Group Structural Realism (Part 2)

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Wigner’s Legacy.

Yuval Ne’eman and Shlomo Sternberg have recorded an old particle physicist’s adage:

Ever since the fundamental paper of Wigner on the irreducible representations of the Poincaré group, it has been a (perhaps implicit) definition in physics that an elementary particle ‘is’ an irreducible representation of the group, G, of ‘symmetries of nature’. (Ne’eman and Sternberg 1991, 327.)

This idea captures much of the physical basis of GSR. Let’s discuss a bit about how one arrives at such a view.

Despite their abstractness, irreducible unitary representations do seem to satisfy our intuitions about elementary particles. Jonathan Bain points out two such intuitions: (1) an elementary particle should be uniquely labeled by a mass and a spin parameter (that is, by the eigenvalues of a total 4-momentum and a total 4-angular momentum operator); and (2) a particle should be invariant up to the group of spacetime symmetries, in order to satisfy “our intuitions concerning the continuity of particle identity through time” (Bain 2000, 402fn). One also wants that, (3) an elementary particle cannot be ‘decomposed’ into further particles; and (4) a particle should be associated with a set of observables that describe its possible states.

One can now observe: Wigner showed that the irreducible unitary representations of the Poincaré group do indeed satisfy (1) and (3) because of irreducibility; they satisfy (2) because they represent the Poincaré group; and (4) follows from the fact that they are unitary.

Although this metaphysical picture of ‘particles as representations’ is often attributed to Wigner, he does not seem to have advocated it in print. The famous 1939 paper (PDF) that Ne’emann and Sternberg refer to actually sets out only to identify the values of physical magnitudes (the so-called ‘quantum numbers’) with parameters labeling group representations, namely, which represent the group of spacetime symmetries. By classifying all the irreducible unitary representations of the Lorentz group, Wigner was able to identify all the possible labels of mass, spin and parity. This provided a deep connection between a symmetry group of nature, and the measurable properties of a quantum system.

A simple textbook example will help illustrate this connection. Take a familiar physical property like angular momentum. Quantum theory assigns a fixed value to some aspects of angular momentum, such as the total angular momentum of an isolated system. Others aspects, such as ‘angular momentum in the z-direction,’ are (prior to measurement) assigned a spectrum of values. How can concrete physical magnitudes like these be properties of a symmetry group?

In the case of angular momentum (ignoring spin, to simplify the example), one begins with the group SO(3) of continuous rotations about a point. The faithful irreducible representations of SO(3) turn out to be representable by groups of complex-valued matrices of odd dimension (2j+1), where j is a positive integer. If desired, a given representation can be thought of as acting on, say, the state space of an electron shell around a Hydrogen atom. However, the imagery of this individual object isn’t required for our construction. Instead, we can skip directly to defining the total angular momentum j = (n − 1)/2, in terms of the dimension n of the representation. The angular momentum operators can then be picked out as elements of the representation, and angular momentum in the z-direction can then be defined and shown to have the usual integer-valued spectrum, {−j, -j+1, …, 0, …, j-1, j}.

In summary: angular momentum is recovered, with all its expected properties, from facts about the symmetry group, and no assumptions about the state ψ of an individual object. Indeed, the construction seems to achieve precisely what Eddington hoped: “In fundamental investigations the conception of group-structure appears quite explicitly as the starting point; and nowhere in the subsequent development do we admit material not derived from group structure” (Eddington 1958, 147). That such a development is possible is a fact about the physics. But it is also what paves the way to a reasonable structuralist metaphysics. Wigner’s approach is just what is needed to allow the group structural realist to speak safely of properties like angular momentum, without recourse to an ontology of individual objects.

Construed this way, GSR seems to lead to some surprisingly informative consequences:

Let’s think about what it would mean if spacetime had a symmetry group other than the Poincaré group. This new group would have different representations, and would thus allow for different properties of quantum systems. On Ne’eman and Sternberg’s definition, this would mean that there are different ‘particles.’ In fact, that is exactly what Bargmann (1954) and Lévy-Leblond (1967) were able to show: the Galilei group gives rise to a theory of ‘Galilei particles,’ which are different (in particular, with respect to the ‘mass’ parameter) than the usual ‘Poincaré particles.’

Let’s take another case: what would it mean for nature to admit more symmetries than just those of spacetime? According to GSR, this larger group would provide richer representations, and hence more properties for quantum particles. This is just what is suggested by the study of so-called ‘internal’ symmetries. For example, Gell-Mann’s (1961) adoption of the symmetry group SU(3) led him to organize a new taxonomy of hadrons (as they are now called) according to the irreducible representations of the new symmetry group.

Of course, in building up quantum theory as it gets used in practice, many other mathematical objects besides groups come into play: vector spaces, commutation relations, Hermitian forms, and on and on. What GSR postulates is that, out of all these tools, group structure is of central metaphysical importance. Other realists might propose a different foundation for the theory, perhaps by arguing (with Geoffrey Sewell) that, “theories of such systems should be based on the algebraic structure of their observables, rather than on particular representations thereof” (Sewell 2002, 18). So why choose GSR over all these other options? Here the two overarching aims of structural realism come into play: groups are thought to do a better job of providing a general programmatic account of science, or of solving specific problems in the interpretation of scientific theory.

We’ll discuss why this is in the next post.

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