Asymptotic Symmetries of SU(2) Yang-Mills-Higgs Theory in Hamiltonian Formulation

We investigate the asymptotic symmetry group of a SU(2)-Yang-Mills theory coupled to a Higgs field in the Hamiltonian formulation. This extends previous work on the asymptotic structure of pure electromagnetism by Henneaux and Troessaert, and on electromagnetism coupled to scalar fields and pure Yang-Mills fields by Tanzi and Giulini. We find that there are no obstructions to global electric and magnetic charges, though that is rather subtle in the magnetic case. Again it is the Hamiltionian implementation of boost symmetries that need a careful and technically subtle discussion of fall-off and parity conditions of all fields involved.


Introduction
The study of asymptotic field-structures and their associated symmetry groups and charges for long-ranging field configurations remains one of the most interesting and challenging problems in all kinds of geometric field theories.The Hamiltonian analysis of these structures has been pioneered by Henneaux and Troessaert in their investigations of various field theories, including the pure electromagnetic case [1].They showed how to simultaneously ensure the existence of a Hamiltonian phase space, a proper symplectic two-form, and a Hamiltionian action of the Poincaré group.Their analysis made perfectly clear that one has to expect severe fine-tuning conditions to be imposed on the fields and hence the analytical characterisation of one's phase space in order to meet these conditions.Subsequently it was shown by Tanzi and Giulini in [2] that such conditions prevent globally charged states to exist in pure SU(N )-Yang-Mills theory for N ≥ 2, which is a quite unexpected result.In [3] these authors also showed that no such surprises exist in scalar electromagnetism and the abelian Higgs model.
The present paper can be seen as a direct continuation of the mentioned work on pure Yang-Mills as well as that on electromagnetism coupled to scalar fields.The latter led to the result that there are no obstructions to globally charged states in case of massive fields or those with Higgs-type potentials, whereas the massless case seems to contain irremovable obstructions.Facing this situation, it seems natural to ask what happens if non-abelian gauge-fields are coupled to scalar fields.This is what we shall focus on here in case of SU (2) with a Higgs fields in the adjoint representation.It turns out that this case cannot be understood by simple generalisations of the previous cases, which essentially means that all calculations need to be perfomed again from scratch in order not to miss out on any of the new terms appearing in the couplings arising from the Higgs field (not present in [2]) and the non-abelian self-coupling (not present in [3]).In addition to previous studies, we also attempt to add some mathematical rigour in the justification of boundary terms of the relevant functions on phase space1 , which are essential in a twofold way: they are necessary for the existence of a corresponding Hamiltonian flow, and they also influence the value of the function on those phase-space points corresponding to the long-ranging field configurations.

Notation and conventions
For the Minkowski metric we use the (−, +, +, +) signature convention.Greek indices µ, ν, ... denote spacetime components.Latin indices i, j, ... denote spatial components while ā, b, ... denote angular components, e.g.ā ∈ {θ, ϕ}.Bold symbols like d, L, ∧ denote operators in phase space while the corresponding non-bold ones denote operators on space or spacetime in the ususal fashion.If a function F on phase space vanishes identically after restriction to the constraint surface C, i.e. if F C = 0, we will sometimes simply write F ≈ 0 (see section 3) following Dirac's notation from [4].The notation δ δφ will be used in the following way: with the Euler-Lagrange derivative δ EL2 .We also use the notation δ for other kinds of variations, e.g.δ ǫ for infinitesimal gauge transformations.We make use of the Bachmann-Landau O-and o-notation.For the characterisation of asymptotic fall-off or growth behaviour we shall employ the Bachmann-Landau notation involving the upper case O and lower case o.As we will encounter asymptotic expansions in the radial parameter r with terms proportional to some r −n for some n ∈ N, we can define an operator O r −n which just picks out the term proportional to r −n in that expansion.Likewise we write o(r −N ) and O r −N for a function that falls off faster then r −n ∀n ∈ N and the projection on that fast fall-off part.Finally, we shall use the symbol ⊲ for the formal action of the gauge group on the phase space.

Mathematical setting
In this section we shall essentially follow standard notions of Hamiltonian field, as e.g.generally outlined in [5] and whose notation we shall follow.From that we recall that the mathematical structure of a Hamiltonian system consists of a symplectic manifold, the phase space (P, Ω), together with a class of functions F ∈ Obs ⊂ C ∞ (P), the observables of the system, with a distinguished observable, the Hamiltonian function H which determines the time evolution as the flow of the vector field X H which, in turn, is defined by dH = −i X H Ω. (3.1) Generalizing the equation (3.1) to an observable F , it defines the corresponding Hamiltonian vector field X F .Conversely, given a Hamiltonian vector field X, the corresponding observable is called generator, which in non-gauge theory is determined up to an additive constant.In contrast, in gauge theory, the Hamiltonian is only defined on a submanifold C ⊂ P, called the constraint surface, which is defined to be the set of unique-valued Hamiltonian with corresponding (non-unique) Hamiltonian vector fields being tangential to C3 .The non-uniqueness of X H is given by the degeneracy directions of i * C Ω.The Hamiltonian vector fields that span the degeneracy directions are called infinitesimal proper gauge transformations.The generators of those Hamiltonian vector fields are functions that vanish on C. Physical states are defined to lie in C and two states that are connected by a gauge transformations are the same physical state.
We are considering a Hamiltonian description of a field theory which has infinite degrees of freedom.For the definition of the phase space we need the notion of a infinite dimensional symplectic manifold.A suitable notion is a smooth manifold modelled onto a locally convex vector space.For the class of locally convex vector spaces there still exists a notion of derivatives for functions between those spaces that fulfills usual properties that are necessary to build a smooth structure on the manifold.For details on the definition of smooth manifolds and the so called Bastiani calculus I refer to [6].
In the framework of [6] one can also define differential forms on such manifolds.
Definition 3.1.A symplectic form Ω on a smooth manifold P is a differential 2-form with dΩ = 0 which is weakly non-degenerate, i.e.
In distinction to the finite-dimensional case, Ω p does in general not induce an isomorphism Ω ♭ p : T p P −→ T * p P as in general T p P and T * p P are not isomorphic as topological vector spaces.Hence for the existence of the Hamiltonian vector field corresponding to F ∈ Obs we have the requirement (dF ) p ∈ Im(Ω ♭ p ) ∀p ∈ P. The phase space P in Hamiltonian field theory will be some subset of the space of pairs of smooth sections in some vector bundle E → Σ over a 3 manifold Σ, i.e. (φ, π) ∈ P ⊂ Γ(E) × Γ(E).This subset will usually be characterized by fall-off and other boundary conditions.Γ(E) is equipped with the C ∞ -topology [6,Example II.1.4],while P might be equipped with a C ∞ -topology induced from the C ∞ -topology of fields on a compactification Σ of Σ, which gives rise to certain fall-off conditions (see A). Observables are local functionals, i.e. of the form where α is a multiindex with |α| ≤ k for some k ∈ N. The symplectic form in such a field theory has the formal gestalt4 For the requirement (dF ) p ∈ Im(Ω ♭ p ) ∀ p ∈ P to be fulfilled for local functionals it might be necessary to add a suitable boundary term to the integral expression (see also [5,Chapter 4.1]).
The addition of boundary terms has important physical consequences.Take an observable G with G| C = 0.Such functions are called constraint functions.The vanishing on C will sometimes to be denoted by G ≈ 0. For it to have a corresponding X G we might have to add a boundary term G ∂ with G ∂ | C = 0.In that case, X G is not an infinitesimal proper gauge transformation.We call it infinitesimal improper gauge transformation.In practice, if the integrand of G has a particular slow fall-off this long-reaching gauge transformation might actually be improper.If additionally X G is a symmetry, i.e.Let (M, 4 g) be Minkowki space.The field in Yang-Mills theory is a connection on a SU(2) principal bundle on M : The bundle is trivial because every principal bundle over a contractible space is trivial 5 .A connection on P is given by a Lie algebra valued 1-form on P .The Lie algebra of SU(2 where × is the vector product on R 3 .On a trivial principal bundle the connection is completely determined by the Yang-Mills field A ∈ C ∞ (M, (R 3 , ×)), which is the pull back of the connection under a section s ∈ Γ(P ).
The Higgs field lives in the adjoint representation of SU(2), i.e. consider the associated vector bundle with the adjoint representation Ad : su(2) carries a natural inner product, the Killing form, which is on (R 3 , ×) given by the euclidean scalar product notated by a dot • .The curvature 2 form is the exterior derivative of the connection 1 form on the principal bundle.Pull back via a section s onto the base manifold defines the Yang-Mills field strength: Now we are able to write down the Yang-Mills-Higgs action: The Higgs field is characterized by the Mexican-hat potential where a > 0. Note, one could also consider a family V λ ( φ) := λ( φ 2 − a 2 ) 2 .The investigation presented here only applies for the cases λ > 0. In that regime, λ is only a rescaling, so we can assume w.l.o.g.λ = 1.The boundary term is yet to be chosen to make the Hamiltonian theory to exist.Let's assume its existence for a moment, but it will be dealt with later on.
5 Hamiltonian of SU(2) Yang-Mills-Higgs theory In this section we review the Hamiltonian formulation of SU(2) Yang-Mills-Higgs theory on a non-dynamical Minkowski spacetime.We take a similar notation and line of argument as [2], where the free Yang-Mills case is discussed.The Hamiltonian formulation of SU(2) Yang-Mills-Higgs theory is also discussed in [8].Take the Hamiltonian formulation with respect to the (3+1)-split M ∼ = Σ×R ∼ = R 3 ×R with the corresponding split of the Minkowski metric where g = δ is the standard Euclidean metric.The action takes the form where is the Lagrangian function with respect to this space-time split.Define canonical momenta via: π 0 ≈ 0 is a primary constraint 6 .By a straightforward calculation, one obtains the Hamiltonian7 : It has to be checked whether the primary constraint π 0 ≈ 0 is time dependent.This is indeed the case and this enforces a secondary constraint, called Gauss constraint: One easily checks that the Gauss constraint is preserved by time evolution and hence no further constraints a present in the theory.The extended Hamiltonian with respect to the constraint algebra is: where the Lagrange multiplier λ absorbed A 0 .The (formal) symplectic form on the not yet specified phase space is: where d and ∧ are, respectively, the exterior derivative and the wedge product in phase space.

Constraint algebra and gauge transformations
The constraints are first class, which means that the set of local constraint functions π (0) and G is a subalgebra of the pointwise Poisson algebra of local functionals.One calculates easily, where ⊗ is the tensor product for vectors in R 3 and (. × .) is the structure constant of the Lie algebra (R 3 , ×).Gauge transformations are Hamiltonian vector fields generated by first-class constraints.The Hamiltonian vector field generated by the constraint π 0 is on the constraint surface C everywhere vanishing, hence there is no degeneracy vector field of the pullback of Ω to C. This enables us to state π 0 = 0 as a global condition without loosing non-degeneracy of Ω. From now on the symplectic form is (5.11) Let's define (5.12) For the Hamiltonian vector field corresponding to G λ to exist, the expression might be corrected by a boundary term B such that λ that have the property Ḡ λ ≈ 0 are called improper gauge transformations, while λ with Ḡ λ ≈ 0 are called proper gauge transformations.Proper gauge transformations are the redundancies in the description of the theory, while improper gauge transformations are symmetries that actually change the physical state.The only possible boundary term that has to be added to G λ to have a finite and functionally differentiable Ḡ λ is (5.14) Proper gauge transformations are generated by Ḡ ǫ , where ǫ has a fast enough fall-off such that O 1 ( ǫ • π r ) = 0. Then: (5.15) Whether improper gauge transformations exist depends on the fall-off-and boundary conditions for the canonical variables.
6 Fall-off and boundary conditions

Fall-off conditions for the Yang-Mills variables
There are three reasons for demanding fall-off-and boundary conditions.For a Hamiltonian description of a relativistic field theory to exist, the Hamiltonian has to be finite, the symplectic form has to be finite and there has to be an action of the Poincaré algebra on phase space.Let's start with the Hamiltonian At first we ensure that the non-interacting Yang-Mills part d 3 x (2 is finite.For that we choose the following fall-off conditions: where we used the notation ∼ for an asymptotic expansion.These fall-off conditions can be justified by geometrical reasons (see appendix A).
Remark 6.1.With these fall-off conditions the symplectic form is logarithmically divergent.This divergence will be handled by parity conditions (10).

Fall-off and boundary conditions for the Higgs variables
Inspired by appendix A we choose the the fall-off This alone does not make the terms of the Hamiltonian that contain φ finite.First of all, let's consider the potential term d 3 x V ( φ).Using the ansatz (6.4) the integrand has to be O(r −4 ).
Proposition 6.1.Under the assumption of the fall-off behaviour being (6.4), Remark 6.2.In principle one might eliminate higher order terms in V ( φ) by using parity conditions, such that after performing the radial integral, the spherical integral is zero.But as ( φ ∞ • φ (1) ) 2 and φ ∞ 2 − a 2 are even functions on S 2 , this is not possible.
It remains to set a boundary condition for the canonical momentum of the Higgs field.In order to make the term Σ d 3 x 1 2 Π • Π of the Hamiltonian finite and to let the Poincaré transformations 7.2 leave the fall-off (6.4) invariant we have to choose necessarily (6.7)

Further boundary conditions
The fall-off conditions (6.2) for A i and (6.4) for φ do not make the term d 3 x 1 2 D i φ •D i φ of the Hamiltonian finite.For that to be the case it has to be D i φ ∈ O(r −2 ), while in general The terms of order r −1 have to vanish.Therefore and by taking the cross product with φ ∞ from the left, the equation can be solved for In spherical coordinates r and xā ∈ {θ, ϕ}, ā ∈ {1, 2} it follows (see also 6.4): Until this point we made sure that the part of the Hamiltonian that does not generate improper or proper gauge transformations is finite, i.e.
There are no boundary terms of infinite values, because π i fulfills the fall-off conditions (6.3) and ǫ will fulfill fall-off conditions that prevent this scenario.Infinitesimal gauge transformations ǫ have to preserve the fall-off conditions (6.2), (6.3), (6.4) and (6.7).By the transformation rules (5.15), While the first equation gives a concrete boundary condition at this point, the second equation will be solved later on when the action of the Poincaré group forces futher boundary conditions on the fields (section 7.2).
All of the fall-off conditions and boundary conditions together guarantee a finite Hamiltonian.To generate a well defined Hamiltonian vector field the Hamiltonian has to be differentiable (see section 3).Indeed we do not have to introduce further boundary terms to H, apart from d . The two terms of H that could in principle also need additional boundary terms are d The latter one does not lead to a boundary term, because this term would come from the integral d ), which gives a zero boundary term.For d 3 x 2 −1 D i φ • D i φ it is a little bit more involved, because the finiteness of this integral is not achieved by a fall-off behaviour.Indeed, where we used D r (0) φ ∞ = 0 which is equivalent to A (0) r = Ār φ ∞ (6.10) and φ ∞ • φ (1) = 0 (6.1).Having that, the fall-off conditions (6.2), (6.3), (6.4) and (6.7) together with the boundary conditions lead to a set of canonical variables with a differentiable and finite Hamiltonian function.
If the symplectic form would be finite at this point one could determine the Hamiltonian vector field.But we can do one intresting remark about the time development at this point.The Higgs part of the symplectic form is Ω Higgs = d 3 x d Π • ∧d φ.Later we will see that φ and Π fulfill the right boundary conditions such that Ω Higgs is finite.Using that without further characterization, we can calculate the equations of motion for the Higgs vacuum φ ∞ .Proposition 6.2.The time development of the Higgs vacuum φ ∞ is a gauge transformation Proof.By the Hamiltonian equation coming from the symplectic form, (6.17) Remark 6.5.At this point it is not clear whether ∂ t φ ∞ is a proper or improper gauge transformation.This depends on the boundary term d 2 x ǫ (0) • π r (0) .As it turns out in 7.2, we have to choose the boundary condition π r (0) = πr φ ∞ which makes only the ǫ (0) that are parallel to φ ∞ to improper gauge transformations.Then ∂ t φ ∞ is obviously a proper gauge transformation.

Fall-off in spherical coordinates
For a lot of the discussions in this work it is convenient to use spherical coordinates on Σ ∼ = R 3 .Spherical coordinates are defined via The fall-off of the Yang-Mills field expressed in spherical coordinates is For the transformation of the canonical momenta it has to be recalled that these are 1densities, i.e.
, where E i is a vector field and g is the flat Riemannian metric on Σ.In spherical coordinates this metric has the form: where γ is the round metric on S 2 .⇒ √ g = r 2 √ γ.It follows (assuming the fall-off behaviour where x ā ∈ {θ, ϕ}.

Asymptotic electromagnetic structure with magnetic charge
With the boundary conditions (6.9) for the asymptotic Yang-Mills field there are distinguished special kinds of gauge transformations, i.e. gauge transformations that stabilize the Higgs vacuum.They give rise to infinitesimal U(1) gauge transformations of the free components Āi of the asymptotic Yang-Mills field: i and therefore: ).Then, under the assumption Proof.See equation (6.25).
As the asymptotic componenent of the Yang-Mills momentum has to be π i (0) = πi φ ∞ , which will be shown in the section 7.2, δ ǭ πi = 0, like in electromagnetism.
The spatial Yang-Mills curvature is getting the form of a slightly modified spatial field-strenght tensor of of electromagnetism: The first term of Fij matches with the spatial field strenght tensor of electromagnetism.
The second term is actually the asymptotic field of a magnetic monopole.
In a theory of electromagnetism where the electric field is dual to the magnetic field, the coupling of the magnetic charge current to the electromagnetic field is descibed by the equation dF = j mag , with the 4 dimensional electromagnetic field strenght tensor F [11, Equation 2.9].The magnetic charge in the Cauchy hypersurface Σ is Taking asymptotically F ij = Fij , the magnetic charge in the SU(2) Yang-Mills-Higgs theory with adapted units is where ǫ āb is the volume two form on S 2 and ∇ the Levi civita connection of the round metric γ āb (see 6.4).In the appendix B it will be shown that m is actually a topological charge, i.e. it is the winding number of the vacuum Higgs field φ ∞ : S 2 −→ S 2 .In the section 9 it will be shown that m labels the connection components of the phase space.
As in electromagnetism, states can also have a electric charge, which is the generator of asymptotically constant U(1) gauge transformations, i.e. with adapted units With the result π r (0) = πr φ ∞ of section 7.2 we get the electric charge of electromagnetism, i.e. q = (4π) −1 d 2 x πr .
Depending on further boundary conditions (section 10) we will investigate whether a non-zero electric charge is actually allowed in phase space.We will see that this is the case.
In the Hamiltonian study of electromagnetism in the Henneaux/Troessaert paper [1] it is also found an infinite number of non-zero soft charges, which are ) being an even function under parity x → −x.Soft charges corresponding to odd ǭ are also allowed, but they are calculated by a different integral expression (see also section 11.3).

Example: Dyon-solution
The fall-off-and boundary conditions we stated should allow for known solutions of SU( 2) Yang-Mills-Higgs theory.The most important solution is the Julia-Zee Dyon, discovered by Bernard Julia and Anthony Zee in 1975 ( [12]).
The Dyon has a certain asymptotic behaviour.Using the notation of [13], the equations (2.5) and [12, Section III.], as well as [11, equation 4.26], one finds: where α, β and b 1 are some constants.The vector space of the coordinate chart x for Σ ∼ = R 3 is identified with su(2) ∼ = R 3 by chosing an orthonormal basis for su (2).In this identification is n = x r and e i the i-th basisvector.These boundary condition fulfill the fall-off and boundary conditions stated above.The boundary condition which fulfills the boundary condition (6.9) with Āi = 0.
The Dyon solution has the magnetic charge m = 1 and the electric charge q = b 1 .m = 1 is easily calculated with the equation (6.28).For the calculation of the electric charge we need the radial asymptotic electric field πr .While using (6.31) and (6.32) one finds π r

Poincaré transformations
As SU(2) Yang-Mills-Higgs theory is a relativistic field theory, the phase space P has to carry an canonical action of the Poincaré group (which is then automatically Hamiltonian as the Poincaré algebra is perfect, see [14,Chapter 3.2,3.3].A necessary condition for such an action is, that there is a canonical action of the Poincaré algebra, which is a Lie algebra homomorphism X : poin −→ Γ(T P), such that L X(poin) Ω = 0 and i X(p) Ω = dP p , where P p is a differentiable local functional for every p ∈ poin.Whether the action of the Poincaré algebra is globally integrable to an action of the Poincaré group is in this infinite dimensional setting a rather technical question, which will not be answered in this paper.
In this chapter we check whether the Poincaré transformations preserve the fall-offand boundary conditions from the previous chapter and state further boundary conditions that are invariant under Poincaré transformations.In the first section the infinitesimal Poincaré transformations of the canonical variables are calculated.

Action of the Poincaré algebra
The action of the Poincaré algebra on Minkowski-space is represented by the vector-fields rotations, (a ⊥ , a i ) are space-time translations (see also [1]).These vector fields define certain Cauchy-hypersurface deformations.Hypersurface deformations are in general locally parametrized by a lapse function N and a shift function N .With an analogous calculation like in [3, Chapter 2], the phase-space generator of the action of the deformations on the canonical variables is determined to be where and B (N,N ) is a boundary term to make the expression functionally differentiable.B is to be determined after the complete specification of the fall-off and boundary conditions.
Remark 7.1.The gauge parameter associated to the Poincaré transformations will be denoted by ζ = ξ ⊥ λ.The fact that the gauge transformation goes along only with the boosts and the time translations is an artefact of the geometrical derivation of (7.2).Indeed we could combine any Poincaré transformation (of spatial-or non-spatial type) with a gauge transformation and the result is a different but also appropriate action of the Poincaré algebra, as a gauge transformation does not change the physical state.But caution: Depending on the asymptotic behaviour, this gauge transformation can actually be improper, hence resulting in a different physical transformation.When those improper gauge transformations are present in the theory the Poincaré algebra will not be a symmetry algebra anymore, rather the symmetry algebra is the asymptotic symmetry algebra asym with an embedding poin ֒→ asym, which is not unique.
Let's determine the local Poincaré transformations of the canonical variables, assuming that we have found the right boundary terms B (ξ ⊥ ,ξ) , with the formula This calculation results in

Asymptotic Poincaré transformations
The aim of this section is to check whether the Poincaré transformations leave the fall-off conditons (6.2), (6.3), (6.4), (6.7), (6.12) and the boundary conditions (6.16) invariant.If this is not the case we have to strengthen them to guarantee an action of the Poincaré algebra on phase space.We will see indeed that this has to be done.Further on we will see that we can state certain additional parity conditions on the leading orders of asymptotic expansions of the fields without breaking the Poincaré invariance.This conditions will be necessary to make the symplectic form finite (section 10).
For the asymptotic analysis it is useful to switch to the spherical coordinates (r, xā ) on Σ.The transformation generators (ξ ⊥ , ξ) expressed in spherical coordinates are (see also [1,Chapter 2.2]) where T, b, W and Y ā are the following functions of the angles (θ, ϕ): where Additionally, let's introduce some notation.In this section it is useful to make an asymptotic expansion of the covariant derivative in orders of r −n , i.e.
One can pull the index of D First of all, the structure of the fall-off behaviour of the fields, being an asymptotic series in r −n , is preserved by the transformation rules (7.6), because ξ ⊥ = rb + T, ξ r = W, ξ ā = Y ā + 1 r γā b∂bW while (7.6)only involves the fields, derivatives (which are ∼ r −1 for fields having an asymptotic series in r −n ), ξ ⊥ and ξ.
The next step is to examine whether any Poincaré transformation of a variable might have higher order terms as allowed by the fall-off conditions.In one strike the invariance of the boundary conditions (6.16) is checked.
The Poincaré transformation of the leading component of A r is: r fulfills the boundary condition we have to propose the boundary condition: Let's calculate the asymptotic Poincaré-transformations of the angular components: .16)This transformation has to preserve the boundary condition (7.17) As calculated before, which together with δ (0,ξ) φ ∞ = Y ā∂ ā φ ∞ and (7.16) shows that the boundary condition is preserved, while δ (0,ξ) Āā = Y b∂b Āā .Concerning the boosts, With the decomposition and With (7.20) and (7.21), (7.17) is fulfilled if and only if This can only be the case, if there is a further boundary condition Remark 7.2.Note that the first two terms of (7.22) are the transformation behaviour from electromagnetism, while the third term is cancelled by a term of the transformation of −(ea ) ⊥ would mirror this fact.Let's consider the Poincaré transformation of the canonical momenta of the Yang-Mills field.These have to preserve the newly found boundary conditions π i = πi φ ∞ and the falloff conditions.This leads to additional conditions for the next to leading order of A i , which have to be invariant under Poincaré transformations too.Subsequently this leads to additional conditions for the next to leading order of π i , resulting in a bootstrap.The result of this bootstrap performed on the constraint surface is presented in the following proposition.
where C is the set of all canonical variables fulfilling all previous stated boundary conditions and the constraints.
Proof.We show this statement per induction.The base case is the boundary condition (6.9) which is equivalent to We have to show The assumption is equivalent to ). Taking the cross product with φ and using φ × O(r −(n+2) ) = O(r −(n+2) ): where This condition has to be invariant under Lorentz boosts Before conclude a condition for π i we have to show an intermediate proposition: Proof: We use induction again, until we reach the order k = n + 1 in O r −k (D i φ).The base case follows from the fact that the boundary condition φ (1) • φ ∞ = 0 has to be invariant under boosts, i.e.
Deploy this in the order ∼ r −(k ′ +2) of the equation (7.29) which gives The only term on the right hand side that is not purely a gauge transformation is the first one.Hence where we used that the term ξ ⊥ π i in δ ξ ⊥ A i is independent of the choice of Poincaré action parameterized by the gauge parameter ζ.
Let's study the different orders of the constraint equation using Π Using (7.33) and (7.28), By bootstrapping this up to the order ∼ r −(n+1) while using at every step Π Applying the constraint at this order (7.34) we can conclude Going further, this canonical momentum gives rise to a boost transformation with Additionally, since Π (n−2) = 0 it has to be ∂ t Π (n−2) = 0, where Via a bootstrap starting from ∂ t Π (0) = 0 and going to ∂ t Π (n−3) one concludes that all expressions O r −l ( φ 2 − a 2 ) for l ≤ n − 1 are already zero.
Together with the equation (7.40), and therefore equation (7.29) gets the shape ). (7.46) ). (7.47)This condition has to be invariant under time evolution.In general The general form of the Yang-Mills curvature F (by the definition 2.1 in [15]) is Using (7.28), ). (7.50) Back to the invariance of (7.47) under the time evolution, i.e. it has to be

.53)
As D m F mi is already parallel to φ up to the desired order and ∀k ≤ n such that e ζ × π i includes the time evolution of φ, it has to be ). (7.55) ). (7.57) As we have seen before as a result of the intermediate proposition, With the same bootstrap argument as it was used at equation ( 7.41) we can conclude Clearly by considering also the  Before investigating the consequences of the proposition 7.1 and the corollaries 7.1.1 and 7.1.2we finish this section by examining the Poincaré transformations of the leading components of the fields to find useful properties.
ā × with ∇ being the Levi-Civita connection of the round metric γ.A well known identity from Riemannian geometry is: For a antisymmetric (2, 0) tensor F ik , a Riemannian metric (or Lorentzian metric) g and the corresponding Levi-Civita connection ∇ the following identity holds9 : In our case we have ). Additionally we have We use this for the Poincaré transformations of the leading order Yang-Mills momenta.The leading order Poincaré transformations are: We have seen before that these transformations are consistent with the boundary conditions Let's look for important properties of the Poincaré transformations: (i) It is easy to see that the radial and angular sector of Āi and πi do not mix under Poincaré transformations.
(ii) The Poincaré transformations leave certain parity conditions invariant.A parity transformation is given by the map This map is completely determined by a map ) is called even under parity ⇔ Da(X) = X and odd under parity ⇔ Da(X) = −X.A one form α ∈ Ω 1 (S 2 ) is called even under parity ⇔ a * α = α and odd under parity ⇔ a * α = −α.In this way one generalizes this easily on all tensor fields.
Let's make the following observation.In carthesian coordinates {x i } on R 3 it is a * (dx i ) = −dx i , hence the components α i of a one form α = α i dx i are odd (even) ⇔ α is even (odd).The same is true for the components of a vector field.The components ω ij of a two form ω are odd (even) ⇔ ω is odd (even).In these coordinates the derivative operator ∂ i always changes the parity, because additionaly to a * (dx i ) = −dx i , a * (dλ) = d(a * λ) for some form λ. Inspired by this observation we make the definition: Definition 7.2.Call α ā, X ā, ω āb parity odd (even) ⇔ α i , X i , ω ij are parity odd (even). 10his is a well posed definition, because the actual objects are the tensor fields with their parity conditions.This definition is just a different code for the same notion.In this code the derivative operator ∂ ā always changes the parity.
It is easy to see that b and Y ā are odd under parity.Let's assume the asymptotic fields { Ā, π} have a definite parity, such that the parities of Ār and πr , as well as Āā and πā , are opposite.Take a look at equation (7.67).Since Y ā∂ ā is an parity-preserving operator and δ A (0) r = (δ Ār ) φ ∞ + Ār (δ φ ∞ ) you can take φ ∞ out of the equation and with the property that b and Y ā are odd, it is clear that this transformation preserves the parity of Ār .
Take a look at equation (7.68).The same argument as before makes sure, that the first three terms preserve the parity.The last term might be a improper gauge transformation.If so, ζ has to be of opposite parity then Āā .Alternatively we can allow a degree of freedom of opposite parity that has the form of a improper gauge transformation.We will expand on that in the next chapter.
One can write the transformation of the radial momentum as (7.75) The first term preserves the parity conditions.The second term is more delicate to handle and will be treated in the section 10.

The Higgs mechanism
Let's study the consequences of the proposition 7.1 and the corollaries 7.1.1 and 7.1.2.The statements of the corollaries expressed differently are: (i) The dynamics of physical states of the Higgs field in SU( 2) Yang-Mills-Higgs theory in 4 dimensions falls off faster then any inverse monomial in the radius r.
(ii) The physical non-abelian degrees of freedom of the Yang-Mills field in SU( 2) Yang-Mills-Higgs theory in 4 dimensions fall off faster then any inverse monomial in the radius r.
Both of these statements are the consequence of demanding Poincaré invariance of the theory.Physical means that these statements are only true for states on the constraint surface.
As we actually only care about physical states and the results of the corollaries have the right shape (to not induce degeneracies of the symplectic form), we demand to be boundary conditions, where the ⊥ is defined to be the projection perpendicular to φ.The conditions for π i and A i are already Poincaré invariant, while the condition for Π has to be expanded by yet another condition, which is because by proposition 7.1, It is possible to demand even stronger boundary conditions, because we find out in the following that any order φ (n) of the asymptotic expansion of φ for n ≥ 1 is a proper gauge degree of freedom.
Proof.Let's perform a sequence of infinitesimal proper gauge transformations starting at φ ′ := φ ∞ + o(r −N ) which fixes φ ∞ .At first order, inf , e ǫ ′ (2) × φ ∞ =: φ (2)⊥ inf , ..., where the "inf" is a label for infinitesimal.This is justified because e ǫ ′ (1) × φ ∞ lies in the tangent space .. .Let's compute the second order: inf has the property e( ǫ . This is the infinitesimal version of the fact that by We would like to make sure that by the successive application of a gauge transformation δ ǫ ′ every possible value of φ with fixed φ ∞ can be reached. By the condition φ 2 = a 2 + o(r −N ), at every order ∼ r −k , φ (k) • φ ∞ is determined by lower orders of φ, while the φ (k)⊥ are the free parameters 12 .Every possible value of φ Proof.The state of the system at time t does not depend on the initial values of φ (n) (n ≥ 1) at time t = 0, because: • The time evolution of the φ (n) is a proper gauge transformation, because Π (n) = 0 ∀n ∈ N.
• By the proposition 8.1, the state at t = 0 does not depend on the φ (n) .This is a necessary condition for a variable to be a proper gauge degree of freedom.
Having that we can sharpen the boundary conditions by demanding This conditions restricts the allowed set of gauge parameters to Let's collect the results and reframe them with respect to the usual notion of the Higgs mechanism.We found that the physical degrees of freedom of the Higgs field and the nonabelian part of the Yang-Mills field with the exception of the vacuum degrees of freedom ( φ ∞ , −(ea 2 ) −1 ( φ ∞ × ∂ i φ ∞ )) fall off faster then any inverse monomial in r.This fall-off suggests by comparison to the behaviour of a massive scalar field [3, Section 3.1] that these degrees of freedom are massive.
Actually the mass of a field is defined via the prefactor of the quadratic term of the field in the Lagrangian.So, let's make a precise statement about the massive fields in this theory.Let's decompose φ = φ ∞ + φ φ ∞ + φ ⊥ , where φ, φ ⊥ ∈ o(r −N ) and φ ⊥ ⊥ φ ∞ .Here we have set the partial gauge φ (n) = 0 ∀n ≥ 1.The potential term in the Lagrangian takes the form The first term is the mass term.It should be noted that only the component pointing in the direction of the vacuum has a mass term.The perpendicular component φ ⊥ is the massless Goldstone boson of the spontaneously broken SU(2).φ ⊥ is actually massless, because it has no mass term in V ( φ), as well as no mass term in D i φ • D i φ.A possible mass term is not an interaction term, so we can assume for a moment that to calculate These are only the kinetic energy terms.Hence φ ⊥ is massless.
Proof.Clearly there exists a smooth U : Hence the Goldstone boson is no physical degree of freedom, but pure proper gauge.
Let's find the mass term of the non-abelian part of the Yang-Mills field in the Lagrangian.For that take a look to the terms which are quadratic in A i in The term a 2 A ⊥ i • A i⊥ is the mass term of the non-abelian part of the Yang-Mills field.
Proof.As the first step we see that every tangent space T φ∞ F { φ∞} consists only of proper gauge transformation generators.
for some m ∈ Z.Take a tangent vector v ∈ T φ∞ F (m) { φ∞} . Using theorem 7.6.and corollary 7.6 in [17], where π : T S 2 −→ S 2 is the natural projection.Hence v can be represented as a smooth function v : . Take an asymptotic proper gauge transformation generated by ǫ ⊥ (0) that has the property ǫ . Then there exists a smooth curve γ : Every tangent vector to the curve is a field dependent generator of a proper gauge transformation.By arguments used in the proof of 9, there exists a smooth homotopy between φ ∞ and φ ′ ∞ .Let U : R 3 −→ SU(2) be a gauge transformation that is constant in radial direction.Take the pullback i * S 2 (U ) : S 2 −→ SU(2).The action on the Higgs vacuum is given by a rotation matrix D(x) such that (i remains to show that this map is bijective.This is easy to see for the variables { φ, Π, π i }, because of the easy form of the action.For example D π i = π i φ ′ ∞ + D π i⊥ with D π i⊥ ⊥ φ ′ ∞ , which is a bijective map D : (π i , π i⊥ ) → (π i , D π i⊥ ) onto its image.Exactly similar it is for the variables φ and Π. Let's take a look at the action on A i .We do not compute the action directly, but use the action on F ij to get the information we need.D ⊲ F ij = D F ij .With the boundary conditions (8.12) we have where Using that we get the equation: Hence by the equation above, ) and A gau does not include an exact term. 14Additionally ∂ i Φ = 0 because we chose the distinguished lift of the curve γ in the bundle that has X ǫ ⊥ (0) as tangent vectors.A term ∂ i Φ would only be generated if we had added a gauge transformation that stabilizes the fibre P proto γ(t) for some t , where A gau i is fixed by D. This map is clearly bijective onto its image.Proof.We use the same argument as in the proof of the corollary 8.1.1.The time evolution of φ ∞ is a proper gauge transformation and by the proposition 9.1, any two φ ∞ , φ ′ ∞ that lie in the same connection component of F φ∞ are connected via a proper gauge transformation.

Parity conditions
Let's take a look at the formal symplectic potential on P proto .This is at a point p = (A, π, φ, Π) ∈ P proto : Evaluating the potential on a tangent vector v ∈ T p P proto of a curve γ(s) with γ(0) = p while using the boundary conditions (8.12): The non-finite term is logarithmically divergent.Consider the non-finite term of the symplectic potential: r-component: ā-component: (10.4) 14 One can show also geometrically that the two form The class of functions that makes the integral d vanish might be rather non trivial, but it certainly vanishes if the integrand is an odd function on S 2 .We will stick with choosing the integrand to be an odd function, because as we will see this will actually be necessary in the angular components (10.7), it is compatible with Poincaré transformations (section 7.2) and it it allows for physically relevant solutions.
Let's propose parity conditions on Ār , Āā , πr and πā .Consider the r-component first.πr ∂ s Ār has to be odd.A constraint on the possible choices is that the electrically charged Dyon should be a state in phase space.
Therefore one chooses: Since Ār as well as πr are gauge invariant, these are appropriate boundary conditions.These conditions also determine what the improper gauge transformations of the theory are: i.e. ǭimproper = ǭeven .
Let's consider the angular components (10.4).We choose the condition πā even = 0. ( This condition might actually necessary 15 .The condition πā even = 0 has to be invariant under boosts.In general δ b πā involves a term ∂b(bVol bā φ∞ ).If Vol bā not odd this breaks the parity condition.But apparently there is no problem at all, which we will see in the remainder of this section.Let's regard the two cases m = 0 and m = 0. m=0: By the corollary 9.1.1,for each connection component (P proto ) (m) the base manifold F (m) { φ∞} parametrizes a proper gauge degree of freedom.Let's phrase the parity condition firstly in the simple gauge φ ∞ = τ 3 = (0, 0, 1), where In 15 This is because of the following reasons.In the appendices B.1 and B.2 of [1] it was shown that in electromagnetism this condition is necessary to have not a divergence in the magnetic field as one approaches null infinity by an infinite sequence of boosts of the Cauchy surface.The same argumentation applies here to the case φ∞ = const, because in that case the asymptotic form of the action is indistinguishable from the electromagnetic action.By a proper gauge transformation (prop.9.1) this applies to the whole m = 0 sector.It is a conjecture that this argumentation also applies to the m = 0 sector.order to make Ω τ 3 < ∞, while πā even = 0, choose Āā = Āeven ā + ∂ āΦ even and πā = πā odd . (10.8) Then it follows F āb (0) = 2 ∇[ā Āb ] even τ 3 ⇒ δ b πā stays odd.These conditions let the angular integral over (10.4) vanish, because the second term of (10.4) vanishes in this gauge and ∂ ā πā = 0 is a boundary condition (see (6.16) and use Π = 0).
Transfer these conditions onto another fibre P proto φ∞ ⊂ (P proto ) (0) by a proper gauge transformation U ⊲, being the canonical lift of the mapping τ 3 → φ ∞ (which is exactly the U from the proof of proposition 9.1).Then (using the proof of prop.9.1), where A gau ā fulfills dA gau = −(ea 2 ) −1 a −2 Vol φ∞ while A gau does not include any term of the form dλ by the choice of the canonical lift.
With these parity conditions it is easy to see that  (7.2) this is equivalent to Vol φ∞ being odd as a two form.m = −1: Choose a northpole and a southpole p N , p S ∈ S 2 (antipodal points) with the corresponding coordinates (h, ϕ), where h : S 2 −→ [−1, 1] is the projection onto the axis connecting p N and p S when S 2 is embedded into R 3 , while ϕ : S 2 \ Geo p N ,p S −→ (0, 2π) is the angular coordinate with respect to this axis, where Geo p N ,p S is some closed geodesic segment connecting p N and p S .Define φ ∞ via the map f : (−1, 1) × (0, 2π) −→ (−1, 1) × (0, 2π), (10.11) The map φ ∞ | S 2 \Geop N ,p S := (h, ϕ) −1 • f • (h, ϕ) extended to S 2 by mapping Geo p N ,p S with the identity map onto Geo p N ,p S on the target sphere does hit every point of S 2 exactly once.Look at the derivative: (10.12) Since (h, ϕ) is a orientation preserving diffeomorphism, the winding number has to be [ φ ∞ ] = −1.
Now show that Vol φ∞ is odd under parity.By definition (B.2), Vol φ∞ = φ * ∞ ω where ω is the standart volume form on S 2 .Let's choose coordinates {y ā} on a region R \ L ⊂ S 2 , where a(R) ⊂ R and ω = dy 1 ∧ dy 2 16 .Now choose those coordinates on the target sphere ({y ā T } on R T \ L T ) as well as on the domain sphere ({y where it is obvious that det[D φ ∞ | a(x) ] and det[Da| x ] are coordinate-independent expressions.It is easy to see that det Hence Vol φ∞ is odd under parity.m = 2: Let's stay in the coordinates (h, ϕ).Let .16)f is extended to h = 0 via φ ∞ ((h, ϕ) −1 ({(0, ϕ)} ϕ∈(0,2π) )) = {p S }.It is easy to see that φ ∞ is smooth 17 .With S 2 N being the closed northern hemisphere with respect to the coordinates (h, ϕ) and S 2 S being the closed southern hemisphere,

S
are injective and have S 2 in their image respectively.As by (10.15) . As (10.13) holds in general for every φ ∞ ∈ F { φ∞} , (10.14) is again applicable which gives the desired result in this case.m = −2: Take the f from m = 2 and reverse the orientation of the image like in the case m = −1. 16An example for such a region R would be some ring segment around the equator bounded by two latitudes which are their mutual image under the antipodal map.L is some line segment of a longitude.It is easy to see that such coordinates exist, by segmenting R into regular equal-volume elements that are cut by longitudes and latitudes. 17The only problematic points could be the points of h = 0.By the definition of φ∞, ∂ϕ φ∞| h=0 = 0, and ∂ h φ∞| h=0 = 4h.You have to change the coordinates on the target sphere, to see that the derivative is continuous in the neighbourhood of the h = 0 set.With that it follows clearly, that φ∞ is smooth.
m ∈ N \ {0}: Define f , using a polynomial g : [−1, 1] −→ [−1, 1] of m'th order that has m − 1 stationary points h i ∈ (−1, 1) (h i < h i+1 ∀i) that are maxima or minima with g(h i ) ∈ {±1}.Additionaly g is an even function if m is even and g is an odd function if m is odd.Let Theorem 10.2.If m = 0, assuming the boundary condition (8.12), a sufficient condition for the symplectic potential to be finite is: where dA . Take the corresponding Vol φ∞ .As Vol φ∞ is a 2-form it can always be written as the sum of a exact form and a residual part: Vol φ∞ = Vol  .Lets consider a variation paramerized by s: which is an odd expression with the exception of a 2 πā ∂ āΦ even , which vanishes by using partial integration and the boundary condition ∂ ā πā = 0.
By that proposition and the result in the case m = 0, for every φ ∞ ∈ F { φ∞} the symplectic form Ω φ∞ exists under the assumption of the above given parity conditions.
Let's state the boundary-and fall-off conditions at this point.In order for the symplectic form to be finite we add to the conditions 8.12 the additional conditions: Definition 10.1.The set of smooth functions {(A, π, φ, Π)} that fulfills the conditions 8.12 and the additional parity conditions above, denoted by (P, Ω), is called the unbroken phase space.
11 The boost problem

The Lorentz boost of the symplectic form
In order to have a well stated phase space of a relativistic field theory, the Poincaré group has to act as a Hamiltonian action.By the result in [14], chapter 3.2 and 3.3, for the Poincaré group to act Hamiltonian it is sufficient to show that the group acts via symplectomorphisms.At the Lia algebra level this is L X (ξ ⊥ ,ξ) Ω = 0, where X (ξ ⊥ ,ξ) is the fundamental vector field of the group action, corresponding to the Lie-algebra elements (ξ ⊥ , ξ) 19 .It is easy to see that L X (0,ξ) Ω = 0, but a straightforeward calculation shows: . ( is exactly the boundary term that appears in U(1) electromagnetism [1].This term is treated in the section 11.3.Let's take a look at the term of (1) that corresponds to A gau ā .By the equation (10.21) (choosing λ = 0), where the term involving φ ′ ∞ does vanish if and only if m = 0.In the case where it does not vanish, the φ ′ ∞ is such that (Vol φ ′ ∞ ) āb is odd.Using (10.21) again we can conclude ) is even.Hence sticking this expression in the integral (1) gives 0. From a 2 d 2 x √ γ d Ār ∧ ∇ā (bdA gau ā ) it remains the expression a , which does cancel with the integral (2).As φ ∞ and also Vol φ∞ does in general not fulfill any parity condition, the integral B 2 := (3) does not vanish.
The integrals B 3 := (4) do not vanish for any field configuration, if and only if ζ is field dependent.A general gauge transformation that fulfills the fall-off and boundary conditions 8.12 to accompany the boost depends at least on φ ∞ .

Solving the boost problem I
In the previous section we found that the Lorentz boosts do not act canonical on phase space.Rather for a general φ ∞ ∈ F { φ∞} and an Āā that has an odd component, L X (ξ ⊥ ,0) Ω = B 1 + B 2 + B 3 = 0. B 1 = 0 happens when one allows for improper gauge transformations ∂ āΦ even in the theory (see (10.1)).This boundary term can be handled by introducing a new degree of freedom ψ on the sphere at infinity and a boundary term for the symplectic form.It will turn out that this new degree of freedom is actually the missing improper gauge transformation that was locked out by the condition πr odd = 0 (10.6)before.This method to handle B 1 will be carried out in the section 11.3.
The boundary term B 2 is made uncritical by a conceptual approach.We use the fact that inside each connection component of P, φ ∞ is only a proper gauge degree of freedom (corollary 9.1.1).If we would work on the reduced phase space which is the set of actual physical states, i.e. the quotient C/Gau, there would be no boundary term B 2 , because this term involves d φ ∞ , which is not anymore present in the theory.But working on the reduced phase space is not realistic because it is unclear whether the quotient C/Gau has any manifold structure.
Rather we take the partial gauge that fixes some φ ∞ in each connected component F where φ ∞ is some element in F {∞} .Proposition 11.1.There is an action of poin on P that acts tangential to all fibres P φ∞ which is the composition of the usual poin action (7.6) with an infinitesimal proper gauge transformation.
Proof.Certainly the action of ξ does not leave φ ∞ invariant is ξ as δ Finding a correcting proper gauge transformation that brings the translation part of δ (0,ξ) φ ∞ to zero was already done before (8.1).Let's now choose a proper gauge transformation ǫ ⊥ such that δ Y φ ∞ + δ ǫ ⊥ φ ∞ = 0.This equation is fulfilled by the choice ǫ ⊥ = (ea 2 ) −1 φ ∞ × Y ā∂ ā φ ∞ .It is now easy to see that δ (0,ξ) + δ ǫ ⊥ acts tangential to Hence poin acts on P pre .The generators of gauge transformations on Ppre are Choosing the space Ppre is associated to the treatment of φ ∞ as a non-dynamical background field.The symplectic form on P φ∞ for some φ ∞ is with d φ ∞ = 0: where B 3 does not vanishes if ζ is field dependenent in some P φ∞ .In the next subsection we will see that we actually have to choose B 3 field dependent for the method in the next subsection to work.

Solving the boost problem II
This section is an exact reprise of the method by Henneaux and Troessaert for electromagnetism in [1], because they have to treat exactly the same boundary term.On the pre-final phase space Ppre the Lorentz boost of the symplectic form in the background φ ∞ is: For the boosts to be canonical, this expression has to vanish.By 10.1 Ār is odd under parity and hence the term including Āeven ā vanishes, while the term including the improper gauge transformation ∂ āΦ even does not vanish.Let's treat this term at first and we will see that the remaining terms will also be treated by this ansatz.is f.d.w.r.t.Ω φ∞ .The corresponding gauge transformation is δ µ ψ = µ where the improper part is given by μodd .Hence up to ψodd , ψ is a proper gauge degree of freedom.
The last adjustment in this section is to choose the boost-accompanying gauge parameter ζ, the boost transformations of ψ and π ψ and an adjustment of the transformation of π i that vanishes on the constraint surface: (11.17 12 Asymptotic symmetries In the last section in order for the boosts to act canonically we introduced an additional gauge degree of freedom parametrized by µ that is generated by Ḡ µ .Together with Ḡ ǫ this combines to Ḡ( ǫ, µ) = d 3 x ( ǫ • G + µ • π ψ ) − a 2 d 2 x (ǭπ r + √ γ μ Ār ).(12.1) The pair ( ǫ, µ) with the asymptotic component (ǭ even , μodd ) generate improper gauge transformations.ǭeven and μodd combine to a single smooth function on S 2 .It is easy to see that Ḡ( ǫ, µ) actually generate symmetries, i.e. { Ḡ ǫ, µ , H} ≈ 0 ∀( ǫ, µ).
Remark 12.1.At first sight the introduction of the new physical variable ψodd seems adhoc, but as δ μ ψodd = μodd this new variable brings only the odd part of ǭ back as a physical degree of freedom.Also Ār is a gauge-invariant conserved quantity.By the introduction of μ we get a corresponding symmetry.
The generators of the Poincaré transformations have the general form:

Conclusions
The study of SU(2) Yang-Mills-Higgs theory in the Hamiltonian framework, starting from first principles, led us to clear insights into symmetry breaking and the Higgs mechanism at a classical level.Symmetry breaking happens as a consequence of a finite Hamiltonian.The Higgs mechanism is a consequence of the Poincaré invariance of this theory.It forces the non-abelian degrees of freedom of the theory to become massive (fast fall-off) while asymptotically electromagnetic degrees of freedom as well as a topological charge -the magnetic charge -remain.The result is similar to the abelian Higgs mechanism [3].
Because of the presence of the magnetic charge it was not completely straightforward to apply the results of Henneaux and Troessaert [1] to the asymptotic degrees of freedom.Instead we had to carefully extract proper gauge degrees of freedom from the asymptotic structure and had to apply an asymptotic partial gauge in every magnetic sector as a prerequisite to the definition of the phase space in order for the boosts acting canonically.
By the same method as in [1] we found the asymptotic symmetry algebra of electromagnetism in every magnetic sector, including global angle-dependent u(1) transformations.
We highlight that the extraction of the improper gauge transformations depends in a subtle way on the underlying functional analytic structure of the phase space.In a future work we will focus on developing a rigorous mathematical background for all these Hamiltonian theories.
In a recent work [20] by Fuentealba, Henneaux and Troessaert the fall-off conditions for the gauge parameter is weakened.It is found that by the extention to include ∼ log(r) and ∼ r growing terms in the angle-dependent u(1) transformations together with conjugacy relations between these new terms with the terms of O 1 and O r −1 , C ∞ (S 2 , u(1)) will be decoupled from poin in asym.This yields in particular that one can give a definition of the angular momentum that is free from ambiguities coming from C ∞ (S 2 , u(1)).
As the asymptotic structure in the SU(2) Yang-Mills-Higgs case is that of electromagnetism of [1], with the only difference being the presence of topological charges, we are optimistic that this generalization works out also for SU(2) Yang-Mills-Higgs theory.
Another interesting generalization would be to consider other gauge groups together with symmetry-breaking fields.Does the method presented here generalize to these other cases?A case of particular interest is obviously the electroweak gauge group SU(2) L × U(1) Y .
Last but not least we mention the possibility of a different spacetime dimension.It is known that the behaviour found in [2] critically depends on that dimension being four.In the case considered here the choice of dimension certainly influences the topological structure of the Higgs vacuum, the impact of which remains to be analysed.

A Fall-off from compactification
The general idea is to compactify the Minkowski space and the fields should be smooth at the boundary.The usual method to compactify Minkowski space is conformal compactification of the null directions [16,Chapter 11.1].Compactification means to embed Minkowki

4 SU( 2 )
always true) it will be called an asymptotic symmetry.Definition 3.2.If all infinitesimal symmetries are fundamental symmetries of a group action of the symmetry group Sym and all proper gauge transformation form a normal subgroup Gau ⊂ Sym, then the asymptotic symmetry group is Asym = Sym/Gau.(3.5) Yang-Mills-Higgs theory where D : SU(2) → SO(3) is the representation as a rotation matrix on R 3 .The Higgs field is a section φ ∈ Γ(E) ∼ = C ∞ (M, (R 3 , ×)).The connection defines a covariant derivative on Γ(E) via the adjoint representation, i.e.Dφ := dφ + e[(Ad * ) e (A)](φ) = dφ + eA × φ. (4.3) and ( a) i = a i with the b jk , b i , a i and a ⊥ from the equation (7.1).

ā
up by γā b.

Proposition 7 . 1 .
With the assumption that the local action (7.6) of the Poincaré algebra leaves the boundary and fall-off conditions invariant, it follows:
smooth, and because of the homotopy, D lies in the identity component of C ∞ (S 2 , SO(3)).Take the distinguished lift (along the fibres) of γ into P proto which is characterized by the tangent vectors X ǫ ⊥ (0) (the action of the gauge generators on the canonical variables), where ǫ ⊥ (0) are tangent to γ. ⇒ as all the ǫ ⊥ (0) are proper gauge generators, D represents a proper gauge transformation with D ⊲ (P proto φ∞ )

(10. 17 )
With a such defined smooth f (same argument as before) we find again det[Df | −h ] = det[Df | h ] ∀h.Now execute the same argumentation as in the case m = 2 to find [ φ ∞ ] = m and Vol φ∞ being odd.m ∈ −N \ {0}: Again, reverse the orientation of f .