NUPHB114685114685S0550-3213(19)30171-310.1016/j.nuclphysb.2019.114685The AuthorQuantum Field Theory and Statistical SystemsEquivalence of helicity and Euclidean self-duality for gauge fieldsLeonardGrossgross@math.cornell.eduDepartment of Mathematics, Cornell University, Ithaca, NY 14853-4201, United States of AmericaDepartment of MathematicsCornell UniversityIthacaNY14853-4201United States of AmericaDepartment of Mathematics, Cornell University, Ithaca, NY 14853-4201Editor: Hubert SaleurAbstractIn 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 together with (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 establish two further properties that support this view.1IntroductionA workable notion of helicity for quantized Yang-Mills fields has been sought in many works in recent years. For a broad perspective on this work see the extensive review by Leader and Lorcé [39]. Any notion of helicity for Yang-Mills fields must be gauge invariant, Lorentz invariant and reduce to the standard notion of helicity for electromagnetic fields. This paper proposes a notion of helicity for non-abelian gauge fields based on a Euclidean interpretation of helicity in electromagnetism. For the latter, it will first be shown that the Poisson extension of the initial gauge potential of an electromagnetic field to a half space in Euclidean space-time sets up an equivalence between helicity, which is a Minkowski space notion, and (anti-)self-duality, which is a Euclidean space notion.For classical electromagnetic fields helicity is customarily defined in terms of plane waves, which do not mesh well with a non-linear theory such as Yang-Mills, [1]. To make the jump from electromagnetism to Yang-Mills fields we are going to describe helicity in the electromagnetic case in a manner that allows a natural extension to the Yang-Mills case. To begin, we will first describe the precise configuration space for the free electromagnetic field that reflects the unique Lorentz invariant norm of Bargmann and Wigner, [5]. Second, given a real valued gauge potential A(x) on R3 we will show that its Poisson extension to a half space in R4 not only captures the Lorentz invariant norm of Bargmann and Wigner, but is also self-dual if and only if A is the initial data in Minkowski space of some solution to Maxwell's equations of negative helicity. Similarly for anti-self-dual and positive helicity.To be more explicit, suppose that A(x):=∑j=13Aj(x)dxj is a real valued gauge potential on R3 and that, for each s≥0, a(x,s)=∑j=13aj(x,s)dxj is another gauge potential on R3 such that the function a satisfies the Maxwell-Poisson equation with initial value A:(1.1)a″(s)=curl2a(s),a(0)=A. We may regard a as a 1-form in temporal gauge on the half-space R+4:=R3×[0,∞). Its four dimensional curvature F is then given by F=ds∧a′+da, where d denotes the three dimensional exterior derivative. Equation (1.1) is the Euler equation for minimization of ∫R+4|F(x,s)|2d3xds, subject to the initial condition a(0)=A. It happens that this minimum is the square of the unique Lorentz compatible norm on configuration space. We will show that A is the initial potential of some solution to Maxwell's equations with only positive helicity plane waves in its plane wave decomposition if and only if F is anti-self-dual. Similarly for negative helicity and self-duality. The Maxwell-Poisson equation (1.1) thereby sets up a correspondence between helicity on the Minkowski side and (anti-)self-duality on the Euclidean side. The result is a decomposition of the electromagnetic configuration space C into two orthogonal subspaces C±, determined by helicity, or equivalently, by (anti-)self-duality.For a non-abelian gauge field the corresponding Poisson-like equation is the Yang-Mills-Poisson equation. It is a nonlinear, degenerate elliptic differential equation for which existence and uniqueness of solutions has not yet been proven. We are going to assume both, however, and show how the Yang-Mills-Poisson equation sets up a similar decomposition of the Yang-Mills configuration space C, which is no longer a linear space, into two submanifolds C± corresponding to anti-self-dual and self-dual solutions. The similarity of this procedure to the electromagnetic case suggests that the submanifolds C± are the “correct” non-abelian analogs of the classical helicity subspaces for Maxwell's theory. But we will give more support to this interpretation in Section 6 by showing that this decomposition exhibits two additional properties which are definitive in the electromagnetic case.The representation of a gauge potential as the initial data of a Poisson-like equation automatically induces a Riemannian metric on the set of potentials (modulo gauge transformations), whose Poisson action ∫R+4|F(x,s)|2d3xds is finite. This in turn gives a quantitative meaning to phase space T⁎(C) for both electromagnetic and Yang-Mills fields. It turns out, remarkably, that the norm defined on the electromagnetic phase space in this way is exactly the unique Lorentz invariant norm first discovered by Bargmann and Wigner [5] in their fundamental work on unitary representations of the inhomogeneous Lorentz group. The norm induced on configuration space and phase space by our procedure is therefore forced on us if we wish to have a Lorentz invariant theory. We will show this in the course of establishing the equivalence of helicity with (anti-)self-duality for electromagnetism. In the non-abelian case the Riemannian metric induced from the Euclidean side by the same procedure is presumably Lorentz invariant in some appropriate sense. But this has not been explored.In accordance with the canonical formalism (as in, say, [7, Chapter 14]) the electric field E at time zero is canonically conjugate to A and therefore the pair {A,E} defines a point in phase space. We will show that if A is in C+ and E is in TA⁎(C+) then the unique solution to Maxwell's equations with initial data A,E contains only positive helicity plane waves in its plane wave decomposition (and conversely). Similarly T⁎(C−) corresponds to negative helicity plane waves. In this way Euclidean (anti-)self-duality provides a completely geometric characterization of helicity in electromagnetic theory.We will reinforce this geometric interpretation of helicity further by starting over again in [25] with the quantized theories, each of which has a generally accepted “forward component of angular momentum” operator used for defining helicity in their respective quantum field theories. In the electromagnetic case we will see that the operator simply “quantizes” the decomposition C±. The Schrödinger representation of the quantized electromagnetic field will be instrumental in implementing this view. In the non-abelian case, where configuration space is not a linear space, we will see that the presumed “forward component of angular momentum” operator similarly quantizes the two submanifolds C±, thereby reinforcing again the link between helicity and Euclidean (anti-)self-duality for non-abelian gauge fields. The Schrödinger representation of the quantized Yang-Mills field will again be indispensable in implementing this view, and for this, the Riemannian metric introduced in this paper will enable us to use the gradient operator on functions over configuration space instead of momentum space annihilation operators, which have no gauge invariant meaning for the (intermediate) non-abelian fields.There are many unsettled purely mathematical issues in the body of this paper that we will ignore. They are in the nature of existence and uniqueness theorems. In the last section we will discuss the open problems raised by these structures.2Plane waves, helicity, configuration space in electromagnetism2.1Review of helicity for plane wavesThis subsection is a review of circularly polarized light in a form that will be useful for our purposes. Nothing in this subsection should be construed to be less than 165 years old. See for example Born and Wolfe [8] or Jackson [34].Notation 2.1For 0≠k∈R3 define(2.1)k⊥={u∈C3:u⋅k=0}. k⊥ is a two dimensional complex vector space closed under complex conjugation u↦u¯. Its real part kreal⊥:=k⊥∩R3 is a two dimensional real vector space. Define(2.2)Cku=ik×u,foru∈C3. Cku is in k⊥ for all u∈C3 because k×u is perpendicular to k. The significance of the operator Ck is that it implements curl under Fourier transformation in accordance with the easily verified identity(2.3)curl(ueik⋅x)=(Cku)eik⋅xfor any vectoru∈C3.Discussions of circularly polarized light and, more generally, elliptically polarized light amount to an analysis of the operator Ck acting in the two dimensional subspace k⊥. See for example [34, Chapter 7] or [8, Section 1.4].Lemma 2.2(Properties of Ck). The restriction of Ck to k⊥ satisfies(2.4)a)Ck is Hermitian in k⊥.b)Ck2=|k|2onk⊥c)Ck decomposes k⊥ into two one dimensional complex subspaces k±⊥corresponding to eigenvalues ±|k|. A vector u∈k⊥ is in k+⊥ ifand only if its complex conjugate u¯∈k−⊥. ProofUsing the triple product identity (k×u)⋅v=−u⋅(k×v), which is valid for u and v in C3, we see that(Cku,v)C3=(ik×u)⋅v¯=u⋅(ik×v‾)=(u,Ckv)C3, which proves a). Item b) follows from the identity Ck2u=−k×(k×u)=|k|2u when u⋅k=0. To prove c) let kˆ=k/|k| and let e1 be any unit vector in kreal⊥. Define e2=kˆ×e1. Then kˆ,e1,e2 form a right handed basis of R3. In particular kˆ×e2=−e1. Thus(2.5)Ck(e1+ie2)=i|k|{e2−ie1}=|k|(e1+ie2) Hence (e1+ie2) is an eigenvector for Ck with eigenvalue |k|. Using the same vectors one can compute similarly that e1−ie2 is an eigenvector of Ck with eigenvalue −|k|. Moreover, if u∈k+⊥ then Cku¯=(−Cku)‾=−|k|u‾=−|k|u¯. This proves c).□Definition 2.3Plane wavesLet 0≠k∈R3. A plane wave with wave vector k is a function on R4 of the form(2.6)Ak(x,t)=aei(k⋅x−|k|t)+a¯e−i(k⋅x−|k|t),x,t∈R3+1,a∈k⊥ Clearly Ak(x,t)∈R3 for each x,t and(2.7)divAk(x,t)=0 because of the identity(2.8)div(aeik⋅x)=(ik⋅a)eik⋅x.Lemma 2.4For the plane wave (2.6) define(2.9)Bk(x,t)=curlAk(x,t)(2.10)Ek(x,t)=−(∂/∂t)Ak(x,t). Then Bk,Ek is a solution to Maxwell's equations,(2.11)∇⋅B=0,∇⋅E=0,B˙=−curlE,E˙=curlB. ProofUse the identities (2.8) and (2.3) along with (2.6) and (2.4).□Definition 2.5Helicity for plane wavesThe standard definition of helicity can be stated thus. The plane wave (2.6) has positive helicity if Cka=|k|a. It has negative helicity if Cka=−|k|a.2.2The Lorentz invariant normA small portion of the classic paper [5] of Bargmann and Wigner will be needed to explain our choice of the norm (2.13), which we will later take to be the norm on configuration space in (2.35). The Coulomb gauge fixing that we are going to use here for convenience will be removed when we discuss Yang-Mills fields.Definition 2.6SpacesThe operator curl on vector fields over R3 acts in the real Hilbert space L2(R3;R3)∩{divu=0} because divcurl=0. It is easily seen to be self-adjoint by an integration by parts. Moreover in this space it has a zero nullspace. We will denote by C the operator curl acting in this Hilbert space or some closely related spaces. The square root of C2 will be denoted |C|: Thus(2.12)C=curl,|C|=C2. For a vector field u on R3 with divu=0 let(2.13)‖u‖H1/22=‖|C|1/2u‖L22(2.14)‖u‖H−1/22=‖|C|−1/2u‖L22. Define Hilbert spaces H±1/2 by the condition that the corresponding norm in (2.13) or (2.14) is finite.For vector fields B and E on R3 with divergence zero define(2.15)‖{B,E}‖bw2=‖B‖H−1/22+‖E‖H−1/22. The main assertion of this section is that the norm ‖⋅,⋅‖bw is invariant under Lorentz transformations in the sense that the unique solution to Maxwell's equations with B and E as initial data has, after any Lorentz transformation, initial data with the same norm. The following computations reduce this assertion to a theorem in [5] by showing that ‖⋅,⋅‖bw is equal to the norm introduced in [5] by Bargmann and Wigner. The Hilbert space introduced by Bargmann and Wigner, which we denote by BWmom, is the space of functions on momentum space defined by (2.16) and (2.17) below. We can take its norm to be given by the right side of (2.21). The next theorem shows that the plane wave expansions (2.18)-(2.20) induce a norm preserving map from BWmom onto the space of divergence free pairs {B,E} in H−1/2(R3) with the norm (2.15). The Lorentz invariance of the norm (2.15) then follows from the Lorentz invariance of the norm (2.21), proved by Bargmann and Wigner. The unitarity of this map will be proved in the proof of Theorem 2.9, where the complex structure on the real side will be explained.Theorem 2.7Plane wave decompositionSuppose that a(k) is a C3 valued function on R3 such that(2.16)a.k⋅a(k)=0for allk∈R3and(2.17)b.∫R3|a(k)|2d3k/|k|<∞. Define(2.18)A(x,t)=∫R3(a(k)ei(k⋅x−|k|t)+a(k)‾e−i(k⋅x−|k|t))d3k/|k|.(2.19)B(x,t)=curlA(x,t).(2.20)E(x,t)=−(∂/∂t)A(x,t). Then A,B and E have divergence zero for each t and B(⋅),E(⋅) is a solution to Maxwell's equations (2.11). Moreover for all real t there holds(2.21)‖{B(t),E(t)}‖bw2=‖A(t)‖H1/22+‖E(t)‖H−1/22=2(2π)3∫R3|a(k)|2d3k/|k|. Conversely, suppose that B0 and E0 are divergence free vector fields on R3 and each is in H−1/2(R3). Then there is a unique solution B(t),E(t) to Maxwell's equations (2.11) with initial values B0,E0 and a unique (up to a set of measure zero) function a(⋅):R3→C3 satisfying (2.16) and (2.17) such that B(t) and E(t) are given by (2.18)-(2.20). ProofUpon differentiating under the integral in (2.18) we see from Lemma 2.4 that A(t),B(t) and E(t) all have divergence zero and the pair {B(t),E(t)} satisfies Maxwell's equations.For the proof of (2.21) observe first that, with C=curl we have ‖B(t)‖H−1/2=‖A(t)‖H1/2 because ‖B(t)‖H−1/2=‖CA(t)‖H−1/2=‖|C|−1/2CA(t)‖L2=‖|C|1/2A(t)‖L2. It suffices therefore to prove the second equality in line (2.21). Moreover it suffices to prove it just at t=0 because replacing a(k) by a(k)e−i|k|t in the t=0 equality gives the equality at time t. Let(2.22)α(k)=a(k)+a(−k)‾and(2.23)e(k)=i|k|(a(k)−a(−k)‾). Then from (2.18) and (2.20) we find(2.24)A(x,0)=∫R3α(k)|k|eik⋅xdk.(2.25)E(x,0)=∫R3e(k)|k|eik⋅xdk. Therefore(2.26)‖A(⋅,0)‖H1/22+‖E(⋅,0)‖H−1/22=(2π)3∫R3(||k|1/2α(k)|k||2+||k|−1/2e(k)|k||2)d3k=(2π)3∫R3(|a(k)+a(−k)‾|2+|a(k)−a(−k)‾|2)d3k/|k|=(2π)3∫R3(|a(k)|2+|a(−k)‾|2)d3k/|k|=(2π)32∫R3|a(k)|2d3k/|k|. This proves the equality in line (2.21).Conversely, suppose that B0 and E0 are given, that divB0=divE0=0 and that ‖B0‖H−1/22+‖E0‖H−1/22<∞. Let A0=(curl)−1B0. This defines A0 as a potential with divA0=0 and ‖A0‖H1/2=‖B0‖H−1/2<∞. Define α(k) and e(k) from A0 and E0 by (2.24) and (2.25) respectively. Since A0 and E0 are “real”, i.e. both take their values in R3, we have α(−k)=α(k)‾ and e(−k)=e(k)‾. With (2.22) and (2.23) in mind, define(2.27)a(k)=(1/2)(α(k)+e(k)i|k|). From the hermiticity of α(⋅) and e(⋅) it follows that(2.28)a(−k)‾=(1/2)(α(k)−e(k)i|k|) and from this the relations (2.22) and (2.23) follow. The computation (2.26) now shows that(2.29)‖A0‖H1/22+‖E0‖H−1/22=(2π)32∫R3|a(k)|2d3k/|k| Hence (2.17) holds for the constructed function a(k). Of course (2.16) follows because A0 and E0 are divergence free. The solution B(t),E(t) with initial values B0,E0 can now be constructed from a(k) by (2.18)-(2.20).□Remark 2.8Integral formula for ‖{B,E}‖bw2Since the Laplacian (Δ≡∇2) on vector fields u(x) over R3 is given by −Δu(x)=curl2u(x)−graddivu(x) we may write −Δu(x)=curl2u(x) in the space of divergence free vector fields. The space of divergence free vector fields is invariant under Δ and consequently |curl|u=(−Δ)1/2u on this space. One therefore has on this space |curl|−1=(−Δ)−1/2, which, by Fourier transform, is easily seen to be given by convolution by const.(1/|x|2). Since ‖u‖H−1/22=(|C|−1u,u)L2(R3) on the space of divergence free vector fields u, the Bargmann-Wigner norm (2.15) can be written(2.30)‖{B,E}‖bw2=const.∫R3∫R3B(x)⋅B(y)+E(x)⋅E(y)|x−y|2dxdy. This representation of the Lorentz invariant norm for the electromagnetic field was derived in [23] and used there to show that the norm is invariant under the 15 dimensional conformal group of Minkowski space, which contains transformations to uniformly accelerated coordinate systems. (See e.g. [23, footnote 2].) It was shown in [23] that each of the two terms in (2.30) is separately invariant under the inversion x↦x/|x|2 of R3, which, together with dilations of R4 and the inhomogeneous Lorentz group, generate the conformal group. Unitarity of the representation of the conformal group, as opposed to just norm invariance, requires also proof of invariance of the complex structure, which will be described on the configuration space side in the proof of Theorem 2.9. Invariance of the complex structure under the conformal group was first proved in [36].2.3Helicity and sgn(curl) for solutions of finite actionOur goal in this subsection is to take a step away from the plane wave decomposition (2.18) by formulating helicity entirely on the position space side with the help of the operator curl. This is reflected explicitly in the equivalence between assertions 1.) and 2.) in the next theoremTheorem 2.9Suppose that B(t),E(t) is a solution to Maxwell's equations (2.11) with finite Bargmann-Wigner norm. Let (2.18)-(2.20) be its plane wave decomposition. Then the following are equivalent.1.) Every plane wave in its plane wave decomposition has positive helicity.2.) A(0) and E(0) are in the positive spectral subspace of curl.3.) B(t) and E(t) are both in the positive spectral subspace of curl for some t.4.) B(t) and E(t) are both in the positive spectral subspace of curl for all t5.) A(t) is in the positive spectral subspace of curl for all t.These statements also hold with positive replaced by negative everywhere. ProofThe space of functions a(k) on momentum space satisfying (2.16) and (2.17) represents Bargmann and Wigner's original description of the one-particle photon space, [5]. We continue to denote it by BWmom. On the position space side denote by BW the space of divergence free real vector fields B,E over R3 with finite norm (2.15). Define(2.31)W:BWmom→BW by W:a(k)↦{B(x,0),E(x,0)} where B and E are given by (2.18)-(2.20). Then Theorem 2.7 shows that W is a norm preserving transformation from BWmom onto the direct sum space BW if one uses 2(2π)3∫R3|a(k)|2d3k/|k| for the norm squared on BWmom. Actually, since there is no hermiticity condition on a(k), BWmom is a complex Hilbert space under the usual operation of multiplication by i≡−1. BW itself has a natural complex structure j given by j{B,E}={sgn(C)E,−sgn(C)B}, where sgn(C)=|C|−1C. The latter complex structure is more often expressed in terms of the pair {A,E} in the form j{A,E}={|C|−1E,−|C|A}, which is an equivalent description since B=CA. BW is therefore also a complex Hilbert space. Comparing (2.22) and (2.23) with (2.24) and (2.25) one sees easily that Wia=jWa. W is therefore unitary. Furthermore (2.3), (2.18)-(2.20), show that W intertwines curl with the operator C(⋅), which is multiplication by Ck.(2.32)curlW=WC(⋅). Here we are writing curl{B,E}={curlB,curlE}.Now suppose that B(t),E(t) is a solution to Maxwell's equations (2.11) with finite Bargmann-Wigner norm and with plane wave decomposition given by (2.18)-(2.20). Suppose also that it has only positive helicity plane waves in its plane wave decomposition. According to Definition 2.5 Cka(k)=|k|a(k) for every k≠0 in R3. Therefore a(⋅) is in the positive spectral subspace for the operator multiplication by Ck. Since W is unitary, {B(0),E(0)} is in the positive spectral subspace of curl on BW, (which is easily seen to be equivalent to saying that B(0) and E(0) are each in the positive spectral subspace of curl). Since |C|:H1/2→H−1/2 is an orthogonal transformation and commutes with curl it preserves all spectral properties of curl. In particular A(0) is also in the positive spectral subspace of curl because B0=curlA0. Therefore statement 1.) implies statement 2.). Conversely, if A(0) and E(0) are in the positive spectral subspace of curl then B(0) is in the positive spectral subspace of curl and so therefore is the pair {B(0),E(0)}. By (2.32) and the unitarity of W it follows that a(⋅) is in the positive spectral subspace of the operator of multiplication by Ck. Hence Cka(k)=|k|a(k) for all k≠0. Therefore statement 1.) holds. This proves the equivalence of statements 1.) and 2.). The proof shows that the pair {A(0),E(0)} could be replaced by the pair {B(0),E(0)} in the statement of item 2.). Now for any fixed t the map Wt:a(⋅)↦{B(t),E(t)} is also unitary and intertwines Ck and curl. Therefore the proof of equivalence of 1.) with 2.) also applies to the proof of equivalence of 1.) with 3.) and 1.) with 4.).Concerning the condition 5.), we see that if 1.) holds then by 2.) A(0), and similarly A(t), is in the positive spectral subspace of curl in its own space H1/2. Conversely, if 5.) holds then (d/dt)A(t) is in the positive spectral subspace of curl. In particular A(0) and its time derivative −E(0) are in the positive spectral subspace of curl and we can apply 2.) to see that 1.) holds.□Remark 2.10The operator of multiplication by Ck on BWmom decomposes BWmom into two orthogonal subspaces given respectively by the two identities Cka(k)=|k|a(k) and Cka(k)=−|k|a(k). These are the two spectral subspaces of curl on the momentum space side.2.4Configuration space, phase space and helicityThe Lorentz invariant norm (2.21) has a geometric interpretation based on the canonical formalism for the electromagnetic field: As in [7, Chapter 14], we take configuration space for the free electromagnetic field to be a set of divergence free potentials A on R3 with size of A yet to be specified. This corresponds to the radiation gauge. The momentum canonically conjugate to A is −A˙, which is E. See [7, Eq. (14.10)] for a derivation of this from the Lagrangian formalism. The canonical formalism therefore dictates that the pair A,E is a point in phase space. Thus if A lies in some configuration space C (still to be determined, quantitatively) and TA(C) denotes the tangent space to C at A, then E is a point in its dual space TA⁎(C), as is customary for momentum in classical mechanics. In the next definition we will make a quantitative choice for C and show that the norm on phase space which is automatically induced by this choice of configuration space is the Bargmann-Wigner norm (2.21).Definition 2.11Configuration spaceLet(2.33)C={real valued 1-forms A on R3 such that(2.34)a.divA=0(2.35)b.‖A‖C:=‖|C|1/2A‖L2(R3)<∞}. Here C=curl as before. Define also(2.36)C+={A∈C:Ais in the positive spectral subspace ofcurl}(2.37)C−={A∈C:Ais in the negative spectral subspace ofcurl}. Then C is a real Hilbert space and(2.38)C=C+⊕C−. The subspaces intersect only at A=0 because curl does not have a zero eigenvalue in C. Since C is a vector space its tangent space at a point A can be identified with C itself. The dual space can therefore be identified with C⁎: TA⁎(C)≡C⁎. Consequently the entire phase space can be identified as(2.39)T⁎(C)≡C⊕C⁎. The electric field E is to be identified with an element of the dual space via the pairing(2.40)〈E,A〉=∫R3E(x)⋅A(x)d3x, which is the field theoretic analog of ∑i=1npidqi, when one identifies TA(C) with C. Combined with our choice of norm (2.35), this pairing induces on E the norm ‖E‖C⁎=‖|C|−1/2E‖L2(R3). Since C involves the restriction divA=0, so does the dual space. We can therefore make the natural identification(2.41)C⁎={real valued vector fields E on R3 such that(2.42)a.divE=0(2.43)b.‖E‖C⁎:=‖|C|−1/2E‖L2(R3)<∞}. In the canonical formalism a point in phase space is thereby specified by a pair {A,E} with ‖A‖C2+‖E‖C⁎2<∞ and divA=divE=0.We see from (2.21) at t=0 that this is precisely the Bargmann-Wigner norm on the initial data when we take B=curlA.(2.44)‖{B,E}‖bw2=‖A‖C2+‖E‖C⁎2,divA=0,divE=0. Since C commutes with |C| the annihilator in C⁎ of the positive spectral subspace of curl in C is the negative spectral subspace of curl in C⁎. Similarly with positive and negative reversed. Consequently, writing C+⁎:=(C+)⁎ and C−⁎:=(C−)⁎, we have the natural identifications(2.45)C+⁎={E∈C⁎:Eis in the positive spectral subspace ofcurl}.(2.46)C−⁎={E∈C⁎:Eis in the negative spectral subspace ofcurl}. Analogous to the identification (2.39) we therefore have the identifications(2.47)T⁎(C+)=C+⊕C+⁎,T⁎(C−)=C−⊕C−⁎,T⁎(C)=T⁎(C+)+T⁎(C−).Theorem 2.12Suppose that the pair {A,E} is a point in phase space T⁎(C). Let A(t),E(t) be the unique solution to Maxwell's equations with initial data A(0)=A,−A˙(0)=E(0)=E. Then the plane wave decomposition of the solution consists ofa) positive helicity plane waves if and only if {A,E} is in T⁎(C+).b) negative helicity plane waves if and only if {A,E} is in T⁎(C−). ProofThe assumption that {A,E} lies in T⁎(C) is equivalent, by (2.44), to the assumption that the pair has finite Bargmann-Wigner norm. In view of (2.36), (2.37), (2.45) and (2.46), the present theorem just restates the equivalence between items 1.) and 2.) in Theorem 2.9.□Remark 2.13Primacy of the configuration space decompositionThe helicity character of a solution to Maxwell's equations is not determined by the value of A(0) alone but also requires knowledge of A˙(0). To conclude, for example, that a solution has only positive helicity plane waves in its plane wave decomposition it is not sufficient to know that A(0) is in the positive spectral subspace of curl. One must know also that A˙(0) is in the positive spectral subspace of curl. And yet the decomposition (2.38) of C is responsible for determining helicity in the sense that once this decomposition is known it automatically determines the decomposition of T⁎(C) as in (2.47), allowing Theorem 2.12 to be applied. The decomposition of T⁎(C) thereby plays a secondary role compared to the decomposition (2.38). This is important to observe for two disparate reasons. First, in the quantized theory, where a state of the field is given by a function ψ on C, the Heisenberg field expectations (B(x,t)ψ,ψ) and (E(x,t)ψ,ψ) in the state ψ will be shown in [25] to have only positive helicity plane waves in their plane wave decomposition when ψ depends only on C+. Second, in the non-abelian theory, where the configuration space is a non-linear manifold, the electromagnetic decomposition (2.38) has a natural analog with properties that well imitate those of the electromagnetic case. This will be shown in Section 6.Remark 2.14A bit of historyImmediately after Heisenberg and Pauli published their fundamental paper [27] in 1929, describing the quantized electromagnetic field, Landau and Peierls [38], in 1930, rederived some of the formalism of Heisenberg and Pauli, emphasizing the role of the electromagnetic configuration space. In the formulation of Landau and Peierls E and B were not independent. They introduced a norm equal to one of the two terms in (2.15) (or equivalently (2.30)) for the normalization condition for their configuration space. See e.g. [38, Equations (7)-(10)]. It is now understood that in the canonical formalism A and E are to be regarded as independent variables in phase space. So their suggestion to allow either term in (2.30) as the norm on their configuration space obscured the distinction between configuration space and phase space. Interestingly, they arrived at their norm, not by citing Lorentz invariance, but by arguing that a unit vector in this particular norm represents a single photon. An updated version of their “number of photons” argument for this norm, leading to the full phase space norm (2.30), can be found in Jackson [34, Problem 7.30].3Equivalence of helicity and Euclidean self-duality in electromagnetism3.1The Poisson action and Maxwell-Poisson semigroupThe helicity decomposition (2.38) and its associated decomposition of phase space (2.47) are based on use of the spectral decomposition of the operator curl and were shown to be equivalently characterizable in terms of plane wave expansions (cf. Theorems 2.9 and 2.12). But neither plane wave expansions nor the operator curl mesh well with non-abelian gauge fields. In this section we are going to show that the helicity decomposition of configuration space can be reproduced without using either plane wave expansions or the spectral decomposition of curl. Instead, we will extend the gauge potential A to a half-space in R4 and show that the decomposition is equivalent to (anti-)self-duality of the extension. This method for implementing the decomposition (2.38) in electromagnetism will be shown in succeeding sections to go over to non-abelian gauge fields.To simplify otherwise cumbersome calculations we will make use henceforth of differential form notation. Recall that if (u1,u2,u3) is a vector field on R3 then the corresponding differential form is given by u=∑i=13uidxi, the Hodge ⁎ operator is given on 1-forms by ⁎u=(1/2)∑i,j,kϵijkuidxj∧dxk and on 2-forms by ⁎∑jkvjkdxj∧dxk=∑ijkϵijkvjkdxi, resulting in ⁎2=1. The exterior derivative is given by du=∑ji(∂ui/∂xj)dxj∧dxi and the action of curl on 1-forms is given by curlu=⁎du. The action of div on 1-forms is given by divu=−d⁎u, while the identities divcurl=0 and curlgrad=0 are both consequences of d2=0.Notation 3.1Denote by R+4 the Euclidean half space R3×[0,∞) with coordinates (x,s). Let a be a real valued 1-form on R+4 in temporal gauge. That is,(3.1)a(x,s)=∑j=13aj(x,s)dxj,x∈R3,s∈[0,∞). Then the four dimensional curvature of a is given by(3.2)F=ds∧a′+da, where ′=(∂/∂s) and d is again the three dimensional exterior derivative operator.We wish to minimize the functional(3.3)∫R+4|F(x,s)|2dxds subject to the initial condition(3.4)a(x,0)=A(x),x∈R3, wherein A is a given real valued 1-form on R3. Observe first that(3.5)|F(x,s)|Λ22=|a′(x,s)|Λ12+|da(x,s)|Λ22 for each (x,s) because the first 2-form on the right side of (3.2) is orthogonal to the second one. In (3.5) Λ1 denotes, as usual, the three dimensional space of linear functionals on R3 and Λ2 denotes the three dimensional space of skew symmetric bilinear functionals on R3. The Euler equation for the minimization problem can be derived in the usual way as follows. Let u(x,s)=∑j=13uj(x,s)dxj with each coefficient lying in Cc∞(R3×(0,∞)). Then we find(3.6)∂u∫R+4{|a′(x,s)|2+|da(x,s)|2}dxds=2∫R+4{−(a″(x,s),u(x,s))+(d⁎da(x,s),u(x,s))}dxds after doing an integration by parts in both time and space. The Euler equation is therefore(3.7)a″=d⁎da. This is not quite Poisson's equation because half of the Laplacian, −Δ=d⁎d+dd⁎, is missing from the right hand side. But it is intimately related to Maxwell's theory and we will refer to (3.7) as the Maxwell-Poisson equation. It is easily solved. Since d⁎=⁎d⁎ on 2-forms over R3 we see that d⁎d=(⁎d)(⁎d)=curl2. The equation (3.7) may therefore be written as(3.8)a″(s)=curl2a(s). Writing C=curl, the function a(s)=esCA clearly solves this equation as does also a(s)=e−sCA. But if A is in the strictly positive spectral subspace of curl then ‖a′(s)‖L2(R3) grows exponentially in the first case, making (3.3) infinite, while if A is in the strictly negative spectral subspace of curl then (3.3) is infinite in the second case. The solution in both cases is given, for arbitrary A∈L2(R3,Λ1), by(3.9)a(s):=e−s|C|A,s≥0. It solves the Maxwell-Poisson equation (3.7) because a″=|C|2a=C2a=d⁎da and also satisfies the initial condition a(0)=A. Under reasonable growth restrictions on a(⋅) the solution given by (3.9) is unique. We will always assume uniqueness. The size of the minimum in (3.3) will be discussed in the following. It bears emphasizing that we are not making an assumption that A is in Coulomb gauge in this discussion.Terminology 3.2The unique solution to the Maxwell-Poisson equation (3.7) with initial value A will be referred to as the Poisson extension of A. We define the Poisson action of A by(3.10)P(A)=∫0∞(‖a′(s)‖22+‖da(s)‖22)ds, where a is the Poisson extension of A. We are only interested in those potentials A for which P(A)<∞. We reiterate that P(A) is the minimum of the integral in (3.10) subject to the condition that a(0)=A.Remark 3.3The Poisson action of a gauge potential A can be computed explicitly. We need not assume that A is in Coulomb gauge. From (3.9) we find(3.11)a′(s)=−|C|e−s|C|A‖a′(s)‖22=‖|C|e−s|C|A‖22=(|C|2e−2s|C|A,A)‖da(s)‖22=‖⁎da(s)‖22=‖Ce−s|C|A‖22=(|C|2e−2s|C|A,A). Therefore(3.12)P(A)=2∫0∞(|C|2e−2s|C|A,A)ds=(|C|A,A). A reader concerned about the lack of exponential decrease in (3.9) when A is in the null space of curl should observe that in this case the integrand in (3.10) is identically zero.Remark 3.4The identity (3.12) holds even if A is longitudinal: that is, A=dλ for some real scalar function λ. In this case A is in the null space of C and therefore also in the nullspace of |C|. It follows then from (3.12) that(3.13)P(dλ)=0for any functionλ:R3→R. This could also be derived directly from the definition (3.10) because the solution to the Maxwell-Poisson equation (3.7) is easily verified to be a(s)=dλ.Remark 3.5Side conditionThere is an interesting contrast between the Maxwell-Poisson equation (3.8) and Maxwell's equations with respect to the side condition. In temporal gauge the evolutionary portion of Maxwell's equations for the potential is A¨(x,t)=−curl2A(x,t) in the absence of any current. If there is also no charge then A(x,t) satisfies the side condition divA˙(x,t)=0 and this condition must be imposed, at least at time zero, to ensure that there is no charge. But for the Maxwell-Poisson equation (3.8) the side condition,(3.14)diva′(s)=0,for alls≥0, follows automatically, even without any further condition on A (such as, e.g., divA=0). Indeed the identity curlgrad=0 shows that the self-adjoint operator curl is zero on the subspace of L2(R3;R3) consisting of longitudinal potentials. By the spectral theorem, |curl| is also zero on this subspace. That is, |curl|grad=0 on L2(R3) (scalar functions). The adjoint of this identity is(3.15)div|curl|=0onL2(R3;R3). Apply this identity to (3.11) to find (3.14). (Another short proof of (3.15) follows from Fourier transformation since Lemma 2.2 shows that |k|−1|Ck| is just the projection of C3 onto k⊥.)In the non-abelian case the side condition on the Euclidean side is automatic also. This will be shown in Corollary 5.11, using, of necessity, a different proof.A need not be in Coulomb gauge in any of the preceding identities in this subsection. On the other hand, for A in Coulomb gauge, (3.12) and Definition 2.11 show that(3.16)P(A)=(|C|A,A)2=‖A‖C2ifdivA=0. In particular if A is in Coulomb gauge then P(A)<∞ if and only if A∈H1/2(R3).Remark 3.6A general gauge potential A can be decomposed into its longitudinal part and transverse part:(3.17)A=Along+Atrans,Along=dλ,divAtrans=0. Its Poisson extension is then given by(3.18)a(s)=Along+e−s|C|Atrans. Since |C| has zero nullspace among the transverse potentials, the second term goes to zero as s→∞. Hence(3.19)lims→∞a(s)=Along. The Maxwell-Poisson equation therefore filters out the transverse part of A and leaves the longitudinal part (i.e. pure gauge part) in the limit. For a non-abelian gauge field the pure gauge part cannot be separated out from the initial potential A. Instead we will use the non-abelian analog of the Maxwell-Poisson equation to separate out a “pure gauge piece” of the initial data by an analog of (3.19).Remark 3.7If A is in Coulomb gauge then (3.16) shows that the first term in the Bargmann-Wigner norm (2.44) is equal to the Poisson action of A. Since the full Bargmann-Wigner norm is then determined by the dual space norm as in (2.44), the Poisson action actually determines the full Lorentz invariant norm on the initial data space for Maxwell's equations.3.2Self-duality and helicity for the classical electromagnetic fieldThe four dimensional Euclidean Hodge star operation ⁎e is given on the four dimensional curvature F defined in (3.2) by(3.20)⁎eF=ds∧⁎da+⁎a′, wherein ⁎ denotes the three dimensional Hodge star operation. Therefore the four dimensional curvature F is self-dual if and only if(3.21)ds∧a′+da=ds∧⁎da+⁎a′(self-duality). That is(3.22)a′=⁎da(self-duality). (Keep in mind that ⁎2=Identity over R3.) Similarly F is anti-self-dual if and only if(3.23)a′=−⁎da(anti-self-duality).It is clear that (3.22) and (3.23) each imply the Maxwell-Poisson equation (3.7).Theorem 3.8Suppose that divA=0 and that A∈H1/2(R3). Let a(s) be the Poisson extension of A. ThenA is in the negative spectral subspace of curl if and only if a(⋅) is self-dual.A is in the positive spectral subspace of curl if and only if a(⋅) is anti- self-dual.ProofWriting C=curl=⁎d we may write (3.22) and (3.23) in the form(3.24)a′=Ca(self-dual)(3.25)a′=−Ca(anti-self-dual). If A lies in the negative spectral subspace of curl then e−s|C|A=esCA and therefore the solution (3.9) reduces to a(s)=esCA. Consequently a′(s)=Ca(s) and therefore a(⋅) is self-dual.Conversely, suppose that a(⋅) is self-dual. Then by (3.24) we have a(s)=esCA and therefore ‖a′(s)‖L2(R3)=‖CesCA‖L2(R3). If A has a non-zero spectral component for C in the spectral subspace [ϵ,∞) with some ϵ>0 then, by the spectral theorem, ‖CesCA‖L2(R3)≥cϵexp(ϵs) for some strictly positive constant c. Therefore a(⋅) does not have finite Poisson action and A is not in H1/2. So A must lie in the spectral subspace (−∞,0] for C if a is self-dual.A similar argument applies to show that the Poisson extension of A is anti-self-dual if and only if A lies in the positive spectral subspace of curl.□The following corollary is the main result of this section. Corollary 3.9Suppose that A is in the electromagnetic configuration space C and a is its Poisson extension. ThenA is in C+ if and only if a is anti-self-dual.A is in C− if and only if a is self-dual. ProofThis just restates Theorem 3.8 in view of Definition 2.11.□4Yang-Mills fieldsBoth the Lorentz invariant norm and helicity for the electromagnetic field can be characterized in terms of either plane waves, the operator curl, or the Maxwell-Poisson semigroup, as we saw in Sections 2 and 3. Since the classical Yang-Mills equations are non-linear, there is no useful gauge invariant representation of these fields in terms of plane waves. In this section we are going to describe the Yang-Mills analog of the Maxwell-Poisson semigroup and then use it to produce both a Riemannian metric on configuration space and a decomposition of configuration space into positive and negative “helicity” submanifolds. Here is an outline of the transition steps from electromagnetism to Yang-Mills.i) The Lorentz compatible norm on the (linear) configuration space of the electromagnetic field will be replaced by a Riemannian metric on the (non-linear) configuration space C of the Yang-Mills field. The Yang-Mills-Poisson equation will be instrumental in this construction (Section 5).ii) The helicity subspaces C± in the linear decomposition (2.38) will be replaced by submanifolds C± determined by (anti-)self-duality of solutions to the Yang-Mills-Poisson equation (Section 6.1).iii) The orthogonal projections onto the helicity subspaces C± in the linear decomposition (2.38) will be replaced by flows along orthogonal vector fields over the manifold C, yielding non-linear analogs of these projections (Section 6.2).iv) The spectral behavior of the operator curl in the electromagnetic helicity spaces will be replaced by a corresponding behavior of the gauge covariant curl operator acting not on the configuration space C but on the tangent spaces T(C±) (Section 6.2).5The configuration space for Yang-Mills fields5.1The Poisson action and configuration spaceDenote by K a compact Lie group contained in the orthogonal or unitary group of a finite dimensional inner product space V. k will denote its Lie algebra, on which we assume given an AdK invariant real inner product 〈⋅,⋅〉k. Let(5.1)a(s):=∑j=13aj(s)dxj,s≥0 be a k valued 1-form on R3 for each (Euclidean) time s≥0. Here each coefficient aj(x,s) lies in k. Denote by(5.2)b(s):=da(s)+a(s)∧a(s) its three dimensional curvature and denote by a′(s) its derivative with respect to s. We may regard a as a 1-form on a half space of R4 in temporal gauge. That is, it has no ds component. Its four dimensional curvature is then given by(5.3)F=ds∧a′(s)+b(s). Since the two summands are mutually orthogonal at each point (x,s) we have(5.4)|F(x,s)|Λ2⊗k2=|a′(x,s)|Λ1⊗k2+|b(x,s)|Λ2⊗k2 in analogy with (3.5). We are interested in the minimization of the functional(5.5)∫0∞(‖a′(s)‖L2(R3)2+‖b(s)‖L2(R3)2)ds subject to an initial condition a(0)=A, where A is a given k valued 1-form on R3. In view of (5.4) this functional of a is also given by the familiar expression (3.3) but with F now given by (5.3).The Euler equation for this minimization problem is the Yang-Mills-Poisson equation(5.6)a″(s)=da(s)⁎b(s),a(0)=A. Here we are using dA to denote the gauge covariant exterior derivative, which is given on any k valued p-form on R3 by dAω=dω+[A∧ω], while da(s)⁎:L2(R3;Λ2⊗k)→L2(R3;Λ1⊗k) denotes the adjoint of da(s) in (5.6). The suggestive notation [u∧v] for k valued forms means to use commutator of the coefficients with wedge product of the differentials. For example [(∑iuidxi)∧(∑jvjdxj)]=∑ij[ui,vj]dxi∧dxj. Later we will also need to use the adjoint of this product, given by 〈[u⌟v],ω〉=〈v,[u∧ω]〉, which must hold for each point in R3 and for all k valued forms ω such that degu+degω=degv. 〈⋅,⋅〉 is the inner product on forms induced by the given inner product on k.Remark 5.1The Yang-Mills-Poisson equation (5.6) is the Euler equation for the minimization of the functional ∫R+4|F(x,s)|2dxds for forms in temporal gauge. In the absence of the assumption that a is in temporal gauge the Euler equation for minimization is the full Yang-Mills equation da(⁎e)F=0, where ⁎e is the Euclidean Hodge star operator and F is the curvature of a. This reduces to (5.6) in temporal gauge because F is then given by (5.3) and da(⁎e)F=−a″+da⁎b in this case. The equation da(⁎e)F=0 is a degenerate elliptic non-linear partial differential equation both in the general case and in the case of temporal gauge. The general case has been investigated for a compact four manifold with boundary in ground breaking work by A. Marini and T. Isobe [31–33,41–46]. For a bounded open set in R4 they have proven existence and non-uniqueness of solutions to the minimization problem if one allows the solution to take on the prescribed boundary values in a weak sense, namely equality only up to local gauge transformations. Our half space problem can be conformally transformed into their setting. We need a stronger notion of attainment of the boundary value at s=0, however, namely actual equality, but we can allow a singularity at the point of their boundary corresponding to s=∞. We also need to impose the temporal gauge. Thus at the present time it seems not inconsistent with their work to assume existence and uniqueness for solutions to the Yang-Mills-Poisson equation (5.6) under some appropriate conditions on A. Moreover R. Maitra, in her 2007 Ph.D. thesis, [40], established the existence of solutions for the full Yang-Mills equations in a half space without conformal transformation to a bounded region. The temporal gauge was not of interest in her program, which is aimed at finding a good approximation to the ground state wave function for the quantized Yang-Mills theory. In a recent work, [54], Maitra, Marini and Moncrief elaborated on her direct half-space proof, allowing initial data in H1,loc(R3). Imposition of the temporal gauge would interfere with their methods, which rely crucially on use of K. Uhlenbeck's good gauge theorem in balls, [67]. Nevertheless a proof of existence of a minimum for forms in temporal gauge might be deducible from their results by a simple gauge transformation similar to that used in Example 5.5, after reducing their countably many charts to one global chart with the use of simple connectedness of the half-space. In any case, a proof of existence in temporal gauge for H1/2 initial data will very likely require a direct construction over the half space.11The author thanks the referee for pointing out the paper [54] and its applicability to our problem.We will assume throughout the remainder of this paper that for the potentials A of interest to us existence and uniqueness holds for the boundary value problem (5.6) with finite value of the action functional in (5.5). In Example 5.5 we will see how instantons give rise to such solutions.Notation 5.2Poisson actionDefine(5.7)P(A)=∫0∞(‖a′(s)‖L2(R3)2+‖b(s)‖L2(R3)2)ds, wherein a is the solution to the boundary value problem (5.6). We will refer to a as the Poisson extension of A and to P(A) as the Poisson action of A.A gauge function g:R3→K acts on a potential A via the action A↦Ag:=g−1Ag+g−1dg. The Poisson action is a gauge invariant function of A. That is,(5.8)P(Ag)=P(A). To see this, suppose that a is a solution to the Yang-Mills-Poisson equation (5.6). Then the function(5.9)ag(s):=g−1a(s)g+g−1dg is again a solution to (5.6), as we can see from the identities (d/ds)nag(s)=g−1(d/ds)na(s)g, n=1,2 and dag(s)⁎bg(s)=g−1(da(s)⁎b(s))g. Of course ag(0)=Ag. And since bg(s)=g−1b(s)g we see also that the functional in (5.7) has the same value for ag. This proves (5.8).As in the electromagnetic case P(A) is zero on pure gauges (cf. (3.13)). Indeed, if A is a pure gauge, say A=g−1dg, then its Poisson extension is given by a(s)=g−1dg for all s≥0 since both a′(s)=0 and b(s)=0 for this function, which shows that (5.6) holds and also that(5.10)P(g−1dg)=0.We wish to use the function P(A) to construct the relativistically correct configuration space as we did in the electromagnetic case. Nominally, the configuration space is given by C=A/G for some appropriate set A of gauge potentials on R3 and corresponding gauge group G. But (5.10) shows that the restriction P(A)<∞ provides no control over A in some directions (the longitudinal directions). Defining A simply by the condition P(A)<∞ would lead therefore to an unsatisfactory candidate for A, a circumstance which we avoided in the electromagnetic case by imposing the Coulomb gauge on A. In our non-abelian case we want to avoid such a gauge choice because of the ever lurking Gribov ambiguity. We will instead put an indirect size restriction on the gauge potentials allowed into our space A by making use of the filtering action of the Yang-Mills-Poisson equation.Let‖ω‖H1/2=‖(−Δ)1/4ω‖L2(R3) for any k valued 1-form on R3. Here we are using the Laplacian −Δ:=d⁎d+dd⁎ on k valued 1-forms over R3. We have chosen the Sobolev index 1/2 because in the electromagnetic case the size restriction P(A)<∞ gives exactly H1/2 gauge potentials when the Coulomb gauge is imposed, as we saw in (3.12). We are going to impose an H1/2 restriction on the “filtered longitudinal part” of A as follows.Suppose that a(s) is a solution to (5.6) and that(5.11)a∞:=lims→∞a(s) exists (for example in the sense of L2(R3)loc). We saw in (3.19) that a∞ exists in the electromagnetic case and is precisely the longitudinal part (i.e. the pure gauge part) of the initial potential A.Notation 5.3R, A and CLet(5.12)R={solutions a to the differential equation in (5.6) such thatP(a(0))+‖a∞‖H1/22<∞}. The gauge potentials of interest to us will be the set of initial values of such solutions:(5.13)A={a(0):a∈R}. We will say that a gauge potential A on R3 is in the soft Coulomb gauge if it lies in A, and in the asymptotic Coulomb gauge if in addition its Poisson extension satisfies a∞=0. The examples we construct from instantons below will be in the asymptotic Coulomb gauge.Having now chosen the space A in a way that parrots the electromagnetic case, the gauge group G appropriate for this choice must be chosen so as to preserve A as a set and also to be isometric for the as yet to be determined Riemannian metric on A. If a is the Poisson extension of A then ag is the Poisson extension of Ag as we have seen in the proof of (5.8). Since lims→∞ag(s)=(a∞)g we needa∞g∈H1/2whenevera∞∈H1/2 in order for A to be a gauge invariant set under the gauge function g. In view of (5.9) we therefore need g−1dg to be itself in H1/2. With this as motivation we define(5.14)G={gauge functions g:R3→K such that g−1dg is in H1/2and g(x)→I as x→∞}. The second condition reflects the fact that we are interested in charge zero. The relation between the behavior at infinity of a gauge function on the one hand and total charge on the other results from the well known relation between gauge invariance and conservation of charge. This will be elaborated on in a future work.It happens that G is a complete topological group in its natural metric under pointwise multiplication. It is also the critical gauge group for three spatial dimensions in the sense that for any ϵ>0, the set of gauge functions for which g−1dg lies in the (locally) larger space H1/2−ϵ is no longer a group: It is not closed under multiplication. See [26] for further properties and proofs. Combining the set A of “H1/2” gauge potentials with the matching group G of “H3/2” gauge functions, we define the configuration space for the non-abelian gauge field to be(5.15)C=A/G. We will make A into a Riemannian manifold with a G invariant metric in the next two sections. To this end we will gather information on the derivatives of P in the next section. This produces, of course, a Riemannian metric on C also.Remark 5.4In addition to the fundamental mathematical question as to the existence and uniqueness of solutions to the Yang-Mills-Poisson equation (5.6), these definitions raise the question as to whether the limit a∞ actually exists for some appropriate class of gauge potentials A. For the Yang-Mills heat equation a similar question has been addressed in many works. See e.g. [28–30], [10], [35], [51] and references therein.5.2Restriction of instantonsExample 5.5Restriction of instantonsFinite action solutions to the Yang-Mills-Poisson equation can be constructed from instantons by gauge transforming an instanton to the Euclidean temporal gauge and then restricting it to a half-space. Take, for example, K=SU(2). The simplest instanton is given byAμ(x)=σμ,νxν/(|x|2+ρ2),x=(x,s)∈R4, where σμ,ν=−σν,μ∈k=su(2) and ρ>0 and constant. Its curvature is given byFμ,ν=−4σμ,νρ2(|x|2+ρ2)2. See e.g. [6] or [4] or [63, Eq. (21.26)] or [68].We want to gauge transform A into the Euclidean temporal gauge, i.e., so that its new fourth component is equal to zero. For a gauge function g:R4→SU(2) we have (Ag)4=g−1A4g+g−1∂sg. So we want to choose g so that A4+(∂sg)g−1=0. We may writeA4(x,s)=us2+a2, where u=∑j=13σ4,jxj∈k and a2=|x|2+ρ2. The solution which is the identity element of SU(2) at s=0 is given by Bitar and Chang [6, Equ. (4.4)] as gbc(x,s)=exp(−(u/a)tan−1(s/a)). For 0≤s<∞, gbc(⋅,s) lies in our critical gauge group G, but gbc(⋅,∞) does not. We need to use the solution which is the identity element at s=∞. It is given byg(x,s)=exp(ua{(π/2)−tan−1(s/a)}),(x,s)∈R4. Then (Ag)4=0 andaj(x,s)≡(Ag)j(x,s)=g(x,s)−1Aj(x,s)g(x,s)+g(x,s)−1∂jg(x,s),j=1,2,3. Since g(x,∞)=I and Aj(x,∞)=0 we see that (Ag)j(x,∞)=0. So Ag is in Euclidean temporal gauge and also in asymptotic Coulomb gauge. Since it is the four dimensional gauge transform of an instanton it satisfies dAg(⁎e)F(Ag)=0, which is Poisson's equation (5.6) for a. Its curvature is square integrable over all of R4 and therefore over the half-space R+4. Hence a has finite action and is in asymptotic coulomb gauge. In particular its initial value, a(0), lies in A.Since, by definition, an instanton has finite action, this construction applies to any instanton as long as the time dependent gauge transformation g(x,s) needed to transform the instanton to temporal gauge can be chosen to be the identity of the group K at s=∞.5.3Derivatives of the Poisson actionNotation 5.6Variational equationIf A(t) is a time dependent family of gauge potentials on R3 and if u:=(d/dt)A(t)|t=0 then u is a k valued 1-form on R3 and measures the variation of A(t) at t=0. For any function f on A the derivative of f at A in the direction u is given, in accordance with the chain rule, by(5.16)(∂uf)(A)=(d/dt)f(A(t))|t=0, where A=A(0).Suppose that for each t the Poisson extension of A(t) is at(s). The Poisson equation at″(s)=dat(s)⁎bt(s) can be differentiated with respect to t at t=0 to find the variational equation associated to the variation u. Define u(s)=(∂/∂t)at(s)|t=0. Then we find for u(s) the variational equation(5.17)u″(s)=da(s)⁎da(s)u(s)+[u(s)⌟b(s)]. Here a(s)≡a0(s) is the Poisson extension of A≡A(0) and u:[0,∞)→L2(R3;Λ1⊗k) is a function satisfying u(0)=u in addition to (5.17). The last term in (5.17) has been defined in the paragraph after Eq. (5.6). In the following we will always assume that for a variation u of interest to us there is a unique solution to the variational equation (5.17) with initial value u and such that u(s)→0 as s→∞. We will refer to u(s) as the Poisson extension of u along a.We can compute the derivative of the Poisson action as follows.Lemma 5.7Let u be a variation of a point A in A, i.e. u is a k valued 1-form on R3. If a(s) is the Poisson extension of A then(5.18)∂uP(A)=−2(u,a′(0))L2(R3;Λ1⊗k). ProofDenote by u(s) the Poisson extension of u. Suppose that A(t) is a curve with A(0)=A and (d/dt)A(t)|t=0=u and that at(s) is the Poisson extension of A(t). Then we have, at t=0,(d/dt)‖at′(s)‖2=2(a′(s),u′(s))and(d/dt)‖bt(s)‖2=2(b(s),da(s)u(s)). Therefore, using the Poisson equation (5.6) in the third line below, we find(5.19)∂uP(A)=2∫0∞((a′(s),u′(s))+(b(s),da(s)u(s)))ds=2∫0∞({(d/ds)(a′(s),u(s))}−(a″(s),u(s))+(da(s)⁎b(s),u(s)))ds=2(u(s),a′(s))|0∞=−2(u(0),a′(0)).□Remark 5.8Longitudinal and transverseThe decomposition of an electromagnetic field into longitudinal and transverse parts is not a gauge invariant decomposition. Nor is there any gauge invariant analog of this decomposition for non-abelian gauge fields. But for the tangent spaces to the manifold A there is a well known gauge invariant decomposition analogous to the Coulomb gauge. The terminology, vertical and horizontal, for the differential analogs of longitudinal and transverse goes back to Ambrose and Singer [2]. This will be reviewed here in our special context to help establish notation.Notation 5.9Vertical and horizontalIf λ:R3→k is a reasonable function then g(t,x):=exp(tλ(x)) defines a curve t→g(t) in the gauge group G. Given a point A∈A the curve t↦Ag(t) is a curve in A. Since, at t=0 one has (d/dt)Ag(t)=[A,λ]+dλ, the tangent to this curve in A at t=0 is given by the 1-form dAλ:=dλ+[A,λ]. The vertical subspace of TA(A) is by definition the set of tangent vectors to these curves at A. They consist therefore of k valued 1-forms dAλ with λ running over the Lie algebra of G. The subspace of L2(R3;Λ1⊗k) orthogonal to the vertical subspace is called the horizontal subspace. Thus a k valued 1-form u on R3 is(5.20) vertical at Aifu=dAλfor some scalar function λ.(5.21) horizontal at AifdA⁎u=0. This is the standard terminology associated to the Coulomb connection for the G bundle A, [65]. In this discussion we have made the usual identification of a tangent vector to A with a k valued 1-form on R3. The corresponding action of a gauge function on such a form is given by u↦ug:=g−1ug.Lemma 5.10If f:A→C is gauge invariant then(5.22)∂vf(A)=0 for any vertical vector v=dAλ. ProofLet g(t,x))=exp(tλ(x)). By assumption f(Ag(t))=f(A). But at t=0, (d/dt)Ag(t)=v. Therefore (∂vf)(A)=0 by the chain rule.□Corollary 5.11If a(⋅) is a solution to the Yang-Mills-Poisson equation of finite action and a(0)=A then a′(0) is horizontal at A. That is,(5.23)dA⁎a′(0)=0. Moreover(5.24)da(s)⁎a′(s)=0∀s≥0. ProofSince P is gauge invariant Lemma 5.10 shows that ∂vP(A)=0 for any vertical vector v, say v=dAλ. But the derivative of P is given by (5.18), from which we can conclude that (dAλ,a′(0))L2=0 for all λ. That is, (λ,dA⁎a′(0))L2=0 for all λ. Hence (5.23) holds. Now for any k valued 2-form ω we have the identity (dA⁎)2ω=[B⌟ω], which follows by duality from the Bianchi identity (dA)2ϕ=[B,ϕ] for scalar functions ϕ. Hence(d/ds)(da(s)⁎a′(s))=da(s)⁎a″(s)+[a′(s)⌟a′(s)]=da(s)⁎da(s)⁎b(s)+0=[b(s)⌟b(s)]=0. Therefore da(s)⁎a′(s) is constant in s. Since, by (5.23), it is zero at s=0 it is identically zero.□Theorem 5.12Second derivatives of the Poisson actionSuppose that A has finite Poisson action and that u and v are k valued 1-forms on R3. Denote by a the Poisson extension of A and by u,v the respective Poisson extensions of u and v along a. Then(5.25)(1/2)∂v∂uP(A)=∫0∞{(u′(s),v′(s))L2+(da(s)u(s),da(s)v(s))L2+([u(s)∧v(s)],b(s))L2}ds and also(5.26)−(1/2)∂v∂uP(A)=(u,v′(0))=(v,u′(0)). In particular,(5.27)(1/2)∂u2P(A)=∫0∞{‖u′(s)‖L22+‖da(s)u(s)‖22+([u(s)∧u(s)],b(s))}ds and also(5.28)∂u2P(A)=−2(u,u′(0))=−(d/ds)‖u(s)‖L22|s=0. ProofDifferentiate (5.18) with respect to A in the direction v to find(5.29)−(1/2)∂v∂uP(A)=(u,v′(0))L2. Using the variational equation (5.17) we get(5.30)(d/ds)(u(s),v′(s))=(u(s),v″(s))+(u′(s),v′(s))=(u(s),da⁎dav(s)+[v(s)⌟b(s)])+(u′(s),v′(s))=(dau(s),dav(s))+(u(s),[v(s)⌟b(s)])+(u′(s),v′(s))=(dau(s),dav(s))+([v(s)∧u(s)],b(s))+(u′(s),v′(s)). Assuming that (u(s),v′(s)) goes to zero as s→∞ we may integrate this identity over [0,∞) to find, using [u∧v]=[v∧u],(5.31)−(u(0),v′(0))=∫0∞{(dau(s),dav(s))+([u(s)∧v(s)],b(s))+(u′(s),v′(s))}ds. (5.25) and (5.26) now follow from (5.29), (5.31) and the symmetry in u and v in (5.31). Set v=u in (5.25) and (5.26) to find (5.27) and (5.28).□Remark 5.13Expansion of P near zeroThe behavior of P(A) for small A is given by(5.32)P(u)=(|C|u,u)L2(R3;Λ1⊗k)+O(u3), where u is a small variation of A at A=0. To see this observe that the Poisson extension of A=0 is a(s)=0. The variational equation (5.17) is therefore u″(s)=d⁎du(s), which is the Maxwell-Poisson equation for the k valued 1-form u(s). The solution, as in the real valued case, (3.9), is u(s)=e−s|C|u, where C is the curl acting on each k component of u. Hence u′(0)=−|C|u. We find therefore, P(0)=0, (∂uP)(0)=−2(u,a′(0))=0 and (1/2)(∂u2P)(0)=(|C|u,u), from which (5.32) follows.Remark 5.14It may be useful to observe that a′(s) is itself a variational field along a, and if u is also a variational field along a then(5.33)(u(0),a″(0))=(u′(0),a′(0)). ProofLet v(s)=a′(s). Thenv″(s)=(d/ds)a″(s)=(d/ds)da⁎b=da⁎daa′(s)+[a′(s)⌟b(s)]=da⁎dav(s)+[v(s)⌟b(s)]. Therefore a′ is a variational field along a. Upon taking v(s)=a′(s) in (5.26) we find (5.33).□5.4The Riemannian metrics on A and CWe want to define a Riemannian metric on A which is gauge invariant, captures a non-linear version of the electromagnetic norm identity (3.16), and, unlike (3.16), does not require the Coulomb gauge. We aim to do this by using the first variation of the Poisson action to define a norm on horizontal vectors and then using an explicitly defined norm on vertical vectors that's commensurate with our choice of gauge group, G.Notation 5.15Riemannian metric on ASuppose that A is in A and that a is its Poisson extension. If u is any k valued 1-form on R3 and u is its Poisson extension along a define(5.34)‖u‖2A=∫0∞(‖u′(s)‖22+‖da(s)u(s)‖22)ds. Suppose that w is a k valued 1-form on R3 which decomposes into horizontal and vertical components:w=u+v,dA⁎u=0,v=dAλ. Define(5.35)‖w‖A2=‖u+v‖A2=‖u‖2A+‖(−ΔA)1/4v‖L2(R3)2, where ΔA=∑j=13(∂jA)2 and ∂jA=∂j+[Aj,⋅]. We define the horizontal subspace at A by(5.36)u∈HAif{dA⁎u=0and‖u‖A<∞. It seems likely that the norm ‖u‖A is non-degenerate on HA and we will assume this in the following. But the integral in (5.34) is meaningful even if u is not horizontal at A. However A‖u‖ is not a norm since it can be degenerate on non-horizontal elements, as will be shown in the next example. For this reason we have added the second term in (5.35). We will see that it is commensurate with our choice of G, defined in (5.14).In the case of electromagnetism, where K=U(1), (3.16) shows that P(A)=‖A‖H1/22 when A is in Coulomb gauge. Identifying, as usual, the tangent space to A with A itself in this linear theory, we see that the norm ‖u‖A in (5.35) reduces to the H1/2 norm of electromagnetism when u is horizontal. One should think of ‖u‖A as an “H1/2”-like norm on HA in the non-abelian case.Example 5.16DegeneracyIf A=0 then, for any k valued function λ on R3, dλ is a vertical vector at A and we have(5.37)A‖dλ‖=0. This is easily seen because the Poisson extension of A:=0 is a(s)≡0. Therefore if we define u(s)=dλ for all s≥0 then u′(s)=u″(s)=0, while d⁎du(s)=d⁎d2λ=0 and b(s)=d2λ=0, from which we see that the variational equation (5.17) is satisfied. Since also u(0)=dλ, u(s) is the Poisson extension of dλ. But then the right hand side of (5.34) is zero.Lemma 5.17Gauge invariance of the metricLet w∈TA(A) and let g∈G. Suppose that w=u+v with u horizontal at A and v vertical at A. Then wg=ug+vg with ug horizontal at Ag and vg vertical at Ag. Moreover(5.38)‖wg‖Ag=‖w‖AProofHorizontal and vertical are preserved under gauge transformations because dAg⁎ug=(dA⁎u)g and (dAλ)g=dAgλg. In view of the definition (5.35) it suffices to show that (5.38) holds for u and v separately.Concerning the horizontal component, let a be the Poisson extension of A and let u be the Poisson extension of u along a. Then ag is the Poisson extension of Ag, as we saw in Section 5.1, and ug(s):=g−1u(s)g is the Poisson extension of ug, as one sees by a similar argument. Since ‖(ug)′(s)‖2=‖g−1u′(s)g‖2=‖u′(s)‖2 and dag(s)ug(s)=g−1da(s)u(s))g, (5.38) follows from (5.34) for the horizontal component of w.Concerning the vertical component, observe that the gauge covariant Laplacian ΔA commutes with gauge transform in the sense that ΔAgvg=g−1(ΔAv)g for any k valued 1-form v. Since Adg−1 is unitary in L2(R3;Λ1⊗k), it follows from the spectral theorem that (−ΔAg)1/4vg=g−1((−ΔA)1/4v)g. Therefore ‖(−ΔAg)1/4vg‖L2=‖(−ΔA)1/4v‖L2.□Corollary 5.18(5.39)|∂uP(A)|≤2P(A)‖u‖A,u∈HA(A)(5.40)|(u,a′(0))|≤P(A)‖u‖A,u∈HA(A). ProofFrom (5.19) and the Schwarz inequality we see that(5.41)|∂uP(A)|≤2P(A)∫0∞(‖u′(s)‖22+‖da(s)u(s)‖22)ds, which is (5.39). (5.40) follows from (5.18).□Note: Both inequalities hold for all u∈TA because ∂vP(A)=0 for vertical v, by Lemma 5.10.Notation 5.19Riemannian metric on CSince G acts isometrically on A it induces a metric on C:=A/G, which makes the projection into a submersion of Riemannian manifolds. We take this induced metric as the Riemannian metric on C.Remark 5.20Kinematic and dynamic normsA point q in configuration space is a G orbit, q:=AG, through a point in A. The tangent space to such an orbit is the set of vertical vectors dAλ at A and the tangent space to C at the orbit q can be identified with the set of horizontal vectors at A. Since the norm defined in (5.35) is gauge invariant in the sense of (5.38), it descends to a norm on Tq(C). But the simple L2 norm ‖u‖L2(R3;Λ1⊗k) on HA is also gauge invariant and also descends to a norm on Tq(C). Unlike the first norm the L2 norm does not depend in any way on the dynamics, which was used e.g. in (5.6). The L2 norm is entirely kinematic and should appropriately be referred to as the kinematic Riemannian metric for configuration space, in contrast to the dynamic Riemannian metric (5.35). We already saw in the electromagnetic case that the norm on phase space (2.15), induced from the dynamic norm on configuration space by the duality set up by the kinematic norm, gives the unique Lorentz invariant norm. Both the kinematic and dynamic norms thereby appear naturally in considerations of phase space. The kinematic metric is needed to construct the Coulomb connection. See e.g. [48,64,65]. Other Sobolev norms have been used in some works to ensure that only irreducible connections need be considered, thereby ensuring in turn that Green functions exist. See e.g. [3,47]. These strong norms enter in a technical way only.5.5Phase spaceThe phase space for the gauge field is T⁎(C), as usual in classical mechanics. The natural Riemannian metric on C to use here is the one defined in Notation 5.19, which gives the Lorentz invariant norm in the electromagnetic case. An electric field is a member of TA⁎(A) for some point A∈A. In the electromagnetic case the space TA⁎(A) can be taken to be independent of A because A is a linear space. The electric field can therefore be, and customarily is, taken to be independent of A. But not in a non-abelian gauge theory.If an electric field E in TA⁎(A) annihilates all vertical vectors at A, that is, 〈dAλ,E〉L2(R3)=0 for all k valued scalar functions λ, then dA⁎E=0. Otherwise one has dA⁎E=−4πρ for some k valued charge distribution ρ. One can see already from this that charge is connected to vertical vectors by some duality and therefore also to the dual space of the Lie algebra of the gauge group. At an informal level this statement just reflects the well recognized fact that conservation of charge is a consequence of gauge invariance. See e.g. [55, pages 210-211]. But the charge will not show up in the structure of T⁎(C) because the projection of E into T⁎(C) depends on evaluation of E on horizontal vectors only. To incorporate charge, from either external sources or additional fields, one must make further use of A and the gauge group G, beyond their roles in forming the configuration space A/G. The incorporation of charge into our phase space framework will be addressed quantitatively in a future work. But in the present paper we are only concerned with helicity and therefore we will always take the charge to be zero.As already noted in Remark 2.13, only the decomposition of configuration space must be specified in order to define helicity, whether in the classical or quantum theory, because the associated decomposition of T⁎(C) is automatically determined by this decomposition in the classical theory and is not needed in the quantum theory. In the remainder of this paper we will be concerned only with decomposition of the non-abelian configuration space C into submanifolds C± analogous to those of the electromagnetic case. We limit further discussion of phase space to the following remarks.Remark 5.21Lorentz invarianceIn the case of electromagnetism we saw how the phase space norm gives the Bargmann-Wigner Lorentz invariant norm (cf. (2.21)). For non-abelian fields the configuration space C is not linear and one must replace what was previously a norm on C⊕C⁎ (namely (2.21)) by a Riemannian metric on T⁎(C). This space automatically inherits a Riemannian metric from the Riemannian metric on C, defined in Notation 5.19. It remains to be seen whether this Riemannian metric is invariant under the action of the Lorentz group. Invariance under spatial rotations and spatial translations is clear. But invariance under time translation is not. It is not at all clear that the Riemannian metric on T⁎(C) needs to be Lorentz invariant in order for the configuration space itself to play its usual role in the quantized theory, where quantum time evolution is not so simply related to the non-linear classical evolution.Remark 5.22History, symplectic form, complex structureCandidates for the phase space associated to linear and non-linear wave equations have been investigated from many different viewpoints. One expects a symplectic structure on the phase space and in some cases one can hope also for a Riemannian structure on this infinite dimensional manifold. In field theory the symplectic structure is an infinite dimensional analog of the canonical 2-form ∑j=1ndpj∧dqj. There is a large literature devoted to the role of the symplectic structure in quantization of the field theory. See e.g. [18–22,52,53,56–58,60,61,69] for samples. The phase space is usually not chosen in a quantitative way, however, as we have done above. But the references [52,53,57,58] do discuss the symplectic form in combination with a Riemannian metric on the phase space. See especially [58] for a discussion of phase space for Yang-Mills fields and a proposed quantitative candidate for it. In the context of the Yang-Mills theory the symplectic form is given by the 2-form,(5.42)ω〈X1,X2〉=∫R3(〈u1(x),v2(x)〉−〈u2(x),v1(x)〉)d3x where Xj:=uj,vj, j=1,2 are two elements of TA,ET⁎(A) and 〈u(x),v(x)〉=〈u(x),v(x)〉Λ1⊗k. If A evolves by the hyperbolic Yang-Mills equation and Xj(t) evolves by the induced flow then it is known that(d/dt)ω〈X1(t),X2(t)〉=0. Moreover ω is gauge invariant, Lorentz invariant and descends to a non-degenerate symplectic form on T⁎(C). See e.g. [56–61] and especially [58] for early discussions of these structures. The 2-form ω is continuous in the uj and vk because they are in dual spaces. But our particular choice of the metric on configuration space gives the Sobolev indices ±1/2 for these spaces, which match fortuitously with the expectation that the electric field should be one derivative less regular than the potential.The complex structure on the linear phase space for electromagnetism, used in the proof of Theorem 2.9, has a natural analog in the non-abelian theory. If u varies A and v varies E then the map j:{u,v}↦{vˆ,−uˆ}, where TA,E⟷ˆTA,E⁎ is the natural isometry, generalizes the electromagnetic complex structure given in the proof of Theorem 2.9.In a series of papers, [12–15], C. Cronström has described a candidate for phase space that, unlike most other treatments, does not discard the fourth component of the four potential.6Decomposition of Yang-Mills configuration space by (anti-)self-duality6.1Definition of C±The four dimensional curvature of the 1-form a on the half space R3×[0,∞) is given by (5.3). Its Euclidean dual is given by ⁎eF=ds∧⁎b(s)+⁎a′(s), where ⁎ is the three dimensional Hodge star operator. As in the discussion in Section 3.2 the conditions for Euclidean self-duality or anti-self-duality, ⁎eF=±F, of the curvature of a(⋅) can be written in terms of a as follows.Definition 6.1A function a:(0,∞)→L2(R3;Λ1⊗k) is self-dual ifa′(s)=⁎b(s),0<s<∞. It is anti-self-dual ifa′(s)=−⁎b(s),0<s<∞.The following lemma restates for temporal gauge a well known fact. Lemma 6.2If a(⋅) is self-dual or anti-self-dual then it satisfies the Yang-Mills-Poisson equation (5.6). ProofIf a′(s)=⁎b(s)ϵ with ϵ=±1 thena″=(d/ds)⁎b(s)ϵ=⁎da(s)a′(s)ϵ=⁎da(s)⁎b(s)ϵ2=da(s)⁎b(s) since ⁎da(s)⁎=da(s)⁎ when acting on 2-forms.□Lemma 6.3Let g:R3→K be a gauge function. If a is (anti-)self-dual then so is ag. ProofSince the three dimensional curvature of ag(s) is g−1b(s)g we have, for ϵ=±1,(d/ds)ag(s)=g−1a′(s)g=g−1⁎b(s)ϵg=⁎g−1b(s)gϵ=⁎Curvag(s)ϵ, where Curv means the three dimensional curvature.□In the following notation we decompose the spaces R, A and C defined in Notation 5.3.Notation 6.4R±, A±, C±Denote by R+ the set of anti-self-dual solutions in R and by R− the set of self-dual solutions in R. R+ and R− are each gauge invariant sets by Lemma 6.3. Let(6.1)A+:={a(0):a∈R+},(6.2)A−:={a(0):a∈R−}. A± are gauge invariant sets because if a is the Poisson extension of A then ag is the Poisson extension of Ag. We define submanifolds of configuration space by(6.3)C+=A+/G,C−=A−/G. The definitions of C± given in Notation 6.4 in the non-abelian case are precisely analogous to those for the electromagnetic field and reduce to them when K=U(1). We saw that in the electromagnetic case these manifolds control helicity in the sense that a solution to Maxwell's equations with initial data in T⁎(C+) has only positive helicity plane waves in its plane wave decomposition. Similarly for T⁎(C−). In the next section and in [25] we will provide further justification for the interpretation of the manifolds C± as characterizers of helicity in the non-abelian case.Remark 6.5In the Riemannian metric on A defined in Section 5.4 the subsets A± are closed in A. Since G acts as isometries on A the quotient spaces C± are also closed submanifolds of C.6.2Flows of the helicity vector fields and positivity of curlAWe will give support in this section for the interpretation of the submanifolds C± as the non-abelian analogs of the electromagnetic helicity subspaces. For non-abelian fields we have used (anti-)self-duality to define the submanifolds C± in Section 6.1. Now we are going to show how this decomposition of configuration space relates to the signature of the covariant curl. We will also show that the orthogonal direct sum (2.38), that holds in electromagnetism, has an analogous product decomposition in the non-abelian case.The transition from the electromagnetic linear space theory to the Yang-Mills non-linear theory will be guided by the following observations, which will gradually shift the paradigm from functional analysis over the electromagnetic Hilbert space C to infinite dimensional differential geometry over the analogous non-abelian configuration space. Denote again by C the operator curl.a. In electromagnetism the positive spectral subspace of C is the null space of the operator |C|−C.b. The function(6.4)h+(A):=−(|C|−C)A can be interpreted as a vector field on the electromagnetic Hilbert space C. As such, it has a flow exp(th+). It will be shown that the flow converges as t→∞ to the orthogonal projection P+ of C onto C+.c. The operator |C|, whose definition as C2 has a functional analytic character, can be replaced in electromagnetism by use of the Maxwell-Poisson semigroup, which has a PDE character. Indeed we see from (3.9) that |C|A=−a′(0), where a(s) is the solution to the Maxwell-Poisson equation with initial value A.d. The second term in the vector field h+(A) in (6.4) is just the magnetic field curlA in its usual representation as a 1-form (which is better written as ⁎B with B=dA). Thus the vector field h+ can be specified in electromagnetism by the expression(6.5)h+(A)=a′(0)+⁎B, thereby bypassing the functional analytic construction |C|=C2.e. (Decomposition of configuration space) In the non-abelian theory we want to avoid making any additional gauge choice, such as the Coulomb gauge. The expression (6.5) defines a vector field on A in the non-abelian case. We will show that it flows exactly onto A+ as t→∞, analogous to the abelian case, thereby producing a non-linear map P+:A→A+. Similarly for A−. Upon quotienting by the gauge group, we arrive at a decomposition C=C+×C−. The one-to-one ness of this map needs to be verified, however.f. (Positivity of the covariant curl.) In electromagnetism one has (curlA,A)L2(R3)≥0 for all A∈C+ by (2.36). In the non-abelian case curlA, the gauge covariant curl, does not act on connection forms A but on tangent vectors to A. We will show that (curlAu,u)L2(R3)≥0 for all A∈A+ and for all tangent vectors u to A+ at A. The form of the last statement reflects the change often needed when replacing a linear space by a non-linear manifold, where one can no longer identify the manifold with its tangent spaces.Definition 6.6For a point A in A denote by B its curvature: B=dA+A∧A and by a(s) the Poisson extension of A. Define(6.6)h+(A)=a′(0)+⁎B(6.7)h−(A)=a′(0)−⁎B. Note that h±(A) are 1-forms and therefore can be identified as tangent vectors to A at A. h± therefore can and will be identified as vector fields on A. Moreover they are both gauge covariant:h±(Ag)=g−1h±(A)g.Theorem 6.7a. The vector fields h±(A) are horizontal at A and mutually orthogonal in L2(R3;Λ1⊗k) for each point A∈A.b. h± is zero exactly on A±.c. The flows expth± generated by the vector fields h± for t≥0 satisfy(6.8)P+(A):=limt→∞(expth+)Alies inA+.(6.9)P−(A):=limt→∞(expth−)Alies inA−.(6.10)P±(A)=AifA∈A±.d. Suppose that the Poisson action is convex at A. That is,(6.11)∂u2P(A)≥0for allu. If A∈A+ and u is tangential to A+ at A then(6.12)(curlAu,u)L2≥0. If A∈A− and u is tangential to A− at A then(6.13)(curlAu,u)L2≤0.The proof depends on the following lemmas. Lemma 6.8(6.14)a.)dA⁎h±(A)=0∀A∈A.(6.15)b.)(h+(A),h−(A))L2(R3;Λ1⊗k)=0∀A∈A.(6.16)c.)h+(A)=0if and only ifA∈A+andh−(A)=0if and only ifA∈A−. ProofFrom (5.23) we see that dA⁎a′(0)=0. Moreover dA⁎(⁎B)=−(⁎dA⁎)⁎B=−⁎dAB=0 by the Bianchi identity. We have used here the identity dA⁎=−⁎dA⁎ when acting on 1-forms. Both terms in h±(A) are therefore horizontal at A. This proves (6.14).To prove (6.15) we will first establish the identity(6.17)‖a′(s)‖22=‖b(s)‖22∀s≥0. The computation(1/2)(d/ds)(‖a′(s)‖22−‖b(s)‖22)=(a″(s),a′(s))−(b′(s),b(s))=(da(s)⁎b(s),a′(s))−(da(s)a′(s),b(s))=(b(s),da(s)a′(s))−(da(s)a′(s),b(s))=0 shows that ‖a′(s)‖22−‖b(s)‖22 is constant in s. Since P(A)<∞ it follows that the constant is zero. This proves (6.17). Using (6.17) we find(h+(A),h−(A))2=(a′(0),a′(0))2−(⁎b(0),⁎b(0))2=‖a′(0)‖22−‖b(0)‖22=0. This proves (6.15).For the proof of (6.16) observe that if A∈A+ and a is its Poisson extension then a′(s)+⁎b(s)=0 for all s≥0 and in particular h+(A)=a′(0)+⁎b(0)=0. Similarly, if A∈A− then h−(A)=0.To prove the converses let α(s)=a′(s)+⁎b(s). Then(d/ds)α(s)=a″(s)+⁎daa′(s)=da⁎b(s)+⁎daa′(s)=⁎da(⁎b(s)+a′(s))=⁎da(s)α(s). This is a linear homogeneous first order differential equation in α(s). Therefore if α(0)=0 then α(s)≡0. So if h+(A)=0 then a′(s)+⁎b(s)=0 for all s≥0. Therefore a is anti-self-dual and A∈A+. Similarly if h−(A)=0 then the function β(s)≡a′(s)−⁎b(s) is zero at s=0 and, since it satisfies the differential equation (d/ds)β(s)=−⁎da(s)β(s), it is identically zero. So a is self-dual and A∈A−.□Lemma 6.9For t≥0, P(exp(th±)A) is a non-increasing function of t for each A∈A and for each of the two flows. ProofIt suffices to compute the derivative of P(exp(th±)A) at t=0 because exp(th±) are semigroups. At t=0 we have, by (5.18),(6.18)(1/2)(d/dt)P(exp(th+)A)|t=0=(1/2)∂uP(A)|u=h+(A)=−(u,a′(0))|u=h+(A)=−(a′(0)+⁎B,a′(0))=−‖a′(0)‖22−(⁎b(0),a′(0)). But from the Schwarz inequality and (6.17) we find|(⁎b(0),a′(0))|≤‖b(0)‖2‖a′(0)‖2≤‖a′(0)‖22. Hence(d/dt)P(exp(th+)A)|t=0≤0. A similar proof applies to exp(th−).□Lemma 6.10Let A∈A.If h+(A)P(A)=0 then h+(A)=0.If h−(A)P(A)=0 then h−(A)=0. ProofThe hypotheses are short for ∂uP(A)|u=h±(A)=0. By (6.18) we have0=(1/2)h+(A)P(A)=−‖a′(0)‖22−(⁎b(0),a′(0)). Therefore−(⁎b(0),a′(0))=‖a′(0)‖22. But ‖−⁎b(0)‖2=‖a′(0)‖2 by (6.17). Therefore −⁎b(0)=a′(0) by saturation of the Schwarz inequality. That is, h+(A)=0. In the case of h− one finds (⁎b(0),a′(0))=‖a′(0)‖22 and therefore ⁎b(0)=a′(0). Hence h−(A)=0.□Proof of Theorem 6.7Items a. and b. of the theorem are proved in Lemma 6.8.Heuristic argument for (6.8): The existence of(6.19)A+:=limt→∞(expth+)A in the L2(R3) sense of convergence is reasonable because P is decreasing on the orbit of the flow by Lemma 6.9, while {A:P(A)≤const.} (modulo gauge transformations) is compact in L2 norm over each bounded set in R3, since P has nominally a Sobolev H1/2 strength (modulo gauge transformations). Assuming then the existence of the limit, P(A+) cannot decrease anymore under the flow. Hence h+(A+)P(A+)=0. Therefore h+(A+)=0 by Lemma 6.10. So A+∈A+ by (6.16). Conversely, if A∈A+ then h+(A)=0 by (6.16). Therefore exp(th+)A=A. So A+=A if A∈A+. A similar argument applies to A−. This proves item c. of the theorem. Of course this ensures that the maps P±:A→A± are surjective.For the proof of (6.12) suppose that A∈A+ and that u is tangential to A+ at A. Since h+ is identically zero on A+ its tangential derivative ∂uh+(A) is zero. That is, by (6.6),(6.20)u′(0)+⁎dAu=0, where u(s) is the Poisson extension of u along a. So curlAu=⁎dAu=−u′(0). From the second derivative formula (5.28) we find ∂u2P(A)=−2(u,u′(0)) for all A and u. In particular(6.21)∂u2P(A)=2(u,curlAu)ifA∈A+and u is tangential toA+. From our convexity assumption on the Poisson action at A it now follows that (6.12) holds. In case A∈A− we find curlAu=u′(0) and therefore (6.13) holds.□Remark 6.11The intersection A+∩A− consists of pure gauges. Indeed if A lies in this intersection then h+(A)=h−(A)=0. Therefore ⁎B=0 and a′(0)=0. Since B=0, A is a pure gauge.6.3The flows in the electromagnetic caseIn the electromagnetic case the vector fields h± are given by(6.22)h±(A)=(−|C|±C)A, as we see from the argument showing equality between (6.4) and (6.5). We can operate in Coulomb gauge in this example. The limits of the flows of h± are given in the following theorem.Theorem 6.12Electromagnetic caseDenote by P± the orthogonal projections of C onto C± respectively. Then(6.23)limt→∞exp(th±)A=P±AforA∈C. ProofThe flow equations for these vector fields are the linear differential equations(d/dt)A(t)=(−|C|±C)A(t), whose solutions with initial value A are correctly given byA(t)=et(−|C|±C)A,t≥0 because −|C|±C≤0 in both cases. But −|C|+C is zero on C+ and is −2|C| on C−. Therefore, writing A=P+A+P−A, we findA(t)=et(−|C|+C)(P+A+P−A)=P+A+e−2t|C|P−A, which converges in C norm to P+A as t→∞ because |C|≥0 and has trivial null space in our Coulomb gauge. This proves (6.23) in the case of h+. The case of h− is similar because −|C|−C is zero on C− and is −2|C| on C+.□7Open questions1. Existence and uniqueness of solutions to the Yang-Mills-Poisson equation (5.6) with H1/2 boundary data needs to be proven. We have assumed the validity of such a result throughout the sections on the Yang-Mills helicity theory. If existence and uniqueness for H1/2 initial data can be proven then the map Ts:A↦a(s) becomes a non-linear semigroup, possibly useful for gauge invariant regularization of the initial potential A. At present, gauge invariant regularization by the Yang-Mills heat equation [9–11,24,26] seems to be potentially easier to use. A variant of the Yang-Mills-Poisson equation (5.6) has been investigated in deep work by Marini and Isobe, cited in Section 5.1, as well as in the works [40] and [54]. See Remark 5.1 for further discussion.2. The soft Coulomb gauge and asymptotic Coulomb gauge were defined in Section 5.1 under the assumption that lims→∞a(s) exists in some useful sense. It needs to be proven that this limit actually exists for H1/2 initial data for the Yang-Mills-Poisson equation and is a pure gauge. Such a limit theorem has been established for the Yang-Mills heat equation in various cases. See Remark 5.4.3. The flows of the vector fields h± have been assumed to exist in Theorem 6.7 for all positive time. This needs to be proven along with the existence of the limits as time goes to infinity. The resulting limit maps P± are gauge invariant and therefore produce a map P+×P−:C→C+×C−. For electromagnetism this gives the linear decomposition C=C+⊕C−. In the non-linear case it remains to be seen whether this map is one-to-one.4. The region of convexity of the Poisson action function P should be understood better. Is there a neighborhood of A=0 on which P is convex? Is there a boundary to this region which somehow reflects the Gribov ambiguity?5. The Lorentz invariance of the natural metric on phase space should be proven or disproven in the non-abelian theory, as already pointed out in Remark 5.21. The first step would be to prove invariance under time translation, just at an informal level for, say, smooth initial data. If such invariance should hold then the metric itself would be a candidate for a useful invariant for proving short and long time existence of solutions to the hyperbolic Yang-Mills equation with H1/2⊕H−1/2 initial data. Existence of solutions to this equation has not yet been proven for these critical initial data over three dimensional space. See e.g. [16,17,37,49,50,59,62,66]. Invariance under Lorentz boosts requires gauge transforming a boosted solution to temporal gauge as part of the boost transformation.References[1]AlfredActorClassical solutions of SU(2) Yang-Mills theoriesRev. Mod. Phys.5131979461525MR 541884Alfred Actor, Classical solutions of SU(2) Yang-Mills theories, Rev. Modern Phys. 51 (1979), no. 3, 461–525. MR 541884[2]W.AmbroseI.M.SingerA theorem on holonomyTrans. Am. Math. Soc.751953428443MR 0063739W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443. MR 0063739[3]M.AsoreyP.K.MitterRegularized, continuum Yang-Mills process and Feynman-Kac functional integralCommun. Math. Phys.80119814358MR 623151M. Asorey and P. K. Mitter, Regularized, continuum Yang-Mills process and Feynman-Kac functional integral, Comm. Math. Phys. 80 (1981), no. 1, 43–58. MR 623151[4]M.F.AtiyahGeometry of Yang-Mills Fields1979Scuola Normale Superiore PisaPisaMR 554924M. F. Atiyah, Geometry of Yang-Mills fields, Scuola Normale Superiore Pisa, Pisa, 1979. MR 554924[5]V.BargmannE.P.WignerGroup theoretical discussion of relativistic wave equationsProc. Natl. Acad. Sci. USA341948211223MR 0024827 (9,553d)V. Bargmann and E. P. Wigner, Group theoretical discussion of relativistic wave equations, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 211–223. MR 0024827 (9,553d)[6]Khalil M.BitarShau JinChangVacuum tunneling of gauge theory in Minkowski spacePhys. Rev. D (3)1721978486497MR 0462356Khalil M. Bitar and Shau Jin Chang, Vacuum tunneling of gauge theory in Minkowski space, Phys. Rev. D (3) 17 (1978), no. 2, 486–497. MR 0462356[7]James D.BjorkenSidney D.DrellRelativistic Quantum Fields1965McGraw-Hill Book Co.New York, Toronto, London, SydneyMR 0187642 (32 #5092)James D. Bjorken and Sidney D. Drell, Relativistic quantum fields, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0187642 (32 #5092)[8]MaxBornEmilWolfPrinciples of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Lightthird revised edition1965Pergamon PressOxfordWith contributions by A. B. Bhatia, P.C. Clemmow, D. Gabor, A.R. Stokes, A.M. Taylor, P.A. Wayman and W.L. Wilcock. MR 0198807 (33 #6961)Max Born and Emil Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, With contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman and W. L. Wilcock. Third revised edition, Pergamon Press, Oxford, 1965. MR 0198807 (33 #6961)[9]NeliaCharalambousLeonardGrossThe Yang-Mills heat semigroup on three-manifolds with boundaryCommun. Math. Phys.31732013727785MR 3009723Nelia Charalambous and Leonard Gross, The Yang-Mills heat semigroup on three-manifolds with boundary, Comm. Math. Phys. 317 (2013), no. 3, 727–785. MR 3009723[10]NeliaCharalambousLeonardGrossNeumann domination for the Yang-Mills heat equationJ. Math. Phys.5672015073505MR 3405967Nelia Charalambous and Leonard Gross, Neumann domination for the Yang-Mills heat equation, J. Math. Phys. 56 (2015), no. 7, 073505, 21. MR 3405967[11]NeliaCharalambousLeonardGrossInitial behavior of solutions to the Yang-Mills heat equationJ. Math. Anal. Appl.45122017873905MR 3624771Nelia Charalambous and Leonard Gross, Initial behavior of solutions to the Yang-Mills heat equation, J. Math. Anal. Appl. 451 (2017), no. 2, 873–905. MR 3624771[12]ChristoferCronströmCanonical quantization of non-abelian gauge theoryCzechoslov. J. Phys.321982388Christofer Cronström, Canonical quantization of non-abelian gauge theory, Czechoslovak Journal of Physics - CZECH J PHYS 32 (1982), 388–388.[13]ChristoferCronströmThe generalization of the Coulomb gauge to Yang-Mills theoryActa Phys. Slovaca4931999337344Christofer Cronström, The generalization of the Coulomb gauge to Yang-Mills theory, Acta Physica Slovaca 49 (1999), no. 3, 337–344.[14]ChristoferCronströmCanonical structure of Yang-Mills theoryActa Phys. Slovaca4951999811822Christofer Cronström, Canonical structure of Yang-Mills theory, Acta Physica Slovaca 49 (1999), no. 5, 811–822.[15]ChristoferCronströmHamiltonian formulation and boundary conditions in Yang-Mills theoryActa Phys. Slovaca5032000369379Christofer Cronström, Hamiltonian formulation and boundary conditions in Yang-Mills theory, Acta Physica Slovaca 50 (2000), no. 3, 369–379.[16]Douglas M.EardleyVincentMoncriefThe global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness propertiesCommun. Math. Phys.8321982171191MR 649158Douglas M. Eardley and Vincent Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171–191. MR 649158[17]Douglas M.EardleyVincentMoncriefThe global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proofCommun. Math. Phys.8321982193212MR 649159Douglas M. Eardley and Vincent Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof, Comm. Math. Phys. 83 (1982), no. 2, 193–212. MR 649159[18]P.L.GarcíaA.Pérez-RendónSymplectic approach to the theory of quantized fields. ICommun. Math. Phys.1319692444MR 0256681P. L. García and A. Pérez-Rendón, Symplectic approach to the theory of quantized fields. I, Comm. Math. Phys. 13 (1969), 24–44. MR 0256681[19]Pedro L.GarcíaGauge algebras, curvature and symplectic structureGroup Theoretical Methods in Physics (Proc. Third Internat. Colloq., Centre Phys. Theor., Marseille, 1974), vol. 11974Centre Nat. Recherche Sci., Centre Phys. Theor.Marseille6371MR 0448418Pedro L. García, Gauge algebras, curvature and symplectic structure, Group theoretical methods in physics (Proc. Third Internat. Colloq., Centre Phys. Theor., Marseille, 1974), Vol. 1, Centre Nat. Recherche Sci., Centre Phys. Theor., Marseille, 1974, pp. 63–71. MR 0448418[20]Pedro L.GarcíaThe Poincaré-Cartan invariant in the calculus of variationsSymposia Mathematica, vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973)1974Academic PressLondon219246MR 0406246Pedro L. García, The Poincaré-Cartan invariant in the calculus of variations, Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, 1974, pp. 219–246. MR 0406246[21]Pedro L.GarcíaReducibility of the symplectic structure of classical fields with gauge-symmetryDifferential Geometrical Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975)Lecture Notes in Math.vol. 5701977SpringerBerlin365376MR 0445542Pedro L. García, Reducibility of the symplectic structure of classical fields with gauge-symmetry, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Springer, Berlin, 1977, pp. 365–376. Lecture Notes in Math., Vol. 570. MR 0445542[22]Pedro L.GarcíaTangent structure of Yang-Mills equations and Hodge theoryDifferential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979)Lecture Notes in Math.vol. 8361980SpringerBerlin, New York292312MR 607703Pedro L. García, Tangent structure of Yang-Mills equations and Hodge theory, Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., vol. 836, Springer, Berlin-New York, 1980, pp. 292–312. MR 607703[23]LeonardGrossNorm invariance of mass-zero equations under the conformal groupJ. Math. Phys.51964687695MR 0164632Leonard Gross, Norm invariance of mass-zero equations under the conformal group, J. Mathematical Phys. 5 (1964), 687–695. MR 0164632[24]LeonardGrossStability for the Yang-Mills heat equationhttp://arxiv.org/abs/1711.00114201799 ppLeonard Gross, Stability for the Yang-Mills heat equation, http://arxiv.org/abs/1711.00114, (2017), 99 pages.[25]Leonard Gross, Helicity in the Schrödinger representation of quantized gauge fields, 2019, in preparation, 35 pp.[26]LeonardGrossThe Yang-Mills heat equation with finite actionMem. Am. Math. Soc.2019in press, 142 pp.http://arxiv.org/abs/1606.04151Leonard Gross, The Yang-Mills heat equation with finite action, Memoirs of the AMS, to appear (2019), 142 pages, http://arxiv.org/abs/1606.04151.[27]WernerHeisenbergWolfgangPauliZur Quantendynamic der WellenfelderZ. Phys.5611929161Werner Heisenberg and Wolfgang Pauli, Zur Quantendynamic der Wellenfelder, Zeits. f. Phys. 56 (1929), no. 1, 1 – 61.[28]Min-ChunHongGangTianAsymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connectionsMath. Ann.33032004441472MR MR2099188 (2006h:53063)Min-Chun Hong and Gang Tian, Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections, Math. Ann. 330 (2004), no. 3, 441–472. MR MR2099188 (2006h:53063)[29]Min-ChunHongGangTianGlobal existence of the m-equivariant Yang-Mills flow in four dimensional spacesCommun. Anal. Geom.121–22004183211MR MR2074876 (2005e:53103)Min-Chun Hong and Gang Tian, Global existence of the m-equivariant Yang-Mills flow in four dimensional spaces, Comm. Anal. Geom. 12 (2004), no. 1-2, 183–211. MR MR2074876 (2005e:53103)[30]Min-ChunHongGangTianHaoYinThe Yang-Mills α-flow in vector bundles over four manifolds and its applicationsComment. Math. Helv.901201575120MR 3317334Min-Chun Hong, Gang Tian, and Hao Yin, The Yang-Mills α-flow in vector bundles over four manifolds and its applications, Comment. Math. Helv. 90 (2015), no. 1, 75–120. MR 3317334[31]TakeshiIsobeAntonellaMariniOn topologically distinct solutions of the Dirichlet problem for Yang-Mills connectionsCalc. Var. Partial Differ. Equ.541997345358MR MR1450715 (99f:58045)Takeshi Isobe and Antonella Marini, On topologically distinct solutions of the Dirichlet problem for Yang-Mills connections, Calc. Var. Partial Differential Equations 5 (1997), no. 4, 345–358. MR MR1450715 (99f:58045)[32]TakeshiIsobeAntonellaMariniSmall coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. IJ. Math. Phys.5362012063706MR 2977701Takeshi Isobe and Antonella Marini, Small coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. I, J. Math. Phys. 53 (2012), no. 6, 063706, 39. MR 2977701[33]TakeshiIsobeAntonellaMariniSmall coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. IIJ. Math. Phys.5362012063707MR 2977702Takeshi Isobe and Antonella Marini, Small coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. II, J. Math. Phys. 53 (2012), no. 6, 063707, 39. MR 2977702[34]John DavidJacksonClassical Electrodynamicsthird ed.1999John Wiley & Sons Inc.New YorkMR MR0436782 (55 #9721)John David Jackson, Classical electrodynamics, third ed., John Wiley & Sons Inc., New York, 1999. MR MR0436782 (55 #9721)[35]AdamJacobThe limit of the Yang-Mills flow on semi-stable bundlesJ. Reine Angew. Math.7092015113MR 3430873Adam Jacob, The limit of the Yang-Mills flow on semi-stable bundles, J. Reine Angew. Math. 709 (2015), 1–13. MR 3430873[36]Hans PlesnerJakobsenMicheleVergneWave and Dirac operators, and representations of the conformal groupJ. Funct. Anal.241197752106MR 0439995Hans Plesner Jakobsen and Michele Vergne, Wave and Dirac operators, and representations of the conformal group, J. Functional Analysis 24 (1977), no. 1, 52–106. MR 0439995[37]S.KlainermanM.MachedonFinite energy solutions of the Yang-Mills equations in R3+1Ann. Math. (2)1421199539119MR MR1338675 (96i:58167)S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in R3+1, Ann. of Math. (2) 142 (1995), no. 1, 39–119. MR MR1338675 (96i:58167)[38]L.D.LandauR.PeierlsQuantum electrodynamics in configuration spaceZ. Phys.621930188199Trans. in ter Haar (1965) pp. 19–30L. D. Landau and R. Peierls, Quantum electrodynamics in configuration space, Zeits. f. Phys. 62 (1930), 188–199 (Trans. in ter Haar (1965) pp. 19–30.[39]ElliotLeaderCédricLorcéThe angular momentum controversy: What's it all about and does it matter?Phys. Rep.54132014163248Elliot Leader and Cédric Lorcé, The angular momentum controversy: What's it all about and does it matter? Physics Reports 541 (2014), no. 3, 163–248.[40]RachelLash MaitraMathematically Rigorous Quantum Field Theories with a Nonlinear Normal Ordering of the Hamiltonian OperatorProQuest LLC, PhD thesis2007Yale UniversityAnn Arbor, MIMR 2711186Rachel Lash Maitra, Mathematically rigorous quantum field theories with a nonlinear normal ordering of the Hamiltonian operator, ProQuest LLC, Ann Arbor, MI, 2007, Thesis (Ph.D.)–Yale University. MR 2711186[41]AntonellaMariniBoundary value problems for Yang-Mills connectionsC. R. Acad. Sci., Sér. 1 Math.31271991503508MR MR1099681 (92c:58020)Antonella Marini, Boundary value problems for Yang-Mills connections, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 7, 503–508. MR MR1099681 (92c:58020)[42]AntonellaMariniDirichlet and Neumann boundary value problems for Yang-Mills connectionsCommun. Pure Appl. Math.458199210151050MR MR1168118 (93k:58059)Antonella Marini, Dirichlet and Neumann boundary value problems for Yang-Mills connections, Comm. Pure Appl. Math. 45 (1992), no. 8, 1015–1050. MR MR1168118 (93k:58059)[43]AntonellaMariniElliptic boundary value problems for connections: a non-linear Hodge theoryWorkshop on the Geometry and Topology of Gauge FieldsCampinas, 1991Mat. Contemp.vol. 21992195205MR MR1303162 (95k:58162)Antonella Marini, Elliptic boundary value problems for connections: a non-linear Hodge theory, Mat. Contemp. 2 (1992), 195–205, Workshop on the Geometry and Topology of Gauge Fields (Campinas, 1991). MR MR1303162 (95k:58162)[44]AntonellaMariniA topological method for finding non-absolute minima for the Yang-Mills functional with Dirichlet dataGeometry, Topology and PhysicsCampinas, 19961997de GruyterBerlin191199MR MR1605228 (99k:58042)Antonella Marini, A topological method for finding non-absolute minima for the Yang-Mills functional with Dirichlet data, Geometry, topology and physics (Campinas, 1996), de Gruyter, Berlin, 1997, pp. 191–199. MR MR1605228 (99k:58042)[45]AntonellaMariniThe generalized Neumann problem for Yang-Mills connectionsCommun. Partial Differ. Equ.243–41999665681MR MR1683053 (2000c:58025)Antonella Marini, The generalized Neumann problem for Yang-Mills connections, Comm. Partial Differential Equations 24 (1999), no. 3-4, 665–681. MR MR1683053 (2000c:58025)[46]AntonellaMariniRegularity theory for the generalized Neumann problem for Yang-Mills connections—non-trivial examples in dimensions 3 and 4Math. Ann.31712000173193MR MR1760673 (2001i:58020)Antonella Marini, Regularity theory for the generalized Neumann problem for Yang-Mills connections—non-trivial examples in dimensions 3 and 4, Math. Ann. 317 (2000), no. 1, 173–193. MR MR1760673 (2001i:58020)[47]P.K.MitterC.-M.VialletOn the bundle of connections and the gauge orbit manifold in Yang-Mills theoryCommun. Math. Phys.7941981457472MR 623962 (83f:81056)P. K. Mitter and C.-M. Viallet, On the bundle of connections and the gauge orbit manifold in Yang-Mills theory, Comm. Math. Phys. 79 (1981), no. 4, 457–472. MR 623962 (83f:81056)[48]M.S.NarasimhanT.R.RamadasGeometry of SU(2) gauge fieldsCommun. Math. Phys.6721979121136MR MR539547 (84k:58050)M. S. Narasimhan and T. R. Ramadas, Geometry of SU(2) gauge fields, Comm. Math. Phys. 67 (1979), no. 2, 121–136. MR MR539547 (84k:58050)[49]Sung-JinOhGauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in H1J. Hyperbolic Differ. Equ.11120141108MR 3190112Sung-Jin Oh, Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in H1, J. Hyperbolic Differ. Equ. 11 (2014), no. 1, 1–108. MR 3190112[50]Sung-JinOhFinite energy global well-posedness of the Yang-Mills equations on R1+3: an approach using the Yang-Mills heat flowDuke Math. J.1649201516691732MR 3357182Sung-Jin Oh, Finite energy global well-posedness of the Yang-Mills equations on R1+3: an approach using the Yang-Mills heat flow, Duke Math. J. 164 (2015), no. 9, 1669–1732. MR 3357182[51]Sung-JinOhDanielTataruThe Yang-Mills heat flow and the caloric gaugehttp://arxiv.org/abs/1709.085992017Sung-Jin Oh and Daniel Tataru, The Yang-Mills heat flow and the caloric gauge, (2017), http://arxiv.org/abs/1709.08599.[52]S.M.PaneitzI.E.SegalQuantization of wave equations and Hermitian structures in partial differential varietiesProc. Natl. Acad. Sci. USA7712198069436947part 1. MR 603063S. M. Paneitz and I. E. Segal, Quantization of wave equations and Hermitian structures in partial differential varieties, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 12, part 1, 6943–6947. MR 603063[53]Stephen M.PaneitzEssential unitarization of symplectics and applications to field quantizationJ. Funct. Anal.4831982310359MR 678176Stephen M. Paneitz, Essential unitarization of symplectics and applications to field quantization, J. Funct. Anal. 48 (1982), no. 3, 310–359. MR 678176[54]AntonellaMariniRachelMaitraVincentMoncriefA euclidean signature semi-classical programhttp://arxiv.org/abs/1901.02380201974 ppAntonella Marini, Rachel Maitra and Vincent Moncrief, A euclidean signature semi-classical program, (2019), 74 pages, http://arxiv.org/abs/1901.02380.[55]Silvan S.SchweberAn Introduction to Relativistic Quantum Field Theory1961Row, Peterson and CompanyEvanston, Ill., Elmsford, N.Y.Foreword by Hans A. Bethe. MR 0127796Silvan S. Schweber, An introduction to relativistic quantum field theory, Foreword by Hans A. Bethe, Row, Peterson and Company, Evanston, Ill.-Elmsford, N.Y., 1961. MR 0127796[56]I.E.SegalQuantization of nonlinear systemsJ. Math. Phys.11960468488MR 0135093I. E. Segal, Quantization of nonlinear systems, J. Mathematical Phys. 1 (1960), 468–488. MR 0135093[57]I.E.SegalStability theory and quantizationDifferential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979)Lecture Notes in Math.vol. 8361980SpringerBerlin, New York375382MR 607709I. E. Segal, Stability theory and quantization, Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., vol. 836, Springer, Berlin-New York, 1980, pp. 375–382. MR 607709[58]I.E.SegalThe phase space for the Yang-Mills equationsDifferential Geometric Methods in Mathematical Physics (Proc. Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978)Lecture Notes in Phys.vol. 1391981SpringerBerlin, New York101109MR 612994I. E. Segal, The phase space for the Yang-Mills equations, Differential geometric methods in mathematical physics (Proc. Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978), Lecture Notes in Phys., vol. 139, Springer, Berlin-New York, 1981, pp. 101–109. MR 612994[59]IrvingSegalThe Cauchy problem for the Yang-Mills equationsJ. Funct. Anal.3321979175194MR 546505Irving Segal, The Cauchy problem for the Yang-Mills equations, J. Funct. Anal. 33 (1979), no. 2, 175–194. MR 546505[60]IrvingSegalQuantization of symplectic transformationsMathematical Analysis and Applications, Part BAdv. in Math. Suppl. Stud.vol. 71981Academic PressNew York, London749758MR 634267Irving Segal, Quantization of symplectic transformations, Mathematical analysis and applications, Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 749–758. MR 634267[61]Irving E.SegalMathematical problems of relativistic physicsProceedings of the Summer Seminar, Boulder, ColoradoLectures in Applied Mathematicsvol. 19601963American Mathematical SocietyProvidence, R.I.With an appendix by George W. Mackey. MR 0144227Irving E. Segal, Mathematical problems of relativistic physics, With an appendix by George W. Mackey. Lectures in Applied Mathematics (proceedings of the Summer Seminar, Boulder, Colorado, vol. 1960, American Mathematical Society, Providence, R.I., 1963. MR 0144227[62]SigmundSelbergAchenefTesfahunNull structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gaugeJ. Eur. Math. Soc.188201617291752MR 3519539Sigmund Selberg and Achenef Tesfahun, Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 8, 1729–1752. MR 3519539[63]M.ShifmanAdvanced Topics in Quantum Field TheoryA Lecture Course2012Cambridge University PressCambridgeMR 2896444M. Shifman, Advanced topics in quantum field theory, Cambridge University Press, Cambridge, 2012, A lecture course. MR 2896444[64]I.M.SingerSome remarks on the Gribov ambiguityCommun. Math. Phys.6011978712MR MR500248 (80d:53025)I. M. Singer, Some remarks on the Gribov ambiguity, Comm. Math. Phys. 60 (1978), no. 1, 7–12. MR MR500248 (80d:53025)[65]I.M.SingerThe geometry of the orbit space for nonabelian gauge theoriesPhys. Scr.2451981817820MR MR639408 (83a:81055)I. M. Singer, The geometry of the orbit space for nonabelian gauge theories, Phys. Scripta 24 (1981), no. 5, 817–820. MR MR639408 (83a:81055)[66]TerenceTaoLocal well-posedness of the Yang-Mills equation in the temporal gauge below the energy normJ. Differ. Equ.18922003366382MR MR1964470 (2003m:58016)Terence Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Differential Equations 189 (2003), no. 2, 366–382. MR MR1964470 (2003m:58016)[67]Karen K.UhlenbeckThe Chern classes of Sobolev connectionsCommun. Math. Phys.10141985449457MR MR815194 (87f:58028)Karen K. Uhlenbeck, The Chern classes of Sobolev connections, Comm. Math. Phys. 101 (1985), no. 4, 449–457. MR MR815194 (87f:58028)[68]StefanVandorenPetervan NieuwenhuisenLectures on instantonshttp://arxiv.org/abs/0802.1862v12008118 ppStefan Vandoren and Peter van Nieuwenhuisen, Lectures on instantons, hep-th (2008), 118 pages, http://arxiv.org/abs/0802.1862v1.[69]N.M.J.WoodhouseGeometric Quantizationsecond ed.Oxford Mathematical Monographs1992The Clarendon Press, Oxford University PressNew YorkOxford Science Publications. MR 1183739N. M. J. Woodhouse, Geometric quantization, second ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992, Oxford Science Publications. MR 1183739