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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.

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Notes

  1. So-called quantum contextuality—where measurement outcomes are dependent on the measurement context—can be considered a special case of the contexts central to contextual emergence.

  2. This is not to imply that the spacetime of special relativity reduces smoothly to the spacetime of Newtonian mechanics in this limit.

  3. The coarser contextual topology is compatible with the original, finer topology if they and their dynamics are topologically equivalent with each other (Appendix A.3).

  4. On should not confuse human states of knowledge with epistemic states. Epistemic states describe those properties of systems that can be measured. Human knowledge describes what we know after a measurement has taken place.

  5. This is to say that a \(C^{*}\)-algebra, \({\mathcal {A}}\), is a \(W^{*}\)-algebra if there exists a Banach space \({\mathcal {A}}_{*}\) such that \(({\mathcal {A}}_{*})^{*}={\mathcal {A}}\), where \(({\mathcal {A}}_{*})^{*}\)is the dual Banach space of \({\mathcal {A}}_{*}\). The Banach space \({\mathcal {A}}_{*}\) is the predual of \({\mathcal {A}}\). An important example of a commutative \(\mathrm {W}^{*}\)-algebra is the Banach space \(L_{\infty }\) where the predual space is the separable Banach space \(L_{1}\).

  6. Dynamics on the dual \({\mathcal {A}}\)* is also well-defined, but the time evolution of non-normal states are not guaranteed to have desirable continuity properties.

  7. While what follows is based on results from quantum statistical mechanics, since the end result always leads to a quantum/classical algebra (Sect. 3.2.2) with temperature as a classical observable, it makes no difference to the case for contextual emergence whether one discusses classical or quantum statistical mechanics.

  8. The knot-type of DNA is also dispersion-free, hence, is a classical observable that has actual-world consequences (e.g., [36]).

  9. This implies that any talk of a universal wavefunction as a fundamental entity is incoherent if anything more is meant than the basic fact that quantum mechanics contributes necessary conditions defining the space of physical possibility [18, Sect. 6.3]. Any wavefunctions are already coming to expression in some concrete context. What is more, as noted above, any physical environment already has classical thermal and electromagnetic states.

  10. An atom is a minimal nonzero element of a lattice, which is to say that it cannot be decomposed into two proper subsets: For \(\alpha \subseteq {\mathcal{A}}\), \(\alpha\) is an atom iff for every \(\beta\), either \(\beta \wedge \alpha =\alpha\) or \(\beta \wedge \alpha =0\).

References

  1. Allefeld, C., Atmanspacher, H., Wackermann, J.: Mental states as macrostates emerging from EEG dynamics. Chaos 19, 015102 (2009)

    ADS  MathSciNet  Google Scholar 

  2. Amann, A.: The gestalt problem in quantum chemistry: generation of molecular shape by the environment. Synthese 97, 125–156 (1993)

    MathSciNet  Google Scholar 

  3. Anderson, P.W.: More is different: broken symmetry and the nature of the hierarchical structure of science. Science 177(4047), 393–396 (1972)

    ADS  Google Scholar 

  4. Araki, H., Kastler, D., Takesaki, M., Haag, R.: Extension of KMS states and chemical potential. Commun. Math. Phys. 53, 97–134 (1977)

    ADS  MathSciNet  Google Scholar 

  5. Atkins, P.: Creation Revisited: The Origin of Space, Time and the Universe. Penguin, New York (1994)

    Google Scholar 

  6. Atmanspacher, H.: Is the ontic/epistemic distinction sufficient to describe quantum systems exhaustively? In: Montonen, C., Laurikainen, K., Sunnarborg, K. (eds.) Symposium on the Foundations of Modern Physics 1994, pp. 15–32. Editions Frontieres, Gif-sur-Yvette (1994)

    Google Scholar 

  7. Atmanspacher, H. (ed.): Hans Primas: Knowledge and Time. Springer International, Cham (2017)

    Google Scholar 

  8. Atmanspacher, H., Graben, P.: Contextual emergence of mental states from neurodynamics. Chaos Complex. Lett. 2(2/3), 151–168 (2007)

    Google Scholar 

  9. Atmanspacher, H., Kronz, F.: Relative onticity. In: Atmanspacher, H., Amann, A., Müller-Herold, U. (eds.) On Quanta Mind and Matter. Hans Primas in Context. Fundamental Theories of Physics, pp. 273–294. Kluwer, Dordrecht (1999)

    Google Scholar 

  10. Balasubramanian, K.: Combinatorial methods in ESR spectroscopy. J. Magn. Reson. 91, 45–56 (1991)

    ADS  Google Scholar 

  11. Balasubramanian, K.: Combinatiorics of NMR and ESR spectral simulations. J. Chem. Inf. Model. 32, 296–298 (1992)

    Google Scholar 

  12. Bedau, M., Humphreys, P. (eds.): Emergence: Contemporary Readings in Philosophy and Science. MIT Press, Cambridge (2008)

    Google Scholar 

  13. beim Graben, P.: Contextual emergence in neuroscience. In: El Hady, A. (ed.) Closed Loop Neuroscience, pp. 171–184. Elsevier, Amsterdam (2016)

    Google Scholar 

  14. Bishop, R.C.: Patching physics and chemistry together. Philos. Sci. 72, 710–722 (2005)

    Google Scholar 

  15. Bishop, R.C.: The hidden premiss in the causal argument for physicalism. Analysis 66(1), 44–52 (2006)

    Google Scholar 

  16. Bishop, R.C.: Whence chemistry? Reductionism and neoreductionism. Stud. Hist. Philos. Sci B 41(2), 171–177 (2010)

    Google Scholar 

  17. Bishop, R.C.: Fluid convection, constraint and causation. Interface Focus 2(1), 4–12 (2012)

    Google Scholar 

  18. Bishop, R.C.: The Physics of Emergence. IOP Concise Physics Series. Morgan & Claypool Publishers, San Rafael, CA (2019)

    Google Scholar 

  19. Bishop, R.C., Atmanspacher, H.: Contextual emergence in the description of properties. Found. Phys. 36(12), 1753–1777 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  20. Born, M., Oppenheimer, R.: Zur Quantentheorie der Molekeln. Ann. Phys. 389(20), 457–484 (1927)

    MATH  Google Scholar 

  21. Buchholz, D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85, 49–71 (1982)

    ADS  MathSciNet  MATH  Google Scholar 

  22. Compagner, A.: Thermodynamics as the continuum limit of statistical mechanics. Am. J. Phys. 57, 106–117 (1989)

    ADS  MathSciNet  Google Scholar 

  23. Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)

    ADS  MATH  Google Scholar 

  24. d'Espagnat, B.: Veiled Reality: An Analysis of Present-Day Quantum Mechanical Concepts. Addison-Wesley, Reading, MA (1994)

    MATH  Google Scholar 

  25. Drossel, B., Ellis, G.: Contextual wavefunction collapse: an integrated theory of quantum measurement. N. J. Phys. 20, 113025 (2018)

    Google Scholar 

  26. Ellis, G.: How Can Physics Underlie the Mind? Top-Down Causation in the Human Context. Springer, Berlin, Heidelberg (2016)

    Google Scholar 

  27. Ellis, G.F.R., Meissner, K.A., Nicolai, H.: The physics of infinity. Nat. Phys. 14(8), 770 (2018)

    Google Scholar 

  28. Emch, G.G., Liu, C.: The Logic of Thermo-statistical Ssytems. Springer-Verlag, Berlin (2002)

    Google Scholar 

  29. Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1992)

    MATH  Google Scholar 

  30. Haag, R., Hugenholtz, N.M., Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Haag, R., Kastler, D., Trych-Pohlmeyer, E.B.: Stability and equilibrium states. Commun. Math. Phys. 38, 173–193 (1974)

    ADS  MathSciNet  Google Scholar 

  32. Koopman, B.O.: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. USA 17(5), 315–318 (1931)

    ADS  MATH  Google Scholar 

  33. Koopman, B.O., von Neumann, J.: Dynamical systems of continuous spectra. Proc. Natl. Acad. Sci. USA 18(3), 255–263 (1932)

    ADS  MATH  Google Scholar 

  34. Kryachko, E.S., Ludña, E.V.: Energy Density Functional Theory of Many-Electron Systems. Kluwer Academic, Dordrecht (1990)

    Google Scholar 

  35. Kubo, R.: Statistical-mechanical theory of irreversible processes. I. general theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957)

    ADS  MathSciNet  Google Scholar 

  36. Lim, N.C.H., Jackson, S.E.: Molecular knots in biology and chemistry. J. Phys.: Condens. Matter 27(35), 1–35 (2015)

    Google Scholar 

  37. Martin, P.C., Schwinger, J.: Theory of many-particle systems. I. Phys. Rev. 115(6), 1342–1373 (1959)

    ADS  MathSciNet  MATH  Google Scholar 

  38. McGuinness, B. (ed.): Ludwig Boltzmann: Theoretical Physics and Philosophical: Problems Selected Writings. D Reidel, Dordrecht, Boston (1974)

    Google Scholar 

  39. Misra, B.: Lyapunov variables, and ergodic properties of classical systems. Proc. Natl. Acad. Sci. USA 75, 1627–1631 (1978)

    ADS  Google Scholar 

  40. Misra, B., Prigogine, I., Courbage, M.: From deterministic dynamics to probability descriptions. Physica A 98, 1–26 (1979)

    ADS  MathSciNet  MATH  Google Scholar 

  41. Müller-Herold, U.: Disjointness of β-KMS states with different chemical potential. Lett. Math. Phys. 4, 45–48 (1980)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Nagel, E.: The Structure of Science: Problems in the Logic of Scientific Explanation. Harcourt, Brace, and World, New York (1961)

    Google Scholar 

  43. Primas, H.: Theory reduction and non-Boolean theories. J. Math. Biol. 4, 281–301 (1977)

    MathSciNet  MATH  Google Scholar 

  44. Primas, H.: Chemistry, Quantum Mechanics and Reductionism: Perspectives in Theoretical Chemistry. Number 24 in Lecture Notes in Chemistry. Springer-Verlag, Berlin, second, corrected edition (1983)

  45. Primas, H.: Mathematical and philosophical questions in the theory of open and macroscopic quantum systems. In: Miller, A.I. (ed.) Sixty-Two Years of Uncertainty, pp. 233–257. Plenum, New York (1990)

    Google Scholar 

  46. Primas, H.: Endo- and exo-theories of matter. In: Atmanspacher, H., Dalenoort, G.J. (eds.) Inside Versus Outside. Endo- and Exo-concepts of Observation and Knowledge in Physics, Philosophy, and Cognitive Science, pp. 163–193. Springer, Berlin (1994)

    Google Scholar 

  47. Primas, H.: Emergence in exact natural sciences. Acta Polytech. Scand. Ma. 91, 83–98 (1998)

    MathSciNet  Google Scholar 

  48. Schaffner, K.: Approaches to reduction. Philos. Sci. 34, 137–147 (1967)

    Google Scholar 

  49. Scheibe, E.: The Logical Analysis of Quantum Mechanics. Pergamon Press, New York (1964)

    Google Scholar 

  50. Sklar, L.: Types of inter-theoretic reduction. Br. J. Philos. Sci. 18, 109–124 (1967)

    Google Scholar 

  51. Sutcliffe, B.T., Woolley, R.G.: Atoms and molecules in classical chemistry and quantum mechanics. In: Hendry, R.F., Needham, P., Woody, A.I. (eds.) Philosophy of Chemistry. Handbook of the Philosophy of Science, vol. 6, pp. 388–426. Elsevier BV, Amsterdam (2012)

    Google Scholar 

  52. Takesaki, M.: Disjointness of the KMS-states of different temperatures. Commun. Math. Phys. 17, 33–41 (1970)

    ADS  MathSciNet  Google Scholar 

  53. Takesaki, Masamichi: Theory of Operator Algebras I. Springer, New York (2002)

    MATH  Google Scholar 

  54. von Neumann, J: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)[1932]

  55. Weinberg, S.: Dreams of a Final Theory. Pantheon Books, New York (1993)

    Google Scholar 

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We thank two referees for helpful comments that have clarified and strengthened the manuscript.

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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).

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.Footnote 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.

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 \(L^{2}\).

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].

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. (1)

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

  2. (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.

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Bishop, R.C., Ellis, G.F.R. Contextual Emergence of Physical Properties. Found Phys 50, 481–510 (2020). https://doi.org/10.1007/s10701-020-00333-9

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Keywords

  • Contextual emergence
  • Reductionism
  • Stability conditions
  • Equivalence classes
  • Temperature
  • Chemical potential
  • Molecular structure