The Gauge Argument: A Noether Reason

Abstract

The hole argument of general relativity threatens a radical and pernicious form of indeterminism. One natural response to the argument is that points belonging to different but isometric models should always be identified, or 'dragged-along', by the diffeomorphism that relates them. In this paper, I first criticise this response and its construal of isometry: it stumbles on certain cases, like Noether's second theorem. Then I go on to describe how the essential features of Einstein's 'pointcoincidence' response to the hole argument avoid the criticisms of the 'drag-along response' and are compatible with Noether's second theorem.Here is the prospectus for this paper: In Section 1.1 I will (very) briefly summarise the hole argument; this Section is not meant as a thorough introduction to the hole argument: there are already many sources for that (see e.g. (Gomes & Butterfield, 2023b;Norton, 2019;Pooley, 2022;Pooley & Read, 2022) and references therein). In Section 1.2 I will similarly summarise the 'drag-along' response to the hole argument. Then in Section 2, I will similarly summarise Noether's second theorem in the case of general relativity, emphasising its comparison of isomorphic models by a map that is not the isomorphism that relates them. Finally, in Section 3, I will trace a path to Noether's theorem that goes between the Scylla of a non-trivial Lie derivative and the and Charybdis of the hole argument.

Should physical symmetries guide metaphysics? Two reasons why they should maybe not

“…First, the gauge argument works. It has worked incredibly well to formulate empirically successful interacting field theories and lies at the core of the many gauge theories we adopt in particle physics today (remarkably, quantum electrodynamics) (see Roberts et al, 2021). Second, the gauge argument is very powerful, but it is in companion with other assumptions, as it was well noted.…”

Identity, individuality and indistinguishability in physics and mathematics

“…According to the critics of the received view, the presence of gauge symmetries cannot be reduced to a mere redundancy of the mathematical representation, but rather encodes some deep fact about the ‘logic of nature’ [ 21 ] that has to be properly understood [ 53 , 118 , 125 ]. The main argument to justify the search for a deep meaning of gauge symmetries is that these symmetries seem to have non-trivial physical consequences, notably the relation between gauge symmetries and fundamental physical interactions encoded in the heuristic gauge principle [ 113 ] (see also [ 21 , 117 , 121 , 126 , 127 ]), 13 the role played by gauge symmetries in renormalization theory [ 128 ], the role played by gauge symmetries in the definition of the so-called topological solutions —like instantons—associated with fibre bundles that are topologically non-trivial, and the supposed ‘physical reality’ of the gauge potential in the Aharonov–Bohm effect and similar escenarios [ 129 – 134 ]. …”

“…the relation between gauge symmetries and fundamental physical interactions encoded in the heuristic gauge principle [ 113 ] (see also [ 21 , 117 , 121 , 126 , 127 ]), 13…”