci constructed explicitly in the proof of Proposition 3.9 are all different: Ci ≠ Cj (i ≠ j). This observation leads to the following question.

Let (Ai, Bi)(i = 1, … n)be a finite set of pairs of events in(X, S, p)that are correlated [Corrp(Ai, Bi) ＞ 0 for every i]. We say that C is a common common cause of these correlations if C is a Reichenbachian common cause of the correlated pair (Ai, Bi)for every i. Does every set of correlations in a classical probability space have a common common cause?

In view of the generality of this question one may surmise that the answer to it is “yes”; that is to say, one may conjecture that given any two correlations there can always exist a Reichenbachian common cause which is a common cause for both correlations, since, one may reason, we just have to refine our picture of the world by adding more and more events to the original event structure in a consistent manner, and finally we shall find a single common cause that explains both correlations.