Contextual Emergence of Physical Properties

Abstract

Contextual emergence was originally proposed as an inter-level relation between different levels of description to describe an epistemic notion of emergence in physics. Here, we discuss the ontic extension of this relation to different domains or levels of physical reality using the properties of temperature and molecular shape (chirality) as detailed case studies. We emphasize the concepts of stability conditions and multiple realizability as key features of contextual emergence. Some broader implications contextual emergence has for the foundations of physics and cognitive and neural sciences are given in the concluding discussion. Relevant facts about algebras of observables are found in the appendices along with an abstract definition of Kubo-Martin-Schwinger states.

Appendices

Appendices

The Appendices cover *-Algebras (Appendix A.1), Strong/Weak Topologies (Appendix A.2), and Structural Stability and the Topological Equivalence of Dynamical Systems (Appendix A.3). The final Appendix defines KMS states (Appendix A.4).

1.1 A.1: *-Algebras

A $^{*}$-algebra admits an involution $^{*}:{\mathcal {A}}\rightarrow {\mathcal {A}}$ with the usual properties. A $^{*}$-algebra is normed, if there is a mapping $||.||:{\mathcal {A}}\rightarrow {\mathbf {R}}_{+}$ with the usual properties. A complete normed $^{*}$-algebra is a Banach $^{*}$-algebra. A $C^{*}$-algebra is a Banach $^{*}$-algebra ${\mathcal {A}}$ with the additional property $||x^{*}x||=||x||^{2}$ for all $x\in {\mathcal {A}}$ [53, chap. I.1]. The associated concept of a state is introduced in terms of positive normalized linear functionals defined over ${\mathcal {A}}$. For a fundamental theory in physics, the state space is chosen such that only the most basic assumptions are required for its definition.

Algebras can be classified by their central decompositions or factors. A factor is of type I if it contains an atom.¹⁰ It is of type II if it is atom-free and contains some nonzero finite projection. It is of type III if it does not contain any nonzero finite projection [53, p.296]. Every factor of type I has normal pure states, though for type $\text{I}_{\infty }$ not all pure states are normal. Factors of types II and III lack normal pure states. For the example of statistical mechanics/thermodynamics, mechanical observables used to develop statistical ensembles and their expectation values reside in a type I $W^{*}$-algebra, while the contextual $W^{*}$-algebra defined through the KMS condition is of type III, meaning temperature cannot be reducible to statistical mechanics in any straightforward sense (Sect. 4).

The center of an algebra contains elements that commute with the rest of the elements in the center. A center is trivial when these elements are simply multiples of the identity operator.

1.2 A.2: Strong/Weak Topologies

Algebras of observables are related to the topologies of the state spaces over which they are defined. Topologies define the convergence properties for a sequence of elements in a space, and can be characterized as strong/fine or weak/coarse. For instance, in a Banach space the $\parallel \cdot \parallel$ norm induces a topology $\tau$, while its dual, the set of all continuous linear functions, induces a topology $\sigma$ on the Banach space. The latter topology is weak while the former is strong; that is to say, $\sigma \subseteq \tau$.

The differences between strong and weak topologies can be illustrated by means of series expansions [13]. An example of convergence in a strong topology would be uniform convergence of a Taylor series of a function within its convergence radius. An example of convergence in a weak topology would be the Fourier series of a function, which converges only in quadratic norm $L^{2}$.

1.3 A.3: Structural Stability and Topological Equivalence of Dynamical Systems

A fundamental notion of stability for a dynamical system is the stability of a point $x^{*}\in {\mathfrak {X}}$ under the flow $\Phi ^{t}:x^{*}=\Phi (x^{*})$. This means $x^{*}$ is a fixed-point attractor for the flow. The technique of Poincaré sections can be used to relate limit cycles or higher-order tori as attractors to fixed points. More generally, attractors are invariant sets $A\subset {\mathfrak {X}}$, such that $\Phi (A)=A$ and $\Phi ^{-1}(A)\subseteq \ A$. This invariance property of A extends to probability measures $\mu$, where $\mu (\Phi ^{-1}(A)=\mu (A)$, which are called stationary or invariant measures. Similarly, a statistical state $\rho _{\mu }$ over the algebra of continuous functions assigned to the measure $\mu$ has the invariance property. The invariance of thermal equilibrium states is the first condition for KMS states given in Haag et al. [31].

Structural stability refers to perturbations in the function space of the flow map $\Phi$. A system $({\mathfrak {X}},\Phi )$ is structurally stable if there is a neighborhood ${\mathcal {N}}$ of $\Phi$ such that all $\Psi \in {\mathcal {N}}$ are topologically equivalent to $\Phi$. Two maps $\Phi$ and $\Psi$ are topologically equivalent, or conjugated, if there is a homeomorphism h such that $h\circ \Phi =\Psi \circ h$. As Haag et al. [31] pointed out, structural stability is closely related to ergodicity: An invariant probability measure $\mu$ is said to be ergodic under the flow $\Phi$ if an invariant set A, has either measure zero or one, $\mu (A)\in \{0,1\}$. If $\mu$ is non-ergodic, there is an invariant set A with $0<\mu (A)<1$ corresponding to an accidental degeneracy. Such degeneracies are not stable under small perturbations. Therefore, non-ergodic systems are in general not structurally stable [31].

1.4 A.4: Defining KMS States

Consider a $C^{*}$-dynamical system with an associated algebra of observables. Suppose ${\mathcal {A}}$ be a $C^{*}$-algebra and $t\rightarrow \tau _t$ a strongly continuous group of automorphisms of ${\mathcal {A}}$. An element $A \subseteq \mathcal{A}$, is analytic if there exists a strip $I_{\eta } = \{z {\in {\mathbb {C}}}:\mid {\mathfrak{G}}(z)\mid <\eta \}$ and a function $f:I_\eta \ \rightarrow {\mathcal {A}}$ such that

• (1) $f(t)=\tau _t (A)$ for all $t\in {\mathbb {R}}$

• (2) $z\rightarrow f(z)$ is analytic for $z\in I_\eta$

For the $C^{*}$-dynamical system (${\mathcal {A}},\tau ,{\mathbb {R}})$, a state $\phi$ defined over ${\mathcal {A}}$ is a $\tau$-KMS state with value $\beta \in {\mathbb {R}}$ if $\phi (A_{\beta }(B))=\phi (BA)$ for all A, B in a norm-dense, $\tau$-invariant $^{*}$-subalgebra of ${\mathcal {A}}_\tau$, where $\beta$ is inverse temperature.