Equivalence of helicity and Euclidean self-duality for gauge fields

In the canonical formalism for the free electromagnetic field a solution to Maxwell's equations is customarily identified with its initial gauge potential (in Coulomb gauge) and initial electric field, which together determine a point in phase space. The solutions to Maxwell's equations, all of whose plane waves in their plane wave expansions have positive helicity, thereby determine a subspace of phase space. We will show that this subspace consists of initial gauge potentials which lie in the positive spectral subspace of the operator curl together with initial electric fields conjugate to such potentials. Similarly for negative helicity. Helicity is thereby characterized by the spectral subspaces of curl in configuration space. A gauge potential on three-space has a Poisson extension to a four dimensional Euclidean half space, defined as the solution to the Maxwell-Poisson equation whose initial data is the given gauge potential. We will show that the extension is anti-self dual if and only if the gauge potential lies in the positive spectral subspace of curl. Similarly for self dual extension and negative spectral subspace. Helicity is thereby characterized for a normalizable electromagnetic field by the canonical formalism and (anti-)self duality. For a non-abelian gauge field on Minkowski space a plane wave expansion is not gauge invariant. Nor is the notion of positive spectral subspace of curl. But if one replaces the Maxwell-Poisson equation by the Yang-Mills-Poisson equation then (anti-)self duality on the Euclidean side induces a decomposition of (now non-linear) configuration space similar to that in the electromagnetic case. The strong analogy suggests a gauge invariant definition of helicity for non-abelian gauge fields. We will provide further support for this view.


Plane wave solutions
Suppose that a and b are real numbers, that ω > 0 and that e 1 , e 2 , e 3 are the standard basis of R 3 . Then the vector fields E (x, y , z, t) = a(sin ψ)e 1 + b(cos ψ)e 2 , ψ = ω(t − z) (1) B(x, y , z, t) = −b(cos ψ)e 1 + a(sin ψ)e 2 (2) are solutions to Maxwell's equations. These are plane wave solutions. Power propagates in the direction of the Poynting vector One should think of this solution as a (very rough) approximation to a flashlight beam. The beam is pointing upward along the z axis.

Linearly polarized light (for sunglasses)
For fixed x, y , z the endpoint of the vector E clearly moves around the ellipse x 2 /a 2 + y 2 /b 2 = 1.
This really is an ellipse if a = 0 and b = 0. But if b = 0 then we see from (1) that E just moves back and forth along the x axis while B moves back and forth along the y axis. This is the prototype of horizontally polarized light.
Light from the sun that bounces off the curved roof of the car in front of you becomes horizontally polarized in this reflection. Fortunately, your polarized sunglasses allow only vertically polarized light thru, thereby cutting out this source of glare.

Circularly polarized light (for us)
In case a = ±b the tip of the electric vector moves around in a circle as time increases. So does the tip of the magnetic field vector. This is the case at any fixed point (x, y , z). The direction of rotation depends on whether a = b or a = −b. Looking down on one of these circles from above the light beam is called left circularly polarized if the electric vector is moving counter clockwise and right circularly polarized if the electric vector is rotating clockwise. Looking down from above is the same as looking up the flashlight beam. Physicists usually refer to left circularly polarized light as having positive helicity and right circularly polarized light as having negative helicity. Here is some good news.

Our problem
Whereas, Helicity in electromagnetism is defined in terms of plane wave expansions, and Whereas, there are no plane wave expansions for the non-linear YM hyperbolic equations, Therefore, we may ask whether helicity has any gauge invariant meaning in non-abelian gauge theories Method: a. In EM show equivalence between the plane wave definition of helicity and (anti-)self duality of gauge fields. b. In YM use the latter for a definition and then justify it.

The canonical formalism: Quantitative version
Recall that there is a gauge potential In terms of A, Maxwell's equations in empty space read A general class of solutions, given as a plane wave expansion, is The q's and p's of the canonical formalism are given by A = A(·, 0) and E = −Ȧ(·, 0).

The electromagnetic phase space
Then in the pairing So we can identify Note: Theorem (Bargmann and Wigner, 1948) If A(x, t) is the unique solution to Maxwell's equations in empty space with initial data then any Lorentz transformation of the solution leaves (10) invariant. The norm square (10) is the unique norm with this property.

Characterization of helicity in EM.
Facts about the operator curl.
1. The operator curl acts in C as a self-adjoint operator. (Do an integration by parts.) 2. It has a zero nullspace in C. (If div A = 0 and curl A = 0 then A = 0. (Size constraints are needed.)) 3. Let C + denote the positive spectral subspace of curl in C and let C − denote the negative spectral subspace. Then Otherwise said: Any pair {A, E } in phase space is a unique sum: with A ± ∈ C ± and E + a conjugate momentum to some element of C + and E − a conjugate momentum to some element in C − .
(Equivalence of helicity with sgn curl) Moral: The operator curl decomposes configuration space into two subspaces, which automatically decompose phase space into two subspaces.

Theorem
(Equivalence of helicity with sgn curl) Suppose that A(x, t) is a solution to Maxwell's equations with Then its plane wave expansion is composed entirely of plane waves of positive helicity iff {A, E } ∈ T * (C + ).
Its plane wave expansion is composed entirely of plane waves of negative helicity iff {A, E } ∈ T * (C − ).

Theorem
If A ∈ C then the M-P equation has a unique solution with finite Poisson action. Moreover where * e is the four dimensional Euclidean Hodge star operator.
Leonard Gross Equivalence of helicity and Euclidean self duality for gauge field Equivalence of helicity with Euclidean (anti-) self duality Theorem If A ∈ C and a(x, s) is its Poisson extension then Proof. Yang-Mills theory K : compact connected Lie group with lie algebra k.
g acts on A by A g = g −1 Ag + g −1 dg .
Yang-Mills-Poisson equation a is a 1-form on R 3 for each s. Equivalently, it is a 1-form on R 4 + in temporal gauge. d is the 3 dimensional exterior derivative b is the 3 dimensional curvature of a for each s. F is the 4 dimensional curvature of a (regarded as a 1-form on R 4 + in temporal gauge).

The Yang-Mills-Poisson equation on
Initial condition a(0) = A.

Definition
The Poisson action of A is Fact: P(A g ) = P(A). Therefore the Poisson action descends to a function on C.
Examples: Any instanton on R 4 restricts to a solution on the half-space with finite Poisson action.
Theorem : For any given A ∈ H 1/2 (R 3 ) the YMP equation has a unique solution with finite action.
Status of proof:

Outline of proof
In the abelian case, i.e. EM, the subspaces C ± are the nullspaces of the operators C ∓ |C |. Let's focus on C + . It's the null space of C − |C |. This operator is terribly big and negative on C − . Consequently This functional analytic method is no good for our nonlinear configuration space. Let's change viewpoint a little. The exponential is the solution of the differential equation h(A) is a k valued 1-form for each A and is therefore a tangent vector to A at A. h is a gauge covariant vector field on A and therefore descends to a vector field on C.
Replace the linear vector field curl A − |curl| A by h(A).
Hint of proof: Use P(A) as a Liapounov function.