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The Mathematical Description of a Generic Physical System

Abstract

When dealing with a certain class of physical systems, the mathematical characterization of a generic system aims to describe the phase portrait of all its possible states. Because they are defined only up to isomorphism, the mathematical objects involved are “schematic structures”. If one imposes the condition that these mathematical definitions completely capture the physical information of a given system, one is led to a strong requirement of individuation for physical states. However, we show there are not enough qualitatively distinct properties in an abstract Hilbert space to fulfill such a requirement. It thus appears there is a fundamental tension between the physicist’s purpose in providing a mathematical definition of a mechanical system and a feature of the basic formalism used in the theory. We will show how group theory provides tools to overcome this tension and to define physical properties.

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Notes

  1. For our purposes, type theory is very similar to set theory: the type/token relation is the analogue of the set/element relation. The reason why I choose the language of type theory rather than of set theory will become clear in the next section.

  2. This point on identity is precisely one of the deepest differences between type theory (contextual identity) and set theory (absolute identity). It is because of this difference that the language of type theory seems better adapted to our discussion.

  3. For an explicit treatment of these formalisms, see for example Gazeau (2009, pp. 13–18).

  4. Indeed, when working in the category of all representations (of a given C*-algebra) “equivalence of representations” is just another name for the general concept of “isomorphism”.

  5. A ‘symplectomorphism’ is an isomorphism in the category of symplectic manifolds.

  6. As sets, these two spaces are different. Whence, sensu stricto, they are not identical.

  7. A ray of a Hilbert space is a one-dimensional subspace.

  8. In Quine (1960), the author introduced the distinction between absolute and relative discernibility. The third term was introduced in Quine (1976), but there he changed “relative discernibility” into “moderate discernibility”. However, I follow the terminology that has been adopted in the philosophy of physics literature (Saunders 2006; Dieks 2014).

  9. In the original, Weyl writes: “A conceptual fixation of points by labels […] that would enable one to reconstruct any point when it has been lost, is here possible only in relation to a coordinate system, or frame of reference, that has to be exhibited by an individual demonstrative act” (Weyl 1949, p. 75).

  10. Adams (1979) introduced a distinction between “thisness” and “suchness”. Intuitively, the thisness (or haecceity) is the property of an object that allows one to point at it and say in a meaningful way ‘this object’. On the other hand, “suchness” is a synonym of “qualitative property”—and also, in this paper, of “objective property” and “structural property”.

  11. Given the left action of a group G on a set E, the orbit O x of an element x is the subset of elements of E to which x can be transformed by some element of G.

  12. See footnote 9.

  13. By "numerical multiplicity” I mean a multiplicity of elements that are only weakly discernible. A mathematical structure is a numerical multiplicity if the action of its group of automorphisms is transitive.

  14. Hence, in this third option, a system is described by a tuple (H, G, X, ρ X , ρ H , π) where ρ X is the action of G on X, ρ H is the action of G on H and π is a C*-algebra morphism from C0(X, \({\mathbb{C}}\)) to B(H). For a modern introduction to Mackey’s approach, see Landsman (2006) and Varadarajan (2008).

  15. In Castellani (1998), the author considers the problem of constitution of physical objects: “What kind of properties and prescriptions do we need in order to construct an object?”, and then studies “the group-theoretic approach to the problem […] grounded on the idea of invariance.” (p. 182).

  16. In Strocchi’s words: “The abstract algebra generated by (abstract) elements U(α), V(β), α, βR […] satisfying U(α) V(β= V(β) U(α) exp(−iαβ), U(α) U(β= U(α + β) and V(α) V(β= V(α + β) is called the Weyl algebra” (Strocchi 2005, pp. 58–59) Notice his insistence on the abstract character of this definition.

  17. To see this, consider an element g of the group G. The unitary representation (G, H, ρ′) defined by ρ′(g′) = ρ(g)ρ(g′)ρ(g −1) for any g′ in G is equivalent to (G, H, ρ), and the intertwining operator that achieves the isomorphism is precisely ρ(g).

  18. An ‘index’ is a number that takes different values for different representations.

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Acknowledgments

This work has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement No 263523). I also want to thank Gabriel Catren, Julien Page, Christine Cachot, Michael Wright and Fernando Zalamea for helpful discussions and comments on earlier drafts of this paper.

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Correspondence to Federico Zalamea.

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Zalamea, F. The Mathematical Description of a Generic Physical System. Topoi 34, 339–348 (2015). https://doi.org/10.1007/s11245-015-9322-7

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Keywords

  • Group theory
  • Individuation
  • Quantum mechanics
  • Structuralism