Kerner equation for motion in a non-Abelian gauge field

,

Two years after Kerner's pioneering paper Wong [2], who ignored all about Kerner' work and used a different, field-theoretical framework, completed the Kerner equations (II.17) below which describe motion in ordinary space-time, with one for the dynamics of the isospin, eqn.(II.18).
As physical illustration, we derive conserved quantities for Wu-Yang monopoles [40] and for diatomic molecules [41,42].This review celebrates the 80th birthday of Richard Kerner by recounting the fascinating story of isospin-carrying particles initiated by him when his given name was still "Ryszard".

II. GAUGE THEORY AND THE KALUZA-KLEIN FRAMEWORK A. Yang-Mills theory
The concept of isotopic spin (in short: isospin) was introduced by Heisenberg in 1932 [43], who argued that a proton and a neutron should be viewed as two different states of the same particle, related by an "internal" SU(2) rotation 2 .
Let us recall that electrodynamics is an Abelian gauge theory: it is described by a real 1-form A = A µ dx µ called the vector potential which is however determined only up to a 1 To study the Non-Abelian Aharonov-Bohm effect was suggested to one of us (PAH) in the early eighties by Tai Tsun Wu, who also insisted that we should study the original paper of Yang and Mills [27].We are grateful for his advices and would like to congratulate also him on his 90th birthday. 2The fascinating story of gauge theory is recounted by O'Raifeartaigh [44].
gauge transformation, where g(x µ ) is an U(1)-valued function on space-time.
Twenty years later, Yang and Mills (YM) generalized Maxwell's theory to non-Abelian fields which take their values in the Lie algebra G = su(2) and can thus be acted upon by G = SU(2)-valued gauge transformations [27,45].In detail, YM fields are described by the Yang-Mills potential represented either by a 3-vector A = (A a µ ), a = 1, 2, 3 or alternatively, by antihermitian su(2) matrices, A µ = A a µ 1 2i σ a , where the sigmas are the Pauli matrices The Lie algebra carries a metric given by the trace form, g ab A a B b = −2tr(AB) we denote also by A • B. For su(2), g ab = δ ab .
The fundamental property of Yang-Mills theory is its behavior under an SU(2)-valued gauge transformation [27], where g(x µ ) ∈ SU(2).A particle is coupled to the electromagnetic field A µ by minimal coupling, which amounts to replacing ordinary derivatives by gauge-covariant derivatives [45], where the electric charge was scaled to one.In [1] Kerner argued that in a YM gauge field, this prescription should be replaced by an expression which (i) describes the properties of proton/neutron type "particles with internal YM structure" (ii) couples such a particle to the non-Abelian gauge potential : the rule (II.4) should be by generalized to acting on fields in the fundamental representation.The non-Abelian coupling constant is scaled to unity.
What is the dynamics of such an isospin-carrying particle (also called a particle with internal YM structure) ?Kerner answers the question by considering a non-Abelian generalization of Kaluza-Klein (KK) theory [46,47].

B. Abelian Kaluza-Klein theory
Electromagnetism and gravitation have been unified into a geometrical framework (now called fiber bundle theory) by Kaluza [46] and by Klein [47] about 100 years ago 3 .
It is assumed that the world has four spatial dimensions but one of the them we denote by x 5 has curled up to form a circle so small as to be unobservable.The basic assumption is that the correct vacuum is M 4 × S 1 R , the product of four dimensional Minkowski space with coordinates x µ , µ = 0, 1, 2, 3, with an internal circle of radius R.
Then general relativity in five dimensions contains a local U(1) gauge symmetry arising from the isometry of the hidden fifth dimension.The extra components of the metric tensor constitute the gauge fields and could be identified with the electromagnetic vector potential.
The theory is invariant under general coordinate transformations that are independent of x 5 .In addition to ordinary four dimensional coordinate transformations, we have a U (1) local gauge transformation under which the g µ5 component transforms as a U(1) gauge field, We write the metric with indices A = µ, 5 as, i.e., (II.9) Expressing the 5-dimensional scalar curvature R 5 in 4-dimensional terms, R 5 = R 4 + 1 4 F µν F µν where R 4 is the 4-dimensional curvature, the effective low-energy theory is described by the four-dimensional action where G = G K /2πR is Newton's constant.The internal radius R is determined by the electric charge.The motion is given by a five-dimensional geodesic, The KK space-time possesses a Killing vector, namely which implies that is a constant of the motion identified with the conserved electric charge.The remaining equations of motion then take the form, where Γ µ αβ is the Levi-Civita connection constructed from the four dimensional metric g µν .On the right we recognize the Lorentz force. of electromagnetism.

C. Non-Abelian generalization
Kerner, in his groundbreaking paper [1], proposed to derive the dynamics of an isospincarrying particle in a Yang-Mills (YM) field by generalizing the Abelian KK framework to non-Abelian gauges.His framework was further generalized [4] and applied later to particle motion in a Yang-Mills field by projecting the geodesic motion to 4D space [15].His clue [eqn.#(12) of [1]] is to replace the internal circle U(1) in the 5th dimension by the non-Abelian gauge group, SU(2) and the gauge potential in (II.8) by its non-Abelian counterpart.
The key new ingredient w.r.t. electromagnetism is the isospin, represented by an su (2) matrix, which couples the particle to the YM field introduced in sec.II A, A a µ and F a αβ , respectively.The covariant derivative is (II.16) The su(2)-valued YM potential is implemented on the isospin Q ∈ su(2) by commutation.
In a judicious coordinate system chosen by Kerner [1], the equations of motion for a test particle in the combined gravitational and gauge fields simplify to his eqn.# (34), (II.17) Generalizing the gauge group from U(1) of electromagnetism to the Yang-Mills gauge group SU(2) has a price to pay, though: unlike the electric charge in the electromagnetic theory which is a conserved scalar, the isospin has indeed its own dynamics : it is not a constant but a vector which (as Kerner puts it) "rotates, depending on the external field".
The equations for the motion of the isospin, [where the "dot" is d/ds] were spelt out two years later by Wong [2].In a geometric language, the isospin is parallel transported along the space-time trajectory, x(t) = (x µ ).Written in terms of the covariant derivative (II.16), this equations says that the isospin is covariantly (but not ordinarily) conserved.Eqn.
(II.18) is consistent with Kerner's words, though, and also with what Yang and Mills say in their [27], where they mention "isospin rotation".
One can wonder why did Kerner not spelt out the equations of motion for the isospin explicitly.A real answer can be given only by him, however one can try to guess what he might have had in his mind.One good reason might well have been that considering the isospin as a non-constant non-Abelian analog of the constant electric charge could have appeared too radical and even shocking, and be therefore discarded 4 .
There might exist also other, subtle reasons related to the gauge invariance and the consequent problems of physical interpretation [49][50][51].Another one could come from the experimental side.
Wong's approach [2] is radically different from that of Kerner : instead of generalizing the classical dynamics of a charged particle moving in a curved space, he "dequantizes" the Dirac equation.Balachandran et al. [5,6], studied particles with internal structure which were then recast in a symplectic framework by Sternberg [7][8][9], by Weinstein [10], and by Montgomery [14].Duval [11][12][13] extended Souriau's approach [52] to particles with spin [12] -hitting yet another shocking idea: physicists, referring to Landau-Lifshitz, were firmly convinced that classical spin just does not exist and rejected Souriau's ideas [52] rooted in the representation theory.
Gauge fields with spontaneous symmetry breaking admit finite-energy static solutions with magnetic charge referred to as non-Abelian monopoles [19].For an isospin-carrying particle in the field of a selfdual monopole [20][21][22] Fehér found, moreover, that outside the monopole core, where the SU(2) symmetry is spontaneously broken to U(1), the dynamics of a particle with isospin reduces to that of an electrically charged particle in the field of a Dirac monopole, combined with specific scalar potentials, familiar from the Abelian theory [53,54].

D. Fibre bundles and a symplectic framework
Trautman [3], and Cho [4] reformulated the non-Abelian KK theory in terms of fibre bundles [55] : for gauge group G, the field is described by a Lie algebra-valued connection form α on a principal bundle P with structure group G over space-time, M .The YM potential A in sec.II is the pull-back to M of the connection 1-form by a section M → P of the bundle.A gauge transformation amounts to changing the section and results in (II.3).
Choosing a section yields a local trivialisation P = M × G and the YM connection form is written as, Recall that the Maurer-Cartan form g −1 dg takes its values in Lie algebra G of G. Using fiber bundles for gauge theory was advocated by T. T. Wu and C.N. Yang [24,25,56] in the monopole context5 ; see also [57,58].
A comprehensive KK unification of non-Abelian gauge fields with gravity in principal fibre bundle terms was put forward by Cho in [4], who derived a unified Einstein-Hilbert action in (4+n)-dimensions both in the basis used by Kerner and also in a horizontal-lift basis which diagonalizes the KK metric and generalizes (II.10).
Duval et al [12,13] proposed an alternative, symplectic version "à la Souriau" [52], reminiscent of but different from the Kaluza-Klein approach.Both theories use a higherdimensional, fiber bundle extension of the conventional space-time structure.Below we summarize the main features of the Souriau framework : 1.The system is described by a fiber bundle V over space-time M called an evolution space -Souriau's "espace d'évolution"; 2. The dynamics is discussed in terms of differential forms.The main tool is a 1-form ω on V whose exterior derivative Ω = dω is, in Souriau's language, "presymplectic", i.e., a closed 2-form which has constant rank, dim Ker Ω = const.Then the motions are the projections onto M of the integral submanifolds of the characteristic foliation of Ker Ω. Factoring out Ker Ω yields U, the space of motions (an abstract substitute for phase space -Souriau's "espace des mouvements" [52]).The presymplectic form Ω projects onto U as a symplectic form i.e., one which is closed and has no kernel, as illustrated in FIG. 2.
3. A group S is a symmetry for a system if it acts on the space of motions U by preserving the symplectic structure.

4.
A system is elementary with respect to a symmetry group S if the action of the latter on U is transitive.Souriau's orbit construction [52] applies to an arbitrary symmetry group: the space of motions of an elementary system is, conversely, a (co)adjoint orbit O = g −1 Q 0 g g ∈ S of a basepoint Q 0 chosen in the Lie algebra ∈ S of the symmetry group.O is endowed with its canonical symplectic form, In particular, applying the general the construction to the gauge group, G, endows the orbit in dual of the Lie algebra G with its canonical symplectic form.
5. The symmetry group S w.r.t. which the system is elementary can be viewed itself as evolution space, V = S [59]; S is a principal fiber bundle over its (co)adjoint orbit O. Now we spell out a simplified form of the Souriau-Duval framework in flat space.For further details the reader is advised to consult [12,13].
• A massive free relativistic particle.The Poincaré group (P ) is a fiber bundle over Minkowski spacetime M with the Lorentz group as structure group [12,59].We Eqn.(II.26) describes the geodesic motion in Minkowski spacei.e., the motion of a free relativistic particle with no spin6 .
• The free theory based on the Poincaré group P is readily extended to a (still free) relativistic particle with internal structure: enlarging the evolution space and 1-form, P and ωm , respectively, to where g takes its values in the gauge group G and the basepoint is Q 0 ∈ G (the Poincaré part being understood).
The kernel of Ω in (II.21) implies the free equation (II.26), supplemented by that for the isospin, (II.19), whose properties will be further studied in sec.III.In geometric language, the isospin belongs to the associated bundle P × G O 0 , where O 0 is the (co)adjoint orbit of endowed with the projection of Ω in (II.21).
For G = U(1) the free charged particle is recovered, with Q identified with the constant electric charge.
• minimal coupling to a Yang-Mills field amounts, in bundle language, to generalize (II.27) on P by, with Q 0 ∈ G, which, in view of (II.20), is indeed the geometric form of (II.5).In local coordinates, The 2-form Ω = dω is, by the Cartan structure equations [55], p. 78, where the last term involves, in addition to the Lie bracket, also the wedge product of the differential forms.Its kernel projects to the Kerner-Wong equations (II.17)-(II.18)[13].

III. PHYSICAL MEANING OF ISOSPIN DYNAMICS
Limiting our investigations to flat Minkowski space, the Kerner equations (II.17To what extent is the isospin vector, Q, an analog of the constant electric charge ?We argue that Q = const would be inconsistent with gauge invariance: if we had Q = 0, the rhs of (II.18) would change, under a gauge transformation, as, and there is no reason for the rhs to vanish.The situation improves, though, if the gauge transformation is non-trivially implemented on the isospin 8 , Then the rhs of (II.18) would transform as 7 The equations (III.1)-(II.18)were also studied by refining the field-theoretical arguments of Wong [49].
The classical isospin is the expectation value of the non-Abelian field, Q a = 1 2 ψ † σ a ψ . 8The covariant transformation rule (III.2) is consistent with the geometric status of the isospin viewed as a section of the associated bundle P × G O [13,63].
The first term in the curly bracket would be perfect but the 2nd one would vanish only for g = const.However the terms coming from d(g −1 Q g)/ds cancel the unwanted terms, leaving us with the desired covariant transformation law cf.(III.2), Further insight into isospin dynamics can be gained by assuming, for simplicity, that the curvature of the connection form (in physical terms, the Yang-Mills field) is zero, F = Dα = 0 which is a gauge-independent statement by (II.3), and the space-time motion is free.Do we have also Q = 0 ?The answer is: yes and no.Let us explain.In topologically trivial situations 9 , F = Dα = 0 implies that one can find a gauge where A µ = 0 and then Q = 0 follows obviously from the isospin equation (II.18).This is a gauge-dependent statement, though : We are allowed to apply a gauge transformation by an arbitrary G = SU(2)-valued function g(x µ ) which changes A µ = 0 to a pure gauge A µ = g −1 ∂ µ g -but it rotates also the isospin, (III.2). d g −1 Q g /ds ̸ = 0 in general; the gauge-covariant statement is that the isospin is covariantly conserved, (II.19).
What is then the physical meaning of the isospin vector ?First we note that is gauge invariant, and deriving it implies, using (III.3),that the length |Q| is conserved, The isospin is thus constrained to lie on an adjoint orbit of the gauge group G in its Lie algebra G -in our case, to a sphere, It is this fact that is behind the Souriau-type construction of isospin-extended models [12,13,63].
Which components of Q do have a gauge invariant physical meaning ?-the question leads to the so-called "color problem" [60][61][62].The point is the subtle difference between gauge transformations and internal symmetries [50,51].
In physical terms: can we implement an element of the gauge group on the physical fields ?And if we can, will it be a symmetry in the usual sense [65] ?In bundle language, a gauge transformation acts on the fibers from the right [55], -while a symmetry should act 9 The topologically non-trivial case is studied in [26,63,64].
from the left [50,51].Can we transfer the right-action to a left action ?In geometric terms, "implementable" means that the G = SU(2) bundle P should be reducible, and "symmetry" requires that the connection form α in (II.20) which represents the YM potential should also be reducible to the reduced bundle.
When the underlying topology is non-trivial (as non-Abelian monopoles [19]), there can be an obstruction : global color can not be defined", as it is put in refs.[60][61][62].Another physical instance is provided by the Non-Abelian Aharonov-Bohm effect [24], for which there is no obstruction but there is an ambiguity of how it should be implemented [28].

IV. CONSERVATION LAWS WITH ISOSPIN A. van Holten's covariant framework
The Hamiltonian of a point particle of unit mass carrying isospin ⃗ Q = (Q a ) which moves in a static YM field is, where we scaled the coupling constant again equal to one.Defining the covariant Poisson bracket as [68], where the f abc are the structure constants of the Lie algebra, and the covariant phase-space derivative is, The nonzero Poisson brackets are, Then the Hamilton equations with allow us to recover the flat-space Kerner-Wong equations, equivalent to (III.1) and (II.18).
Following van Holten [68][69][70][71], constants of the motion can be sought for by expanding into powers of the covariant momentum, Skipping the Abelian case, we move directly to the non-Abelian one.Requiring q to Poisson-commute with the Hamiltonian then yields a series of constraints, eqn.# (70) in [68].
The expansion (IV.8) can be truncated at a finite order when the covariant Killing equation is satisfied at some order n.When we have a Killing tensor, for all p ≥ n, and find a constant of the motion of the polynomial form, . .π i k (IV.10) [68].Referring to the literature for details [68][69][70][71] we mention that in the Abelian theory ⃗ Q is just a constant identified with the electric charge.
The van Holten algorithm can be generalized by adding a static scalar potential which may depend also on the isospin.The Hamiltonian (IV.1) then becomes Comparison with (IV.6) then shows that (IV.12a) picks up a covariant force term.Note also that when V does depend on ⃗ Q the isospin is not more parellel transported.
Generalizing (IV.7) to isospin-dependent coefficients, the constraints (IV.8) are also generalised [69], New, gradient-in-V terms thus arise even when the potential does not depend on the isospin, V = V (r).These terms play a rôle for self-dual Wu-Yang monopoles [40], and for diatoms [41], as it will be seen in subsections IV B and IV C, respectively.
1.When C i (r) is a Killing vector then we have p = 2 and the expansion can be reduced to a linear expression, allowing us to recover the conserved momentum and angular momentum [68].Focusing our attention at the latter, we choose a unit vector ⃗ n; then is a Killing vector for rotations around ⃗ n and thus generates the conserved angular momentum, ⃗ J. van Holten's recipe can be applied also to a Dirac monopole of charge q = eg, recovering the angular momentum vector which includes the celebrated radial "spin from isospin" term [69,72].
2. Similarly, choosing again a unit vector ⃗ n, is a Killing tensor of order 2 which generates the well-known Runge-Lenz vector of planetary motion, [68][69][70][71], More generally, the framework applies also to the so-called "MIC-Zwanziger" system [53,54], which combines a Dirac monopole of charge q with an arbitrary r −1 Newtonian and a fine-tuned inverse-square potential, (IV.20) The combined system generalizes the well-known dynamical O(4)/O(3,1) symmetry of planetary motion spanned by the angular momentum and the Runge-Lenz vector, ⃗ J in (IV.17) and ⃗ K, respectively [53,54].The relations then imply that the motion is a conic section, as depicted in FIG. 3.

FIG. 3:
The conservation of the monopole angular momentum ⃗ J implies that a particle moves on a cone, whose axis is ⃗ J.The O(4)/O(3, 1) dynamical symmetry generated by the angular momentum and the Runge-Lenz vector ⃗ K implies in turn that the trajectory lies in the plane perpendicular to J and is therefore a conic section.
Spin can also be considered [23].
3. The van Holten algorithm applies also to quantum dots, Hénon-Heiles and Holt systems, with Killing tensors whose rank ranges from one to six are studied in [70,71].

B. Motion in the Wu-Yang monopole field
The Wu-Yang monopole [40] is given by the non-Abelian gauge potential with a "hedgehog" magnetic field, The terminology is justified by presenting the field strength as The projection of the Wu-Yang magnetic field onto the "hedgehog" direction is thus which shows that the Wu-Yang magnetic field is that of a Dirac monopole of unit charge, embedded into isospace.The remarkable feature of this expression is that the external and internal coordinates are correlated.
Let us consider an isospin-carrying particle moving in a Wu-Yang monopole field augmented with a rotationally invariant scalar potential V (r), and inquire about conserved quantities.
• A most important observation says that, for an arbitrary radial potential V (r), we can choose C = ⃗ Q • r which is covariantly constant, and the (IV.14) are satisfied with C i = C ij = ... = 0 .The van Holten algorithm then applies, proving that the projection of the isospin onto the radial direction, is a constant of the motion [68].
The Wu-Yang Ansatz (IV.22) played an important rôle in later developments as it prefigured the finite-energy non-Abelien monopoles [17][18][19].The "hedgehog" is the large-r behavior of the Higgs field, and (IV.26) is identified with the electric charge outside the monopole core.See e.g.[19] or [80] for comprehensive reviews.
• Applied to the Killing vector (IV.16),we get the conserved angular momentum [68], which looks formally identical to the Abelian expression (IV.17).Remember however that q here is not a universal constant but the (conserved) projection of the isospin onto the "hedgehog" direction r, which mixes internal and external coordinates.Thus we have the familiar radial term -but now in the non-Abelian context.
• We now inquire about quantities which are quadratic in the momentum.Inserting (IV.18) into (IV.14),from the 2nd-order equation we find, For which potentials do we get a quadratic conserved quantity ?Referring to [68,69] for details, we just record the answer: where α and β are arbitrary constants.The coefficient of the r −2 term is correlated with the conserved charge q (IV.26) as (IV.20) for MIC-Zwanziger [53,54,78].Collecting our results, is a conserved Runge-Lenz vector for an isospin-carrying particle in the Wu-Yang monopole field combined with the fine-tuned potential V (r) in (IV.29).
The conserved quantities ⃗ J and ⃗ K span an O(4)/O(3, 1) dynamical symmetry which allow us to describe the large-r motion both classically and quantum mechanically [21,22].The trajectories are again conic sections as for MIC-Zwanziger in FIG. 3.
This generalizes the Abelian result to an isospin-carrying particle outside the core of a self-dual non-Abelian monopole [81].This "coincidence" is explained as follows : for large The projection of the isospin onto Φ, q in (IV.26), is thus conserved, and outside the core the motion is that of an electric charge in the MIC-Zwanziger field [21,22,53,54,78].The isospin-dependent dynamical symmetry is analyzed in [81].

C. Diatomic molecules
In Ref. [41] Moody, Shapere and Wilczek have shown that nuclear motion in a diatomic molecule can be described by the effective non-Abelian gauge field, Dropping scalar potential V (r) we return to the Hamiltonian of a spinless particle with non-Abelian structure, (IV.1), Inquiring about conserved quantities, we note first that when κ ̸ = 0, then q is not covariantly conserved in general, implying that q in (IV.26) is not conserved for κ ̸ = 0, unless the isospin is also radial.The bracketed quantity in (IV.35) is indeed the non-alignedwith-the-field piece of the isospin.When the isospin spin and the magnetic field happen to be aligned, then q in (IV.26) is conserved.
Nor is the length of the to-become-charge q is conserved in general, whereas le the length of the isospin, ⃗ Q 2 , is conserved, {H, ⃗ Q 2 } = 0. Thus electric charge nonconservation comes from isospin precession, as in the non-Abelian Aharonov-Bohm effect [24,28,41].For κ = 0 we recover the Wu-Yang case when q is conserved as we have seen in sec.IV B.
The gauge field (IV.33) is rotationally symmetric and an isospin-carrying particle submitted to it has, nevertheless conserved angular momentum [41,42].Its form is, however, somewhat unconventional.
Our starting point is the first-order condition in (IV.14).We consider first V = 0 ; then with F a jk in (IV.33), the equation to be solved is In the Wu-Yang case, κ = 0, we have C = −⃗ n • q r, but for κ ̸ = 0 the to-be electric charge, q, is not conserved.Using (IV.36) allows us to infer [69] that The conserved angular momentum is, therefore, ⃗ J = r × ⃗ π − ⃗ Ψ, (IV.40) consistently with the results in [41,42].Note however that the spin-from-isospin contribution changes, w. making manifest the "spin from isospin term" which is however not aligned with the "hedgehog" magnetic field.Consistently with (IV.35), the non-conservation of q in (IV.26) comes precisely for this non-alignement.
Restoring the potential, we see that, again due to the non-conservation of q, D j V ̸ = 0 in general.The zeroth-order condition ⃗ C • ⃗ DV = 0 in (IV.14) is nevertheless satisfied if V is a radial function which is independent of ⃗ Q, V = V (r), since then ⃗ DV = ⃗ ∇V , which is perpendicular to infinitesimal rotations, ⃗ C. Alternatively, a direct calculation, using the same formulae allow us to confirm that ⃗ J commutes with the Hamiltonian, {J i , H} = 0.
Multiplying (IV.43) by r yields by, once again, ⃗ J • r = −q (IV.44) as in the Wu-Yang case.This is, however, less useful as before, since q is not a constant of the motion anymore so that the angle between ⃗ J and the radius vector, r(t), is not constant either: the motion is not confined to a cone anymore.
Our attempts to find a conserved Runge-Lenz vector for the diatomic system have failed.
In addition to the conceptual works above, we underline that the Kerner-Wong model [1,2] admits important physical applications.

1 .
which satisfy [σ a , σ b ] = 2iϵ abc σ c .In what follows we shall use mainly the matrix-formalism.Space-time indices will be denoted by greek letters, typically µ, ν, . . .etc. Latin characters a, b . . .are used for the internal, isospin indices.The Lie bracket in su(2) is [A, B] a = ϵ a bc A b B c .The field strength of a Yang-Mills field is

FIG. 2 :
FIG. 2: Souriau's framework: the worldline in M is the projection of a characteristic sheet of the 2-form Ω = dω on the evolution space, V. Factoring out the characteristic foliation tangent to ker ω, V projects to the space of motions, U, to which the 2-form dω projects as a symplectic form Ω and correspond to the worldlines in M .

1 
where the 4 × 4 matrix L belongs to the Lorentz subgroup and x = (x µ ) ∈ M .Then Poincaré/Lorentz = M = Minkowski .(II.22)Moreover, we choose the basepoint Q m in the Poincaré Lie algebra, 23) where m = const interpreted as rest mass.Writing the Maurer-Cartan form as g −1 dg = on the Poincaré orbit O m of Q m , ωm = mI µ dx µ ⇒ Ωm = mdI µ ∧ dx µ (II.25)where I µ , a component of the Lorenz matrix, is future pointing and belongs to the unit tangent bundle of M [12, 13, 59].Then the characteristic foliation projects, in a suitable parametrisation, to M onto a curve, which is a solution of ẋµ = I µ , İµ = 0 .(II.26) isospin equation (II.18) 7 .

r,
the gauge field of a self-dual non-Abelian monopole of charge m [19] is of the radially symmetric Wu-Yang form, eqn.(IV.22), completed with a "hedgehog" Higgs field, , (IV.33) respectively, where κ is a real parameter.For κ = 0, (IV.33) is the field of the Wu-Yang monopole[40], (IV.22).For other values of κ, it is a truly non-Abelian configuration (except for κ = ±1, when the field strength vanishes and (IV.33) is a gauge transform of the vacuum).