✎ Fiber Bundles [The Physics Travel Guide]

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====== Fiber Bundles ======

* The best intuitive introduction to fiber bundles is "Fiber Bundles and Quantum Theory" by  Herbert J. Bernstein and Anthony V. Phillips. (Can be found for free at Google.)

----

[{{ :advanced_tools:liftingpathfiber.png?nolink |Source: Fiber Bundles and Quantum Theory by Bernstein and Phillips}}]

* A nice introduction to fiber bundles with many pictures is [[https://arxiv.org/pdf/1607.03089.pdf|Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results]] by Adam Marsh 
  * A good textbook on the topic is Topology, Geometry, and Gauge Fields: Foundations by Naber
  * For a quick overview with many pictures, see http://gregnaber.com/wp-content/uploads/FIBER-BUNDLES-IN-MATHEMATICS-AND-PHYSICS.pdf

**Bundles in Field Theory**
<blockquote>A field configuration on a given spacetime Σ is meant to be some kind of quantity assigned to each point of spacetime (each event), such that this assignment varies smoothly with spacetime points. For instance an electromagnetic field configuration is at each point of spacetime a collection of vectors that encode the direction in which a charged particle passing through that point will feel a force (the Lorentz force). This is readily formalized: If

$$
F \in SmthMfd
$$
is the smooth manifold of "values"  that the the given kind of field may take at any spacetime point, then a field configuration $\Phi$ is modeled as a smooth function from spacetime to this space of values:

$$
\Phi
    \;\colon\;
   \Sigma
     \longrightarrow
   F
  \,.
$$
It will be useful to unify spacetime and the space of field values into a single space, the Cartesian product

$$
E \; = : \; \Sigma \times F
$$
and to think of this equipped with the projection map onto the first factor as a [[advanced_tools:fiber_bundles|fiber bundle]] of spaces of field values over spacetime

[...]

This is then called the field bundle, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the fiber of this fiber bundle, it is sometimes also called the field fiber.<cite>https://ncatlab.org/nlab/show/field+%28physics%29#AFirstIdeaOfQuantumFields</cite></blockquote>

<tabbox Abstract> 
[[http://xavirivas.com/cloud/Differential%20Geometry/Eguchi,%20Gilkey,%20Hanson%20-%20Gravitation,%20Gauge%20Theories%20And%20Differential%20Geometry%20(Pr%201980).pdf|Gravitation, Gauge Theories and Differential Geometry by Tohru Eguchi et. al.]]

<tabbox Why is it interesting?> 
<blockquote>All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of the standard lore deals just with the local and infinitesimal aspects -- the perturbative aspects_ and fiber bundles play little role there. But they are the all-important structure that govern the global -- the non-perturbative -- aspect. Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.

For instance the gauge fields in Yang-Mills theory, hence in EM, in QED and in QCD, hence in the standard model of the known universe, are not really just the local 1-forms $A^a_μ$ known from so many textbooks, but are globally really connections on principal bundles (or their associated bundles) and this is all-important once one passes to non-perturbative Yang-Mills theory, hence to the full story, instead of its infinitesimal or local approximation.<cite>https://physics.stackexchange.com/a/77412/37286</cite></blockquote>

Fiber bundles are the appropriate mathematical tool to describe, for example, the physics around a magnetic monopole or also instanton effects. (This is described very nicely in chapter 1 of Topology, Geometry and Gauge fields - Part 1 Foundations by G. Naber). Moreover, non-local aspects like [[advanced_notions:quantum_field_theory:gribov_ambiguities|Gribov ambiguities]] can be understood much more clearly with fiber bundles.

They are also useful to show the similarity between general relativity and Yang-Mills theory, which we use in the [[models:standard_model|standard model]].  The reformulation of the notions that we usually use in quantum field theory, like the notion of a quantum field, in the fibre bundle formalism, allows us to understand them geometrically. This makes them, contrary to what one may think at the beginning, much less abstract and lets us view at them from a completely new perspective.

See also: https://physics.stackexchange.com/questions/77368/intuitively-why-are-bundles-so-important-in-physics

<blockquote> Although [[advanced_tools:gauge_symmetry|gauge theory]] is introduced in the above inductive manner for historical and pedagogical reasons it is clear that the essential ingredients -the gauge potential, the gauge field, and the covariant derivative - have an intrinsic mathematical structure which is independent of the context.This structure has been well studied by mathematicians, in the context of differential geometry. In this context transformations g(x) are identified as sections of principal bundles, with Minkowski space J (as base and the Lie groups G as fibres, the scalar and fermion fields are identified as sections of vector bundles with base Jl, the gauge potential as a connection form for G{x), and i^„(x) as the components of the curvature. A review of these aspects is given by [[http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.52.175|Daniel and Viallet (1980)]]. **The fibre-bundle formulation is not necessary for dealing with those aspects of gauge theory which are local** in M, **but it becomes important for** understanding problems, such as the axial anomaly (next section) and the gauge-fixing ambiguity (Gribov, 1977; Singer, 1978) which are of **global** origin. **It also shows that gauge theory, and thus the theory of strong, weak and electromagnetic interactions, is basically a geometrical theory. This is not only aesthetically pleasing but brings the unification of weak,electromagnetic and strong interactions with gravitation a step closer.**
<cite>Group Structure of Gauge Theories
by Lochlainn O’Raifeartaigh</cite> </blockquote>

<tabbox History> 
  * [[http://www.math.ucla.edu/~vsv/papers/paper.pdf|VECTOR BUNDLES AND CONNECTIONS IN PHYSICS AND MATHEMATICS: SOME HISTORICAL REMARKS]] by V. S. Varadarajan
</tabbox>

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