aa r X i v : . [ m a t h - ph ] D ec Theory of Classical Higgs Fields. I. Matter Fields
G. SARDANASHVILY, A. KUROVDepartment of Theoretical Physics, Moscow State University, Russia
Abstract
Higgs fields are attributes of classical gauge theory on a principal bundle P → X whosestructure Lie group G if is reducible to a closed subgroup H . They are represented bysections of the quotient bundle P/H → X . A problem lies in description of matter fieldswith an exact symmetry group H . They are represented by sections of a composite bundlewhich is associated to an H -principal bundle P → P/H . It is essential that they admit anaction of a gauge group G . Higgs fields are attributes of classical gauge theory on a principal bundle P → X if itssymmetries are spontaneously broken [2, 6, 7]. Spontaneous symmetry breaking is a quantumphenomenon, but it is characterized by a classical background Higgs field [8]. Therefore, sucha phenomenon also is considered in the framework of classical field theory when a structure Liegroup G of a principal bundle P is reduced to a closed subgroup H of exact symmetries.One says that a structure Lie group G of a principal bundle P is reduced to its closed subgroup H if the following equivalent conditions hold: • a principal bundle P admits a bundle atlas with H -valued transition functions, • there exists a principal reduced subbundle of P with a structure group H .A key point is the following. Theorem 1.
There is one-to-one correspondence between the reduced H -principal subbundles P h of P and the global sections h of the quotient bundle P/H → X possessing a typical fibre G/H . In classical field theory, global sections of a quotient bundle
P/H → X are treated as classicalHiggs fields [2, 7].In general, there is topological obstruction to reduction of a structure group of a principalbundle to its subgroup. In particular, such a reduction occurs if the quotient G/H is diffeomorphicto a Euclidean space R m . For instance, this condition is satisfied if H is a maximal compactsubgroup H of a Lie group G [9].Given a principal bundle P → X whose structure group is reducible to a closed subgroup H , one meets a problem of description of matter fields which admit only an exact symmetrysubgroup H . Here, we aim to show that they adequately are represented by sections of thecomposite bundle Y (5) which is associated to an H -principal bundle P → P/H . A key point isthat Y also is a P -associated bundle (Theorem 3), and it admits the action (15) of a gauge group G . In the case of a pseudo-orthogonal group G = SO (1 , m ) and its maximal compact subgroup H = SO ( m ), we obtain this action in the explicit form (24).Forthcoming Part II of our work is devoted to Lagrangians of these matter fields and Higgsfields. 1n accordance with Theorem 1, we consider a reduced H -principal subbundles P h of P . Let Y h = ( P h × V ) /H (1)be an associated vector bundle with a typical fibre V which admits a group H of exact symmetries.One can think of its sections s h as describing matter fields in the presence of a Higgs fields h andsome principal connection A h on P h .For different Higgs fields h , h ′ , the corresponding reduced H -principle bundles P h , P h ′ and,consequently, the associated bundles Y h , Y h ′ however fail to be isomorphic. Lemma 2.
If the quotient
G/H is isomorphic to a Euclidean space R m , all H -principalsubbundles P h of a G -principal bundle P are isomorphic to each other [9]. Nevertheless, if there exists an isomorphism between different reduced subbundles P h and P h ′ , this is an automorphism of a G -principal bundle P which sends h to h ′ [2, 7]. Therefore, a V -valued matter field can be regarded only in a pair with a certain Higgs field h . A goal thus isto describe the totality of these pairs ( s h , h ) for all Higgs fields h .For this purpose, let us consider a composite bundle P → P/H → X, (2)where P Σ = P π P Σ −→ P/H (3)is a principal bundle with a structure group H andΣ = P/H π Σ X −→ X (4)is a P -associated bundle with a typical fibre G/H where a structure group G acts on the left.Note that, given a global section h of Σ → X (4), the corresponding reduced H -principalbundle P h is the restriction h ∗ P Σ of the H -principal bundle P Σ (3) to h ( X ) ⊂ Σ. Herewith, anyatlas Ψ h of P h defined by a family of its local sections { U, z h } also is an atlas of a G -principalbundle P and that of a P -associated bundle Σ → X (4) with H -valued transition functions.With respect to this atlas Ψ h of Σ, a global section h of Σ takes its values into the center of thequotient G/H .With the composite bundle (2), let us consider the composite bundle π Y X : Y π Y Σ −→ Σ π Σ X −→ X (5)where Y → Σ is a P Σ -associated bundle Y = ( P × V ) /H (6)with a structure group H . Given a global section h of the fibre bundle Σ → X (4) and thecorresponding reduced principal H subbundle P h = h ∗ P , the P h -associated fibre bundle (1) isthe restriction Y h = h ∗ Y = ( h ∗ P × V ) /H (7)of a fibre bundle Y → Σ to h ( X ) ⊂ Σ. 2s a consequence, every global section s h of the fibre bundle Y h (7) is a global section of thecomposite bundle (5) projected onto a section h = π Y Σ ◦ s of a fibre bundle Σ → X . Conversely,every global section s of the composite bundle Y → X (5) projected onto a section h = π Y Σ ◦ s of a fibre bundle Σ → X takes its values into the subbundle Y h (7). Hence, there is one-to-onecorrespondence between the sections of the fibre bundle Y h (1) and the sections of the compositebundle (5) which cover h .Thus, it is the composite bundle Y → X (5) whose sections describe the above mentionedtotality of pairs ( s h , h ) of matter fields and Higgs fields in classical gauge theory with spontaneoussymmetry breaking [2, 6, 7]. In particular, one can show the following [2, 7]. • An atlas { z Σ } of an H -principal bundle H → Σ and, accordingly, of an associated bundle Y → Σ yields an atlas { z h = z Σ ◦ h } of an H -principal bundle P h and, consequently, Y h . • An H -principal connection A Σ on a fibre bundle Y → Σ yields a pullback H -principalconnection A h on Y h .A key point under consideration here is the following. Theorem 3.
The composite bundle π Y X : Y → X (5) is a P -associated bundle with astructure group G . Its typical fibre is an H -principal bundle π W H : W = ( G × V ) /H → G/H (8) associated with an H -principal bundle π GH : G → G/H. (9)
Proof.
Let us consider a principal bundle P → X as a P -associated bundle P = ( P × G ) /G, ( pg ′ , g ) = ( p, g ′ g ) , p ∈ P, g, g ′ ∈ G, whose typical fibre is a group space of G which a group G acts on by left multiplications. Thenthe quotient (6) can be represented as Y = ( P × ( G × V ) /H ) /G, ( pg ′ , ( gρ, v )) = ( pg ′ , ( g, ρv )) = ( p, g ′ ( g, ρv )) = ( p, ( g ′ g, ρv )) , ρ ∈ H. It follows that Y (6) is a P -associated bundle with the typical fibre W (8) which the structuregroup G acts on by the law g : ( G × V ) /H → ( gG × V ) /H. (10)This is a familiar induced representation of G [5]. It is an automorphism of the fibre bundle W → G/H (8). Given an atlas Ψ GH = { ( U α , z α : U α → G ) } of the H -principal bundle G → G/H (9), the induced representation (10) reads g : ( σ, v ) = ( z α ( σ ) , v ) /H → ( σ ′ , v ′ ) = ( gz α ( σ ) , v ) /H =( z β ( π GH ( gz α ( σ ))) ρ, v ) /H = ( z β ( π GH ( gz α ( σ ))) , ρv ) /H, g ∈ G, (11) ρ = z β ( π GH ( gz α ( σ ))) − gz α ( σ ) ∈ H, σ ∈ U α , π GH ( gz α ( σ )) ∈ U β . H is a Cartan subgroup of G , an example of the induced representation (11) is the well-knownnon-linear realizations [1, 2, 3].A problem however lies in the existence of an atlas of Y both as P -and P Σ -associated bundles.Given an atlas Ψ = { U i , z i : U i → P } of P , we have the trivialization charts ψ i : π − Y X ( U i ) → U i × W (12)of an associated bundle Y → X . An atlas Ψ GH of G → G/H in turn yields the trivializationcharts ψ α : π − W H ( U α ) → U α × V (13)of the fibre bundle W (8). Then the compositions of trivialization morphisms (12) and (13) definefibred coordinate charts( O ; x µ , σ m , v A ) , O = ψ − i ( U i × π − W H ( U α )) → U i × π − W H ( U α ) → U i × U α × V, (14)of Y → Σ as a fibred manifold, but not a fibre bundle. They possess the transition functions( σ, v ) → ( σ ′ ( x, σ ) , v ′ ( x, σ, v )) (11).If G → G/H and, consequently, W → G/H are trivial bundles, then U α = G/H and thecharts (14) are fibre bundle charts O = ψ − i ( U i × W ) = π − Y X ( U i ) → U i × W → U i × G/H × V, O = π − Y Σ ( U i × G/H ) , of Y as a P Σ -associated bundle. Lemma 4.
By the well known theorem [9], a fibre bundle W → G/H always is trivial overa Euclidean base
G/H . Any principal automorphism of a G -principal bundle P → X , being G -equivariant, alsois H -equivariant and, thus, it is a principal automorphism of a H -principal bundle P → Σ.Consequently, it yields an automorphism of the P Σ -associated bundle Y (5). Accordingly, every G -principal vector field ξ on P → X also is an H -principal vector field on P → Σ. It yield aninfinitesimal gauge transformation υ ξ of a composite bundle Y seen as a P - and P Σ -associatedbundle. It reads υ ξ = ξ λ ∂ λ + ξ p ( x µ ) J mp ∂ m + ϑ aξ ( x µ , σ k ) I Aa ∂ A , (15)where { J p } is a representation of a Lie algebra g of G in G/H and { I a } is a representation of aLie algebra h of H in V .In view of Lemma 2 and Lemma 4, we restrict our consideration to groups G and H such thatthe quotient G/H is an Euclidean space. As was mentioned above, this is the case of a maximalcompact subgroup H ⊂ G .An additional condition on G and H is motivated by the following fact [2]. Theorem 5.
Let a Lie algebra g of G be the direct sum g = h ⊕ f (16) of a Lie algebra h of H and its supplement f obeying the commutation relations [ f , f ] ⊂ h r , [ f , h r ] ⊂ f . e.g., H is a Cartan subgroup of G ). Let A be a principal connection on P . The h -valuedcomponent A h of its pullback onto a reduced H -principal subbundle P h is a principal connectionon P h . Moreover, there is a principal connection on an H -principal bundle P → Σ so that itsrestriction to very reduced H -principal subbundle P h coincides with A h . If the decomposition (16) holds, the well-known non-linear realization of a Lie group G possess-ing a subgroup H exemplifies the induced representation (11) [1, 3]. In fact, it is a representationof the Lie algebra of G around its origin as follows [2, 8].In this case, there exists an open neighbourhood U of the unit ∈ G such that any element g ∈ U is uniquely brought into the form g = exp( F ) exp( I ) , F ∈ f , I ∈ h r . Let U G be an open neighbourhood of the unit of G such that U G ⊂ U , and let U be an openneighbourhood of the H -invariant center σ of the quotient G/H which consists of elements σ = gσ = exp( F ) σ , g ∈ U G . Then there is a local section z ( gσ ) = exp( F ) of G → G/H over U . With this local section,one can define the induced representation (11) of elements g ∈ U G ⊂ G on U × V given by theexpressions g exp( F ) = exp( F ′ ) exp( I ′ ) , g : (exp( F ) , v ) → (exp( F ′ ) , exp( I ′ ) v ) . (17)The corresponding representation of a Lie algebra g of G takes the following form. Let { F α } , { I a } be the bases for f and h , respectively. Their elements obey the commutation relations[ I a , I b ] = c dab I d , [ F α , F β ] = c dαβ I d , [ F α , I b ] = c βαb F β . Then the relation (17) leads to the formulas F α : F → F ′ = F α + X k =1 l k [ . . . k [ F α , F ] , F ] , . . . , F ] − (18) l n X n =1 [ . . . n [ F, I ′ ] , I ′ ] , . . . , I ′ ] ,I ′ = X k =1 l k − [ . . . k − [ F α , F ] , F ] , . . . , F ] , (19) I a : F → F ′ = 2 X k =1 l k − [ . . . k − [ I a , F ] , F ] , . . . , F ] , (20) I ′ = I a , (21)where coefficients l n , n = 1 , . . . , are obtained from the recursion relation n ( n + 1)! = n X i =1 l i ( n + 1 − i )! . Let U F be an open subset of the origin of the vector space f such that the series (18) – (21)converge for all F ∈ U F , F α ∈ f and I a ∈ h . Then the above mentioned non-linear realization ofa Lie algebra g in U F × V reads F α : ( F = σ β F β , v ) → ( F ′ = σ ′ β F β , I ′ v ) , I a : ( F = σ β F β , v ) → ( F ′ = σ ′ β F β , I ′ v ) , F ′ and I ′ are given by the expressions (18) – (20). In physical models, the coefficients σ α of F = σ α F α are treated as Goldstone fields.A problem is that the series (18) – (21) fail to be summarized in general, and one usuallyrestrict them to the terms of second degree in σ α .In the case of a pseudo-orthogonal group G = SO (1 , m ) and its maximal compact subgroup SO ( m ), one however can bring the expressions (18) – (21) into an explicit form. Let us notethat, as it was required above, the quotient SO (1 , m ) /SO ( m ) is homeomorphic to R m , and wehave the Lie algebra decomposition (16) such that so (1 , m ) = so ( m ) + f . (22)For instance, this is the case of a Lorentz group SO (1 ,
3) and its subgroup SO (3) of spatialrotations.A key point is that there is a monomorphism of a Lie algebra so (1 , m ) to a real Cliffordalgebra C ℓ ( m ) modelled over a pseudo-Euclidean space ( R m , η ) with a pseudo-Euclidean metric η of signature (+ , − ... − ) [4].Given the generating basis { γ i } , γ i γ k + γ k γ i = 2 δ lk , for a Clifford algebra C ℓ ( m ), the above mentioned monomorphism reads I k = γ k , I ik = 14 [ γ k , γ i ] , (23)where { I k , I ik } is a basis for a Lie algebra so (1 , m ) and, accordingly, { I ik } is a basis for asubalgebra so ( m ) and { F k = γ k = I k } is a basis for its complement f . Substituting theseexpressions into the formulas (17) – (21), we obtainexp( σ k γ k ) = cosh σ + σ i σ sinh σγ i , σ = δ lk σ i σ k ,F m = I m = u km ∂∂σ k = σ k σ m σ (cid:18) − σ cosh(2 σ )sinh σ (cid:19) ∂ k + 2 σ cosh(2 σ )sinh(2 σ ) ∂ m , (24) I ′ = 2 σ k σ tanh σ I km ,I ik = σ i ∂ k − σ k ∂ i , I ′ = I ik . References [1] Coleman, S., Wess, J., and Zumino, B. (1969) Structure of phenomenological Lagrangians.I, II
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