Matchete`

CD

CD[ind,expr]

returns the covariant derivative(s) of a given expression expr. The argument ind can either be a single Lorentz index or a list of Lorentz indices.

Details and Options

CD automatically distributes over sums and applies the product rule.
The exact representation of the covariant derivative is only implicitly defined through the gauge quantum numbers of single field or composite operator that the derivative is acting on.
When acting on a single instance of Field the the CD head will disappear and its indices will be added to the last argument of the Field.

Examples

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Basic Examples (2)

Defining a scalar field:

Since the scalars does not transform under any gauge group, the covariant derivative is equivalent to a partial derivative here (D_μ=∂_μ).

Apply a single derivative to one field:

Apply two derivatives to one field:

For a gauged field we instead have D_μ = ∂_μ -  T^a, where the sum runs over all representations of the fields with the corresponding generators (T^a) and gauge fields ).

The CD object does not carry any information about the gauge representations. This information is exclusively stored in the Field.

Scope (1)

Also works with FieldStrength (or FS):

Properties & Relations (1)

CD automatically applies the product rule:

Possible Issues (1)

Covariant derivatives always apply the product rule. Therefore, the Warsaw basis operator (HH)(HH)=(HH)∂²(HH) cannot be kept in its unevaluated form in Matchete.

CD