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
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.
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.
For an explicit treatment of these formalisms, see for example Gazeau (2009, pp. 13–18).
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”.
A ‘symplectomorphism’ is an isomorphism in the category of symplectic manifolds.
As sets, these two spaces are different. Whence, sensu stricto, they are not identical.
A ray of a Hilbert space is a one-dimensional subspace.
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).
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).
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”.
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.
See footnote 9.
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.
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).
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).
In Strocchi’s words: “The abstract algebra generated by (abstract) elements U(α), V(β), α, β ∈ R […] 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.
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).
An ‘index’ is a number that takes different values for different representations.
References
Abraham R, Marsden J (1978) Foundations of mechanics. Addison-Wesley, Redwood City
Adams RM (1979) Primitive thisness and primitive identity. J Philos 76:5–26
Ashtekar A, Lewandowski J (2004) Background independent quantum gravity: a status report. Class Quantum Grav 21:53–152
Ashtekar A, Schilling TA (1997) Geometrical formulation of quantum mechanics. In: Harvey A (ed) On Einstein’s path: essays in honor of Engelbert Schücking. Springer, New York
Awodey S (2004) An answer to Hellman’s Question: ‘does category theory provide a framework for mathematical structuralism?’. Philos Math 12:54–64
Awodey S (2013) Structuralism, invariance, and univalence. Philos Math 22(1):nkt030
Black M (1952) The identity of indiscernibles. Mind 61(242):153–164
Castellani E (1998) Galilean particles: an example of constitution of objects. In: Castellani E (ed) Interpreting bodies: classical and quantum objects in modern physics. Princeton University Press, Princeton
Catren G (2009) A throw of the quantum dice will never abolish the Copernican revolution. Collapse Philos Res Dev 5:453–500
Catren G (2014) On the relation between gauge and phase symmetries. Found Phys 44:1317–1335
Dieks D (2014) Weak discernibility and the identity of spacetime points. In: Fano V, Orilia F, Macchia G (eds) Space and time: a priori and a posteriori studies. De Gruyter, Berlin
Dorato M, Laudisa F (2015) Realism and instrumentalism about the wave function: how should we choose? In: Gao S (ed) Protective measurements and quantum reality: toward a new understanding of quantum mechanics. Cambridge University Press, Cambridge
Eddington AS (1939) The philosophy of physical science. Macmillan, New York
Esfeld M, Lam V (2009) Structures as the objects of fundamental physics. In: Feest U, Rheinberger H-J (eds) Epistemic objects. Max Planck Institute for the History of Science, preprint 374
Gazeau JP (2009) Coherent states in quantum physics. Wiley, Wanheim
Haag R (1996) Local quantum physics. Fields, particles, algebras. Springer, Berlin
Landsman NP (1998) Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics. arXiv:math-ph/9807030
Landsman NP (2006) Lie groupoids and lie algebroids in physics and non-commutative geometry. J Geom Phys 56:24–54
Quine WV (1960) Word and object. Harvard University Press, Cambridge
Quine WV (1976) Grades of discriminability. J Philos 73(5):113–116
Rédei M (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud Hist Philos Mod Phys 27:493–510
Resnik MD (1990) Between mathematics and physics. In: Proceedings of the Biennial meeting of the Philosophy of Science Association. University of Chicago Press, Chicago
Rodin A (2011) Categories without structures. Philos Math 19:20–46
Rovelli C (2004) Quantum gravity. Cambridge University Press, Cambridge
Saunders S (2006) Are quantum particles objects? Analysis 66(289):52–63
Souriau J-M (2005) Les groupes comme Universaux. In: Kouneiher J et al (eds) Géométrie au XXe siècle, 1930–2000. Histoire et horizons. Hermann, Paris
Strocchi F (2005) An introduction to the mathematical structure of quantum mechanics. World Scientific, Singapore
Varadarajan VS (2008) George Mackey and his work on representation theory and foundations of physics. Contemp Math 449:417–446
von Neumann J (1955) Mathematical foundations of quantum mechanics. Trans. by R.T. Beyer. Princeton University Press, Princeton
Weyl H (1949) Philosophy of mathematics and natural science. Princeton University Press, Princeton
Weyl H (1950) The theory of groups and quantum mechanics. Dover, New York
Wigner E (1939) On unitary representations of the inhomogeneous Lorentz group. Ann Math 40:149–204
Wigner E (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New York
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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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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DOI: https://doi.org/10.1007/s11245-015-9322-7