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.
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.
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:
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.
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:
The mapping
The inverse 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
One example that arises in high-energy physics is the
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.
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
The inverse image
For each
Commutative diagram used in the definition of fiber bundles.
Note that
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
Definition 4. A local section of
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
Definition 5. If we consider a local section
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
Example 1. Let
The 1st-order jet space
The 2nd-order jet space
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
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
using the Euler-Lagrange equations and performing some computations, we ultimately get a conserved current:
Theorem 2 (Noether’s Second Theorem). Local gauge symmetries that leave the Lagrangian density
where
that imply the existence of certain conserved quantities.
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
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].
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.