It is possible to construct an invariant action for the connection field itself. We want an action that only has two derivatives (since actions with higher derivatives are not unitary). Define the quantity:  formula_31  formula_32  formula_33. The unique action with only two derivatives is given by:  formula_34. Therefore, one can derive electromagnetic theory from arguments based solely on symmetry. We now generalize the above reasoning to general gauge groups. One begins with the generators of some Lie algebra:  Let there be a fermion field that transforms as  Again the derivatives of formula_37 are not covariant under this transformation. We introduce a covariant derivative  with connection field given by  We require that formula_40 transforms as:  We define the field strength operator  As formula_43 is covariant, this means that the formula_44 tensor is also covariant:  Note that formula_46 is only invariant under gauge transformations if formula_47 is a scalar, that is, only in the case of electromagnetism. We can now construct an invariant action out of this tensor. Again we want an action that only has two derivatives. The simplest choice is the trace of the commutator:  The unique action with only two derivatives is given by:  This is the action for Yang-mills theory. We consider a change of representation in the quantum Maxwell gauge theory. The idea is to introduce a basis of states labeled by loops formula_50 whose inner product with the connection states is given by  The loop functional formula_52 is the Wilson loop for the abelian formula_53 case. We consider for simplicity (and because later we will see this is the relevant gauge group in LQG) an formula_54 Yang–Mills theory in four dimensions. The field variable of the continuous theory is an formula_54 connection (or gauge potential) formula_56, where formula_57 is an index in the Lie algebra of formula_54. We can write for this field  where formula_60 are the formula_61 generators, that is the Pauli matrices multiplied by formula_62. note that unlike with Maxwell's theory, the connections formula_40 are matrix-valued and don't commute, that is they are non-Abelian gauge theories. We must take this into account when defining the corresponding version of the holonomy for formula_54 Yang–Mills theory. We first describe the quantum theory in terms of connection variable.