Understanding the field strength tensor $F_{\mu\nu}$ as a commutator

Question

As far as I understand, one way to define the field-strength tensor is by using the commutator of covariant derivatives as follows: $$-igT^aF^a_{\mu\nu} = [D_\mu, D_\nu]$$ where $T^a$ is a basis for the Lie algebra, $$D_\mu = I\partial_\mu - igT^aA^a_\mu$$ is the covariant derivative, $g$ is a coupling constant, and $A_\mu^a$ are the components of the gauge field.

By comparing coefficients of basis elements, one obtains that the field strength tensor is given by the following relation: $$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc}A_\mu^b A_\nu^c$$ where $f^{abc}$ are structure constants of the Lie algebra being studied.

I understand that by interpreting the gauge field $A$ as a Lie algebra-valued 1-form on a principle bundle that we can define the field-strength tensor as the exterior derivative of $A$, i.e. $F = dA$. However this doesn't seem to be how physicists first discovered the generalized version of the field-strength tensor (generalized to more than electromagnetism).

1. What was the first motivation for looking at the commutator $[D_\mu, D_\nu]$, before physicists were aware of its geometric interpretation as a measure of curvature?
2. The geometric interpretation of the commutator, based on an answer to this phys.SE quesetion, is that it measures the extent to which the of the covariant derivatives $D_\mu$ and $D_\nu$ fail to commute. That is we consider an infinitesimally small square with edges $e_\mu$ and $e_\nu$ and see how parallel transporting along this square in different orders differs. Is this all that is going on? This is reminiscent of how one defines the full Riemann curvature tensor, but here we omit the Lie bracket term, which with an abuse of notation I write as $D_{[\mu, \nu]}$. Why is this term omitted?

Answer 1

1. Levi-Civata defined rotation free parallel transport of the local tangent field $e_\mu/(x) \to e_\mu (x + v^i e_i dt)$ and found that all tensor products of basis tangent vectors need to be differentiated, too, by the product rule, yielding a transformed vector field, known by the names of Christoffel symbols

$$\nabla_i (v^k(x) e_k(x) ) = (\nabla_i v^k(x)) e_k(x) +v^k(x) \nabla_i e_k(x) = (\partial_i v^k(x)) \ e_k(x) + v^k \sum_s \ e_s \ \mathbf (\Gamma_i)^s{}_k\ $$

If the $\mathbf \Gamma$ symbols are symmetric wrt to the lower differential and Lie-algebra index, the frame transport is rotation free, a true parallel transport. A torsion free Levi-Civita connection $\nabla$ with connection matrix 1-form $\mathbf \Gamma$ repects the scalar product as a constant.

$$\nabla_i G(e_k,e_l) = G(\nabla_i e_k, e_l) + G( e_k,\nabla_i e_l) $$

In a sense Levi-Civita defined parallel transport of a general smooth $GL(n)$ bundle over a Riemann manifold, since local coordinate transforms as Jacobi matrices don't have any restrictions except existence of the inverse.

2. The electromagnetic gauge group $U(1)$ is an abelian group, the Lie algebra is 1-d, no commutator element necessary to close the parallelogram of tangent vectors in the Lie-algebra for a loop integral.

The evolution of covariant derivatives, including the commutator as the curvature tensor is due to the Italian group of differential geometers; Ricci and mostly his student Levi-Civita, the first to use parallel transport of tensors and their covariant derivative on a metric manifold.

Levi-Civita (tensor analyis) and the silent mathematical parents of general relativity in Göttingen, Felix Klein (geometry), David Hilbert (PDE) and Emmy Nöther (symmetries and abstract algebra), put Einsteins crude ideas on the mathematical rails.

The exterior derivative calculus, based on Grassmanns geometric algebra, is largely the work of E. Cartan and his school.

Extension to group manifolds and their tangent space came by the group of algebraists around Nöther, Weyl, Hopf spreading to the Bourbaki group in Paris.

The extension of the electromagnetic gauge group to introduce the electroweak, nonabelian group $U(1) \times SU(2)$ was a long process, interplay with experiments, to explain the unforseen massless 2-component neutrino state separation, breaking parity symmetry, and the symmetry conserving but mass generating shift of the ground states by the Higgs mechanism.