Noether’s first theorem, in its modern form, does not establish a one-way explanatory arrow from symmetries to conservation laws, but such an arrow is widely assumed in discussions of the theorem in the physics and philosophy literature. It is argued here that there are pragmatic reasons for privileging symmetries, even if they do not strictly justify explanatory priority. To this end, some practical factors are adduced as to why Noether’s direct theorem seems to be more well-known and exploited than its converse, with special attention being given to the sometimes overlooked nature of Noether’s converse result and to its strengthened version due to Luis Martínez Alonso in 1979 and independently Peter Olver in 1986.

This chapter has three main aims. First, it gives a pedagogical introduction to Noether’s two theorems and their implications for energy conservation in general relativity, which was a central point of discussion between Hilbert, Klein, Noether, and Einstein. Second, it introduces and compares two proposals for gravitational energy and momentum, one of which is very influential in physics, and neither of which has been discussed in the philosophical literature. Third, it assesses these proposals in connection with recent philosophical discussions of energy and momentum in general relativity. After briefly reviewing the debates about energy conservation between Hilbert, Klein, Noether, and Einstein, the chapter shows that Einstein’s gravitational energy-momentum pseudo-tensor, including its superpotential, is fixed, through Noether’s theorem, by the boundary terms in the action. That is, the freedom to add an arbitrary superpotential to the gravitational pseudo-tensor corresponds to the freedom to add boundary terms to the action without changing the equations of motion. This freedom is fixed in the same way for both problems. The chapter also includes a review of two proposals for energy and momentum in GR: one is a quasi-local alternative to the local expressions, and the other builds on Einstein’s local pseudo-tensor approach.

This chapter sketches the contents of Noether’s 1918 article, ‘Invariante Variationsprobleme’, as it may be seen against the background of the work of her predecessors and in the context of the debate on the conservation of energy that had arisen in the general theory of relativity.

This chapter examines the assumptions behind Noether’s theorem connecting symmetries and conservation laws, taking an algebraic approach to compare classical and quantum versions of this theorem. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether’s theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics, this mapping is multiplication by the square root of ?1. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a ‘dynamical correspondence’ and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. This expresses a relation between quantum and statistical mechanics, closely connected to the principle that ‘inverse temperature is imaginary time’.

Using the converse of Noether’s first theorem, this chapter shows that the Bessel-Hagen-type transformations are uniquely selected in the case of electrodynamics, which powerfully dissolves the methodological ambiguity at hand. It then considers how this line of argument applies to a variety of other cases, including in particular the challenge of defining an energy-momentum tensor for the gravitational field in linearised gravity. Finally, the search for proper Noether energy-momentum tensors is put into context with recent claims that Noether’s theorem and its converse make statements on equivalence classes of symmetries and conservation laws. The aim is to identify clearly the limitations of this latter move, and to develop this position by contrast with recent philosophical discussions about how symmetries relate to the representational capacities of theories.

In approaching ‘Invariante Variationsprobleme’ as a contribution to mathematical physics (which it undeniably was), one might easily regard it as a singularity within Noether’s corpus of collected works. This impression quickly dissipates, however, if one shifts the focus to the mathematical methods she employed. Beyond Lie’s theory of differential equations, Noether also made use of formal methods in the calculus of variations, ideas first set forth by Riemann and Lipschitz. This chapter shows the importance of these methods for understanding Noether’s broader agenda in 1917-18. It highlights two competing approaches to the study of differential invariants before, during, and after the advent of Einstein’s general theory of relativity. Noether’s expertise in invariant theory made her an ideal assistant to Felix Klein in his explorations of older literature relating to the mathematical foundations of special and general relativity. Klein argued that Christoffel’s purely algebraic methods for deriving differential invariants were essentially inferior to those based on formal variational methods. The former-as championed by Ricci and later taken up by Grossmann and Einstein-thus stood in opposition to Noether’s work from this period.

A fundamental tenet of gauge theory is that physical quantities should be gauge-invariant. This prompts the question: can gauge symmetries have physical significance? On one hand, the Noether theorems relate conserved charges to symmetries, endowing the latter with physical significance, though this significance is sometimes taken as indirect. But for theories in spatially finite and bounded regions, the standard Noether charges are not gauge-invariant. This chapter’s argument is that gauge-variance of charges is tied to the nature of the non-locality within gauge theories. It will flesh out these links by providing a chain of (local) implications: local conservation laws è conserved regional charges Ô Non-separability Ô direct empirical significance of symmetries.

Famously, Klein and Einstein were embroiled in an epistolary dispute over whether General Relativity has any physically meaningful conserved quantities. This chapter explores the consequences of Noether’s second theorem for this debate and connects it to Einstein’s search for a ‘substantive’ version of general covariance as well as his quest to extend the Principle of Relativity. The chapter’s argument is that Noether’s second theorem provides a clear way to distinguish between theories in which gauge or diffeomorphism symmetry is doing real work in defining charges, as opposed to cases in which this symmetry stems from Kretchmannization. Finally, a comment is made on the relationship between this Noetherian form of substantive general covariance and the notion of ‘background independence’.

Why is gauge symmetry so important in modern physics, given that one must eliminate it when interpreting what the theory represents? This chapter offers a discussion of the sense in which gauge symmetry can be fruitfully applied to constrain the space of possible dynamical models in such a way that forces and charges are appropriately coupled. It reviews the most well-known application of this kind, known as the ‘gauge argument’ or ‘gauge principle’; discusses its difficulties, and then reconstructs the gauge argument as a valid theorem in quantum theory. The chapter then presents what the authors take to be a better and more general gauge argument, based on Noether’s second theorem in classical Lagrangian field theory, and argues that this provides a more appropriate framework for understanding how gauge symmetry helps to constrain the dynamics of physical theories.

This chapter provides a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases, the answer is ‘it depends’: that is, it depends on the details of the symmetry in question. A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry, it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices.

Given a Lie group acting on the space of independent (spacetime) and dependent (field) variables, it is proved that a divergence invariant variational problem is equivalent to a strictly invariant variational problem if and only if a certain associated cohomology class in the invariant variational bicomplex vanishes. This result is illustrated by several examples, starting with the free particle Lagrangian that appeared in Emmy Noether’s original paper, and includes derivations of associated conservation laws through application of her First Theorem. The chapter concludes with some speculations as to the role such cohomology classes might play in fundamental physics, based on the construction of suitable invariant Lagrangians.