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Noether’s Theorem: A Differential Geometry Perspective

Published onMar 24, 2024
Noether’s Theorem: A Differential Geometry Perspective
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Abstract

In the following article, we explain the meaning of symmetry and its role in mathematical physics. We provide motivation for the need for a formulation derived from differential geometry. This leads us to discuss the most important groups in physics and their relevance in high-energy physics. Subsequently, we present the mathematical foundation required to present the two Noether’s theorems.

Symmetries

The significant role of symmetries in the advancement of modern physics cannot be neglected, as they are fundamental to the laws of nature, including the principles of energy, linear momentum, angular momentum, and charge conservation.

Understanding the relationship between symmetries and conservation laws requires distinguishing between the concepts of invariance and conservation. “Invariant” refers to the numerical value of a quantity remaining unchanged under a coordinate transformation, while “conserved” implies that the quantity remains constant throughout a process within a given coordinate system [1]. Although one might assume that this connection was discovered through physical experiments, it actually emerged from the interplay of physics and mathematics. Noether’s theorem establishes a profound connection between conservation, invariance, and symmetry.

Emmy Noether (1882–1935). Adapted from https://es.wikipedia.org/wiki/Emmy_Noether.

In 1918, Emmy Noether published the article “Invariante Variationsprobleme” [2], which introduced two theorems resulting from mathematical physics. The first theorem focuses on the invariance of a variational problem, thus establishing a general relationship between its symmetries and the conservation laws associated with the corresponding variational equations. This groundbreaking discovery holds important implications for physics. Noether’s second theorem addresses the invariance of a variational problem under the action of a group that includes arbitrary functions, i.e. every variational problem that is invariant under a symmetry group and depends on arbitrary functions possesses only “improper conservation laws”. These improper conservation laws are identities that imply the presence of conserved quantities.

Differential Geometry

At this point, a vital question arises: Why is it necessary to study differential geometry? Typically, one’s initial encounter with differential geometry occurs while studying Einstein’s theory of general relativity. However, the applications of this field extend beyond that scope. For instance, in the study of fluids and thermodynamics, there are substantial advantages to approach these topics from a differential geometry standpoint. In our specific case, we will observe that the Hamiltonian and Lagrangian approaches to classical mechanics are most effectively described in this manner. This is because of the geometry of the system.

If we were to attempt to define the position of a particle in space, we would need to specify its position vector, which can be written in terms of three Cartesian coordinates: xx, yy, and zz. Consequently, for NN particles, we would require 3N3N coordinates. The number of independent quantities that must be specified to define uniquely the position of any system in the phase space is called the number of degrees of freedom. This implies that in phase space, the degrees of freedom correspond to the independent coordinates needed to fully specify a system’s state. Each coordinate in phase space represents one degree of freedom of the system.

The concept of phase space represents an abstract space where the coordinates of each point describe the positions and momenta of particles in a physical system. This space plays a crucial role in comprehending and analysing the dynamics and statistical behaviour of physical systems.

The relationship between phase space and degrees of freedom allows for visualizing how a system evolves as its degrees of freedom change and how conservation principles and physical constraints are manifested in the trajectories within that space.

Phase space representation for a simple pendulum. Adapted from https://acortar.link/aZRQlU.

Gauge transformations

The foundation of Noether’s second theorem lies within the concept of Lie groups, which are differentiable manifolds equipped with a group structure. We can define Lie groups as follows:

Definition 1. Let G be a smooth manifold of dimension n, it is called a Lie group if it possesses a group structure and satisfies the following conditions:

  1. The mapping :G×GG*: G \times G \rightarrow G, defined as (g1,g2)g1g2\left(g_1, g_2\right) \mapsto g_1 * g_2, is a smooth mapping.

  2. The inverse mapping 1{}^{-1}: GGG \rightarrow G, denoted as gg1g \mapsto g^{-1}, is a smooth mapping.

While the elements of a Lie group exist as abstract entities, adopting a more tangible perspective to interpret them and to understand their actions within a system helps us to understand the physics behind them. This concept is formally known as “group representation”. Representations of a Lie group encompass an array of linear transformations that uphold the algebraic structure of the group. One of the most significant among these is the matrix representation, which serves as a pathway for discovering the fundamental groups of physics.

A gauge group is a Lie group that encodes the local symmetry transformations of the fields in a gauge theory. Gauge theory plays a fundamental role in the realm of particle physics. They enable us to describe the interactions between elementary particles. The concept behind this idea involves promoting certain global symmetries within the Lagrangian to local symmetries. It helps to determine the structure of the theory, its interactions, conservation laws, and the behaviour of particles in general.

A gauge group GG is associated with the concept of a principal bundle over spacetime. Gauge fields represent connections on this principal bundle. Gauge transformations act on both, the fields and the bundle structure.

One example that arises in high-energy physics is the U(1)U(1) gauge group, which describes probability-preserving transformations in quantum mechanical systems, but which is also found in the description of the electromagnetic interaction through quantum field theory.

As David Tong would say about gauge symmetry [3]:

Gauge symmetry is, in many ways, an odd foundation on which to build our best theories of Physics. It is not a property of Nature, but rather a property of how we choose to describe Nature. Gauge symmetry is, at heart, a redundancy in our description of the world. Yet, it is a redundancy that has enormous utility and brings subtlety and richness to those theories that enjoy it.

The redundancies refer to the fact that certain degrees of freedom in the mathematical description of the theory do not correspond to physically distinct states. These redundancies allow multiple mathematical descriptions of the same physical state, with the guarantee that physically observable quantities remain invariant. Gauge theories provide a framework to handle these redundancies while preserving the physical predictions and observables of the theory.

Bundle Formulation

In the world of classical physics, the distinction between matter and force was clear. The attributes of matter were understood through everyday, tangible experiences in the macroscopic realm, while the notion of force presented greater challenges and uncertainties. However, this issue was resolved in the 19th century when James Maxwell formulated the well-known Maxwell’s equations and introduced the concept of a field.

In physics, fields are regarded as independent physical entities capable of propagating through space without the need for an underlying medium. These fields originate from the diverse charges linked to elementary particles. However, from a more contemporary and abstract standpoint, fields can be seen as cross-sections of certain vector (or fiber) bundles. This indicates that in order to gain a comprehensive grasp of the foundational Noether’s theorems, an approach based on differential geometry becomes essential.

Fiber bundle scheme. Adapted from: Todd Rowland. Fiber Bundle. URL: https://mathworld.wolfram.com/FiberBundle.html (visited on 03/18/2024).

The concept of a bundle holds fundamental significance in both topology and geometry, serving as the foundational framework for our exploration of jet bundles. This structure extends beyond the more commonplace setup involving pairs of manifolds and maps, enabling the consideration of more complicated topological configurations.

Definition 2. Let F,MF, M, and EE be CC^\infty manifolds, and let π:EM\pi: E \rightarrow M be a surjective CC^\infty map. The quadruple (E,π,M,F)(E, \pi, M, F) is called a (locally trivial) CC^\infty fiber bundle such that for all pMp \in M:

  1. The inverse image π1(p):=σ(p)=FpF\pi^{-1}(p) :=\sigma(p)=F_p \cong F is called the fiber at pp.

  2. For each pMp \in M there is an open set UU containing pp and a diffeomorphism ϕ:π1(U)U×F\phi: \pi^{-1}(U) \rightarrow U \times F such that the diagram in Fig. 1 commutes.

Commutative diagram used in the definition of fiber bundles.

Note that

πUϕ=π (where πU:U×FU is the projection); (1)\pi_U \circ \phi=\pi \text { (where } \pi_U: U \times F \rightarrow U \text { is the projection); } \tag{1}

A local trivialization of a fiber bundle. Adapted from: John M. Lee. Introduction to Topological Manifolds. Springer, 2011. DOI: 10.1007/978-1-4419-7940-7, page 250.

Definition 3. If (E,π,M,F)(E, \pi, M, F) is a smooth fiber bundle, then EE is called the total space, π\pi is called the bundle projection, MM is called the base space and FF is called the typical fiber. For each pMp \in M, the set Fp:=σ(p)F_p:=\sigma(p) is called the fiber over pp.

Definition 4. A local section of E\boldsymbol{E} is a continuous map σ:UE\sigma:U \rightarrow E defined on some open subset UMU \subseteq M and satisfying πσ=IdU\pi \circ \sigma=\operatorname{Id}_U.

Local section of a fiber bundle. John M. Lee. Introduction to Topological Manifolds. Springer, 2011. DOI: 10.1007/978-1-4419-7940-7, page 256.

To emphasize the distinction, a section defined on all of MM is sometimes called a global section. It is important to note that a local section of EE over UMU \subseteq M is the same as a global section of the restricted bundle EU\left.E\right|_U. If MM is a smooth manifold with or without boundary and EE is a smooth vector bundle, a smooth (local or global) section of E\boldsymbol{E} is one that is a smooth map from its domain to EE.

Definition 5. If we consider a local section σ\sigma of a smooth fiber bundle (E,M,π,F)(E, M, \pi, F) with σ(x)=p\sigma(x)=p, the equivalence class of all local sections that have both σ(x)=p\sigma(x)=p and also the same tangent space TpσT_p \sigma is called the jet jpσj_p \sigma with representative σ\sigma. We can also require that further derivatives of the section match the representative, in which case the order of matching derivatives defines the order of the jet, which is also called a k\boldsymbol{k}-jet so that the above definition would be that of a 1 -jet. xx is called the source of the jet and pp is called its target.

Jet with representative σ, source x, and target p. Adapted from Adam Marsh. Smooth Bundles and Jets. URL: https://www.mathphysicsbook.com/mathematics/fiber-bundles/generalizing-tangent-spaces/smooth-bundles-and-jets (visited on 03/18/2024).

Remark 1. Providing a physical interpretation, M becomes the space-time, and EE is also referred to as the space of field configurations. This implies that the values of the field at every point in the space-time determine the observables in the theory.

Example 1. Let ϕ\phi a scalar field, which is a section of a bundle

ϕ:ME(2)\phi: M \longrightarrow E \tag{2}
  1. The 1st-order jet space J1(σ)J^1(\sigma) contains the field and its 1st derivatives (ϕ,μϕ)\left(\phi, \partial_\mu \phi\right).

  2. The 2nd-order jet space J2(σ)J^2(\sigma) contains the field and its 2nd derivatives (ϕ,μϕ,μνϕ)\left(\phi, \partial_\mu \phi, \partial_{\mu\nu} \phi\right). In general, for a jet of order k, where k is a non-negative integer, we can define it as the collection of values of the components in a local system of coordinates of a vector-valued function and its partial derivatives1 up to order k.

These examples can be applied to the Lagrangian formulation of classical field (or particle) physics. What makes this formulation so powerful is that symmetries can be efficiently incorporated, and their connection with conservation laws can be easily demonstrated. The components comprising the Lagrangian L(ϕ,μϕ)\mathcal{L}\left(\phi, \partial_\mu \phi\right) reveal themselves to be jet spaces.

Noether’s theorem

After grasping the concepts of symmetry and their integration into the Lagrangian formulation, as well as comprehending how gauge transformations can be applied to these Lagrangians, we will then proceed to outline Noether’s theorems.

Theorem 1 (Noether’s First Theorem). Under a global transformation of the fields that leaves the Lagrangian invariant

δL=Lϕiδϕi+L(μϕi)μ(δϕi)=0,(3)\delta \mathcal{L}=\frac{\partial \mathcal{L}}{\partial \phi^i} \delta \phi^i+\frac{\partial \mathcal{L}}{\partial\left(\partial_\mu \phi^i\right)} \partial_\mu\left(\delta \phi^i\right)=0, \tag{3}

using the Euler-Lagrange equations and performing some computations, we ultimately get a conserved current:

Jμ=L(μϕi)δϕiμJμ=0.(4)J^\mu=\frac{\partial \mathcal{L}}{\partial\left(\partial_\mu \phi^i\right)} \delta \phi^i \Rightarrow \partial_\mu J^\mu=0. \tag{4}

Theorem 2 (Noether’s Second Theorem). Local gauge symmetries that leave the Lagrangian density L\mathcal{L} invariant up to a total divergence:

δL=μKμ(5)\delta \mathcal{L}=\partial_{\mu} K^\mu \tag{5}

where KμK^\mu is a vector field that depends on the fields and their derivatives. Using the Euler-Lagrange equations and the Bianchi identity, we get Noether’s identities

μ[L(μϕi)δϕiKμ]=0,(6)\partial_\mu\left[\frac{\partial L}{\partial\left(\partial_\mu \phi^i\right)} \delta \phi^i-K^\mu\right]=0, \tag{6}

that imply the existence of certain conserved quantities.

Conclusions

By 1967, the geometrical study of Noether’s theorems began to be understood. The first of such studies consisted of finding an invariant formulation of the first theorem within the framework of the geometry of differentiable manifolds, which means without using local coordinates.

The ideas that enabled the reformulation of Noether’s theorems in a geometric form and their true generalization were, firstly, those of differentiable manifolds and then the concept of a jet bundle of order kk. Additionally, the concept of manifolds of jets of sections of a bundle, and finally, of jets of infinite order. Once these concepts were formulated, it became possible to rigorously define more general results of Noether’s Theorem.

In conclusion, we employ differential geometry because it serves as the specific language utilized in gauge theories, gravity, string theory, and other areas. This is due to its ability to describe theories in a coordinate-independent manner, enabling us to work with curved spaces and effectively characterize systems with numerous degrees of freedom.

There are many resources that I used by myself when writing this article and may provide a deeper understanding, in particular [4][5][6][7][8][9].

Acknowledgments

To my supervisor, MSc. Miroslava Mosso Rojas, who assisted me in comprehending mathematical concepts and guided me in discussing their implications in physics for this project. Above all, I am grateful for always showing me the beauty of physics.


Editor’s note: The background image is taken from: https://commons.wikimedia.org/w/index.php?title=File:Gothic-Rayonnant_Rose-6.jpg&oldid=746455766.

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