What if electromagnetism were a SU(2) Yang-Mills gauge theory?

Let us ignore for a moment the Standard Model, the weak force, and the spontaneous symmetry breaking and try to imagine how would the world look like if electromagnetism would not be present and would have been replaced by a SU(2) Yang-Mills theory.

To get a good grip on this, let's do a very quick review of Lie groups and algebras. A Lie group is a continuous group, and is so named after Sophus Lie, a Norwegian mathematician (no, he was not Chinese).  Like any group, a Lie group has a unit element, and because of the continuity, we can define a tangent space for this element. This tangent space is a Lie algebra. A Lie group can be recovered from the Lie algebra by exponentiation, and the elements of the Lie algebras are called generators.

For SU(2), there are three generators:

\( F_i = \frac{1}{2} \sigma_i\)

where \( \sigma_i\) are the Pauli matrices:

\( F_1 = \frac{1}{2}\left( \begin{array}{cc} 0 &1 \\ 1 &0 \end{array}\right) \), \( F_2 = \frac{1}{2}\left( \begin{array}{cc} 0 &-i \\ i &0 \end{array}\right) \), \( F_3 = \frac{1}{2}\left( \begin{array}{cc} 1 &0 \\ 0 & -1 \end{array}\right) \)

[side note - I think I was a bit overly ambitious to present the derivations of the equations in this post and I will instead restrict to just stating the results]

The field tensor \( F_{\mu \nu}\) has the usual definition in terms of the "electric" and "magnetic" fields \( E \) and \( B \):

\( F_{\mu \nu} = \left( \begin{array}{cccc} 0 & E_1 & E_2 & E_3 \\ -E_1 & 0 & -B_3 & B_2 \\ -E_2 & B_3 & 0 & -B_1 \\ -E_3 & -B_2 & B_1 & 0 \end{array} \right) \)

where:

\( F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu -i q (A_\mu A_\nu - A_\nu A_\mu ) \)

Now the essential thing is that instead of \( A_\mu = (\phi, A_x , A_y , A_z ) \) meaning the electric potential and the magnetic vector field, the four components are no longer scalars, but linear combinations of the generators. And the generators can be naively interpreted as rotations in a 3D space. Naively because for each SO(3) rotation there are two SU(2) elements, what mathematicians call a "double cover". A better physical interpretation comes from quantum mechanics where two linear combination of \(F_1 \) and \(F_2 \) are interpreted as "raising" \( I_+ \) and "lowering" \( I_- \) operators. To get to physics, in the original isospin Yang-Mills paper where SU(2) was applied to protons and neutrons, \( I_+ \) corresponded to a transformation of a neutron into a proton, while \( I_- \) corresponded to the reversed operation. In other words, the quanta of SU(2) interaction carries (an isospin) charge. For ordinary electromagnetism, the photon is not electrically charged, but non-abelian gauge interactions are no longer charge neutral.

Let us work out how \( E_x\) would look like :

\( E_x = E_1 = F_{01} = \partial_0 A_1 - \partial_1 A_0 -i q (A_0 A_1 - A_1 A_0 )  = \frac{\partial A_x}{\partial t} - \frac{\partial \phi}{\partial x}\)

the same way as in standard electromagnetism.

Let us also work out how \( B_x\) would look like :

\( B_x = B_1 = F_{43} = \partial_4 A_3 - \partial_3 A_4 -i q ( A_4 A_3 - A_3 A_4 ) \)

\( B_x =  \frac{\partial A_y}{\partial z} - \frac{\partial A_z}{\partial y} - q A_x \)

Here we pick up an additional term due to the non-commutativity. On top of all this, at each space-time point \( A's \) are no longer scalars, but they are vectors in an internal space which can carry SU(2) charges.

Beside the internal space motion, for space-time motion, the Lorentz force law is the same as in the electromagnetic case, but the inhomogeneous Maxwell's equation:

\( \partial^\mu F_{\mu \nu} = j_\nu \)

generalizes to:

\( \partial^\mu F_{\mu \nu} - iq [A^\mu , F_{\mu \nu}]= j_\nu \)

The (Dirac) current itself generalizes from:

\( j_\nu = q \psi^\dagger \gamma_\nu \psi \)

to 3 currents corresponding to the 3 generators \( F^k \) of SU(2):

\( {(j_\nu )}^k = q \psi^\dagger \gamma_\nu F^k \psi \)

So overall, the Yang-Mills theory it is quite more complicated due to the non-commutativity of the gauge group. But one thing should be clear: the magnetic field is just a very naive simplistic picture of what is going on and this mental picture only works for electromagnetism because U(1) is a commutative Lie group. The real physical objects are the "vector potentials" \( A_\mu \). Then the Bohm-Aharonov effect where measurable changes are produced by changes in vector potential while the the net magnetic field is zero is no longer counter-intuitive. The real explanation of this effect is geometrical. 

Yang-Mills is quite an interesting model and its original intention proved to be not in agreement with reality, but physicists kept studying it and it turned out that the SU(2) gauge theory does describe a physical interaction, that of the weak force responsible for particle decays but the story is a bit more complicated due to symmetry breaking. The generalization to SU(3) is straightforward, one simply change the group generators, but then brand new physics arises in the form of asymptotic freedom which explains why we do not see free quarks in nature.

Yang-Mills theory was only accepted by everyone after the proof of renormalizability was obtained in the 70s showing that a quantum field theory based on Yang-Mills does produce sensible finite predictions and all the infinities can be cured in a mathematical consistent way.