nLab field bundle

Idea

In Lagrangian field theory a field history is a section of some fiber bundle $E \overset{fb}{\to} \Sigma$ over spacetime/worldvolume $\Sigma$. This is then called the field bundle.

The de Rham complex of the jet bundle of the field bundle, equipped with its canonical horizontal/vertical bigrading, is the variational bicomplex of the Lagrangian field theory.

For more see at A first idea of quantum field theory the chapter Fields.

Remark

The notion of field bundle captures only some aspects of the general notion of spaces of physical fields. For instance the configurations of the B-field do not form sections of a fiber bundle (but of a fiber 2-bundle) and hence for instance the magnetic charge anomaly has no real description by field bundles. See at field (physics) in the section The idea of field bundles and its problems for more.

Examples

Example

For $E \to X$ the trivial bundle $X \times \mathbb{R} \to X$ or $X \times \mathbb{C} \to X$ a smooth section is simply a smooth function on $X$ with values in the real numbers or complex numbers, respectively. In physics this is called a scalar field.

Example

More generally, for $V$ a vector space, the trivial field bundle $X \times V \to X$ has as secton $V$-valued functions. These are also called linear sigma-model fields.

Still more generally, if $Y$ is any manifold, then the sections of the trivial field bundle $X \times Y \to X$ are called non-linear sigma-model fields.

Example

For $X$ a smooth manifold and $E \to X$ some tensor product of copies of the tangent bundle $T X \to X$ and the cotangent bundle $T^* X \to X$ a section is a tensor field of the corresponding rank.

Many physical fields are related to tensor fields, but few are genuinely tensor fields. See at field (physics) for more on this.

Related concepts

• fiber bundles in physics

• field (physics)

• extended phase space

References

The traditional idea of field bundle is discussed for instance around section 7.3.3 of

• Laurent Claessens, Field theory from a bundle point of view (2011) (pdf) .

Detailed discussion of field bundles in gauge theory with a fixed instanton sector/principal bundle-class is around section 2.5 of

• Marco Benini, Claudio Dappiaggi, Alexander Schenkel, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (arXiv:1303.2515)

and the issue is highlighted more explicitly in

• Alexander Schenkel, On the problem of gauge theories

  in locally covariant QFT_, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (pdf)

• Urs Schreiber, Higher field bundles for gauge fields, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (web)

The issue was then fixed in

• Marco Benini, Alexander Schenkel, Richard Szabo, Homotopy colimits and global observables in Abelian gauge theory (arXiv:1503.08839)

precisely by retaining to groupoids/stacks of fields, hence using a higher stacky field bundle.

A discussion of the problems of the traditional notion and its rectification in higher geometry is at

• section Fields of geometry of physics