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In "The Large Scale Structure of Space and Time" by Hawking and Ellis the following is stated:

"There will be various fields on $M$, such as the electromagnetic field, the neutrino field, etc., which describe the matter content of space-time"

Here $M$ is meant to be a spacetime (i.e. a smooth $4$-dimensional manifold). From a mathematical perspective, how does one interpret "field" as quoted above. Are the authors saying that these matter fields (such as the electromagnetic field and the neutrino field) are vector fields on $M$?

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For a physicist, a “field” is any function over the manifold taking values in some vector space. Scalar fields are just real/complex functions on $M$, vector fields are just vector-valued functions over $M$, and spinor fields are spinor-valued functions over $M$. These fields are taken together with some action or stress tensor to couple them to the gravitational field.

Mathematically, a “field” is a section of some bundle $E$ over $M$, taken together with some least-action principle or stress tensor. A scalar field is a section of a line bundle, a vector field is a section of the (co)tangent bundle; spinor fields are sections of the “spin bundle”, and the gravitational field is a section of the (0,2) tensor bundle.

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No. The electromagnetic field is a vector field (as is the gluon field and those for the W and Z bosons) but the neutrino fields are spinor fields (as are those for the quarks, and for the electron, muon, and tau). Finally the Higgs field is a scalar field.

In general, “field” here simply means “quantum field” without implying anything about which representation of the local Poincaré group it might transform under.

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