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Top-Down Approaches in Physics

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Abstract

Chapters 6 and 7 contain Primas’ account of the conceptual lessons of quantum physics. They contrast the familiar bottom-up approach (Chap. 6) with a less discussed top-down point of view (Chap. 7).

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Notes

  1. 1.

    Compare Houtappel et al. (1965, p. 596) and Wigner (1964b). The distinction between initial conditions and laws of nature is due to Newton’s (1729) Philosophiae Naturalis Principia Mathematica, even though in an implicit fashion.

  2. 2.

    The term physicalism was introduced into philosophy by Otto Neurath (1932, p. 405, editor’s translation): “There are no ‘explanations’ which would not be physical statements”, and Rudolf Carnap (1932, p. 463): “Each matter of fact in science can be explained as a physical matter of fact, i.e. by quantitatively determinable properties of a space-time position”. Later, Herbert Feigl (1963, pp. 227f) stated that “physicalism claims that the facts and laws of the natural and the social sciences can all be derived—at least in principle—from the theoretical assumptions of physics”.

  3. 3.

    Editor’s note: Although the causal closure of the physical is still defended by influential philosophers of science today (e.g., Spurrett 1999; Papineau 2001), there is an increasing number of cautious and critical voices as well (e.g., Montero 2003; Bishop 2006; Atmanspacher and Filk 2012).

  4. 4.

    From a letter by Pauli to Heisenberg of June 15, 1935 (von Meyenn 1985, p. 404, editor’s translation).

  5. 5.

    This is demonstrated by the Unruh effect (Unruh 1976). Compare the review by Crispino et al. (2008).

  6. 6.

    In his fundamental study of the relation between a W*-algebra (also called von Neumann algebra) and its commutant, Tomita (1967) discovered that each left Hilbert algebra is associated with a one-parameter group of automorphisms of the left von Neumann algebra, and that every faithful normal positive functional \(\rho \) on a von Neumann algebra ℳ induces a left Hilbert algebra structure on ℳ and gives rise to a one-parameter group \(\{\sigma_{\rho ,s}|s\in \mathbb{R}\}\) of automorphisms of ℳ. His original paper was difficult to read and little notice was taken of it until Takesaki (1970) gave a full account of Tomita’s theory.

  7. 7.

    Compare also Araki (1968), Bratteli and Robinson (1981).

  8. 8.

    See Takesaki (1970). A state functional \(\rho \in \mathcal{M}_{*}\) on a W*-algebra ℳ with a \(\sigma \)-weakly continuous one-parameter group of automorphism \(\alpha_{s}\) of ℳ satisfies the KMS condition at the parameter value \(\beta \) (\(0<\beta <\infty \)) with respect to \(\{\alpha_{s}|s\in \mathbb{R}\}\) if for any \(A,B\in \mathcal{M}\) there exists a complex function \(z\mapsto F_{A,B}(z)\) analytic on the strip \(\{z\in \mathbb{C} | 0<\text{Im} z<\beta \}\) and continuous on the closure of this strip such that \(F_{A,B}(s)=\rho \{\alpha_{s}(A) B \} , F_{A,B}(s+i\beta ) = \rho \{B \alpha_{s}(A) \}\) for all \(s\in \mathbb{R}\).

    Since the basic elements of modular theory were discovered independently by mathematicians and physicists, there is still a trivial but unfortunate discrepancy in the formulation of the KMS condition. Mathematicians define the modular flow with opposite sign than physicists, so that there is a sign change in the definition of the KMS condition.

  9. 9.

    See also Landsman’s review of Haag’s book (Landsman 1996, p. 512). Editor’s note: For a compact recent introduction into the basic concepts of algebraic quantum field theory with slightly different emphases than in the following sections see Fredenhagen (2015).

  10. 10.

    The Minkowski space ℳ is the simplest solution of the Einstein field equations in the absence of matter and provides the natural mathematical background of the special theory of relativity. It is a four-dimensional real vector space equipped with a non-degenerate symmetric bilinear form \(g:\mathscr{M}\times \mathscr{M}\to \mathbb{R}\). A point \(\boldsymbol {x}\) in Minkowski space ℳ is called an event and can be described by the Cartesian space-time coordinates \(\mathbf{x}= (x^{0},x^{1},x^{2},x^{3}) = (c t, x,y,z)\), where \(c\) is the speed of light in vacuum. The Minkowski inner product is a non-degenerate symmetric bilinear form of signature \((-,+,+,+)\), defined by \(g(\mathbf{x},\bar{\mathbf{x}})= \sum_{\mu ,\nu =0}^{3}g_{\mu ,\nu }x ^{\mu } \bar{x}^{\nu }\), where \(g_{\mu ,\nu }\) represents the geometry of space-time. The Minkowski metric is given by the diagonal matrix \((g_{\mu ,\nu })\) with elements \(g_{0,0}=- 1\), \(g_{1,1}=g_{2,2}=g _{3,3}=+ 1\), so that \(g(\boldsymbol {x},\bar{\boldsymbol{x}})=- x^{0} \bar{x}^{0}+x^{1} \bar{x}^{1} + x^{2} \bar{x}^{2}+x^{3} \bar{x}^{3}\). The Minkowski norm \(\|\boldsymbol {x}\|\) of a vector \(\boldsymbol {x}\) is \(\|\boldsymbol {x}\|^{2}=-(x^{0})^{2}+(x^{1})^{2}+(x ^{2})^{2}+ (x^{3})^{2}, x^{0}=c t\), which can be positive, negative or zero.

  11. 11.

    The relation between the traditional way to formulate quantum field theory in terms of distribution-valued fields and local algebraic quantum field theory is not quite trivial. Since in general fields are not observables, a physical theory is not characterized by the smeared fields \(\varphi (f)\) but by the net \(\mathscr{O}\to \mathcal{M}(\mathscr{O})\) of W*-algebras \(\mathcal{M(\mathscr{O})}\) of local observables. For details compare Roberts (2004).

  12. 12.

    http://www.imar.ro/~purice/conferences/2011/EUNCG4/Talks/Cadamuro.pdf.

  13. 13.

    For more details, precise formulations and proofs compare Pauli (1955), Lüders and Zumino (1958), Streater and Wightman (1964).

  14. 14.

    Compare the introductions by Horuzhy (1990), Haag (1992), Baumgärtel and Wollenberg (1992), and Araki (1999).

  15. 15.

    Non-pure state functionals are often called “mixed states” but we shall avoid this misleading term since they do not refer to any kind of (classical) mixture.

  16. 16.

    More generally, for every vector state there exist observables localized in complementary wedge-shaped regions in Minkowski space-time that maximally violate Bell’s inequalities (Summers and Werner 1987). For further generalizations compare also Summers and Werner (1988, 1995).

  17. 17.

    This result is related to the Unruh effect (Unruh 1976), which states that to a uniformly accelerated observer the vacuum state in Minkowski space-time appears as a modular \(\beta_{U}\)-KMS equilibrium state, where \(\beta_{U}=2\pi c/\hbar a\), \(a\) is the local acceleration, and \(c\) is the speed of light. The modular KMS equilibrium state is often interpreted as a thermal equilibrium state with Unruh temperature \(T_{U}=1/(k_{B}\beta_{U})\), where \(k_{B}\) is the Boltzmann constant.

  18. 18.

    Recall that Wigner (1939) identified relativistic particle states with irreducible positive energy representations of the Poincaré group.

  19. 19.

    The split property is a consequence of the Buchholz-Wichmann nuclearity condition. See Buchholz and Wichmann (1986) and Buchholz et al. (1987).

  20. 20.

    A binary relation is called an equivalence relation if and only if it is reflexive, symmetric and transitive. If \(a\sim b\) means “\(a\) is equivalent to \(b\)”, then the following independent conditions are valid: (i) reflexivity: \(a\sim a\); (ii) symmetry: \(a\sim b\) implies \(b\sim a\); (iii) transitivity: \(a\sim b\) and \(b\sim c\) imply \(a\sim c\). Compare Eq. (2.3a)–(2.3c).

  21. 21.

    Redlich (1968) objected correctly that this postulate is not a “law”, but he missed the crucial point that the transitivity of a relation between pairs of systems implies the existence of a state function which takes on the same value for all such systems.

  22. 22.

    Carathéodory stated this result without proof. Many later authors who referred to it did tacitly assume certain uniqueness and continuity properties. For a precise formulation see Whaples (1952) and Lenker (1979).

  23. 23.

    For some improvements in the technical clustering conditions compare Bratteli and Robinson (1981, Chap. 5.4). The clustering requirement excludes the mixture of different phases and selects a pure phase.

  24. 24.

    Recall that a normal state \(\rho \in \mathcal{M}_{*}\) of a W*-algebra ℳ is faithful if for every \(M\in \mathcal{M}\) the relation \(\rho (M)=0\) implies \(M=0\). Every faithful normal state \(\rho \) on a W*-algebra ℳ defines a one-parameter family \(\{\sigma_{\rho ,s}|s\in \mathbb{R}\}\) of modular automorphisms \(\sigma_{\rho ,s}\) on ℳ (compare Appendix A.5).

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  1. (deceased), Lab. für Physikalische Chemie, Eidgen. Technische Hochschule Zürich, Zürich, Switzerland

    Hans Primas

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    Harald Atmanspacher

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Primas, H. (2017). Top-Down Approaches in Physics. In: Atmanspacher, H. (eds) Knowledge and Time. Springer, Cham. https://doi.org/10.1007/978-3-319-47370-3_7

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