Fundamentality, Effectiveness, and Objectivity of Gauge Symmetries

ABSTRACT

Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, that is, the fact that they have been chosen from an infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. This is a symmetry argument in line with Curie’s first principle. Further, I argue that this same feature of group theory helps to explain the ‘unreasonable’ effectiveness of (this subfield of) mathematics in (this subfield of) physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries.

Acknowledgements

For their helpful comments I wish to thank Elías Okon, Carl Hoefer, Keizo Matsubara, Kerry McKenzie, María Martínez, Auro Michele, David Rey, Carlos Romero, Giulio Tirabassi, and two anonymous referees of this journal. I am grateful to audiences at the Seminario de investigadores at UNAM and at the CauSci Final Conference in Oslo.

Notes

1. I briefly compare their approaches with mine at the end of the article (last paragraph of section 5 and notes therein).

2. I thus refer to any present or future theory based on what nowadays is our best empirically tested physical theory, quantum field theory. This includes ToEs such as superstring theory in its many variants, the E₈ proposal, and ToEs based on canonical and loop quantum gravity. More details in section 2.

3. Critical assessments of naturalistic metaphysics are a hot topic today. See, for example, Ross, Ladyman, and Kincaid (2013), French and McKenzie (2015), and Morganti and Tahko (2016).

4. In addition to Williams (2015), for a philosophical assessment of the notion of naturalness see Friederich, Harlander, and Karaca (2014). Other discussions around unnaturalness include Borrelli (2011), Feng (2013), Evans et al. (2014), and Fowlie (2014).

5. The gauge principle specifies a procedure for obtaining an interaction term in the Lagrangian which is symmetric with respect to a continuous symmetry. The results of localising (or ‘gauging’) the global symmetry group involves the introduction of additional fields so that the Lagrangian is extended to a new one that is covariant with respect to the group of local transformations.

6. For a detailed presentation of the mathematical machinery behind this, see Cottingham and Greenwood (2007, ch. 16), Griffiths (2008, ch. 8), and Robinson et al. (2008, part 2, esp. 2.2.15).

7. A representation of a Lie group is one of the ways of representing the elements of the group as linear transformations of the group’s Lie algebra, where the elements constitute a vector space. Then, an adjoint representation is defined as a (finite-dimensional irreducible) representation in which the structure constants themselves form a representation of the group. (The ‘structure constants’ of a Lie group determine the commutation relations between its generators in the associated Lie algebra.)

8. The states are added in linear combinations according to the principle of superposition of quantum mechanics. The numerical parameters are required for normalisation.

9. In general, the force carriers correspond to the eigenvectors of the generators, while the eigenvalues of these eigenvectors are the physically measurable charges (colour, in our case).

10. This group is determined by the spatiotemporal dimension, the geometric signature, and the spatial and temporal local orientation of our world. Such group is determined by the large-scale structure of a space–time, which is described by a pseudo-Riemannian manifold (M,g), in our actual world of dimension 4 (Lorentzian manifold, isomorphic to Minkowski space–time), and a metric of 3 spatial + 1 temporal dimensions. More exactly, the group is the so-called restricted Poincaré group and is O(3,1)⋉ℝ^3,1 (McCabe 2007, 16). The simply connected version of O(3,1)⋉ℝ^3,1 is the so-called universal covering group, which is SL(2, ℂ)⋉ℝ^3,1. I am simplifying by not introducing the further restrictions of local space orientation and local time orientation. I refer the interested reader to McCabe (2007, 38).

11. In Griffiths (2008, 303, problem 8.11) it is explained how the situation would look like: the gluon would couple to all baryons with the same strength, not, as the photon does, in proportion to their charge. In the end this would look like as an extra contribution to gravity, contrary to actual evidence.

12. The phenomenon of confinement states that all naturally occurring free particles have to be colour singlets. Correspondingly, the gluons of the octet are not free particles. Instead, is a colour singlet, and if it existed as a mediator it would be a free particle. We do not know a priori that this is not the case, so it is empirical evidence that makes us discard this option. Indeed, if existed it could be exchanged between two colour singlets—a proton and a neutron, say—bringing about a long-range coupling of the colour strength (Griffiths 2008, 286), but this is contrary to actual evidence.

13. The stronger claim of underdetermination, illustrated before with the cases of SU(3) vs. U(3), might not generalise to other groups. We need not: the coming arguments hinge on the weaker feature that I have dubbed as the contingency of gauge symmetry; that is, the contingency in the choice of a specific group from different layers of infinite possibilities.

14. The question could be generalised to the whole of mathematics, asking not only about group theory but also about Hilbert spaces, Riemannian geometry, differential calculus, and so on and so forth. Here, though, the focus is only on group theory—an especially interesting subfield of mathematics due to its predominant role in particle physics and its apparent virtues and privileged status (see section 3).

15. The huge expressive power of group theory is what makes its effectiveness in physics more reasonable. It is not the other way around: it is not that given the abundance of possibilities is even more unreasonable that one of them applies (this line of thought would mirror that of the lottery paradox). It is just because we have at our disposition a rich language (group theory) that is reasonable to expect that one of its many possible expressions (groups) will be capable of describing a certain pattern of nature.

16. They opt to reject this standard association of symmetry with objectivity (called by them ‘invariantism’) and propose their so called ‘perspectival invariantism’, according to which the objectivity is always subject to a choice of an invariance criterion—the choice of which is the given group of transformations—and that choice is a matter of convention.

17. This dichotomy is analogous to what Klein and Lachièze-Rey (1993, 11) set forth regarding the goal of physics: to study ‘what changes in terms of what is permanent’ or ‘what is permanent in terms of what changes’.

18. According to the team led by H. B. Nielsen, all complex Lagrangians lead in the low-energy limit to the symmetries of current physics: Chadha and Nielsen (1983), Froggatt and Nielsen (1991, 2002), and Chkareuli, Froggatt, and Nielsen (2011) (see also Jacobson and Wall 2010 and Mukohyama and Uzan 2013 for the case of Lorentz symmetry). They are considering a fundamental level ruled by an undetermined highly complex behaviour, labelled by them as ‘random dynamics’. Also the (speculative) projects of entropic forces should be mentioned, such as that of Verlinde (2011) or the more elaborated derivation of the Einstein field equations from thermodynamic assumptions of Jacobson (1995).

19. Perhaps the reader has noticed the interesting convergence of the lawless fundamental state of the second branch with the state of absolute dynamical symmetry of the first branch. That is, it could be elsewhere explored that the absolute dynamical symmetry that we would expect following Kosso (2003) amounts to the random dynamics of Froggatt and Nielsen (1991)—in the same sense as the liquid state of water is more symmetric than the frozen snowflake, in spite of the randomness and lack of ordered organisation of the molecules in the liquid state.

20. This conclusion, though, is not intended to discourage the search for ever-more encompassing symmetry groups as a principle of heuristics.

Additional information

Funding

This research has been financially supported by the Instituto de Investigaciones Filosóficas of the Universidad Nacional Autónoma de México through a fellowship from the postdoctoral fellowship program DGAPA-UNAM.