$U(1)$ Yang--Mills Theory as Electromagnetism¶

For an Abelian gauge field, the connection is a spacetime one-form $A$ and its curvature is the electromagnetic two-form

$$F=dA.$$

Maxwell's equations become the Bianchi identity $dF=0$ and the source equation $\delta F=J$. In vacuum, $J=0$.

Minkowski spacetime and form operators¶

We use signature $(-,+,+,+)$. The codifferential is built from the Hodge star and the exterior derivative, so all metric signs enter in a single place.

declare symbol t, x, y, z : MathValue

def N : Integer := 4
def coords : Vector MathValue := [| t, x, y, z |]
def g : Matrix MathValue :=
  [| [| -1, 0, 0, 0 |]
   , [| 0, 1, 0, 0 |]
   , [| 0, 0, 1, 0 |]
   , [| 0, 0, 0, 1 |] |]

def d (X : Tensor MathValue) : Tensor MathValue :=
  !(flip ∂/∂) coords X

def hodge (A : DiffForm MathValue) : DiffForm MathValue :=
  let k := dfOrder A
   in withSymbols [i, j]
        sqrt (abs (M.det g_#_#)) *
        foldl
          (.)
          ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k))
          (map (\n -> g~(i_n)~(j_n)) [1..k])

def δ (A : DiffForm MathValue) : DiffForm MathValue :=
  let k := dfOrder A
   in (-1) ^ (N * k + 1) * hodge (d (hodge A))

A quick basis check fixes the Lorentzian orientation and sign convention: $\star(dt\wedge dx)=-dy\wedge dz$.

hodge (wedge [| 1, 0, 0, 0 |] [| 0, 1, 0, 0 |])

$\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}$

Gauge potential and curvature¶

Let $A=\varphi\,dt+A_x\,dx+A_y\,dy+A_z\,dz$, with arbitrary symbolic component functions. Antisymmetrizing $dA$ exposes the electric and magnetic field-strength combinations.

def ϕ : MathValue := function (t, x, y, z)
def Ax : MathValue := function (t, x, y, z)
def Ay : MathValue := function (t, x, y, z)
def Az : MathValue := function (t, x, y, z)

def potential : DiffForm MathValue := [| ϕ, Ax, Ay, Az |]

dfNormalize (d potential)

$\begin{pmatrix} 0 & \frac{-1}{2} \frac{\partial ϕ}{\partial 2} + \frac{1}{2} \frac{\partial Ax}{\partial 1} & \frac{-1}{2} \frac{\partial ϕ}{\partial 3} + \frac{1}{2} \frac{\partial Ay}{\partial 1} & \frac{-1}{2} \frac{\partial ϕ}{\partial 4} + \frac{1}{2} \frac{\partial Az}{\partial 1} \\ \frac{1}{2} \frac{\partial ϕ}{\partial 2} + \frac{-1}{2} \frac{\partial Ax}{\partial 1} & 0 & \frac{1}{2} \frac{\partial Ay}{\partial 2} + \frac{-1}{2} \frac{\partial Ax}{\partial 3} & \frac{1}{2} \frac{\partial Az}{\partial 2} + \frac{-1}{2} \frac{\partial Ax}{\partial 4} \\ \frac{1}{2} \frac{\partial ϕ}{\partial 3} + \frac{-1}{2} \frac{\partial Ay}{\partial 1} & \frac{-1}{2} \frac{\partial Ay}{\partial 2} + \frac{1}{2} \frac{\partial Ax}{\partial 3} & 0 & \frac{1}{2} \frac{\partial Az}{\partial 3} + \frac{-1}{2} \frac{\partial Ay}{\partial 4} \\ \frac{1}{2} \frac{\partial ϕ}{\partial 4} + \frac{-1}{2} \frac{\partial Az}{\partial 1} & \frac{-1}{2} \frac{\partial Az}{\partial 2} + \frac{1}{2} \frac{\partial Ax}{\partial 4} & \frac{-1}{2} \frac{\partial Az}{\partial 3} + \frac{1}{2} \frac{\partial Ay}{\partial 4} & 0 \\ \end{pmatrix}$

Maxwell tensor¶

To display the field equations in their familiar variables, define the antisymmetric tensor $F_{\mu\nu}$ from symbolic electric and magnetic components. Egison stores a differential two-form as $\tfrac12F_{\mu\nu}dx^\mu\wedge dx^\nu$, so the full antisymmetric component matrix carries an explicit factor of one half.

def Ex : MathValue := function (t, x, y, z)
def Ey : MathValue := function (t, x, y, z)
def Ez : MathValue := function (t, x, y, z)
def Bx : MathValue := function (t, x, y, z)
def By : MathValue := function (t, x, y, z)
def Bz : MathValue := function (t, x, y, z)

def F : DiffForm MathValue :=
  (1 / 2) *
    [| [| 0, Ex, Ey, Ez |]
     , [| -Ex, 0, -Bz, By |]
     , [| -Ey, Bz, 0, -Bx |]
     , [| -Ez, -By, Bx, 0 |] |]

The dual Bianchi expression packages $\nabla\cdot B=0$ and $\nabla\times E=-\partial_tB$.

hodge (d F)

$\begin{pmatrix} \frac{\partial Bz}{\partial 4} + \frac{\partial By}{\partial 3} + \frac{\partial Bx}{\partial 2} \\ \frac{\partial Ez}{\partial 3} - \frac{\partial Ey}{\partial 4} + \frac{\partial Bx}{\partial 1} \\ -\frac{\partial Ez}{\partial 2} + \frac{\partial Ex}{\partial 4} + \frac{\partial By}{\partial 1} \\ \frac{\partial Ey}{\partial 2} - \frac{\partial Ex}{\partial 3} + \frac{\partial Bz}{\partial 1}\\ \end{pmatrix}$

The codifferential packages Gauss's law and the Ampere--Maxwell law. Setting this vector equal to a current one-form gives $\delta F=J$.

δ F

$\begin{pmatrix} -\frac{\partial Ez}{\partial 4} - \frac{\partial Ey}{\partial 3} - \frac{\partial Ex}{\partial 2} \\ -\frac{\partial Ex}{\partial 1} + \frac{\partial Bz}{\partial 3} - \frac{\partial By}{\partial 4} \\ -\frac{\partial Ey}{\partial 1} - \frac{\partial Bz}{\partial 2} + \frac{\partial Bx}{\partial 4} \\ -\frac{\partial Ez}{\partial 1} + \frac{\partial By}{\partial 2} - \frac{\partial Bx}{\partial 3}\\ \end{pmatrix}$

The one-half component convention now agrees with dfNormalize (d potential). The derivative combinations are the homogeneous and sourced Maxwell equations. This is the Abelian $U(1)$ Yang--Mills system, where the nonlinear $A\wedge A$ term vanishes.

$U(1)$ Yang--Mills Theory as Electromagnetism¶

Minkowski spacetime and form operators¶

Gauge potential and curvature¶

Maxwell tensor¶

Links