Asymptotic symmetries of scalar electrodynamics and of the abelian Higgs model in Hamiltonian formulation
aa r X i v : . [ h e p - t h ] J a n Prepared for submission to JHEP
Asymptotic symmetries of scalar electrodynamics andof the abelian Higgs model in Hamiltonian formulation
Roberto Tanzi a and Domenico Giulini a,b a University of Bremen, Center of Applied Space Technology and Microgravity (ZARM), 28359Bremen b Leibniz University of Hannover, Institute for Theoretical Physics, 30167 Hannover, Germany
E-mail: [email protected] , [email protected] Abstract:
We investigate the asymptotic symmetry group of a scalar field minimally-coupled to an abelian gauge field using the Hamiltonian formulation. This extends previouswork by Henneaux and Troessaert on the pure electromagnetic case. We deal with min-imally coupled massive and massless scalar fields and find that they behave differentlyinsofar as the latter do not allow for canonically implemented asymptotic boost symme-tries. We also consider the abelian Higgs model and show that its asymptotic canonicalsymmetries reduce to the Poincar´e group in an unproblematic fashion. ontents
With this paper we wish to continue our previous work [1] on the asymptotic structureand symmetries of Yang-Mills gauge theories, which was motivated by the seminal workof Henneaux and Troessaert on various field theories, including the pure electromagneticcase. We refer to [1] for a more extended list of references. In our previous paper weinvestigated the case of pure SU( N )-Yang-Mills-theory and found that globally-chargedstates cannot exist in that theory if its phase space is to be endowed with well-definedsymplectic structure, a well-defined Hamiltonian vector field driving its evolution, and awell-defined canonical (Poisson-) action of the whole identity-component of the Poincar´egroup. When we established this result, which we initially did not expect, we welcomed– 1 –t as a potential classical analogue (or sign) of confinement. At the same time, we askedourselves whether one must fear that this exclusion of globally-charged states may alsooccur in other field theories and in situations in which physics actually requires such statesto exist. Clearly, if that turns out to happen it would cast serious doubts on the methodwe employed. Hence we set ourselves the goal to check other field theories in order tosee — hopefully — how the Hamiltonian formalism leads to results compatible with theseexpectations.Our overall plan is to start with simple models, gradually including more fields of phys-ical significance. In doing that we follow the Hamiltonian strategy pioneered by Henneauxand Troessaert, who already made a detailed investigation into the electromagnetic case [2](even in higher dimensions [3]). As an obvious generalisation of their work we decided tolook at the case of electromagnetism coupled to a scalar field. This is what the presentpaper is about.More precisely, we deal with two main cases, with two subcases in the first. In the firstmain case we consider what is commonly referred to as “scalar electrodynamics”. That is, ascalar field endowed with a potential which, depending on its precise form, represents eithera massless (first subcase) or a massive (second subcase) scalar field, minimally-coupled tothe electromagnetic fields. Interestingly, the outcome of our analysis crucially depends onwhether or not the scalar field has a mass. We show that a massive field has to decay atinfinity faster than any power-like function in the affine coordinates, so that the behaviourof the electromagnetic fields, as well as the symmetry group, is the same as the one foundby Henneaux and Troessaert in the case of free electrodynamics [2]. On the other hand,a massless scalar field renders the boosts of the Poincar´e transformations non canonicalin a way which is difficult to circumvent, leading either to a trivial asymptotic symmetrygroup or to a non-canonical action of the Poincar´e group. We highlight a connection ofthis problem with the impossibility of a Lorenz gauge-fixing if the flux of charge-current atnull infinity is present, as pointed out by Satishchandran and Wald [4]. All this is derivedin Section 4As our second main case we consider the abelian Higgs model, i.e., a potential of thescalar field which leads to spontaneous symmetry breaking, thereby reducing the U(1)gauge-symmetry group to the trivial group. We show that the asymptotic symmetry groupreduces in a straightforward way to the Poincar´e transformations without any complica-tions. All this is derived in Section 5.Section 2 sets up the Hamiltonian formalism for the present context and shows howto canonically implement the Poincar´e action. Section 3 introduces the scalar-field modelswith a brief digression of the free scalar field for illustrative purposes. The final Section 6concludes and also contains some speculations on what might happen in the physicallyrelevant electroweak case. Appendix A contains the proof of the statement that in themassive case the scalar field as well as its momentum fall-off faster than any power in theaffine coordinates. – 2 – onventions and notation Throughout this paper, we adopt the following conventions. Lower-case Greek indicesdenote spacetime components, e.g. α = 0 , , ,
3, lower-case Latin indices denote spatialcomponents, e.g. a = 1 , ,
3, and lower-case barred Latin indices denote angular compo-nents, e.g. ¯ a = θ, ϕ . We adopt the mostly-plus convention ( − , + , + , +) for the spacetimefour-metric g . In this section, we provide the basic tools that are necessary for the Hamiltonian treatmentof an abelian gauge field minimally-coupled to a scalar field. Specifically, we are going toidentify the canonical fields and momenta, derive the Hamiltonian and the symplecticform, and infer the action of the Poincar´e transformations on the canonical fields. We willpostpone to the next sections the discussion concerning whether or not the derived quantityare well defined, which usually amounts to the correct identification of the phase space —by imposing fall-off and parity conditions — and to the completion of the Hamiltoniangenerators and of the symplectic form by adequate surface terms. For now, we will assumethat these quantities are well defined, in order to allow the following formal manipulations.We start our discussion from the more-familiar Lagrangian picture, in which the actionreads S [ A α , ˙ A α , ϕ, ˙ ϕ ; g ] = Z d x p − g (cid:20) − g αγ g βδ F αβ F γδ − g αβ (cid:0) D α ϕ (cid:1) ∗ D β ϕ − V ( ϕ ∗ ϕ ) (cid:21) ++ (boundary terms) , (2.1)where A is the one-form abelian potential, F := dA is the curvature (or field strength)two-form, ϕ is a complex scalar field, and g is the four-dimensional flat spacetime metric.Moreover, D α ϕ := ∂ α ϕ + iA α ϕ (2.2)is the gauge-covariant derivative in the fundamental representation and the potential V ( ϕ ∗ ϕ ) is explicitly given by the expression V ( ϕ ∗ ϕ ) := − µ ϕ ∗ ϕ + λ ( ϕ ∗ ϕ ) , (2.3)where λ and µ are two real parameters. In this paper, we wish to analyse two specificsituations, which arise depending on the value of these two parameters.The former situation is scalar electrodynamics . Namely, it corresponds to the case inwhich the two parameters appearing in the potential (2.3) are such that m := − µ ≥ λ ≥
0. As we shall see in section 4, there are going to be some important differencesin the asymptotic structure of the theory depending on whether we are dealing with amassless scalar field ( m = 0) or with a massive one ( m > λ > abelian Higgs model . This corre-sponds specifically to the case in which the two parameters appearing in the potential (2.3)are such that µ > λ >
0. This choice leads to the well-known Mexican-hat shapeof the potential and, ultimately, to the spontaneous symmetry break of the U(1) gaugesymmetry. Finally, let us point out that we have included an undetermined boundary term in theaction (2.1), which ought to be chosen such that the variation principle is well defined.For now, we merely assume that a boundary leading to a well-defined action principleexists and postpone to the next sections a thorough discussion about whether or not thisassumption is in fact correct.
The starting point of the Hamiltonian analysis is the (3+1) decomposition of the spacetimeand of its tensor fields. In particular, the four-dimensional spacetime manifold M splitsinto R × Σ, with Σ ∼ R being the three-dimensional manifold of constant time (spatialslice), and the flat four-metric g is (3 + 1)-decomposed into g αβ = − N + g ij N i N j N b N a g ab ! . (2.4)Here we introduced the spacetime functions of lapse N and shift N , whose componentswill be denoted by N a . Certainly, if we were interested only in finding the time evolutionof the fields, it would possible — and simpler — to set N = 1 and N = 0. In this case,the Dirac procedure [5] would lead us from the action (2.1) to the Hamiltonian H , whichwould generate the time evolution of the fields through the Poisson brackets. However,since we wish to derive, other than the time evolution of the fields, also their behaviourunder the Poincar´e transformations, it is more convenient to work with generic lapse andshift. If one does so, indeed, the same procedure that let us derive the Hamiltonian H from the action (2.1) produces a Hamiltonian generator H [ N, N ], which now dependson the lapse and shift. In turn, the new generator H [ N, N ] produces through the Poissonbrackets the transformations of the fields under the hypersurface deformation parametrisedby the vector field ( N, g ij N j ). For simplicity, we will refer to H [ N, H ] as “the generator ofhypersurface deformations”, instead of “the generator of the transformations of the fieldsunder hypersurface deformations”.In this paper, we are interested in two specific cases. First, the case in which thehypersurface deformation coincide with a unit time translation, i.e., N = 1 and N = 0. Inthis case, the generator H [1 ,
0] coincides with the Hamiltonian H and the transformationsof the fields obtained though the Poisson brackets are the equations of motion. Second,the case in which the hypersurface deformation is given by N = ξ ⊥ and N i = g ij ξ j , being ξ ⊥ and ξ the orthogonal and tangential part of the Poincar´e transformations, respectively. Note that the case λ < λ = 0 when µ > – 4 –or the explicit value and the derivation of ( ξ ⊥ , ξ ) see Section 3 of [1] or Section 2 of [2].For the following discussion, it suffices to know that, in radial-angular coordinates, theyare explicitly ξ ⊥ = rb ( x ) + T , ξ r = W ( x ) , ξ ¯ a = Y ¯ a ( x ) + 1 r γ ¯ a ¯ m ∂ ¯ m W ( x ) , (2.5)where b ( x ) is responsible for the Lorentz boosts and is an odd function of the sphere, T isresponsible for the time translations and is a constant, W ( x ) is responsible for the spatialtranslations, and Y ( x ) is the Killing vector field of the metric of the unit two-sphere γ andis responsible for the rotations.Before we actually derive the H [ N, N ], let us note three facts. First, the transforma-tion of the fields under tangential hypersurface deformations (parametrised by N ) can bedetermined from geometrical considerations. Specifically, one needs merely to require thatthe tangential transformations are given by Lie derivatives. Therefore, the ensuing discus-sion and computations would be symplified by setting N = 0. Nevertheless, we prefer topresent the full derivation leaving an arbitrary shift N in the remainder of this section. Forcomparison, see [6, Sec. 3], where H [ N, N ] is derived in the case of free electrodynamicson a spacetime manifold M = R × Σ, being Σ a three-dimensional closed manifold. Thoseresults can be readily applied to our situations up to boundary terms, which are triviallyabsent in [6]. It is worth noting already at this point that obstructions to a well-definedHamiltonian action of the Poincar´e group are usually caused by the boost in the orthogonaldeformation ξ ⊥ , so that one should usually pay more attention to the contribution due to N rather than the one due to N .Secondly, although we are dealing with flat Minkowski spacetime, it is more convenientto leave the three-metric g in general coordinates for now. Later on, we will express it inradial-angular coordinates, but there is no advantage in doing it at this stage. From nowon, spatial indices are lowered and raised using the three-metric g and its inverse.Thirdly, the complex scalar field ϕ can be decomposed into ϕ = 1 √ ϕ + iϕ ) (2.6)where ϕ and ϕ are two real scalar fields. Although this replacement makes some expansionless compact, it also makes clearer which are the actual degrees of freedom, with respect towhich we have to vary the action. In the following discussion, we will express the resultseither in terms of the complex scalar field ϕ or in terms of the two real scalar fields ϕ and ϕ , depending on which of the two approaches is more convenient in each situation.– 5 –he action (2.1) becomes S = R dt L [ A, ˙ A, ϕ, ˙ ϕ ; g, N, N ], where the Lagrangian is L = Z d xN √ g (cid:26) N g ab F a F b + g ab N c N F a F bc − F ab F ab + g ac N b N c N F ab F cd ++ 12 N h ( ˙ ϕ − A ϕ ) + ( ˙ ϕ + A ϕ ) i + − N a N [( ˙ ϕ − A ϕ )( ∂ a ϕ − A a ϕ ) + ( ˙ ϕ + A ϕ )( ∂ a ϕ + A a ϕ )] + − (cid:18) g ab − N a N b N (cid:19) h ( ∂ a ϕ − A a ϕ )( ∂ b ϕ − A b ϕ ) + ( ∂ a ϕ + A a ϕ )( ∂ b ϕ + A b ϕ ) i + − V ( ϕ ∗ ϕ ) (cid:27) + (boundary terms) . (2.7)First of all, let us note that the above expression does not contain any time derivativeof A . As a consequence, we will not include A into the canonical fields, as it doesnot evolve in time, and treat it as a Lagrange multiplier. We will discuss in the nextsection the consequence of this choice in the next subsection. Secondly, the variation ofthe Lagrangian (2.7) with respect to ˙ A a yields the conjugate three-momenta π a := δLδ ˙ A a = √ gN g ab (cid:0) F b + N m F bm (cid:1) , (2.8)which are vector densities of weight +1. Thirdly, the variation of the Lagrangian withrespect to the real scalar fields gives the further three-momentaΠ := δLδ ˙ ϕ = √ gN h ˙ ϕ − A ϕ − N m ( ∂ m ϕ − A m ϕ ) i and (2.9)Π := δLδ ˙ ϕ = √ gN h ˙ ϕ + A ϕ − N m ( ∂ m ϕ + A m ϕ ) i , (2.10)which are scalar densities of weight +1 and can be rewritten in the more compact formΠ := 1 √ (cid:0) Π + i Π (cid:1) = √ gN h D ϕ − N m D m ϕ i . (2.11)Finally, the symplectic form, from which the Poisson brackets ensue, isΩ[ A a , π a , ϕ, Π] = Z d x (cid:16) d π a ∧ d A a + d Π ∧ d ϕ + d Π ∧ d ϕ (cid:17) + (boundary terms) , (2.12)where the bold d and ∧ are, respectively, the exterior derivative and the wedge product inphase space. Note that we are allowing the standard symplectic form to be complementedby a boundary term, which could emerge as a consequence of the boundary term includedin the action. A detailed discussion about boundary terms will be done in the next sec-tions. Before we complete the derivation of the generator H [ N, N ] and provide its explicitexpression, let us briefly discuss the consequences of treating A as a Lagrange multiplier.– 6 – .2 Constraints and constraints’ algebra The consequence to the fact that no time derivative of A appears in the action is that thevariation of the latter with respect to said field takes the form R d x G d A , so that oneis lead to the conclusion that G should equate 0. However, since the equation does notdescribe a time evolution of some field, but rather a condition that the fields need to satisfyat each time, one has to treat it as a constraint equation [5], which identifies a surface inphase space, on which the solution to the equations of motion lies. To stress this fact, oneusually write G ≈
0, instead of G = 0. Explicitly, the variation of (2.7) with respect to A yields G := ∂ a π a + ϕ Π − ϕ Π ≈ , (2.13)which is the well-known Gauss constraint in the presence of a charge density provided bythe complex scalar field.In theory, in order for the Hamiltonian picture to be consistent, the constraints needto be preserved by time evolution — up to constraints — and, in the case of Poincar´einvariant theories, by Poincar´e transformations as well. If this does not happen, one needsto proceed following the Dirac algorithm [5].In practice, this is readily satisfied in our case. Indeed, although we have not giventhe explicit value of the Hamiltonian yet, let us anticipate that the variation of the Gaussconstraint is δ G = { G , H [ N, N ] } = 0 . (2.14)This shows that we have found all the constraints of the theory, namely the Gauss constraint G , as this is preserved not only by time evolution, but also by the Poincar´e transformations.Finally, one can trivially verify that the constraint is first class and, more precisely,satisfy the abelian algebra { G ( x ) , G ( x ′ ) } = 0 . (2.15)Having determined all the constraints of the theory, we can now complete the deriva-tion of H [ N, N ], which includes both the Hamiltonian and the generator of the Poincar´etransformations. The generator H [ N, N ] can be readily obtained by means of the Legendre transformationof the Lagrangian, i.e. H := R d x ( π α ˙ A α + Π ˙ ϕ + Π ˙ ϕ ) − L , in which one replaces ˙ A a with π a by means of (2.8) and ˙ ϕ , with Π , by means of (2.9) and (2.10), respectively. Inaddition, following [6, Sec. 3] up to a sign, we define A ⊥ := 1 N ( A − N m A m ) (2.16)At this point, we are finally able to write down explicitly the generator of hypersurfacedeformations H [ A, π, ϕ, Π; g, N, N ; A ⊥ ] = Z d x h N H + N i H i i + (boundary terms) , (2.17)– 7 –here H := π a π a + Π + Π √ g + √ g F ab F ab − A ⊥ G + √ g g ab (cid:0) ∂ a ϕ ∂ b ϕ + ∂ a ϕ ∂ b ϕ (cid:1) ++ √ gA a (cid:0) ϕ ∂ a ϕ − ϕ ∂ a ϕ (cid:1) + 12 A a A a (cid:0) ϕ + ϕ (cid:1) + √ g V ( ϕ ∗ ϕ ) (2.18)is responsible for the orthogonal transformations and H i := π a ∂ i A a − ∂ a ( π a A i ) + Π ∂ i ϕ + Π ∂ i ϕ (2.19)is responsible for the tangential transformations. Note that the Gauss constraint (2.13)appears in the generator (2.17) multiplied by the Lagrange multiplier A ⊥ .The knowledge of the symplectic form (2.12) and of the generator of hypersurfacedeformations (2.17) allows us to determine how the fields vary infinitesimally under saiddeformation. Specifically, the infinitesimal change of the fields under a transformation isrepresented by a vector field X = ( δA a , δπ a , . . . ) in phase space. Knowing the symplecticform (2.12) and the generator (2.17), the corresponding vector field can be determined bymeans of the equation d H = − i X Ω, which, in general, might be impossible to fulfil due tothe presence of boundary terms in the variation d H which cannot be compensated by theinclusion of boundary terms in the symplectic structure. If we neglect this issue for themoment, as it will be thoroughly discussed in the next sections, we find δA a = N π a √ g + ∂ a ( N A ⊥ ) + L N A a , (2.20) δπ a = ∂ b ( N √ g F ba ) − √ g N Im ( ϕ ∗ D a ϕ ) + L N π a , (2.21) δϕ = N Π √ g − i ( N A ⊥ ) ϕ + L N ϕ , (2.22) δ Π = D a ( √ g N D a ϕ ) + √ g N (cid:16) µ − λ | ϕ | (cid:17) ϕ − i ( N A ⊥ )Π + L N Π , (2.23)where L N is the three-dimensional Lie derivative with respect to N i and we have chosento use the more compact complex notation. The above equations reduce to the equationsof motion when N = 1 — in which case the left-hand sides become the time derivative ofthe fields — and to the Poincar´e transformations when N = ξ ⊥ = rb ( x ) + T and N i = ξ i . The presence of the Gauss constraints (2.13) in the Hamiltonian (2.17) causes the trans-formations (2.20)–(2.23) to include a gauge transformation, whose gauge parameter is thearbitrary function ζ := − N A ⊥ . In order to ensure the uniqueness of solutions despite thearbitrariness of ζ , one needs to treat this transformations as mere relabelling of a physicalstate, i.e., a redundancy in the mathematical description of the theory.The infinitesimal form of the gauge transformations, which we can read from thetransformations (2.20)–(2.23), is δ ζ A a = − ∂ a ζ , δ ζ π a = 0 , δ ζ ϕ = iζϕ , and δ ζ Π = iζ Π . (2.24)– 8 –pecifically, these are generated by G [ ζ ] = Z d x ζ ( x ) G ( x ) (2.25)through the equation d G [ ζ ] = − i X ζ Ω. The left-hand side of this equation can be readilycomputed to be d G [ ζ ] = Z d x h − ∂ a ζ d π a + ζ Π d ϕ − ζ Π d ϕ − ζϕ d Π + ζϕ d Π i ++ lim R →∞ I S R d x k ζ d π k . (2.26)Assuming that the symplectic form (2.12) does not contain any boundary term, the equa-tion d G [ ζ ] = − i X ζ Ω is fulfilled so long as the boundary term in the above expression isactually zero. Whether or not this is the case, and for which class of functions ζ ( x ) thishappens, vastly depends on the asymptotic behaviour of the fields. We will discuss this inthe next sections and we will see that the asymptotic behaviour of the fields changes de-pending on the choice of parameters in the potentials (2.3), i e., on whether we are dealingwith scalar electrodynamics or with the abelian Higgs model. For now, let us note thatthe generator G [ ζ ] can be in general extended to G ext. [ ζ ] = G [ ζ ] − lim r →∞ I S r d x k ζπ k (2.27)whose variation, now, does not contain any boundary term. In the case in which theboundary term in the above expression is non-trivial, the transformations correspondingto G ext. [ ζ ] are not, in fact, proper gauge transformations. Rather they are true symmetriesof the theory relating physically-different states and are commonly referred to as impropergauge transformations , following [7]. In addition, one can define the charge Q [ ζ ] := lim r →∞ I S r d x k ζπ k , (2.28)which implies G ext. [ ζ ] = G [ ζ ] − Q [ ζ ] ≈ − Q [ ζ ], so that one can tell whether a transformationis a proper gauge or an improper one by checking whether the charge is zero or not, respec-tively. Whether or not there is a non-trivial class of functions ζ ( x ) such that impropergauge transformations exist and have a well-defined action on phase space depends, again,on the asymptotic behaviour of the fields, which will be discussed in the next sections.Finally, let us point out that the expressions for the infinitesimal gauge transforma-tions (2.24) can be integrated to get the finite form of gauge transformationsΓ ζ ( A a ) = A a − ∂ a ζ , Γ ζ ( π a ) = π a , Γ ζ ( ϕ ) = e iζ ϕ , and Γ ζ (Π) = e iζ Π , (2.29) Note that, due to the limit in the definition (2.28), the charge depends only on the asymptotic valuesof the fields and of the gauge parameter ζ , which are going to be thoroughly discussed in the next sections.Let us anticipate that the asymptotic part of ζ () is going to be denoted by ζ ( x ), which is a function onthe two-sphere at infinity. Thus, the charge can be written simply as Q (cid:2) ζ (cid:3) , which is usually decomposedinto spherical-harmonics components. Specifically, one defines Q ℓm := Q [ Y ℓm ] in terms of the sphericalharmonics Y ℓm . The component Q corresponds to the global (electric) charge. – 9 –here Γ ζ denotes the action of e iζ ( x ) ∈ U(1) on the fields, i.e. A a A ′ a = Γ ζ ( A a ) forinstance. This clearly show the U(1) nature of the gauge symmetry.In the next sections, we are going to discuss the specific cases of scalar electrodynamicsand of the abelian Higgs model. Before that, in the next section, we are going to brieflydiscuss the case of a free scalar field with the potential (2.3), as this simple situation letus highlight some of the features of the asymptotic structure. Let us study first the behaviour of the complex scalar field when it is not coupled tothe gauge potential. This can be achieved by considering the equations (2.20)–(2.23) andsetting to zero the values of the gauge potential A a , the conjugated momenta π a , and theLagrange multiplier A ⊥ , obtaining δϕ = N Π √ g + L N ϕ , (3.1) δ Π = ∇ a ( √ gN ∂ a ϕ ) + N √ g (cid:16) µ − λ | ϕ | (cid:17) ϕ + L N Π . (3.2)Finally, the generator of hypersurface deformations (2.17) reduces to H scalar [ ϕ, Π; g, N, N ] = Z d x (cid:26) N (cid:20) Π √ g + √ gg ab ∂ a ϕ ∗ ∂ b ϕ + √ g (cid:16) − µ | ϕ | + λ | ϕ | (cid:17)(cid:21) ++ 2 N i Re (cid:0) Π ∗ ∂ i ϕ (cid:1)o + (boundary terms) , (3.3)Let us analyse separately the two different scenarios in the next two subsections. First,we will consider the case in which m := − µ ≥ λ ≥
0. This describes a massive( m >
0) or massless ( m = 0) complex scalar field with ( λ >
0) or without ( λ = 0)a self interaction. This will be useful when studying scalar electrodynamics in section 4.Secondly, we will consider the case in which µ > λ >
0, so that the potential takesthe well-known Mexican-hat shape. This will be relevant in the analysis of the abelianHiggs model in section 5.
Let us first consider the case of a scalar field with squared mass m := − µ ≥
0. The selfinteraction is either present or not, i.e., λ ≥
0. Note that the equation of motion (3.1)–(3.2)contain the trivial solution ϕ (0) ( x ) = 0 and Π (0) ( x ) = 0 , (3.4)which is also the solution that minimises the potential (2.3) and the energy. Indeed,neglecting the boundary, the value of the Hamiltonian for this solution is E (0) := H scalar [ ϕ (0) , Π (0) ; g, N = 1 , N = 0] = 0 , (3.5)whereas the value of the Hamiltonian for any other field configuration is positive.– 10 –t this point, we use a power-like ansatz for the fall-off behaviour of the field and thepotential. In detail, we assume that they behave as ϕ ( x ) = 1 r α ϕ ( x ) + O (cid:0) /r α +1 (cid:1) and Π( x ) = 1 r β Π( x ) + O (cid:0) /r β +1 (cid:1) (3.6)in radial-angular coordinates. Whether or not α and β can be found, such that the fall-offconditions are preserved by the Poincar´e transformations, depends crucially on the value ofthe mass. More precisely, in the massless case, i.e. m = 0, one finds the fall-off conditions ϕ massless ( x ) = 1 r ϕ ( x ) + O (cid:0) /r (cid:1) and Π massless ( x ) = Π( x ) + O (cid:0) /r (cid:1) , (3.7)which also make the symplectic form logarithmically divergent.Before we discuss the massive case, let us point out that, in order to make the sym-plectic form actually finite, one needs to impose parity conditions on the asymptotic partof the fields in addition to the aforementioned fall-off conditions. Specifically, it suffices,for instance, to require that ϕ ( x ) is either an even or an odd function of the sphere underthe antipodal map and, at the same time, that Π( x ) has the opposite parity. In this way,the potentially logarithmically divergent term in the symplectic form is, in fact, zero. It iseasy to check that these parity conditions are preserved by the Poincar´e transformations,which take the asymptotic form δ ξ ϕ = b Π √ γ + Y ¯ m ∂ ¯ m ϕ , (3.8) δ ξ Π = − b p γ ϕ + ∇ ¯ m (cid:16)p γ b∂ ¯ m ϕ (cid:17) − b p γ λ | ϕ | ϕ + ∂ ¯ m (cid:0) Y ¯ m Π (cid:1) , (3.9)where ∇ denotes the covariant derivative of the round unit sphere, b parametrises theLorentz boost, and Killing vector field Y of the round-unit-two-sphere metric γ parametrisesthe rotations. We will come back to the discussion about parity conditions in section 4where we will consider the couple of the scalar field to electrodynamics.In the massive case, the appearance of a new term proportional to m > ϕ and Π need to be function approaching zero at infinity faster thanany power-like function. Hence, we will restrict the phase space by requiring that both ϕ and Π are quickly-falling functions. In details, we will require that the scalar field ϕ , aswell as its spatial derivatives up to second order, and the momentum Π vanish in the limitto spatial infinity faster than any power-like functions (in Cartesian coordinates). Note We remind that the antipodal map, denoted hereafter by Φ : x
7→ − x , consists in the explicit transfor-mation ( θ, φ ) ( π − θ, φ + π ) in terms of the standard spherical coordinates. A generic tensor field T (ora density) is said to be even under the antipodal map if Φ ∗ T = T , being Φ ∗ the pull back of the antipodalmap. Analogously, T is odd if Φ ∗ T = − T . To see how this translate into the exact parity of the componentsof a tensor field (or density) expressed in some coordinates like the standard ( θ, φ ) spherical coordinates,see footnote 2 of [2]. We will refer to this fall-off behaviour of the scalar field and its momentum and to similar behavioursencountered in the remainder of this paper by saying that the fields are “quickly vanishing (at infinity)”. – 11 –hat, due to these fall-off conditions, the Hamiltonian and the generator of the Poincar´etransformations of the massive scalar field are finite and functionally differentiable withrespect to the canonical fields without any need of a boundary term.Finally, let us note that the theory, both in the massless and in the massive case,possesses the global U(1) symmetry (cid:2) Γ ζ ( ϕ ) (cid:3) ( x ) = e iζ ϕ ( x ) , and (cid:2) Γ ζ (Π) (cid:3) ( x ) = e iζ Π( x ) , (3.10)where Γ ζ denotes the action of e iζ ∈ U(1) on the fields. Note that, differently from (2.29),the action is that of the global U(1), i.e., the parameter e iζ ∈ U(1) is the same at eachspacetime point. The infinitesimal version of the above transformations is generated by G [ ζ ] = Z d x ζ h ϕ ( x )Π ( x ) − ϕ ( x )Π ( x ) i , (3.11)where, again, ζ is independent of x . Note that the above generator is always finite anddifferentiable, i.e. d G [ ζ ] = − i X ζ Ω. In additions, it is not proportional to a constraint, asthe theory of a free scalar field (with a potential) does not possess any constraint. It canbe easily verified that the generator above Poisson-commutes with Hamiltonian, i.e. n G [ ζ ] , H [ N = 1 , N = 0] o := i X G (cid:0) i X H Ω (cid:1) = 0 , (3.12)showing that it generates indeed a symmetry.To sum up, in this subsection, we have studied the fall-off conditions of a complexscalar field and its conjugated momentum with or without a quartic self-interaction. Thefall-off behaviour of the field and the momentum crucially depends on whether or not themass is zero. On the one hand, in the massless case, the fall-off conditions are power-likeand, precisely, the ones in (3.7). On the other hand, in the massive case, the scalar field —as well as its spatial derivatives up to second order — and its momentum need to vanishat spatial infinity faster than any power-like function. In addition, we have seen that thetheory possesses a global U(1) symmetry. Let us now consider the case of a scalar field with a Mexican-hat potential µ ≥ λ > λ has to be strictly positive for, otherwise,the potential and, as a consequence, the Hamiltonian are not bounded from below. As inthe cases of a massive and massless scalar field, the equation of motion (3.1)–(3.2) containthe trivial solution ϕ (0) ( x ) = 0 and Π (0) ( x ) = 0 . (3.13)However, this is not any more the solution that minimises the potential V ( ϕ ∗ ϕ ) and theHamiltonian. Indeed, this solution is found at a local maximum of the potential and givesthe value of the Hamiltonian E (0) := H scalar [ ϕ (0) , Π (0) ; g, ,
0] = 0 . (3.14) In the massless case, the generator is finite thanks to the combination of the fall-off and parity conditions.The former alone would make the generator logarithmically divergent. – 12 –n this case, the potential and the Hamiltonian are minimised by the constant solutionsto the equations of motion ϕ ( ϑ ) ( x ) = v √ e iϑ and Π ( ϑ ) ( x ) = 0 , (3.15)where the parameter ϑ belongs to R / π and v := p µ /λ . On all these solutions, neglectingthe boundary, the Hamiltonian takes the same value E ( ϑ ) := H scalar [ ϕ ( ϑ ) , Π ( ϑ ) ; g, ,
0] = − Z d x √ g λ v , (3.16)which diverges to −∞ , since it is the integral of a negative constant over a spatial sliceΣ ∼ R . This means that we would not be able to include the solutions (3.15) if we wishedto have a well-defined, i.e. finite and functionally-differentiable, Hamiltonian.The solution to this issue is quite simple. We merely need to redefine the generator ofhypersurface deformations (3.3) and, as a consequence, the Hamiltonian to H ′ scalar [ ϕ, Π; g, N, N ] = Z d x N (cid:26) Π √ g + √ gg ab ∂ a ϕ ∗ ∂ b ϕ + √ g (cid:16) λ v − µ | ϕ | + λ | ϕ | (cid:17) ++ 2 N i Re (cid:0) Π ∗ ∂ i ϕ (cid:1)o + (boundary terms) . (3.17)This amounts to nothing else than the addition of the constant λv / E ′ ( ϑ ) := H ′ scalar [ ϕ ( ϑ ) , Π ( ϑ ) ; g, ,
0] = 0 , (3.18)whereas the value is positive for any other field configuration. Note that, however, thevalue of the Hamiltonian evaluated on the trivial solution (3.13) is now divergent. As aconsequence, we need to remove this solution from the allowed field configuration, but thisdoes not have a huge impact on the physical side, as (3.13) is on a local maximum of thepotential and, thus, unstable under perturbations.Let us now discuss the fall-off conditions of the field and its conjugated momentum.Although most of the discussion does not differ much from the case of the massive scalarfield discussed in the previous subsection, there are nevertheless a few subtleties that oneshould take into consideration. We will work in radial-angular coordinates.First, let us focus on the terms in the Hamiltonian (3.17) containing the potential V ( ϕ ∗ ϕ ) with the newly-added constant λv /
4. If we wish this part to be finite uponintegration, we need to require the absolute value of the field | ϕ ( x ) | to approach the value v/ √ r → ∞ . In other words, this means that, if we write ϕ ( x ) = 1 √ ρ ( x ) e iϑ ( x ) , (3.19)then ρ ( x ) = v + h ( x ), where h ( x ) vanishes in the limit r → ∞ . Note that, in principle, weallow the phase ϑ ( x ) to be non-constant. Nevertheless, we require that it has a well-definedlimit ϑ ( x ) := lim r →∞ ϑ ( x ) as a possibly non-constant function on the sphere at infinity.– 13 –econdly, let us note that ϕ ( x ) is not vanishing in a neighbourhood of spatial infinity,so that we can always write, and it is convenient to do so,Π( x ) = (cid:16) u ( x ) + iw ( x ) (cid:17) ϕ ( x ) , (3.20)where u ( x ) and v ( x ) are both real. From the transformation of ϕ , one can easily find thetransformations of its absolute value and phase as δρ = ρ Re (cid:18) δϕϕ (cid:19) and δϑ = Im (cid:18) δϕϕ (cid:19) . (3.21)Analogously, the transformations of u and w can be obtained from those of Π and ϕ , as δu = Re (cid:18) ϕ δ Π − Π δϕϕ (cid:19) and δw = Im (cid:18) ϕ δ Π − Π δϕϕ (cid:19) . (3.22)Thirdly, we can show that h ( x ), u ( x ), and w ( x ) need to fall off at infinity faster thanany power-like functions. A precise proof of this statement would require us to proceed asin appendix A. Instead, let us here provide a less rigorous argumentation. Specifically, letus assume the power-like behaviours h ( x ) = 1 r α h ( x ) + o (1 /r α ) , u ( x ) = 1 r β u ( x ) + o (1 /r β ) , w ( x ) = 1 r γ w ( x ) + o (1 /r γ ) . (3.23)Note that α need to be greater than zero, since we requested h to vanish as r tends toinfinity. Now, let us consider only a part of the Poincar´e transformations (2.20)–(2.23)Namely, δ ′ ϕ = ξ ⊥ Π √ g and δ ′ Π = √ g ξ ⊥ (cid:16) µ − λ | ϕ | (cid:17) ϕ , (3.24)from which we can derive the corresponding transformations of h , u , and v using (3.21)and (3.22), obtaining δ ′ h = ξ ⊥ √ g u , δ ′ u = − vξ ⊥ √ gλ (cid:18) h − h v (cid:19) − ξ ⊥ √ g (cid:0) u − w (cid:1) , δ ′ w = − ξ ⊥ √ g uw . (3.25)At this point, we insert (3.23) in the above expressions and expand everything in powersof r , including √ g = r √ γ and ξ ⊥ = rb + T . Requiring that the fall-off conditions (3.23)are preserved, i.e., that the terms on the right-hand side of the above expressions do notfall off slower than the respective field, we find that the exponents in the power-like ansatzneed to satisfy the non-trivial inequalities β + 1 ≥ α , α − ≥ β , γ + 1 ≥ β , β + 2 ≥ , (3.26)where the first inequality comes from the transformation of h , the last from that of w , andthe remaining two from that of u . One sees immediately that the first two inequalities leadto the contradiction α ≤ β + 1 ≤ α − , (3.27)– 14 –hich would lead to the conclusion that h and u are quickly vanishing at infinity, if oneproceeded like in appendix A. Furthermore, the third inequality in (3.26) would lead us tothe conclusion that also w is quickly vanishing.Lastly, let us note that the conditions that we have determined so far show us that Πand h need to fall-off at infinity faster than any power-like function. However, we have stillto determine the fall-off behaviour of the phase ϑ ( x ). To do so, it suffices to consider thetransformation of Π under time evolution, i.e., equation (2.23) at N = 1 and N = 0. Upto terms that are quickly vanishing at infinity, we find δ Π = ϕ h −√ g ∂ a ϑ g ab ∂ b ϑ + i∂ a (cid:16) √ g g ab ∂ b ϑ (cid:17)i + (quickly-vanishing terms) . (3.28)Thus, we have to impose that ∂ a ϑ is quickly vanishing in order to preserve the fall-offcondition of Π. This leads us to two fact. First, the asymptotic part ϑ ( x ) needs to beconstant on the sphere at infinity. We will simply denote it with ϑ . Second, if we write ϑ ( x ) = ϑ + χ ( x ) /v , we will find out that χ ( x ) is quickly vanishing at infinity, as well as itsderivatives up to second order. We will see in section 5 that this situation changes whena gauge potential is present, as in the abelian Higgs model. Finally, note that, from thePoincar´e transformation of ϑδϑ = Im (cid:18) δϕϕ (cid:19) = Im (cid:18) ξ ⊥ Π √ gϕ + L N ϕϕ (cid:19) , (3.29)we infer that ϑ is invariant under the Poincar´e transformations and, in particular, is timeindependent. Indeed, the first summand on the left-hand side of the above expression isclearly quickly vanishing in the limit r → ∞ , while the second summand reduces to L N χ/v which, too, is quickly vanishing.This concludes the derivation of the fall-off conditions of a complex scalar field with aMexican-hat potential. In short, we have shown that, when one considers the Mexican-hatpotential as in the case of the Higgs mechanism, the generator of hypersurface deformationsand the Hamiltonian needs to be modified to (3.17) by adding a constant to the potential,so that the minimal-energy solutions (3.15) to the equations of motion have finite energy.The phase space is then defined by all those fields and momenta, whose difference from oneof the minimum-energy solutions vanishes at infinity faster than any power-like function.As in the case of the scalar massive field, one has to require the quick fall-off of the fieldup to the second-order spatial derivatives. Note that the asymptotic part of the phase ofthe scalar field ϑ needs to be constant on the sphere at infinity and is time-independent.Moreover, the phase ϑ can differ from its constant value at infinity by a function χ/v thatis quickly vanishing. This will not be the case when we reintroduce the gauge potential A a , as we shall see in section 5.Before we move to the study of scalar electrodynamics in section 4 and to that of theabelian Higgs model in section 5, let us make the connection with the usual interpretationof h and χ in high-energy physics. To this end, let us consider the action in the Lagrangianpicture, which can be obtained from (2.1) by setting A α = 0 and adding the constant λv /
4– 15 –o the potential. Rewriting this action in terms of h and χ , we obtain S [ h, χ ] = Z d x (cid:26) − (cid:16) g αβ ∂ α h ∂ β h + 2 µ h (cid:17) − g αβ ∂ α χ ∂ β χ + (interactions) (cid:27) , (3.30)where the interactions include all the terms that are not quadratic in the fields. From theabove expression, we read that h is a scalar field of squared mass m h := 2 µ , whereas χ is a massless scalar field. The latter is precisely the Goldstone boson of the spontaneouslybroken global U(1) symmetry. Indeed, as in the case analysed in the previous subsection,the theory possesses the symmetry (3.10) generated by (3.11). However, in this case, theminimum-energy solutions are not invariant under the action of the symmetry. Rather, thevacuum solution (cid:0) ϕ ( ϑ ) , Π ( ϑ ) (cid:1) is mapped to the different, physically-nonequivalent vacuumsolution (cid:0) ϕ ( ϑ + ζ ) , Π ( ϑ + ζ ) (cid:1) under the action of ζ ∈ U(1). In the abelian Higgs model analysedin section 5, the Goldstone boson χ will turn out to be pure gauge, i.e. physically irrelevant,whereas h will be the Higgs boson. In this section, we will discuss the asymptotic symmetries of scalar electrodynamics, thatis the case of a complex scalar field minimally coupled to electrodynamics. Specifically,this amount to consider the Hamiltonian (2.17) in the case in which the parameters in thepotential (2.3) are such that m := − µ ≥ λ ≥
0. The former parameter representthe (squared) mass of the scalar field and distinguishes between the massive case ( m > m = 0). The latter parameter regulates the magnitude of theself-interaction of the scalar field and is allowed, in principle, λ to be different from zero.The ensuing discussion vastly differs depending on whether the scalar field is massiveor massless. Therefore, we will keep separated the analyses of these two different situations.We will begin our discussion with the massive case, as this is significantly simpler and wewill dedicate to it the first subsection, showing that a well-defined Hamiltonian formulationwith non-trivial asymptotic symmetries can be found.The rest of the section is devoted to the massless case, which presents subtle compli-cations. We will start the discussion of this second case by deriving the fall-off and (strict)parity conditions of the fields and their momenta, which are going to provide a theorywith a finite symplectic form, a finite and functionally-differentiable Hamiltonian, and asymplectic action of the Poincar´e group. However, these conditions are a bit too strong, inthe sense that they do not allow for non-trivial asymptotic symmetries. We will attemptto relax the strict parity conditions and discuss which issues arise during the process, thatmake either the asymptotic symmetry group trivial or the Lorentz boost non-canonical.Finally, we will make the connection between these issues at spatial infinity and someproblems concerning the Lorenz gauge fixing encountered in analyses at null infinity. Let us begin with the derivation of the fall-off conditions of the fields. As it was done forfree electrodynamics in [2], for free Yang-Mills in [1], and for a free complex scalar field– 16 –n section 3, we are going to derive the fall-off conditions by demanding that they are themost general ones preserved by the action of the Poincar´e group (2.20)–(2.23).Focusing on the transformation of ϕ and Π and proceeding as in section 3.1, one canshow that the massive scalar field needs to vanish at infinity faster than any power-likefunction, as it happens in the free case. It is easy to verify, at this point, that the fall-offconditions of A and π are exactly those of free electrodynamics [2], that is A r ( r, x ) = 1 r A r ( x ) + O (1 /r ) , π r ( r, x ) = π r ( x ) + O (1 /r ) ,A ¯ a ( r, x ) = A ¯ a ( x ) + O (1 /r ) , π ¯ a ( r, x ) = 1 r π ¯ a ( x ) + O (1 /r ) , (4.1)where the results are expressed in radial-angular coordinates. In addition, the gauge pa-rameter is required to fall off as ζ ( x ) = ζ ( x ) + O (1 /r ) , (4.2)so that the gauge transformations (2.24) preserve the fall-off conditions of the canonicalfields. Note that this last expression, together with the definition (2.16) the known fall-offof N and N for the Poincar´e transformations, implies the fall-off A ⊥ ( r, x ) = 1 r A ⊥ ( x ) + O (1 /r ) (4.3)for the Lagrange multiplier.Since the scalar field and its momentum quickly vanish at infinity, the asymptoticstructure of the theory is effectively the same as in the free electrodynamics case. Thismeans that proceeding as in [2], one would find a well-defined Hamiltonian formulation ofmassive-scalar electrodynamics with a canonical action of the Poincar´e group, and withnon-trivial asymptotic symmetries, corresponding to an extension of the Poincar´e groupby the angle-dependent-U(1) transformations at infinity. As in the massive case, we begin with the derivation of the fall-off conditions of the fields. Inthis case, it is possible to find a power-law ansatz which is preserved by the Poincar´e trans-formations (2.20)–(2.23). Specifically, this corresponds to merging the fall-off conditions ofthe free massless scalar field (3.7) and of free electrodynamics (4.1). Also in this case, thegauge parameter is required to fall-off as in (4.2), so that the gauge transformations (2.24)preserve the fall-off conditions of the fields. The asymptotic Poincar´e transformations of– 17 –he fields are then found to be δ ξ,ζ A r = b π r √ γ + Y ¯ m ∂ ¯ m A r , (4.4) δ ξ,ζ A ¯ a = b π ¯ a √ γ + Y ¯ m ∂ ¯ m A ¯ a + ∂ ¯ a Y ¯ m A ¯ m − ∂ ¯ a ζ , (4.5) δ ξ,ζ π r = ∇ ¯ m (cid:0) b p γ ∂ ¯ m A r (cid:1) − b p γ | ϕ | A r + ∂ ¯ m ( Y ¯ m π r ) , (4.6) δ ξ,ζ π ¯ a = ∂ ¯ m (cid:0) b p γ F ¯ m ¯ a ) − b p γ Im (cid:16) ϕ ∗ D ¯ a ϕ (cid:17) + ∂ ¯ m ( Y ¯ m π ¯ a ) − ∂ ¯ m Y ¯ a π ¯ m , (4.7) δ ξ,ζ ϕ = b Π √ γ + Y ¯ m ∂ ¯ m ϕ + iζ ϕ , (4.8) δ ξ,ζ Π = − b p γ (cid:16) A r (cid:17) ϕ + D ¯ m (cid:16) b p γ D ¯ m ϕ (cid:17) − b p γ λ | ϕ | ϕ + ∂ ¯ m (cid:0) Y ¯ m Π (cid:1) + iζ Π , (4.9)where ∇ is the covariant derivative of the round unit two sphere and D ¯ m := ∇ ¯ m + iA ¯ m .The fall-off conditions are not enough to provide a finite symplectic form and a sym-plectic action of the Poincar´e group. In particular, the symplectic form (2.12) still containstwo logarithmically-divergent contributions: The first is due to the fall-off conditions of A and π , while the second is due to the fall-off conditions of ϕ and Π. One possible solutionto this issue is quite simple. One merely requires that the asymptotic part of the fieldshave one definite parity (either even or odd) as functions on the two-sphere at infinity and,then, imposes the opposite parity on their conjugated momenta. This way, the potentiallylogarithmically-divergent contributions to the symplectic form are actually zero. We willsee that the presence of the massless scalar field will cause the parity conditions to beslightly more involved.To fully determine the exact form of the parity conditions, let us remind that theyshould be such that, not only do they make the symplectic form finite, but also the Poincar´etransformations symplectic. Specifically, this happens when L X Ω = 0, being L X theLie derivative in phase space with respect to the vector field X defining the Poincar´etransformations (2.20)–(2.23). Using Cartan magic formula and the fact that the symplecticform is closed, one gets L X Ω = d ( i X Ω) = d I d x p γ A r h d ∇ ¯ m (cid:0) b A ¯ m (cid:1) + 2 b Im ( ϕ ∗ d ϕ ) i , (4.10)after having simplified the expression. Note that the first summand in the right-hand sideof the above expression is precisely the term already appearing in free electrodynamics [2],while the second summand appears due to the presence of the massless scalar field. Wewish to impose parity conditions that make the above expression to vanish identically. Tothis end, let us decompose the complex scalar field as ϕ ( x ) = 1 √ ρ ( x ) e iϑ ( x ) . (4.11)The newly-introduced absolute value and the phase of the scalar field need to satisfy thefall-off conditions ρ ( x ) = 1 r ρ ( x ) + O (1 /r ) and ϑ ( x ) = ϑ ( x ) + O (1 /r ) , (4.12)– 18 –n order to be consistent with (3.7). Rewriting (4.10) in terms of these new fields, we seethat the Poincar´e transformations are canonical if L X Ω = d I d x p γ A r h d ∇ ¯ m (cid:0) b A ¯ m (cid:1) + b ρ d ϑ i (4.13)vanishes. This can be achieved in the following way. First, we require the parity conditions A r = A odd r and π r = π r even , (4.14)so that the related part in the symplectic form is finite and Coulomb is included in theallowed fields configurations. Note that this choice of parity for π r implies that gaugetransformations are proper if ζ is an odd function on the sphere and are improper if it isan even function, applying the results of section 2.4. Second, one makes (4.13) to be finiteby requiring that A ¯ m = A even¯ m , π ¯ m = π ¯ m odd , ϑ = ϑ odd , (4.15)and that ρ is of definite parity, either even or odd. Note that the parity conditions of ϑ excludes the improper gauge transformations, such as the constant U (1) at infinity, asthese would shift ϑ by an even function. Finally, in order to make the symplectic formfinite, we decompose also the momentum Π asΠ( x ) = 1 √ R ( x ) e i Θ( x ) , (4.16)which needs to satisfy the fall-off conditions R ( x ) = 1 r R ( x ) + O (1 /r ) and Θ( x ) = Θ( x ) + O (1 /r ) . (4.17)In terms of the absolute values and the phases, the logarithmically-divergent contributionto the symplectic form is Z drr Z d x h cos( ϑ − Θ) (cid:16) d R ∧ d ρ + ρR d ϑ ∧ d Θ (cid:17) − sin( ϑ − Θ) (cid:16) ρ d R ∧ d ϑ + R d ρ ∧ d Θ (cid:17)i , (4.18)which vanishes identically once we require that R has the opposite parity of ρ and that Θis odd. Note that also the parity of Θ, other than that of ϑ , is such that improper gaugetransformations are not allowed. Indeed, in order to preserve these parity conditions, weneed to restrict the gauge parameters such that ζ is an odd function. In turn, this impliesthat the generator (2.25) is finite and differentiable without the need of a surface term.Finally, note that the parity conditions that we have just found are preserved by thePoincar´e transformations. To see this, one only need to use the asymptotic form of thetransformations (4.4)–(4.9) and the equations δϑ = Im (cid:18) δϕϕ (cid:19) , δρ = ρ Re (cid:18) δϕϕ (cid:19) , δ Θ = Im (cid:18) δ ΠΠ (cid:19) , δR = R Re (cid:18) δ ΠΠ (cid:19) . (4.19)To sum up, we have seen that, in the massless case, the fields satisfy power-like fall-offconditions. In order to have a finite symplectic form and a canonical action of the Poincar´e– 19 –roup, the fall-off conditions need to be complemented with some parity conditions. Wehave shown that it is possible to find (strict) parity conditions leading to a well-definedHamiltonian formulation. Specifically, the strict parity conditions of A and π are the sameas those in free electrodynamics [1, Sec. 5]. The parity conditions of the complex scalarfield and its momentum have been found after decomposing them into an absolute valueand a phase. The absolute values of ϕ and Π are required to have opposite parity, whilethe phases need to be both of odd parity. Notably, the parity conditions imposed on thephases, as well as those on A ¯ a , exclude the improper gauge transformations from the theoryand reduces the asymptotic symmetry group to the Poincar´e group. In the next subsection,we will try to solve this problem by relaxing the parity conditions. The solution to reintroduce the possibility of performing improper gauge transformationsis quite simple. Specifically, since the improper gauge transformations are excluded dueto the (strict) parity conditions, we simply need relax them so that they are satisfied upto an improper gauge transformations. Therefore, we require the asymptotic part of thefields that transform non-trivially under gauge transformations to be such that A ¯ a = A even¯ a − ∂ ¯ a Φ even , ϑ = ϑ odd + Φ even , and Θ = Θ odd + Φ even , (4.20)where Φ even ( x ) is an even function on the sphere. At the same time, the other fields arerequired to satisfy the same parity conditions as before, that is A r = A odd r , π r = π r even , π ¯ a = π ¯ a odd , (4.21)while R and ρ are of definite, and opposite, parity.These relaxed parity conditions allow for certain the possibility of performing impropergauge transformations, thus extending the asymptotic symmetry group. However, they alsoreintroduce back in the theory two issues. First, the symplectic form is not finite any more.Indeed, it contains now the logarithmically divergent contributionΩ = Z drr I S d x d h ∂ ¯ a π ¯ a − (cid:16) Π ∗ ϕ (cid:17)i ∧ d Φ even + (finite terms) . (4.22)To solve this issue, we need merely to note that the term in square brackets in the expressionabove is nothing else than the leading contribution in the asymptotic expansion of the Gaussconstraint (2.13). Indeed, it is easy to verify that G = 1 r h ∂ ¯ a π ¯ a − (cid:16) Π ∗ ϕ (cid:17)i + O (1 /r ) =: 1 r G + O (1 /r ) . (4.23)As a consequence, the symplectic form can be made finite by restricting the phase spaceto those fields configurations satisfying the further condition G = 0. This does not excludeany solution to the equations of motion, since they already need to satisfy the full Gaussconstraint G ≈
0. – 20 –he second issue reintroduced after relaxing the parity condition is that the Poincar´etransformations are not canonical any more. This is due to the fact that L X Ω = d I d x p γ A r h d ∇ ¯ m (cid:0) b A ¯ m (cid:1) + 2 b Im ( ϕ ∗ d ϕ ) i (4.24)does not identically vanish any more. In the expression above L X Ω is the Lie derivative (inphase space) of the symplectic form Ω with respect to the vector field X , which identifiesthe Poincar´e transformations. In the case of free electrodynamics, it was shown that it ispossible to make the Poincar´e transformations canonical once again, by introducing a newboundary degree of freedom Ψ and complementing the symplectic form with a boundaryterm ω [2]. Specifically, this works as follows. First, one requires that Ψ transform underthe Poincar´e transformations as δ X Ψ = ∇ ¯ m (cid:0) b A ¯ m (cid:1) + Y ¯ m ∂ ¯ m Ψ and chooses the boundaryterm to be ω = I d x p γ d Ψ ∧ d A r , (4.25)so that L X (Ω + ω ) = 0. Second, one extends the new field Ψ in the bulk and makes Ψ bulk pure gauge. The details can be found in [2], where this method was presented for the firsttime. Here, we are interested in pointing out that a similar attempt in this case wouldnot be as successful. Indeed, on the one hand, one would still be able to compensate thefirst summand in square brackets of (4.24). On the other hand, one would not be able tocompensate also the second summand in square brackets, as this is not an exact form. While studying a similar issue in Yang-Mills, we have shown that it is in general not-easily possible to circumvent this type of problems [1]. In that case, we used a general ansatzwith quite a few free parameters for the boundary degrees of freedom, for the boundaryterm of the symplectic form, and for the Poincar´e transformations of the boundary degreesof freedom and showed that no choice of free parameters was yielding a solution. Inthis paper, we will not pursue a similar tedious path. Rather, we will point out a possibleconnection between obstructions to a canonical Lorentz boost and some issues in the Lorenzgauge fixing when a flux of charge-current at null infinity is present [4], as in the case of acharged massless scalar field with the weakest possible fall-off conditions compatible withthe Poincar´e transformations. To this end, we will first analyse some aspects of the freeelectrodynamics case and, then, deal with the scalar electrodynamics one.
In this subsection, we focus on free electrodynamics and highlights the relation betweencanonical Poincar´e transformations and the Lorenz gauge fixing (at infinity). Since theissues, which were mentioned in the previous subsection, arise because of the boost con-tribution to the orthogonal part of the Poincar´e transformations, we will set, after a littleintroduction, N = 0 and N = ξ ⊥ = rb ( x ) + T in this and in the next subsection. We willonce again start our analysis from the action, but use the knowledge that we have gainedso far in the discussion of fall-off and parity conditions. To see precisely that the form is not exact one could either rely on the decomposition into phase andabsolute value as done in (4.13) or on the decomposition into ϕ , . We will come back to this point insection 4.5. – 21 –et us begin with the action in Lagrangian formulation, which is given by (2.1) whensetting the scalar field ϕ to zero. Note that, in principle, the action in the bulk can becomplemented by a boundary term. Let us write it as(boundary term) = Z dt I d x B ( x ) . (4.26)Note that the above expression implies that we are adding a boundary term at spatialinfinity, as it is more fitted for the ensuing discussion, although different types of boundaryterms can be and have been considered. The function B ( x ) depends on (the asymptoticpart of) the fields and of N .The variation of the bulk action, in general, will produce other boundary terms dueto the necessity of performing some integration by parts while deriving the equations ofmotion. In order to have a well-defined action principle, we need to require that, notonly do the bulk part of the variation vanishes producing the bulk equations of motion,but also that the boundary term (which contains also the contribution due to B ) of thevariation is zero. One way to deal with the boundary term in the variation of the actionis to make it vanish identically by imposing some suitable (fall-off and parity) conditionson the asymptotic behaviour of the fields. Another way is to make sure that, even if itis not identically zero, it produces boundary equations of motion that do not contain anynew information with respect to the bulk ones. If neither one of the two said situationshappens, we end up with some non-trivial equations of motion at the boundary, whichcould affect the physics of the theory, for instance by trivialising some symmetry.Before we actually show explicitly the situation in electrodynamics, let us stress thatwhether or not boundary terms are produced during the variation of the action in the bulkdepends on the asymptotic behaviour of the fields, of the lapse N , and of the shift N . Ifwe had been interested only in the time evolution of the theory, it would have sufficed toconsider the fall-off conditions of N and N to be as strong as in [8, Sec. 2]. However,since we wish to have a well-defined action of the Poincar´e group as well, we need to allowthe lapse and the shift to behave asymptotically as N = ξ ⊥ = rb ( x ) + T and N i = g ij ξ j ,respectively. As a consequence, we need to make sure that the variation of the action iswell-defined also for these lapse and shift. Note that, in the following expressions, a dotabove a quantity represents the change of that quantity under an infinitesimal (orthogonal)Poincar´e transformation, since we are considering the case N = rb ( x ) + T and N = 0. If b = 0 and T = 1, this coincides with the standard time derivative.Explicitly, the variation of the action is d S = Z dt (cid:26)Z d x (cid:20) √ gN g ab F b d ˙ A a + ∂ b (cid:16) N √ gF ba (cid:17) d A a + N ∂ a (cid:18) √ gN g ab F b (cid:19) d A ⊥ (cid:21) ++ I S ∞ d x h − √ g g rm F m d A ⊥ + N √ gF ar d A a + d B i) , (4.27)where the surface integral on S ∞ has to be understood as a surface integral over a sphereof radius R followed by the limit R → ∞ . In the above expression, we have replaced A – 22 –ith A ⊥ using (2.16) and we have already performed the needed integration by parts. Thebulk part of the variation lead to the usual equations of motion and symplectic form, whichwe have already discussed. Thus, let us focus on the boundary part. Inserting the usualfall-off conditions of the field and N = rb + T , the boundary part of the variation becomes Z dt I d x h − p γ ˙ A r d A ⊥ + b p γ ∂ ¯ m A r γ ¯ m ¯ n d A ¯ n + d B i , (4.28)where the limit R → ∞ has already been taken, so that the remaining surface integral iseffectively on a unit two-sphere. Let us assume for the moment that B = 0, i.e., the actionis the usual action of Maxwell electrodynamics without any boundary terms.On the one hand, we could try to make the expression in (4.28) identically zero byimposing parity conditions on the fields similarly as in section 4.2, but this choice wouldexclude the improper gauge transformations from the theory trivialising the asymptoticsymmetry group. On the other hand, in the absence of parity conditions, the above ex-pression would produce the boundary equations of motion˙ A r = 0 and ∂ ¯ m A r = 0 (4.29)so that the variation of the action is well defined for any value of b . The first one ofthe above equations implies that the asymptotic part of the electric field vanish and, as aconsequence, the charges. One way to see this is to expand the expression in (2.8) to get π r = ˙ A r √ γ/b = 0. In addition, the second equation implies that A r is constant on thesphere. Since it is also required to be an odd function, in order to have a finite symplecticform, we must conclude that A r = 0. Thus, also in this case, we end up with trivialasymptotic symmetries.This shows that, if we wish to have a well-defined action principle for Maxwell elec-trodynamics with the Poincar´e transformations and non-trivial asymptotic symmetries, wemust include a boundary term in the original action. A suitable choice is B = p γ ˙ A r A ⊥ − b p γ ∂ ¯ m A r γ ¯ m ¯ n A ¯ n . (4.30)This boundary term is chosen because it moves all the variations in (4.28) to A r and ˙ A r ,so that we obtain one, single boundary equations of motion rather than the two, very-restrictive ones of (4.29). A similar boundary term was already considered by Henneauxand Troessaert in [2, App. B], in order to solve the same issue. At this point, the boundarypart of the variation of the action becomes Z dt I d x hp γ A ⊥ d ˙ A r + p γ ∇ ¯ m (cid:0) bA ¯ m (cid:1) d A r i , (4.31)Two things can be noted. First, when going from the Lagrangian to the Hamiltonianpicture as in section 2, A r has now a conjugated momentum on the boundary, namely √ γ A ⊥ . This means that the usual bulk symplectic form needs to be complemented withthe boundary term ω = I d x p γ d A ⊥ ∧ d A r , (4.32)– 23 –hich coincides with the boundary term (4.25) used by Henneaux and Troessaert in [2],after identifying A ⊥ with Ψ. Note that the need of such a boundary term was first pointedout by Campiglia and Eyheralde in [9, Sec. 4].Second, the boundary equation of motion ensuing from (4.32) is ˙ A ⊥ = ∇ ¯ m (cid:0) bA ¯ m (cid:1) ,which is nothing else than the leading term in the asymptotic expansion of the Lorenzgauge condition ∇ µ A µ = 0, where ∇ is the Levi-Civita connection of the four-metric g given in (2.4).It is possible, although not strictly necessary, to extend the boundary equation ofmotion into the bulk. To do so, one can proceed as in [6, Sec. 4-5] and introduce a newcontribution to the action ˜ S [ A, ψ ] = Z d x p − g ∂ α ψ g αβ A β , (4.33)where ψ is a new scalar field. The variation of the action with respect to ψ yields thedesired Lorenz gauge condition in the bulk. When passing to the Hamiltonian picture, onefinds that the conjugated momentum of ψ is π ψ = −√ gA ⊥ , that is the Lagrange multiplier A ⊥ up to the density weight. In order to make sure that the equations of motion in thebulk are not physically affected in the procedure, one needs merely to impose the constraint ψ ≈
0. Without redoing all the computations, let us simply note that, in this way, weobtain exactly the same solution proposed in [2], after identifying Ψ = A ⊥ = − π ψ / √ g and π Ψ = √ g ψ ≈
0. Note that the constraint π Ψ ≈ A ⊥ by an arbitrary function in the bulk, so that the bulk part ofthe Lorenz condition ∇ µ A µ = 0 can be violated arbitrarily. However, on the boundarythis is not the case since shifting Ψ by an odd function is not a gauge transformation, butrather a true symmetry of the theory. Finally, note that this procedure introduces two newcanonical degrees of freedom: the orthogonal component of the vector potential A ⊥ , whichhas been elevated from being a mere Lagrange multiplier to a true degree of freedom, anda momentum conjugated to it. To sum up, we have shown that the action of Maxwell electrodynamics needs to becomplemented by a boundary term, if one wishes to have a well-defined action principle,which works also with the lapse and shift given by the Poincar´e transformations, featuringnon-trivial asymptotic symmetries. A suitable choice for the boundary term is (4.30),which, once added to the original action, leads to two consequences. First, when derivingthe symplectic form, one finds that it contains the boundary term (4.32) as in [2]. Second,one gets a new, non-trivial boundary equation of motion, which is nothing else than theleading term in the asymptotic expansion of the Lorenz gauge condition. In addition,changing this gauge fixing at infinity by shifting Ψ by an odd function is not a propergauge transformation, but rather a true symmetry of the theory, as thoroughly explainedin [2]. Let us now focus on the case in which a charged massless scalar field is present. In the analysis done in [6], the spatial slices of the spacetime are closed manifolds, i.e., compact andwithout boundary. Nevertheless, the results of the paper can be applied to our situation as well and arecorrect up to boundary terms. Due to the constraint π Ψ and the gauge symmetry ensuing from it, however, the only physically-relevantdegree of freedom that has be introduced is the odd component of A ⊥ . – 24 – .5 The Lorentz boost and the Lorenz gauge: scalar electrodynamics Let us now reintroduce the massless scalar field minimally-coupled to electrodynamics.Proceeding as in the previous subsection, we consider the variation of the action (2.1) andsplit it into a bulk part and a boundary part. The part in the bulk provides the equationsof motion in the bulk, after some integration by parts. The boundary part of the variation,at this point, reads Z dt I d x h − p γ ˙ A r d A ⊥ + b p γ ∂ ¯ m A r γ ¯ m ¯ n d A ¯ n ++ 2 b p γ Re ( ϕ ∗ d ϕ ) − b p γ A r Im ( ϕ ∗ d ϕ ) + d B i , (4.34)where, again, we are allowing the presence of a boundary term B in the action. The first lineof the expression above contains the contribution due to free electrodynamics, which wasamply discussed in the previous subsection. The second line appears due to the presenceof the scalar field and is made up of two contributions. The former does not bring anyissue, as it can be readily absorbed into d B . Indeed,2Re ( ϕ ∗ d ϕ ) = ϕ d ϕ + ϕ d ϕ = d (cid:18) ϕ + ϕ (cid:19) = d ( ϕ ∗ ϕ ) . (4.35)The second term is the one causing all the troubles. Indeed, not only cannot it be rewrit-ten as a total variation, but also it cannot be written as A r times the total variation ofsomething, since 2Im ( ϕ ∗ d ϕ ) = ϕ d ϕ − ϕ d ϕ , (4.36)which is not even a closed one form. If we had been able to rewrite the term in the variationas A r d B ′ , we could have included a term − A r B ′ into B and we would have obtain a single,more relaxed equation of motion at the boundary, as in the previous subsection. Thisequation would have been the Lorenz gauge condition at infinity modified by a contributioncoming from B ′ . When passing to the Hamiltonian formalism, the above issue translatein the fact that the Lorentz boost fails to be canonical due to the presence of a boundaryterm in L X Ω, unless strict parity conditions are impose, de facto trivialising the asymptoticalgebra.We propose a connection between the impossibility of having a canonical Lorentzboost when asymptotic symmetries are allowed, i.e. when we impose the relaxed parityconditions, and some issues related to the Lorenz gauge fixing when a flux of charge-currentat null infinity is present [4]. Although we will not provide a formal proof of this statements,we will provide two indications that this is the case.Before we present the two arguments, let us summarise the relevant results describedby Wald and Satishchandran in [4]. Specifically, they have analysed the case of electrody-namics in four and higher and shown that, due to the fall-off conditions of the fields, itis not possible to find a Lorenz gauge fixing — i.e. it does not exist a gauge parametersatisfying the correct fall-off conditions and bringing the four potential in Lorenz gauge— if the dimension of the spacetime is four and a flux of charge-current at null infinity is– 25 –resent. This setup is expected in our situation, due to the presence of a charged mass-less scalar field satisfying the most general fall-off at spatial infinity, compatible with thePoincar´e transformations. Note that no obstruction to the Lorenz gauge fixing is presentin higher dimension, even in the presence of a charge-flux at null infinity.The first argument which we provide is that, in the case of free electrodynamics, theLorenz gauge fixing at infinity appears as a boundary equations of motion that needs to beimposed if we wish, at the same time, a well-defined action principle (also for the lapse andshift of the Poincar´e transformations) and non-trivial asymptotic symmetries. The sameequation cannot be derived if a massless scalar field is present, as we have seen in the firstpart of this subsection.The second argument is that no issue arises in the variation of the action (or, equiva-lently, in the Lorentz boost being canonical) in higher dimensions. In this paper, we haveworked exclusively in 3 + 1 dimensions, as this is the physically-relevant case. However, itis possible to repeat the same analyses in higher dimensions, as in the case of free electro-dynamics, which was already studied by Henneaux and Troessaert in [3]. So, let us assumefor the remainder of this subsection that the spacetime dimension is n + 1, being n an oddnumber.We can derive also in this case the fall-off conditions of the fields by requiring that theyare power-like and that they are preserved by the Poincar´e transformations, obtaining A r ( r, x ) = 1 r n − A r ( x ) + O (cid:0) /r n − (cid:1) , π r ( r, x ) = π r ( x ) + O (1 /r ) ,A ¯ a ( r, x ) = ∂ ¯ a Φ( x ) + 1 r n − A ¯ a ( x ) + O (cid:0) /r n − (cid:1) , π ¯ a ( r, x ) = 1 r π ¯ a ( x ) + O (cid:0) /r (cid:1) ,ϕ ( r, x ) = 1 r ( n − / ϕ + O (cid:16) /r ( n +1) / (cid:17) , Π( r, x ) = 1 r (3 − n ) / Π + O (cid:16) /r (5 − n ) / (cid:17) , (4.37)The first two lines of the above expression contain precisely the findings of [3]. Two thingscan be noted about them. First, if n >
3, the fall-off conditions are enough to ensure thatthe symplectic form is finite (see [3] for a detailed discussion), so that no parity conditionis needed. Second, A ¯ a contains two relevant asymptotic parts: a zeroth-order contribution,which is a gradient of a function on the ( n − /r n − .If n = 3, as it is in the previous part of this section, the gradient can be reabsorbed in A ¯ a , but this is not possible if n >
3. Finally, the last line of the expression above containsthe fall-off conditions of the scalar field and its momentum. These lead to a logarithmicdivergence in the symplectic form which can be dealt with by means of parity conditions.Ignoring the details about these subtleties, let us show directly that the scalar fielddoes not bring any obstruction to a canonical Lorentz boost. To this end, let us computethe Lie derivative of the symplectic form with respect to the vector field of the Poincar´etransformations. After a few passages, we find L X Ω = I S n − ∞ d n − x h ξ ⊥ √ g d F ra ∧ d A a + 2 √ g ξ ⊥ Re ( d D r ϕ ∧ d ϕ ∗ ) i , (4.38) We work in radial angular components ( r, x ), where x are coordinates on the unit ( n − – 26 –here the integration over S n − ∞ has to be understood as an integration over an ( n − R followed by the limit R → ∞ . Also in the case of higher dimensions, wesee that the Poincar´e transformations fail to be canonical due to a boundary contribution.The first term in square brackets in equation (4.38) is the contribution due to freeelectrodynamics. Using the fall-off conditions (4.37), it reduces to I d n − x n − b p γ γ ¯ m ¯ n d (cid:2) ( n − A ¯ m + ∂ ¯ m A r (cid:3) ∧ d ∂ ¯ n Φ o , (4.39)where the integration is now performed on a unit ( n − R → ∞ hasbeen already taken. This contributions has been already thoroughly analysed in [3] and,basically, can be dealt with by introducing a new boundary degree of freedom, similarly tothe case of free electrodynamics in four dimension, which we have discussed in the previoussubsection. The second term, on the contrary, is the new contribution due to the masslessscalar field. Expanding it with the use of the fall-off conditions (4.37), it reduces tolim R →∞ I S n − R d n − x (cid:26) − R n − b p γ Im (cid:2) d (cid:0) ϕA r (cid:1) ∧ d ϕ ∗ (cid:3)(cid:27) , (4.40)which vanishes if n > n = 3. Thus, wehave shown that no issue is present if n > In this section, we wish to study the asymptotic symmetries of the theory described bythe Hamiltonian (2.17), when µ > λ >
0. This choice of the parameters leads to– 27 –he Mexican-hat potential for the scalar field and to the abelian Higgs mechanism. Let usbegin by determining the fall-off behaviour of the fields.
Let us begin the discussion about the abelian Higgs model by studying the asymptoticbehaviour of the fields and, in particular, their fall-off conditions. As usual, we wish to findthe “largest” phase space which is stable under the action of the Poincar´e transformations.The derivation of the fall-off conditions is very similar to that presented in section 3.2 anddiffers only in the last steps and in the fact that one needs to take into consideration agreater number of fields, as we have to include the abelian one-form potential A a and itsconjugated momentum π a in the discussion. This will have an effect also on the fall-offconditions of the phase of the scalar field, which will turn out to be a bit different fromthose of section 3.2.First of all, let us note that we need the phase space to contain the minimum-energysolutions to the equations of motion, as these are physically relevant solutions. Specifically,this means that the phase space needs to include at least the solutions A a ( x ) = 0 , π a ( x ) = 0 , ϕ ( x ) = ϕ ( ϑ ) ( x ) , and Π( x ) = 0 , (5.1)where the constant solution ϕ ( ϑ ) ( x ) := v/ √ iϑ ) was already defined in equation (3.15).We already know that one consequence of this fact is that the potential (2.3) needs to becorrected by the addition of the constant λv /
4, being v := p µ /λ , so that it becomes V ( ϕ ∗ ϕ ) = λ (cid:18) v − ϕ ∗ ϕ (cid:19) . (5.2)Another consequence is that we have to exclude the trivial solution to the equation ofmotion — i.e. all fields and momenta equal to zero — from phase space, for otherwise theHamiltonian would not be finite.Secondly, the fall-off conditions are expressed more effectively when the ϕ ( x ) is ex-pressed in terms of its absolute value and phase. So, let us write ϕ ( x ) = 1 √ ρ ( x ) e iϑ ( x ) (5.3)At this point, we only need to proceed in the same way as in section (3.2) excluding thelast step, in which the behaviour of ϑ ( x ) was determined. With the same arguments, weconclude also in this case that ρ ( x ) = v + h ( x ), where h ( x ) is quickly vanishing up to thesecond derivative order, and that Π( x ) is quickly vanishing.Thirdly, let us determine the fall-off behaviour of the phase ϑ ( x ). As in section 3.2, letus consider the transformation of Π under time evolution, i.e., equation (2.23) at N = 1and N = 0. Up to terms that are quickly vanishing at infinity, we find δ Π = ϕ n −√ g g ab ( ∂ a ϑ + A a )( ∂ b ϑ + A b ) + iD a h √ g g ab ( ∂ b ϑ + A b ) io +(quickly-vanishing terms) . (5.4)– 28 –he above transformation preserves the fall-off condition of Π so long as ∂ a ϑ + A a is quicklyvanishing together with its first-order derivatives. So, let us write A = − dϑ + ˜ A , where ϑ isonly required to have a well-defined limit ϑ ( x ) = lim r →∞ ϑ ( x ) as a function on the sphereat infinity, whereas ˜ A is a quickly-vanishing function together with its first derivatives.Note that the Lagrange multiplier N A ⊥ needs to satisfy the same fall-off conditions of ϑ .Lastly, we need to determine the fall-off behaviour of π a . To do so, one merely needto demand that the fall-off behaviour of A = − dϑ + ˜ A is preserved by a generic Lorentzboost. One sees that the only possibility is to require that π a is quickly vanishing. In turn,this fall-off behaviour is preserved by the Poincar´e transformations so long as the secondderivatives of ˜ A , too, are quickly vanishing. This concludes the discussion about the fall-offconditions of the fields in the abelian Higgs model.To sum up, we have shown that, if one splits the scalar field into an absolute value anda phase as in (5.3), the former has to be ρ ( x ) = v + h ( x ), where h ( x ) is quickly vanishingup to its second-order derivatives. The phase ϑ ( x ), on the contrary is merely required tohave a well-defined limit ϑ ( x ) = lim r →∞ ϑ ( x ) as a function on the sphere at infinity andthe same holds true for the Lagrange multiplier N A ⊥ . In addition, the one-form A can bewritten as A = − dϑ + ˜ A , where ˜ A is quickly vanishing up to its second-order derivatives.Finally, the momenta π a need to be quickly vanishing. In particular, note that the fall-offbehaviour of the one-form A a and of its momentum π a is substantially different from thatof electrodynamics, either in the free case [1, 2] or when coupled to a scalar field (seesection 4). These fall-off conditions ensure that the Poincar´e transformations have a welldefined action on the phase space. Having derived the fall-off conditions of the fields, we can now provide the well-definedHamiltonian formulation of the abelian Higgs model. In particular, we will provide theexact form of the Hamiltonian, of the generator of the Poincar´e transformations, and of thegenerator of the gauge transformations. Furthermore, we will also identify the asymptoticsymmetries of the theory.To begin with, let us note that the symplectic form (2.12) is finite, thanks to thequick fall-off of the fields. For the same reason, both the Hamiltonian and the generator ofthe Poincar´e transformations are finite and differentiable. These can be inferred from thegenerator H [ A, π, ϕ, Π; g, N, N ; A ⊥ ] = Z d x h N H + N i H i i , (5.5)by setting N = 1, N = 0 (Hamiltonian) and N = ξ ⊥ , N i = ξ i (Poincar´e generator). Inthe above generator H := π a π a + Π + Π √ g + √ g F ab F ab − A ⊥ G + √ g g ab (cid:0) ∂ a ϕ ∂ b ϕ + ∂ a ϕ ∂ b ϕ (cid:1) ++ √ gA a (cid:0) ϕ ∂ a ϕ − ϕ ∂ a ϕ (cid:1) + 12 A a A a (cid:0) ϕ + ϕ (cid:1) + √ g V ( ϕ ∗ ϕ ) (5.6)– 29 –s responsible for the orthogonal transformations and H i := π a ∂ i A a − ∂ a ( π a A i ) + Π ∂ i ϕ + Π ∂ i ϕ (5.7)is responsible for the tangential transformations. Note that the potential is V ( ϕ ∗ ϕ ) = λ (cid:18) v − ϕ ∗ ϕ (cid:19) , (5.8)which differs from the original potential (2.3) due to the addition of the constant λv / G ( x ) — which has the same expres-sion as in (2.13) — appears in the generator (5.6) multiplied by the Lagrange multiplier A ⊥ . In general, gauge transformations are generated by G [ ζ ] := Z d x ζ ( x ) G ( x ) ≈ , (5.9)which is finite and differentiable without the need of any surface term. Note that, in theabove generator, ζ is only required to have a well-defined limit ζ ( x ) = lim r →∞ ζ ( x ), sothat the transformations (2.24) preserve the fall-off conditions identified in the previoussubsection.Two things can be noted at this point. First, the phase ϑ can always be trivialisedby a proper gauge transformation, so that it carries no physical meaning. Specifically,from (2.29), we see that Γ − ϑ ( ϕ ) = ρ/ √
2, without any phase. Second, since (5.9) is alreadyfinite and differentiable without the need of any boundary term, it cannot be extended to agenerator of improper gauge transformations, contrary to the case of electrodynamics [2].As a consequence, the asymptotic symmetries of the theories are trivially the Poincar´etransformations. Indeed, the only generator of asymptotic symmetries is H [ ξ, ξ ] whichsatisfies the algebra (cid:8) H [ ξ ⊥ , ξ ] , H [ ξ ⊥ , ξ ] (cid:9) = H [ ˆ ξ ⊥ , ˆ ξ ] + G [ˆ ζ ] , (5.10)whereˆ ξ ⊥ = L ξ ξ ⊥ − L ξ ξ ⊥ , ˆ ξ m = ˜ ξ m + [ ξ , ξ ] m , ˆ ζ = A m ˜ ξ m + ξ L ξ A ⊥ − ξ L ξ A ⊥ . (5.11)Here, ˜ ξ i := g ij ( ξ ⊥ ∂ j ξ ⊥ − ξ ⊥ ∂ j ξ ⊥ ) (which simplifies the expressions above and the followingdiscussion), L is the Lie derivative on spatial slices, and [ ξ , ξ ] is the commutator of thevector fields ξ and ξ . The above algebra is easily seen to be a Poisson-representationof the Poincar´e algebra up to (proper) gauge transformations, due to the presence of theconstraint on the right-hand side of (5.10). The fact that the Poincar´e algebra is recoveredup to proper gauge transformations is not in general a problem (see e.g. the discussionin [10, Sec. 2]).Before we conclude this section, let us note that the ˆ ζ in the expressions above dependson the canonical fields and, in particular, on A m . As a consequence the transformation Note that this is a complete gauge fixing. We remind that A ⊥ is not a canonical field, but only a Lagrange multiplier. – 30 –enerated by G [ˆ ζ ] slightly differs from the usual gauge transformations. Specifically, itinduces the transformations δA a = − ∂ a ˆ ζ , δπ a = − ˜ ξ a G ≈ , δϕ = i ˆ ζ ϕ , and δ Π = i ˆ ζ Π . (5.12)It is useful to compare the above transformations with those caused by ζ in equations (2.24).Two things emerge. First, A , ϕ , and Π transform in the same way, with the only differencebeing that the parameter ˆ ζ is field-dependent. Secondly, the transformation of π due to ˆ ζ is not trivial any more. Nevertheless, it is proportional to the Gauss constraint and, thus,vanishes on the constraint hypersurface. Note that the transformations above deserveby all means the title of gauge transformations, as one part of them is generated by theconstraints A m G ≈ ξ m , while the other part of them is generated by the usual Gauss constraint G smeared by ξ L ξ A ⊥ − ξ L ξ A ⊥ . Let us neglect this second part, as it is of a well-knownshape, and focus on the first one, whose generator ˜ G [˜ ξ ] := G [ ˜ ξ m A m ] is easily seen to be welldefined and functionally differentiable with respect to the canonical fields, as the fields arerapidly vanishing while approaching spatial infinity. Furthermore, it satisfies the algebra (cid:8) ˜ G [˜ ξ ] , ˜ G [˜ ξ ] (cid:9) = ˜ G [˜ ξ ] , where ˜ ξ = [˜ ξ , ˜ ξ ] . (5.14)This concludes the discussion about the asymptotic symmetries of the abelian Higgsmodel. To sum up, we have shown that the fall-off conditions derived in the previoussubsection are enough to ensure a well-defined Hamiltonian formulation with a canonicalaction of the Poincar´e group. Moreover, we have seen that the phase ϑ can always betrivialised by a proper gauge transformation and that the asymptotic symmetries of theabelian Higgs model are trivial, in the sense that the asymptotic-symmetry group is thePoincar´e group. This was shown by computing the Poisson-algebra of H [ ξ, ξ ], which is aPoisson representation of the Poincar´e algebra up to proper gauge transformations. Be-fore we draw our conclusions, let us briefly comment on the fate of the Goldstone boson,which emerged as a consequence of the spontaneous symmetry break of the global U(1) insection 3.2. At the end of section 3.2, we discussed that, in the case of the spontaneous symmetry breakof the global U(1) symmetry, the action could be rewritten in terms of two real scalar fields:the massive h and the massless χ . The former was identified to be the candidate Higgsfield in the abelian Higgs model, while the latter was recognised as the Goldstone boson.Let us repeat that analysis for the abelian Higgs model using the fall-off conditionsof section 5.1. Proceeding as in section 3.2, let us consider the action in the Lagrangianpicture, which can be obtained from (2.1) by adding the constant λv / h , ϑ , and ˜ A , we obtain S [ h, ϑ, ˜ A ] = Z d x (cid:26) − (cid:16) g αβ ∂ α h ∂ β h + 2 µ h (cid:17) + − (cid:18) g αγ g βδ ˜ F αβ ˜ F γδ + v g αβ ˜ A α ˜ A β (cid:19) + (interactions) (cid:27) , (5.15)where the interactions include all the terms that are not quadratic in the fields. In theexpression above, we have introduced ˜ A := A + ˙ ϑ , whose quickly-falling asymptoticbehaviour can be inferred from that of the momentum Π, and ˜ F := d ˜ A .Three things can be noted from the expression above. First, there is a real scalar fields h of squared mass m h := 2 µ , which corresponds to the Higgs field. Second, the spin-onefield ˜ A becomes massive with a squared mass m A := v . The mass m A of the spin-one fielddepends on the vacuum expectation value v of the complex scalar field ϕ and, in general,on the coupling of the Higgs to the original gauge potential A (in this paper, it was set tothe value of 1). Last, but not least, there is no trace of a massless scalar field, which couldplay the role of the Goldstone boson.The disappearance of the Goldstone boson can be tracked down precisely to the choice A = − dϑ + ˜ A , which we did in section 5.1. On the one hand, this choice makes the fall-offcondition of the momentum Π to be preserved by the Poincar´e transformations. On theother hand, it makes the gauge-covariant derivative of ϕ to be independent of ϑ , so that theaction (5.15) is also independent of the phase ϑ . Therefore, if we wished to reintroducethe Goldstone boson, we would have to modify slightly the fall-off conditions of section 5.1.To this end, let us write the phase ϑ = ϑ ′ + χ/v as the sum of two parts. The former ofthe two, ϑ ′ , is the “power-like” part of ϑ , while the latter, χ/v , is the “quickly-falling” part.The only requirement while performing this split is that χ/v is actually a quickly-fallingfunction. Two things can be noted. First, the fall-off behaviour of Π is preserved by thePoincar´e transformations so long as A = − dϑ ′ + ˜ A , being ˜ A quickly falling. When thischoice is introduced in the action (2.1), we get the expression S [ h, χ, ˜ A ] = Z d x (cid:26) − (cid:16) g αβ ∂ α h ∂ β h + 2 µ h (cid:17) − g αβ ∂ α χ ∂ β χ + − (cid:18) g αγ g βδ ˜ F αβ ˜ F γδ + v g αβ ˜ A α ˜ A β (cid:19) + (interactions) (cid:27) , (5.16)rather than (5.15). The expression above does indeed contain the massless Goldstone boson χ , other than the already-present Higgs field h and massive spin-one field ˜ A . Second, thesplit of ϑ into a power-like part ϑ ′ and a quickly-falling part χ/v is obviously ambiguous.This was not the case in section 3.2, since, in that case, the only allowed power-like partof ϑ was its asymptotic value ϑ on the sphere at infinity, which could be unequivocallyidentified by ϑ := lim r →∞ ϑ . The main consequence of this ambiguity in the splitting of ϑ into ϑ ′ and χ is that Ω becomes degenerate, so that it is a pre-symplectic form rather than We remind that in section 3.2, the role of the Goldstone boson was played by the part χ of the phase ϑ which was quickly falling at infinity. – 32 – symplectic one. Indeed, one can easily check that i Y Ω = 0, if Y is chosen so that δ Y ϑ ′ = ζ , δ Y χ = − vζ , δ Y ˜ A = dζ , and δ Y (other fields) = 0 , (5.17)for any quickly-falling ζ . At this point, one would need to deal with this issue as in [11].In this paper, we have preferred not to pursue this path, since it would introducesome mathematical complications without any advantage on the physical side. Indeed, aswe have seen in section 5.2, the phase ϑ can always be set to zero by means of a propergauge transformation (with gauge parameter − ϑ ). As a consequence, neither ϑ ′ nor χ arephysically-relevant fields.This concludes the discussion concerning the abelian Higgs model. To summarise thissection, we have derived the fall-off conditions of the fields and shown that these lead to awell-defined Hamiltonian formulation of the theory with a canonical action of the Poincar´egroup. As a consequence of the quick fall-off behaviour of the fields, the proper gaugetransformations cannot be extended to improper ones and the asymptotic symmetry grouptrivially coincide with the Poincar´e group. Furthermore, we have seen that the variousfields can be interpreted as a massive spin-one field, a Higgs field, and a Goldstone boson.The latter, although absent due to the chosen fall-off conditions, can be reintroduced bya slight modification of these. Nevertheless, it is physically irrelevant, since it can betrivialised by means of a proper gauge transformation. We consider the results of this paper to be both, interesting and encouraging. The resultsconcerning massive scalar fields were clearly hoped for and it is encouraging to see that thishope was fulfilled in an unambiguous way, thereby providing further confidence into theHamiltonian method for the analysis of asymptotic structures and symmetries. As alreadydiscussed at length in [1], the obvious and characteristic advantage of this method is toembed the discussion on asymptotic symmetries into a formalism of clear-cut rules andinterpretation. The result for the massless case was not a surprise, though we had no firmintuition whether we should expect it. In that sense, we consider it interesting.Although the models considered here are not at the forefront of physical phenomenol-ogy, the abelian model does provides good insight into what to expect in other Higgsmodels, such as the physically-relevant case of the electroweak sector. We provided amplediscussion of these expectations. In fact, one may speculate that similar results hold in thecase of the abelian mechanism of the electroweak theory, that is SU(2) L × U(1) Y → U(1) e.m. ,where SU(2) L is the isospin acting on the left-handed fermions, U (1) Y is generated by hy-percharge, and U(1) e.m. by electric charge. Let W I , I = 1 , ,
3, be the standard componentsof the connection associated to SU(2) L and B the one associate to U(1) Y . Then, one canrewrite A = sin θ W W + cos θ W B , (6.1) Z = cos θ W W − sin θ W B , (6.2) W ± = 1 √ (cid:0) W ∓ iW (cid:1) , (6.3)– 33 –here θ W is the Weinberg angle. Due to the Higgs mechanism, the equations of motion(and the Poincar´e transformations) of Z and W ± will contain an effective mass term, whileno such term will be present in the equations of A .Given that, one may expect to find that A has a power-like behaviour, while Z and W ± are quickly vanishing. This expectations is due to the fact that the behaviour atinfinity seems to depend on whether or not a mass term (or an effective mass term) ispresent, and not on the specific field under consideration. In this paper, this is whathappens to the free scalar field and to the one form A in the abelian Higgs mechanism.A consequence of the above-mentioned fall-off conditions is that the equations of motion(and the Poincar´e transformations) of A and its conjugated momentum π become those offree electrodynamics near spatial infinity. Therefore, one may be led to the conjecture thatthe discussion on parity conditions simply reduces to that already presented by Henneauxand Troessaert.Another way to generalise the field content, that would also be of particular interestin view of our previous results in [1], is to consider the SU(2)-Yang-Mills-Higgs case. Thisis currently under investigation and will follow soon. Acknowledgments
Support by the DFG Research Training Group 1620 “Models of Gravity” is gratefullyacknowledged.
A Fall-off behaviour of massive fields
We would like to show that, in the massive case, the fall-off behaviour of the field and themomentum needs to be decreasing more rapidly than any power-like function. Specifically,let us denote with P the phase space consisting of all the allowed field configurations ( ϕ, Π)and with
Poi the Poincar´e group. In order to have a well-defined relativistic field theory,we need to require that the action of any Poincar´e transformation maps points belongingto the phase space into points belonging to the phase space. In other words, we haveto impose the condition that, for all g ∈ Poi , one has g P ⊆ P . In this appendix, wewish to show that this requirement, together with the finiteness of the Hamiltonian andthe at-most-logarithmic divergence of the symplectic form, implies that, ∀ ( ϕ, Π) ∈ P , ∀ α, β ∈ Z , r α ϕ ( x ) → r β Π( x ) → The phase space P should be though of as a sub-manifold in some infinite-dimensional manifold offunctions which are sufficiently regular so that the explicit expressions of the Poincar´e transformations (3.1)–(3.2) make sense. Note that we require the group
Poi to operate on P by a group action , which means that the map Poi × P → P , ( g, p ) gp , satisfies g ( hp ) = ( gh ) p and ep = p (where e is the group identity) for all g, h ∈ Poi and all p ∈ P . This immediately implies that, for any g ∈ Poi , the map P → P , p gp is a bijection. Hence g P ⊆ P is, in fact, equivalent to g P = P . – 34 –n the limit r := | x | → ∞ . To this end, let us focus only on a part of the full Poincar´e transformations (3.1)–(3.2)and, specifically on δ ′ ϕ = ξ ⊥ Π √ g and δ ′ Π = − ξ ⊥ √ g m ϕ . (A.2)When considering only a Lorentz boost, i.e. ξ ⊥ = r b ( x ), and writing explicitly the depen-dence on the radial and angular coordinates, the above expressions become δ ′ ϕ ( r, x ) = b ( x ) p γ ( x ) Π( r, x ) r and δ ′ Π( r, x ) = − b ( x ) p γ ( x ) m r ϕ ( r, x ) . (A.3)Let us define for all ( ϕ, Π) ∈ P the quantities α ϕ := sup (cid:8) α ∈ Z : r α ϕ ( x ) → (cid:9) and β Π := sup (cid:8) β ∈ Z : r β Π( x ) → (cid:9) . (A.4)First of all, let us note that these quantities are well defined. Indeed, the finiteness of themass term in the Hamiltonian (proportional to ϕ ∗ ϕ ) implies that ϕ ( x ) →
0, whereas thefiniteness of the kinetic term (proportional to Π ) implies that r − Π( x ) →
0. Therefore,the sets on the right-hand sides of the definitions above are not empty and the supremaexist. Note that, with the same argument, we can also conclude that α ϕ ≥ β Π ≥ − ϕ, Π) belonging to the phase space.There are two possibilities for α ϕ and, analogously, for β Π . First, the value of α ϕ may be + ∞ , in which case ϕ is according to the statement (A.1) that we wish to prove.Secondly, it may happen that α ϕ is a finite integer number, in which case the supremum isactually a maximum and r α ϕ +1 ϕ converges to some function on the sphere. In principle,this function on the sphere can be divergent, but need not be identically zero.In order to prove the original statement (A.1), we need to show that, for all ( ϕ, Π) ∈ P ,both α ϕ and β Π are infinite. To this purpose, let us define α := min { α ϕ : ϕ ∈ P | ϕ } and β := min { β Π : Π ∈ P | Π } , (A.5)which are well-defined quantities, since α ϕ ≥ β Π ≥ − ϕ, Π) ∈ P , so thatthe sets on the right-hand sides of the definitions above are non-empty subsets of Z ∪ { + ∞} bounded from below and, as a consequence, the minima exist. Note that the value of α and β can actually be infinite. This happens, respectively, when α ϕ = + ∞ for all ϕ andwhen β Π = + ∞ for all Π. It is easy to see that the statement (A.1) is equivalent to thecase in which both α and β are infinite.Let us assume, ad absurdum , that at least one among α and β is finite. To beginwith, we note that also the other quantity need to be finite. This can be seen as follows.Let us assume that α ∈ Z and let ( ϕ, Π) ∈ P be such that α ϕ = α . After applying a In order to avoid issues in the ensuing proof, we need to assume that the phase space P is not empty.This is easily achieved by assuming that ( ϕ (0) , Π (0) ) ∈ P , being ϕ (0) ( x ) = 0 and Π (0) ( x ) = 0. This fieldconfiguration, other than being the minimum-energy solution to the equations of motion, is also invariantunder the action of the Poincar´e group and satisfies the statement in (A.1). The existence of ( ϕ, Π) ∈ P satisfying α ϕ = α is guaranteed by the fact that α is a minimum. – 35 –oincar´e transformation, we reach the field configuration ( ϕ ′ , Π ′ ) which still belongs to thephase space P due to the hypothesis. From the second equation in (A.3), it follows that Π ′ contains the term δ ′ Π( r, x ) = − b ( x ) p γ ( x ) m r ϕ ( r, x ) , (A.6)which is easily seen to satisfy r α − δ ′ Π →
0, while r α − δ ′ Π does not converge to zero. Sincethis is only one of the terms composing Π ′ , we cannot make an exact statement about thevalue of β Π ′ , but we can nevertheless conclude that β Π ′ ≤ α −
3, which implies β ≤ α − , (A.7)showing that β is finite if α is finite. Analogously, one can show that, if β is finite, also α is finite and satisfies the inequality α ≤ β + 1 . (A.8)The combination of the two inequalities (A.7) and (A.8) readily yields us the contradiction α ≤ β + 1 ≤ ( α −
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