Parallel transport - Lec 23 - Frederic Schuller

Frederic Schuller

21 Sept 2015104:31

EducationalLearning

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TLDRThe transcript delves into the mathematical concept of parallel transport, a fundamental aspect in the study of connections on a vector bundle. It explains how parallel transport allows for the definition of a covariant derivative, which is essential in fields such as general relativity and quantum mechanics. The discussion illustrates the process of lifting a curve from the base manifold to the principal bundle using a connection one-form, leading to the concept of horizontal lift. This is further connected to the idea of holonomy, which depends on the path chosen in the base manifold. The transcript also touches on the application of these concepts to define covariant derivatives on vector bundles, which are associated with principal bundles. The mathematical formalism is rooted in the use of local trivializations and the construction of associated bundles. The summary concludes with a mention of upcoming topics, including curvature, torsion, and their relevance in physics.

Takeaways

📐 **Parallel Transport**: The concept of parallel transport is fundamental in defining connections and covariant derivatives on a vector bundle, especially when considering a curve in the base manifold and its horizontal lift through a principal bundle.
🔄 **Horizontal Lift**: A horizontal lift of a curve is a unique curve in the principal bundle that projects down to the original curve in the base manifold, with specific properties related to the connection one form and horizontal subspaces.
🔗 **Connection One Form**: The connection one form (Ω) is crucial as it defines a horizontal space at each point of the fiber, which is essential for the horizontal lift and parallel transport.
🧵 **Vector Bundles and Associated Bundles**: The formalism of parallel transport is extended from principal bundles to associated bundles, particularly vector bundles, where it can be used to define covariant derivatives.
🔍 **Local Trivialization**: To perform calculations, one must consider a local chart on the base manifold and a local section of the principal bundle, which allows relating the arbitrary curve to the curve in the base manifold and the connection one form to the corresponding Yang-Mills field.
🔬 **Covariant Derivative**: The covariant derivative can be defined on a vector bundle that is associated with a principal bundle carrying a connection, allowing for the comparison of vector values in the same fiber.
🔶 **Matrix Groups**: For matrix groups, the ordinary differential equation (ODE) describing the horizontal lift takes a specific form involving matrix multiplication, which simplifies calculations.
🔄 **Path-Ordered Exponential**: The horizontal lift of a curve is expressed using a path-ordered exponential, which is essential for dealing with the non-commutativity of Lie group elements.
🔃 **Holonomy**: The holonomy group, associated with a loop in the base manifold, provides information about the topology of the manifold and depends on the connection one form on the principal bundle.
🌀 **Curvature and Torsion**: While curvature is a general notion defined for any connection on a principal bundle, torsion is defined for associated bundles with additional structure, known as a soldering form.
⚙️ **Physical Applications**: The mathematical concepts discussed, such as curvature, torsion, and covariant derivatives, have profound implications in physics, including general relativity and quantum mechanics.

Q & A

What is the basic idea behind parallel transport?
-The basic idea behind parallel transport is to move a vector along a curve in a manifold in such a way that the vector remains parallel to its original direction. This is particularly useful in the context of a principal bundle with a connection one form, which defines a horizontal space at each point of the fiber, allowing for a unique 'horizontal lift' of a curve in the base manifold to the bundle.
How does a connection one form induce a horizontal space in each fiber?
-A connection one form, denoted as Omega, induces a horizontal space at each point of a fiber in the principal bundle. This is a subspace of the tangent space of the fiber that is complementary to the vertical space. The horizontal space is defined such that the push forward of any vector in the horizontal space under the projection map of the bundle is a tangent vector to the curve in the base manifold.
What is a covariant derivative and how is it related to parallel transport?
-A covariant derivative is a derivative that generalizes the concept of a total derivative to curved spaces or manifolds. It is related to parallel transport as it can be defined using the isomorphism between fibers along a curve in the base manifold, which is established through parallel transport. Specifically, the covariant derivative of a vector field can be understood as the rate of change of that field along a curve, taking into account the curvature of the manifold.
What is the significance of the horizontal lift of a curve?
-The horizontal lift of a curve is significant because it provides a way to move a point from one fiber to another along a curve in the base manifold while preserving the structure of the bundle. This lift is unique once a starting point in the fiber is chosen and is crucial for defining parallel transport and, consequently, covariant derivatives.
How does the choice of an arbitrary curve in the principal bundle affect the horizontal lift?
-The choice of an arbitrary curve in the principal bundle affects the horizontal lift by determining the path along which the lift is taken. Different arbitrary curves that project down to the same curve in the base manifold can result in different horizontal lifts, as they may pass through different points in the fibers of the bundle.
What is the role of the local section in the construction of the horizontal lift?
-A local section of the principal bundle plays a crucial role in the construction of the horizontal lift by providing a specific point in the fiber over each point of the base manifold. This allows for the generation of an arbitrary curve in the principal bundle that projects down to the original curve in the base manifold, which is then used to find the horizontal lift.
What is the Yang-Mills field and how is it used in the context of parallel transport?
-The Yang-Mills field is a gauge field that arises in the context of non-Abelian gauge theories. It is used in the context of parallel transport to provide an explicit expression for the horizontal lift of a curve. The Yang-Mills field contains all the information about the connection one form and, when used in conjunction with the Maurer-Cartan form of the Lie group, allows for the calculation of the horizontal lift.
How is the path-ordered exponential used in the expression for the horizontal lift?
-The path-ordered exponential is used in the expression for the horizontal lift to account for the non-commutativity of the Lie algebra elements. It ensures the correct ordering of the group elements along the path of the curve in the group manifold, which is essential for obtaining the unique horizontal lift of the curve.
What is the holonomy group and how is it related to parallel transport?
-The holonomy group is a subgroup of the structure group of a principal bundle that represents the set of group elements associated with the horizontal lifts of closed curves based at a point in the base manifold. It is related to parallel transport as it describes the monodromy, or the failure of parallel transport to be periodic, around closed loops in the base manifold.
How does the concept of parallel transport extend to associated bundles?
-The concept of parallel transport extends to associated bundles by defining a horizontal lift of a curve in the associated bundle that passes through a specified point. This allows for the construction of a parallel transport map on the associated bundle, which provides a bijective mapping between fibers of the associated bundle, similar to the case with principal bundles.
What is the significance of the covariant derivative in the context of vector bundles?
-In the context of vector bundles, the covariant derivative allows for the definition of a derivative that is compatible with the geometry of the bundle. It is used to define the rate of change of a section of the vector bundle along a curve in the base manifold, taking into account the parallel transport properties of the bundle. This is particularly important in physical theories such as general relativity and quantum mechanics, where covariant derivatives are used to describe the behavior of fields on curved spaces.

Outlines

00:00

📐 Introduction to Parallel Transport and Connections

The video begins with an introduction to parallel transport, a concept fundamental to understanding connections within a vector bundle. It emphasizes the importance of distinguishing between connections, parallel transport, and covariant derivatives. The concept of parallel transport is illustrated using a principal bundle with a connection one form, which defines a horizontal space at each point of the fiber. The process of lifting a curve from the base manifold to the principal bundle is discussed, highlighting the conditions for a unique horizontal lift.

05:03

🔄 Horizontal Lift and Isomorphism Between Fibers

The paragraph delves into the horizontal lift of a curve, explaining how it can be used to create an isomorphism between fibers along a curve. It discusses how the mapping between fibers depends on the path chosen in the base manifold, introducing the concept of holonomy. The paragraph also outlines how to extend the formalism to associated bundles, particularly vector bundles, and how this enables the definition of a covariant derivative.

10:04

🧮 Explicit Expression for Horizontal Lift

The focus shifts to finding an explicit expression for the horizontal lift of a curve. The strategy involves generating the horizontal lift from an arbitrary curve in the principal bundle that projects down to the original curve. The paragraph introduces the concept of a suitable curve in the Lie group and discusses the need to write down an ordinary differential equation (ODE) for this curve. It also touches on the uniqueness of the horizontal lift and the dependence on the starting point.

15:13

🔬 Strategy for Calculating Horizontal Lift

The paragraph outlines a two-step strategy for calculating the horizontal lift. The first step involves generating the horizontal lift from an arbitrary curve in the principal bundle. The second step is solving the ODE for the curve in the group manifold. The paragraph also discusses the use of a local trivialization to facilitate calculations and the importance of the initial condition in solving the ODE.

20:19

🚀 Solving the ODE for the Horizontal Lift

The paragraph presents the ODE for the curve G in the group, which is used to find the horizontal lift of a curve gamma. It discusses the form of the ODE and its initial condition, emphasizing the role of the arbitrary curve Delta and the connection one form in the principal bundle. The paragraph also highlights the need for a proof to validate the theorem and introduces the concept of a matrix group for simplification.

25:24

🔗 Local Trivialization and the Young-Mills Field

The paragraph discusses the process of local trivialization to facilitate the calculation of the horizontal lift. It explains how choosing a local chart on the base manifold and a section of the principal bundle allows for the expression of the horizontal lift in terms of the Young-Mills field. The paragraph also describes how the arbitrariness in choosing an arbitrary curve is encoded in the section and the Young-Mills field.

30:28

🌀 Path-Ordered Exponential for Horizontal Lift

The paragraph introduces the concept of a path-ordered exponential to express the horizontal lift of a curve. It explains the process of integrating both sides of the ODE and inserting the expression for G of T into itself to remove recursion. The paragraph also discusses the importance of path ordering due to the non-commutative nature of the Lie algebra elements and how this affects the solution of the ODE.

35:29

🔄 Parallel Transport and Its Properties

The paragraph defines the parallel transport map, which is a bijective mapping between fibers of a principal bundle connected by a curve. It discusses the properties of parallel transport, including its dependence on the curve and the connection one form. The paragraph also touches on the case of closed curves and the concept of holonomy, which is the subgroup of the Lie group resulting from parallel transport around loops.

40:33

🔗 Holonomy Group and Associated Bundles

The paragraph explores the holonomy group, which is a subgroup of the Lie group associated with the parallel transport around closed curves. It also discusses the concept of an associated bundle, which is constructed from a principal bundle and a group action. The paragraph explains how the connection one form on the principal bundle can be transferred to the associated bundle, allowing for the definition of parallel transport and covariant derivatives on the associated bundle.

45:34

📐 Covariant Derivative in Vector Bundles

The paragraph concludes with a discussion on the covariant derivative in the context of vector bundles. It explains how a covariant derivative can be defined for a section of a vector bundle, which is an associated bundle of a principal bundle with a connection. The paragraph highlights the importance of the covariant derivative in physics, particularly in general relativity and quantum mechanics, and how it is used to understand the tangent space of the manifold and the behavior of wave functions.

50:43

🔧 Curvature, Torsion, and Further Applications

The final paragraph teases upcoming topics, including the curvature and torsion of a connection, the Cartan structural equation, and further applications in physics. It emphasizes the mathematical foundation laid by the concepts of parallel transport and covariant derivatives and hints at their significance in understanding physical phenomena.

Mindmap

Keywords

💡Parallel Transport

Parallel transport is a concept in differential geometry that refers to moving a vector along a path in a way that is parallel to the original direction. In the video, it is used to define a covariant derivative on a vector bundle, which is crucial for understanding the properties of manifolds and their tangent spaces. The script discusses how parallel transport allows for the creation of an isomorphism between fibers of a principal bundle along a certain curve.

💡Covariant Derivative

The covariant derivative is a concept used in calculus on manifolds and is a generalization of the derivative to vector bundles. It is used to define the rate of change of a vector field in the presence of curvature. In the context of the video, the covariant derivative is defined using parallel transport on a vector bundle, which is essential for the study of manifolds and their associated bundles.

💡Principal Bundle

A principal bundle is a mathematical construct that is used in the field of differential geometry and is central to the study of fiber bundles. It consists of a total space, typically a group, a base space, and a projection map that locally resembles a product space. In the script, the principal bundle is equipped with a connection one-form, which allows for the definition of parallel transport and horizontal lifts of curves.

💡Connection One-Form

A connection one-form is a mathematical object that encodes the curvature and torsion of a connection on a principal bundle. It is used to define horizontal spaces within each fiber of the bundle. The script explains how the connection one-form, denoted by Omega, is used to lift curves from the base manifold to the principal bundle, which is a key step in defining parallel transport.

💡Horizontal Lift

The horizontal lift is a specific type of lift of a curve from the base manifold to the total space of a principal bundle. It is defined such that the lifted curve is horizontal, meaning its tangent vectors lie within the horizontal spaces of the fibers. The script discusses how to calculate the horizontal lift of a curve using an ordinary differential equation (ODE) and the connection one-form.

💡Lie Group

A Lie group is a group that is also a differentiable manifold, with the property that the group operations are differentiable. Lie groups are fundamental in the study of symmetries in physics. The script mentions the Lie group in the context of the principal bundle, where the fibers of the bundle are isomorphic to the Lie group.

💡Yang-Mills Field

The Yang-Mills field is a generalization of the electromagnetic field to non-Abelian gauge symmetries. It is named after Chen-Ning Yang and Robert Mills and plays a central role in modern theoretical physics. In the script, the Yang-Mills field is used in the context of solving the ODE for the horizontal lift, which is crucial for understanding the behavior of the principal bundle under the connection.

💡Holonomy

Holonomy is a concept in differential geometry that describes the transport of a vector around a loop in a manifold and its return to a different point due to curvature. In the video, the notion of holonomy is introduced when discussing how the mapping between fibers depends on the path chosen in the base manifold, leading to a group of transformations called the holonomy group.

💡Vector Bundle

A vector bundle is a topological construction that allows for the definition of vector spaces that vary from point to point on a base space. It is a generalization of the tangent bundle. The script explains how, with a connection on a principal bundle, one can define a covariant derivative on an associated vector bundle, which is essential for the study of manifolds in physics and geometry.

💡Curvature

Curvature is a measure of how much a curve deviates from being a straight line. In differential geometry, curvature can be used to describe the deviation of a manifold from being flat. The script hints at discussing curvature in the context of connections on principal bundles, which is a fundamental concept in understanding the geometry of these bundles.

💡Torsion

Torsion, in the context of differential geometry, is a measure of how much a curve in a manifold twists from a plane as it moves along the curve. Unlike curvature, torsion is not defined for all connections but only for those on associated bundles with additional structure. The script mentions torsion in relation to the structure equation, which is a key concept in the study of manifolds.

Highlights

Introduction to parallel transport and its significance in defining covariant derivatives on a vector bundle.

Explanation of the concept of parallel transport using a principal bundle and a base manifold.

Illustration of how to lift a curve from the base manifold to the principal G bundle using a connection one form.

Description of the horizontal space at each point of a fiber, induced by the connection one form.

Conditions required for a unique horizontal lift of a curve in the principal bundle.

Discussion on the dependency of the mapping between fibers on the chosen path in the base manifold, leading to the concept of holonomy.

Process of writing down a precise definition for the horizontal lift of a curve.

Strategy to calculate the horizontal lift by generating it from an arbitrary curve in the principal bundle.

Introduction of an ordinary differential equation (ODE) for finding the curve in the group manifold.

Theorem stating the ODE for the curve G in terms of the push forward of the one-form Omega and the Yang-Mills field.

Solution to the ODE using a path-ordered integral over the local Yang-Mills field.

Corollary for matrix groups that simplifies the ODE into a form involving matrix multiplication.

Use of local trivializations to express the horizontal lift in terms of a connection one form and a section of the principal bundle.

Discussion on the explicit construction of the horizontal lift using a local chart on the base manifold.

Definition of the parallel transport map T, which provides a mapping between fibers over the initial and final points of a curve.

Explanation of how the parallel transport map induces a bijection between different fibers in the principal bundle.

Introduction to the concept of holonomy and the holonomy group for closed curves in the base manifold.

Definition of the horizontal lift in an associated bundle and its relation to the principal bundle's connection.

Discussion on the definition of a covariant derivative on a vector bundle associated with a principal bundle.

Overview of the applications of covariant derivatives in physics, such as general relativity and quantum mechanics.

Upcoming discussion on curvature, the Cartan structural equation, and their implications in the context of connections on principal bundles.

Transcripts

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Related Tags

Parallel Transport Covariant Derivative Differential Geometry Vector Bundle Holonomy Principal Bundle Connection One Form Tangent Space General Relativity Quantum Mechanics