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Holism as the significance of gauge symmetries

Henrique Gomes

TL;DR

The paper tackles whether gauge symmetries can have direct empirical significance (DES) by reframing the issue in terms of global non-supervenience on subsystems (GNSS). It develops a technical framework that combines a gauge-invariant projection (h) with external sophistication to glue regional states, showing that DES can arise for a finite, rigid subset of gauge transformations, notably in Abelian electromagnetism and perturbatively in non-Abelian Yang–Mills. DES corresponds to relational under-determination of the universal state by regional states, encoded by a residual variety I that, in simple cases, mirrors the charge group and its rigid subgroups. The approach clarifies the debate between BB and GW, arguing that DES emerges from the proper treatment of gauge-invariant regional data and gluing, rather than from boundary-continuity constraints alone, and emphasizes the role of subsystems in constructing the global state. This has implications for holism in gauge theories and links DES to conserved charges via Gauss constraints, offering a nuanced coexistence of local gauge freedom and global relational significance.

Abstract

Not all symmetries are on a par. For instance, within Newtonian mechanics, we seem to have a good grasp on the empirical significance of boosts, by applying it to subsystems. This is exemplified by the thought experiment known as Galileo's ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin, but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore. But the significance of gauge symmetries seems less clear. For example, can gauge transformations in Yang-Mills theory---taken as mere descriptive redundancy---exhibit a similar relational empirical significance as the boosts of Galileo's ship? This question has been debated in the last fifteen years in philosophy of physics. I will argue that the answer is `yes', but only for a finite subset of gauge transformations, and under special conditions. Under those conditions, we can mathematically identify empirical significance with a failure of supervenience: the state of the Universe is not uniquely determined by the intrinsic state of its isolated subsystems. Empirical significance is therefore encoded in those relations between subsystems that stand apart from their intrinsic states.

Holism as the significance of gauge symmetries

TL;DR

The paper tackles whether gauge symmetries can have direct empirical significance (DES) by reframing the issue in terms of global non-supervenience on subsystems (GNSS). It develops a technical framework that combines a gauge-invariant projection (h) with external sophistication to glue regional states, showing that DES can arise for a finite, rigid subset of gauge transformations, notably in Abelian electromagnetism and perturbatively in non-Abelian Yang–Mills. DES corresponds to relational under-determination of the universal state by regional states, encoded by a residual variety I that, in simple cases, mirrors the charge group and its rigid subgroups. The approach clarifies the debate between BB and GW, arguing that DES emerges from the proper treatment of gauge-invariant regional data and gluing, rather than from boundary-continuity constraints alone, and emphasizes the role of subsystems in constructing the global state. This has implications for holism in gauge theories and links DES to conserved charges via Gauss constraints, offering a nuanced coexistence of local gauge freedom and global relational significance.

Abstract

Not all symmetries are on a par. For instance, within Newtonian mechanics, we seem to have a good grasp on the empirical significance of boosts, by applying it to subsystems. This is exemplified by the thought experiment known as Galileo's ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin, but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore. But the significance of gauge symmetries seems less clear. For example, can gauge transformations in Yang-Mills theory---taken as mere descriptive redundancy---exhibit a similar relational empirical significance as the boosts of Galileo's ship? This question has been debated in the last fifteen years in philosophy of physics. I will argue that the answer is `yes', but only for a finite subset of gauge transformations, and under special conditions. Under those conditions, we can mathematically identify empirical significance with a failure of supervenience: the state of the Universe is not uniquely determined by the intrinsic state of its isolated subsystems. Empirical significance is therefore encoded in those relations between subsystems that stand apart from their intrinsic states.

Paper Structure

This paper contains 36 sections, 1 theorem, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the debate and my position within it
  3. External sophistication
  4. Non-technical summary
  5. Direct empirical significance
  6. DES as a transformation of the universe
  7. Two distinctions
  8. The debate re-expressed
  9. My own position in the debate: rigid variety
  10. Finding GNSS
  11. General Notation
  12. DES as GNSS: set-up
  13. Reduction, sophistication, and gluing
  14. The projection $h$ introduced
  15. The projection $h$ introduced
  16. ...and 21 more sections

Key Result

Theorem 1

For electromagnetism as coupled to a Klein-Gordon scalar field in a simply-connected universe: given the physical content of two regions, $[\varphi_\pm]$, for matter vanishing at the boundary but not in the bulk of the regions, the universal state is underdetermined, resulting in a residual variety where the particular action of $\mathrm{U}(1)$ is that which leaves the gauge-fields invariant, but

Figures (1)

  • Figure 1: The two spacetime regions $\Sigma^\pm$, separated by a manifold with codimension one, $S$.

Theorems & Definitions (3)

  • Definition 1: Direct Empirical Significance (DES) as a transformation
  • Definition 2: GSS
  • Theorem 1: Rigid variety for $\mathrm{U}(1)$