Towards non-perturbative quantization and the mass gap problem for the Yang-Mills Field
aa r X i v : . [ h e p - t h ] F e b TOWARDS NON–PERTURBATIVE QUANTIZATION AND THE MASS GAPPROBLEM FOR THE YANG-MILLS FIELD
A. SEVOSTYANOV
Abstract.
We reduce the problem of quantization of the Yang–Mills field Hamiltonian to aproblem for defining a probability measure on an infinite–dimensional space of gauge equivalenceclasses of connections on R . We suggest a formally self–adjoint expression for the quantized Yang–Mills Hamiltonian as an operator on the corresponding Lebesgue L –space. In the case when theYang–Mills field is associated to the Abelian group U (1) we define the probability measure whichdepends on two real parameters m > c = 0. This yields a non–standard quantization ofthe Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian.The corresponding quantized Hamiltonian is a self–adjoint operator in a Fock space the spectrumof which is { } ∪ [ m, ∞ ), i.e. it has a gap. Introduction
The purpose of this short note is to reduce the problem of non–perturbative quantization of theYang–Mills field Hamiltonian to a problem for defining a probability type measure on an infinite–dimensional space of gauge equivalence classes of connections on R . Recall that the Hamiltonian ofthe Yang–Mills field associated to a compact Lie group K with Lie algebra k is quadratic in momentaand its potential is equal to the square of the three–dimensional curvature tensor F with respect to anatural metric < · , · > on the space of k –valued differential forms on R . Our key observation is thatthe k –valued one–form G on R given by the Hodge star operator ∗ in R applied to F , G = ∗ F , isa potential vector field on the space of gauge equivalence classes of connections on R , the potentialbeing the Chern–Simons functional. So that the potential term of the Yang–Mills Hamiltonianbecomes the square of a potential vector field < G, G > on the space of gauge equivalence classes ofconnections on R equipped with the metric < · , · > which plays the role of the configuration spaceof the Yang–Mills field, and the cotangent bundle to it is the corresponding phase space.We show that for a Riemannian manifold M with a Riemannian metric < · , · > any Hamiltonianon the symplectic manifold T ∗ M of the form(1) 12 ( < p, p > + < v ( x ) , v ( x ) > ) , where p ∈ T x M ≃ T ∗ x M is the momentum and v = grad φ is a potential vector field, admits a familyof canonical quantizations of the form(2) 12 n X a =1 ξ ∗ a ( x ) ξ a ( x ) : L ( M, dµ ) → L ( M, dµ ) . Here ξ a ( x ), a = 1 , . . . , dim M is an orthonormal basis of T x M , and ξ ∗ a ( x ) is the operator formallyadjoint to ξ a ( x ) with respect to the canonical scalar product in the space L ( M, dµ ) of squareintegrable functions on M with respect to the measure dµ = ψe − φ dx , where dx is the Lebesgue Key words and phrases.
Yang-Mills field, Gaussian measure. measure on M associated to the Riemannian metric, and ψ is an arbitrary smooth non–vanishingfunction on M .The appearance of the function ψ shows some ambiguity which is permitted by the correspon-dence principle in the course of quantization. We shall see that according to this principle for anysmooth non–vanishing function ψ on M the operator given by expression (2) is a quantization ofthe Hamiltonian ( < p, p > + < v ( x ) , v ( x ) > ). But, of course, the properties of the quantizedHamiltonian depend on the choice of ψ . In practice the choice of ψ should be dictated by experi-mental data and by purely mathematical restrictions. It seems that the freedom of this kind in thequantization of classical Hamiltonian systems has not been used so far. As we shall see the lattertype of restrictions becomes primarily important in the case of the Yang–Mills field.To illustrate the above mentioned ambiguity we are going to consider the situation when M = R with the usual Euclidean metric, and the classical Hamiltonian is < p, p > , i.e. it describes a freeparticle on the line. If ψ ( x ) = 1 then the corresponding operator (2) is − d dx : L ( R , dx ) → L ( R , dx ) , i.e. it is the quantum Hamiltonian of a free particle on the line. It gives rise to a self–adjoint operatorthe spectrum of which is [0 , ∞ ).But one can also choose ψ ( x ) = exp( − x ), and then the corresponding operator (2) becomes − e x ddx e − x ddx : L ( R , exp( − x ) dx ) → L ( R , exp( − x ) dx )which is the Hermite differential operator. It gives rise to a self–adjoint operator on L ( R , exp( − x ) dx )the spectrum of which is the set { , , , , . . . } , and the corresponding eigenfunctions are the Her-mite polynomials (see e.g. [8]). Thus with this choice of ψ we obtain, up to a non–essential constant,the Hamiltonian of a quantum harmonic oscillator, and the spectrum of it has a gap separating itfrom the zero eigenvalue corresponding to the ground state.We show that the Hamiltonian of the Yang–Mills field is of type (1), where M is the space ofgauge equivalence classes of connections on R equipped with the metric < · , · > , and φ is the Chern–Simons functional which we denote by CS . Expressing the corresponding quantized Hamiltonian inform (2) solves the so called normal ordering problem which appears in the course of quantization.Thus the problem of quantization of the Yang–Mills Hamiltonian is reduced to defining a measureon the infinite–dimensional space of gauge equivalence classes of connections on R with “density” ψe − φ . Note that measures on infinite–dimensional spaces are probability measures, and to ensurethat the obtained measure on the space of gauge equivalence classes of connections on R is aprobability measure it is natural to choose ψ = exp( − < G, G > ) which guarantees that ψe − φ decreases at “infinity” in this space.It turns out, however, that even in the Abelian case when K = U (1) this ansatz does not work.If we use the Coulomb gauge fixing condition to describe the space of gauge equivalence classes of U (1)–connections on R as the space of vector fields satisfying the condition div A = 0 then theappropriate choice for ψ is exp( − c < G, G > − ( c + m ) < A, A > ), c, m ∈ R , c = 0, m >
0, andwe show that ψe − φ = exp( − c < G, G > − CS ( A ) −
12 ( c + m ) < A, A > )is the exponent of a negatively defined quadratic expression in A . So that the corresponding proba-bility measure is Gaussian. With this choice of ψ the quantized Abelian Yang–Mills field suggestedin this paper rather resembles the second harmonic oscillator type quantization of the classicalHamiltonian for a free particle on the line considered above.Indeed, we prove that the corresponding quantized Yang–Mills Hamiltonian defined followingrecipe (2) is self–adjoint and its spectrum is { } ∪ [ m, ∞ ), i.e. it has a gap. OWARDS NON–PERTURBATIVE QUANTIZATION OF THE YANG–MILLS FIELD 3
The paper is organized as follows. In Sections 1 and 2 we recall the results on the Lagrangianand the Hamiltonian formulation for the Yang–Mills field. These results are well–known in someform. We formulate them in a form suitable for our purposes. In Proposition 2 we make the keyobservation about the structure of the potential in the Hamiltonian of the Yang–Mills field.In Section 3.1 we discuss quantizations of Hamiltonians of the Yang–Mills type mentioned above,and in Section 3.2 these results are applied to the Yang–Mills Hamiltonian.
Acknowledgements
The results presented in this paper have been partially obtained during research stays at Institutdes Haut ´Etudes Scientifiques, Paris and Max–Planck–Instut f¨ur Mathematik, Bonn. The author isgrateful to these institutions for hospitality.The research on this project received funding from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement677368) during the visit of the author to Institut des Haut ´Etudes Scientifiques, Paris.1.
The Yang-Mills field in Hamiltonian formulation
The Yang-Mills field as a Hamiltonian system with constrains.
In this section fol-lowing [5] we recall the Lagrangian and the Hamiltonian formalism for the Yang-Mills field. Thecanonical variables and the Hamiltonian will be obtained via the Legendre transform starting fromthe Lagrangian formulation.Let K be a compact semisimple Lie group, k its Lie algebra and g the complexification of k . Wedenote by ( · , · ) the Killing form of g . Recall that the restriction of this form to k is nondegenerateand negatively defined. We shall consider the Yang-Mills functional on the affine space of smoothconnections in the trivial K -bundle, associated to the adjoint representation of K , over the standardMinkowski space R , . Fixing the standard trivialization of this bundle and the trivial connectionas an origin in the affine space of connections we can identify this space with the space Ω ( R , , k )of k -valued 1-forms on R , . Let A ∈ Ω ( R , , k ) be such a connection. Denote by F the curvature2-form of this connection, F = d A + [ A ∧ A ]. Here as usual we denote by [
A ∧ A ] the operationwhich takes the exterior product of k -valued 1-forms and the commutator of their values in k . TheYang-Mills action functional Y M evaluated at A is defined by the formula(3) Y M = 12 Z R , ( F∧ , ∗F ) , where ∗ stands for the Hodge star operation associated to the standard metric on the Minkowskispace, and we evaluate the Killing form on the values of F and ∗F and also take their exteriorproduct. The corresponding Lagrangian density L is equal to ∗ ( F∧ , ∗F ),(4) L = 12 ∗ ( F∧ , ∗F ) . Next, following [5], we pass from Lagrangian to Hamiltonian formulation for the Yang-Mills field.To this end one should use the modified Lagrangian density L ′ ,(5) L ′ = ∗ (( d A + 12 [ A ∧ A ] − F ) ∧ , ∗F ) , where A and F should be regarded as independent variables. The equations of motion obtained byextremizing the corresponding action functional are equivalent to those derived from the action (3).Indeed, the equation for F following from (5) is just the definition of the curvature, and the otherequation becomes the usual Yang-Mills equation after expressing F in terms of A . A. SEVOSTYANOV
In order to pass to the Hamiltonian formalism for the Yang-Mills field we introduce a convenientnotation that will be used throughout of this paper. Let Ω ∗ ( R , k ) be the space of k -valued differentialforms on R . We define a scalar product on this space, whenever it is finite, by(6) < ω , ω > = − Z R ( ω ∧ , ∗ ω ) = − Z R ∗ ( ω ∧ , ∗ ω ) d x, ω , ∈ Ω ∗ ( R , k )where ∗ stands for the Hodge star operation associated to the standard Euclidean metric on R , andwe evaluate the Killing form on the values of ω and ∗ ω and also take their exterior product.Let A be k -valued connection 1-form in the trivial K -bundle, associated to the adjoint represen-tation of K , over the standard Minkowski space, F its curvature 2-form. Fix a coordinate system( x , x , x , x ) on R , so that x = t is the time and ( x , x , x ) are orthogonal Cartesian coordinateson R ⊂ R , . We denote by A the “three–dimensional Euclidean part” of A , A = P k =1 A k dx k ,where A k = A k for k = 1 , ,
3. We also introduce the “electric” field E and the “magnetic” field G associated to F as follows: E = P k =1 E k dx k , E k = F k ,G = ∗ F, F = dA + [ A ∧ A ] , i.e. F is the “three-dimensional” spatial part of F .We recall that the covariant derivative d A : Ω n ( R , k ) → Ω n +1 ( R , k ) associated to A is definedby d A ω = dω + [ A ∧ ω ], and the operator formally adjoint to d A with respect to scalar product (6)is equal to − ∗ d A ∗ . We denote by div A the part of this operator acting from Ω ( R , k ) to Ω ( R , k ),with the opposite sign, div A = ∗ d A ∗ : Ω ( R , k ) → Ω ( R , k ) . Using this notation the Lagrangian density (5) can be rewritten, up to a divergence, in thefollowing form:(7) L ′ = − (cid:18) ∗ ( E ∧ , ∗ ∂ t A ) −
12 ( ∗ ( E ∧ , ∗ E ) + ∗ ( G ∧ , ∗ G )) + ( A , div A E ) (cid:19) . For the corresponding action we have(8)
Y M ′ = Z ∞−∞ (cid:18) < E, ∂ t A > −
12 ( < E, E > + < G, G > )+ < A , div A E > (cid:19) dt.
Denote div A E by C , C = div A E , and introduce an orthonormal basis T a , a = 1 , . . . , dim k in k with respect to the Killing form and the components of A , E , A and C associated to this basis, A k = P a A ak T a , E k = P a E ak T a , A = A a T a , C = C a T a . In terms of these components the action(8) takes the form(9) Y M ′ = Z ∞−∞ X k,a E ak ∂ t A ak − h ( E, A ) + X a A a C a d x, where h ( E, A ) = 12 ( ∗ ( E ∧ , ∗ E ) + ∗ ( G ∧ , ∗ G ))is the Hamiltonian density. Denote(10) H ( E, A ) = 12 ( < E, E > + < G, G > ) . From formula (9) it is clear that A ak and E ak are canonical conjugate coordinates and momenta forthe Yang-Mills field, H ( E, A ) is the Hamiltonian, A a are Lagrange multipliers and C a = 0 areconstrains imposed on the canonical variables. OWARDS NON–PERTURBATIVE QUANTIZATION OF THE YANG–MILLS FIELD 5
The Yang-Mills equations become Hamiltonian with respect to the canonical Poisson structure(11) { E ak ( x ) , A bl ( y ) } = δ kl δ ab δ ( x − y ) , and all the other Poisson brackets of the components of E and A vanish. One can also check that(12) { C a ( x ) , C b ( y ) } = X c t abc C c ( x ) δ ( x − y ) , where t abc are the structure constants of Lie algebra k with respect to the basis T a , [ T a , T b ] = P c t abc T c , and that(13) { H ( E, A ) , C a ( x ) } = 0 . This means that the Yang-Mills field is a generalized Hamiltonian system with first class constrainsaccording to Dirac’s classification [4].2.
The structure of the phase space of the Yang-Mills field
Reduction of the phase space.
In this section we collect some facts on the Poisson geometryof the phase space of the Yang-Mills field and related gauge actions. These results are certainly wellknown. But it seems that they are not presented in the literature in the form suitable for ourpurposes (see, however, [13] about the gauge actions).To begin with, we consider the Yang-Mills field as a generalized Hamiltonian system with Hamil-tonian (10) and constraints C = div A E = 0 on the phase space Ω c ( R , k ) × Ω c ( R , k ) equipped withPoisson structure (11). Here Ω c ( R , k ) stands for the space of smooth k -valued 1-forms on R withcompact support. Later the phase space will be considerably extended.The Poisson structure (11) has a natural geometric interpretation. Indeed, consider the affinespace of smooth connections in the trivial K -bundle, associated to the adjoint representation of K ,over R . As in Section 1.1 we fix the standard trivialization of this bundle and the trivial connectionas an origin in the affine space of connections and identify this space with the space Ω ( R , k ) of k -valued 1-forms on R . Let D be the subspace in the affine space of connections isomorphic toΩ c ( R , k ) under this identification. We shall frequently write D instead of Ω c ( R , k ) and call thisspace the space of compactly supported K -connections on R .The space D has a natural Riemannian metric defined with the help of scalar product (6),(14) < E, E ′ > = − Z R ( E ∧ , ∗ E ′ ) , E, E ′ ∈ T A D ≃ D , This metric gives rise to a natural imbedding T D ֒ → T ∗ D induced by the natural imbedding D ֒ → D ∗ ,ω ˆ ω, ˆ ω ( ω ′ ) = < ω, ω ′ >, ω, ω ′ ∈ D . Using this imbedding the tangent bundle T D can be equipped with the natural structure of a Poissonmanifold induced by the canonical symplectic structure of T ∗ D . The Poisson structure (11) on thespace Ω c ( R , k ) × Ω c ( R , k ) ≃ T D can be identified with that induced by the canonical symplecticstructure of T ∗ D .Now let us discuss the meaning of the constrains. First of all we note that the constrains C =div A E infinitesimally generate the gauge action on the phase space T D . More precisely, let K be the group of K -valued maps g : R → K such that g ( x ) = I for | x | ≥ R ( g ), where I is theidentity element of K and R ( g ) > g . K is called the gauge groupof compactly supported gauge transformations. The Lie algebra of K is isomorphic to Ω c ( R , k ). A. SEVOSTYANOV
The gauge group K acts on the space of connections D by(15) K × D → D ,g × A g ◦ A = − dgg − + gAg − , where we denote dgg − = g ∗ θ R , gAg − = Ad g ( A ), and θ R is the right-invariant Maurer-Cartanform on K . This action is free, so that the quotient D / K is a smooth manifold.The action (15) of K on the space of connections D induces an action(16) K × T D → T D ,g × ( E, A ) ( gEg − , − dgg − + gAg − ) , where as before we write gEg − = Ad g ( E ). This action gives rise to an action of the Lie algebraΩ c ( R , k ) of the gauge group K on T D by vector fields. If X ∈ Ω c ( R , k ) then the correspondingvector field V X ( E, A ) is given by(17) V X ( E, A ) = ([
X, E ] , − dX + [ X, A ]) , ( E, A ) ∈ T D ≃ D × D . Here we, of course, identify T ( E,A ) T D ≃ T D ≃ D × D .The action (17) is generated by the constraint div A E in the sense that for X ∈ Ω c ( R , k ) , ( E, A ) ∈ T D we have { < div A E, X >, A ( x ) } = − dX ( x ) + [ X ( x ) , A ( x )] , and { < div A E, X >, E ( x ) } = [ X ( x ) , E ( x )] . Using the language of Poisson geometry and taking into account formula (12) for the Poissonbrackets of the constrains one can say that
K × T D → T D is a Hamiltonian group action, and themap(18) µ : T D → Ω c ( R , k ) ,µ ( E, A ) = div A E is the moment map for this action. In particular, action (16) preserves the symplectic form of T D .We note that action (16) also preserves Riemannian structure (14) of the configuration space D .This follows from the fact that the Killing form on k is invariant with respect to the adjoint actionof K .The properties of the phase space of the Yang-Mills field and of the gauge action discussed aboveare formulated in the following proposition. Proposition 1.
Let D be the space of compactly supported K -connections on R , K the group ofcompactly supported gauge transformations. Then(i) The space D is an infinite dimensional Riemannian manifold equipped with the metric (19) < E, E ′ > = − Z R ( E ∧ , ∗ E ′ ) , E, E ′ ∈ T A D . This metric induces a natural imbedding T D ֒ → T ∗ D , and the tangent bundle T D can be equippedwith the natural structure of a Poisson manifold induced by the canonical symplectic structure of T ∗ D . OWARDS NON–PERTURBATIVE QUANTIZATION OF THE YANG–MILLS FIELD 7 (ii) The gauge action
K × D → D preserves Riemannian metric (19) and gives rise to a Hamil-tonian group action
K × T D → T D with the moment map µ : T D → Ω c ( R , k ) ,µ ( E, A ) = div A E, ( E, A ) ∈ T D ≃ D × D . (iii) The action of the gauge group K on the spaces D and T D is free, and the reduced phase space µ − (0) / K is a smooth manifold. Finally we make a few remarks on the structure of the Hamiltonian of the Yang-Mills field.Since the Hamiltonian H ( E, A ) of the Yang-Mills field is invariant under the gauge action (16)(this fact can be checked directly and also follows from formula (13)) the generalized Hamiltoniandynamics described by this Hamiltonian together with the constrains div A E = 0 is equivalent tothe usual one on the reduced phase space µ − (0) / K (see [1, 4]).The Hamiltonian (10) itself has a very standard structure; H ( E, A ) is equal to the sum of a halfof the square of the momentum, < E, E > , and of a potential U ( A ), U ( A ) = < G, G > . Thepotential U ( A ) is, in turn, equal to a half of the square of the vector field G ∈ Γ( T D ). By definitionthe vector field G is invariant with respect to the gauge action of K , G ( g ◦ A ) = gG ( A ) g − . The valueof this field at each point A ∈ D belongs to the kernel of the operator div A , G ( A ) ∈ Ker div A ∀ A ∈ D .Indeed, from the Bianci identity d A F = 0, the definition of G = ∗ F and the formula ∗∗ = id itfollows that div A G = − ∗ d A ∗ ∗ F = − ∗ d A F = 0 . The vector field G has one more important property: it is potential with the potential functionequal to the Chern–Simons functional. Recall that this functional is defined by(20) CS ( A ) = 12 < A, ∗ dA > + 13 < A, ∗ [ A ∧ A ] > . This functional is gauge invariant and its gradient is equal to G .Now we summarize the properties of the Hamiltonian of the Yang-Mills field. Proposition 2. (i) The generalized Hamiltonian system on the Poisson manifold T D with theHamiltonian H ( E, A ) , H ( E, A ) = ( < E, E > + < G, G > ) , G = ∗ F , F = dA + [ A ∧ A ] , and theconstrains div A E = 0 describes the Yang-Mills dynamics on T D .(ii) The Hamiltonian H ( E, A ) is invariant under the gauge action K × T D → T D and the gener-alized Hamiltonian dynamics described by this Hamiltonian together with the constrains div A E = 0 is equivalent to the usual one on the reduced phase space µ − (0) / K .(iii) The vector field G is invariant with respect to the gauge action of K , G ( g ◦ A ) = gG ( A ) g − .The value of this field at each point A ∈ D belongs to the kernel of the operator div A , G ( A ) ∈ Ker div A ∀ A ∈ D .(iv) The vector field G is potential with the potential equal to the gauge invariant Chern–Simonsfunctional (20). The structure of the reduced phase space.
In Propositions 1 and 2 we formulated all theproperties of the Yang-Mills field which are important for our further consideration. In this sectionwe study an arbitrary Hamiltonian system satisfying these properties.First we consider a phase space equipped with a Lie group action of the type described in Propo-sition 1. Actually the Riemannian metric introduced in that proposition is only important for thedefinition of the Hamiltonian of the Yang-Mills field. This metric is not relevant to Poisson geome-try. We used this metric in the description of the phase space in order to avoid analytic difficultiesarising in the infinite-dimensional case. Now let us forget about the metric for a moment and discussthe geometry of the reduced space.
A. SEVOSTYANOV
The Poisson structure described in Proposition 1 is an example of the canonical Poisson structureon the cotangent bundle, and the group action on this bundle is induced by a group action on thebase manifold. Thus we start with a manifold M and a Lie group G freely acting on M . Thecanonical symplectic structure on T ∗ M can be defined as follows (see [1]).Denote by π : T ∗ M → M the canonical projection, and define a 1-form θ on T ∗ M by θ ( v ) = p ( dπv ), where p ∈ T ∗ x M and v ∈ T ( x,p ) ( T ∗ M ). Then the canonical symplectic form on T ∗ M isequal to dθ .Recall that the induced Lie group action G × T ∗ M → T ∗ M is a Hamiltonian group action witha moment map µ : T ∗ M → g ∗ , where g ∗ is the dual space to the Lie algebra g of G . The momentmap µ is uniquely determined by the formula (see [10], Theorem 1.5.2)(21) ( µ ( x, p ) , X ) = θ ( bb X )( x, p ) = p ( b X ( x )) , where b X is the vector field on M generated by arbitrary element X ∈ g , bb X is the induced vectorfield on T ∗ M and ( , ) stands for the canonical paring between g and g ∗ .Formula (21) implies that for any x ∈ M the map µ ( x, p ) is linear in p . We denote this linearmap by m ( x ), m ( x ) : T ∗ x M → g ∗ ,(22) m ( x ) p = µ ( x, p ) . Next, following [1], Appendix 5, with some modifications of the proofs suitable for our purposes,we describe the structure of the reduced space µ − (0) /G . We start with a simple lemma. Lemma 3.
The annihilator T x O ⊥ ∈ T ∗ x M of the tangent space T x O to the G -orbit O ⊂ M at point x is isomorphic to Ker m ( x ) , T x O ⊥ = Ker m ( x ) .Proof. First we note that the space T x O ⊥ is spanned by the differentials of G -invariant functionson M . But from the definitions of the moment map and of the Poisson structure on T ∗ M we have(23) L b X f ( x ) = { ( X, µ ) , f } ( x ) = ( X, m ( x ) df ( x )) , where b X is the vector field on M generated by element X ∈ g , f ∈ C ∞ ( M ), and ( , ) stands for thecanonical paring between g and g ∗ .Formula (23) implies that f is G -invariant if and only if df ( x ) ∈ Ker m ( x ). This completes theproof. (cid:3) Proposition 4.
The action of the group G on T ∗ M is free, and the reduced phase space µ − (0) /G is a smooth manifold. Moreover, we have an isomorphism of symplectic manifolds, µ − (0) /G ≃ T ∗ ( M /G ) , where T ∗ ( M /G ) is equipped with the canonical symplectic structure. Under this isomor-phism T ∗O x ( M /G ) ≃ T x O ⊥ x , where O x is the G -orbit of x .Proof. Let O x be the G -orbit of point x ∈ M and π : M → M /G the canonical projection, π ( x ) = O x . Denote by Ξ the foliation of the space M by the subspaces T x O ⊥ . Since the foliation Ξis G -invariant and Ker dπ | T x M = T x O x we can identify the subspace T x O ⊥ x with the tangent space T ∗O x ( M /G ) by means of the dual map to the differential of the projection π . But the definition ofthe moment map µ and Lemma 3 imply that µ − (0) = { ( x, p ) ∈ T ∗ M : p ∈ T x O ⊥ x } . Therefore thequotient µ − (0) /G is diffeomorphic to T ∗ ( M /G ), the diffeomorphism being induced by the canonicalprojection π .From the definitions of the Poisson structures on T ∗ ( M /G ) and on the reduced space µ − (0) /G it follows that the diffeomorphism µ − (0) /G ≃ T ∗ ( M /G ) is actually an isomorphism of symplecticmanifolds. (cid:3) OWARDS NON–PERTURBATIVE QUANTIZATION OF THE YANG–MILLS FIELD 9
Using the last proposition one can easily describe the space Γ T ∗ ( M /G ) of covector fields on M /G . Corollary 5.
The space Γ T ∗ ( M /G ) is isomorphic to the space of G -invariant sections V ∈ Γ T ∗ M such that V ( x ) ∈ T x O ⊥ x for any x ∈ M . Such covector fields will be called vertical G -invariantcovector fields on M . We denote this space by Γ ⊥ G T ∗ M , Γ ⊥ G T ∗ M ≃ Γ T ∗ ( M /G ) . Now we discuss the class of Hamiltonians on T ∗ M we are interested in. First, recalling Proposition1 we equip the manifold M with a Riemannian metric <, > and assume that the action of G on M preserves this metric. Using this metric we can establish an isomorphism of G -manifolds, T M ≃ T ∗ M . We shall always identify the tangent and the cotangent bundle of M and the spacesof vector and covector fields on M by means of this isomorphism. The tangent bundle T M will beregarded as a symplectic manifold with the induced symplectic structure. Under the identification T M ≃ T ∗ M the subspace T x O ⊥ ⊂ T ∗ x M is isomorphic to the orthogonal complement of the tangentspace T x O in T x M . Note also that since T ∗O x ( M /G ) ≃ T x O ⊥ x and the metric on M is G -invariant T ∗O x ( M /G ) has a scalar product induced from T x O ⊥ x , i.e. M /G naturally becomes a Riemannianmanifold. We shall also identify T ∗ ( M /G ) ≃ T ( M /G ) by means of the metric. Denote by Γ ⊥ G T M the space of G -invariant vertical vector fields on M . By Corollary 5 we have an isomorphism,Γ ⊥ G T M ≃ Γ T ( M /G ).On the symplectic manifold T M we define a Hamiltonian of the type described in Proposition 2.In order to do that we fix a G -invariant vertical vector field V on M . Then we put l ( x, p ) = 12 ( < p, p > + < V ( x ) , V ( x ) > ) , p ∈ T x M . This Hamiltonian is obviously G -invariant and gives rise to a Hamiltonian l red on the reduced space µ − (0) /G ≃ T ∗ ( M /G ). Since by Corollary 5 V can be regarded as a (co)vector field on M /G wehave(24) l red ( O x , p ⊥ ) = 12 ( < p ⊥ , p ⊥ > + < V ( x ) , V ( x ) > ) , p ⊥ ∈ T x O ⊥ x ≃ T ∗O x ( M /G ) . Now we can apply the above obtained results in the case of the Yang–Mills field. The reducedphase space of the Yang–Mills field is of the type considered in Lemma 3 and Proposition 4 with M = D and G = K . In the infinite–dimensional case we have to distinguish between T D and T ∗ D .But according to Proposition 1 for the description of the Yang–Mills dynamics it suffices to consider T D and equip it with the Poisson structure induced by the imbedding T D ⊂ T ∗ D with the help ofmetric (19). Then the action of K of T D becomes Hamiltonian, and in the notation of Lemma 3 m ( x ) = div A .Let O A be the gauge orbit of a connection A ∈ D . By Lemma 3 the space T O A D / K is isomorphicto the kernel of the operator div A in T A D . The metric (19) induces a Riemannian metric on D / K which we denote by the same symbol.According to Proposition 2 the vector field G on the space D is K –invariant and horizontal.Hamiltonian (10) is of type (24). Therefore from formula (24) and Proposition 4 we infer thatHamiltonian (10) gives rise to the Hamiltonian(25) H red ( O A , E ⊥ ) = 12 ( < E ⊥ , E ⊥ > + < G, G > ) , E ⊥ ∈ T O A ( D / K ) ≃ Ker div A on the reduced phase space µ − (0) / K ≃ T D / K .Based on the results of this section we can also make two remarks on the structure of the gaugeorbit space D / K . Remark 6.
The Riemannian geometry of the space D / K is nontrivial. In particular, its curvaturetensor is not identically equal to zero (see [13] ). This is the main peculiarity of non-abelian gaugetheories. Remark 7.
The quotient D / K can not be realized as a cross-section for the gauge action of K on D . For any local cross-section of this action there are K -orbits in D which meet this cross-sectionmany times. This phenomenon is called the Gribov ambiguity (see [14] ).The Riemannian manifold D / K can not be realized as a cross-section for the action of K on D even locally. This is due to the fact that the foliation Ξ of D by the subspaces Ker div A ⊂ T A D is notan integrable distribution, and therefore the subspaces Ker div A are not tangent to a submanifold in D . Indeed, the components C a of the constraint div A E regarded as an Ω ( R , k ) -valued 1-form on D do not form a differential ideal. Therefore the conditions of the Frobenius integrability theoremare not satisfied. In Poisson geometry constrains of this type are called non–holonomic. Quantization of the Hamiltonian of the Yang–Mills field
Quantization of Yang–Mills type Hamiltonians: a model case.
Let M be n –dimensionalRiemannian manifold with a metric < · , · > . For simplicity we denote the pairing between T x M and T ∗ x M and the induced scalar product on T ∗ x M by the same symbol as the metric on M . As beforewe can identify T ∗ M and T M using the metric.Consider a Hamiltonian of type (24) on T ∗ M ≃ T M ,(26) h ( x, p ) = 12 ( < p, p > + < v ( x ) , v ( x ) > ) , x ∈ M, p ∈ T x M, where v is a vector field on M . So M plays the role of M /G in this section.Assume that the vector field v is potential with a potential function φ , so v = grad φ .Let ξ a ( x ), a = 1 , . . . , n be an orthonormal basis in T x M , < ξ a ξ b > = δ ab . Let T ξ a = < ξ a , p > − i < ξ a , v > , T ∗ ξ a = < ξ a , p > + i < ξ a , v > . From this definition and from the definition of the basis ξ a it follows immediately that(27) h ( x, p ) = 12 n X a =1 T ∗ ξ a T ξ a . Now let x , . . . , x n be a local coordinate system on M defined on an open subset of M , ξ ia thecoordinates of ξ a with respect to this coordinate system, so ξ a = P ni =1 ξ ia ∂∂x i . Denote by g ij thecomponents of metric tensor of the metric < · , · > in terms of the coordinates x , . . . , x n . We alsohave p = P ni =1 p i dx i .Let L ( M, ψ ) be the Hilbert space of complex–valued functions on M such that Z M | f | ψdµ < ∞ , where µ is the Lebesgue measure on M associated to the Riemannian metric, and ψ ∈ C ∞ ( M ) is asmooth non–vanishing function on M . The scalar product on L ( M, ψ ) is given by the usual formula( f, f ′ ) ψ = Z M f ¯ f ′ ψµ. According to the canonical quantization philosophy and the correspondence principle after quan-tization p i becomes the operator i ∂∂x i in L ( M, ψ ), and any function of x becomes the multiplicationoperator by that function in L ( M, ψ ), so T ξ a becomes the operator i ξ a − i ( ξ a , v ) = − i ∇ ξ a , where ∇ ξ a = ξ a + < ξ a , v > .We would like to define a self–adjoint operator in L ( M, ψ ) which is a quantization of the Hamil-tonian h ( x, p ). According to the canonical quantization philosophy we have to ensure that the OWARDS NON–PERTURBATIVE QUANTIZATION OF THE YANG–MILLS FIELD 11 quantized Hamiltonian becomes a self–adjoint operator in L ( M, ψ ). In order to fulfill this require-ment we have to require that after quantization T ∗ ξ a becomes the operator adjoint to i ξ a − i ( ξ a , v )in L ( M, ψ ). In terms of the local coordinates the operator formally adjoint to i ξ a − i < ξ a , v > takes the form f i (cid:18) − √ g ψ − ∂∂x i ( ξ ia √ gψf )+ < ξ a , v > f (cid:19) = i ∇ ∗ ξ a f, where g = | det g ij | , so a natural candidate for a quantized Hamiltonian is the self–adjoint operator h defined by the expression(28) 12 n X a =1 ∇ ∗ ξ a ∇ ξ a . One straightforwardly verifies that, after applying reversely the correspondence principle accord-ing to which the operator i ∂∂x i becomes p i , and the multiplication operator by a function in L ( M, ψ )becomes this function in the classical limit, expression (28) becomes Hamiltonian (26) in the classicallimit.Note that the operator of multiplication by e φ gives rise to a unitary equivalence L ( M, ψ ) → L ( M, ψe − φ ), and the operator h in L ( M, ψe − φ ) unitarily equivalent to h , h = e φ h e − φ , isdefined using the expression(29) e φ n X a =1 ∇ ∗ ξ a ∇ ξ a e − φ = 12 n X a =1 ξ ∗ a ξ a , where as above in local coordinates ξ a = P ni =1 ξ ia ∂∂x i , and ξ ∗ a is the operator formally adjoint to ξ a with respect to the scalar product in L ( M, ψe − φ ).A formal definition of the self–adjoint operator h can be given using its bilinear form. Clearly,expression(29) defines a non–negative symmetric operator on L ( M, ψe − φ ), with the domain beingthe space C ∞ of smooth complex–valued compactly supported functions on M . Thus one can applythe Friedrichs extension method to define its self–adjoint extension (see [11], Theorem X.23). Thisyields the following statement. Theorem 8.
The non–negative bilinear form ( f, f ′ ) h = P na =1 ( ξ a f, ξ a f ′ ) ψe − φ , with the domainbeing the space C ∞ of smooth complex–valued compactly supported functions on M , is closable on L ( M, ψe − φ ) with a domain D and its closure defines a non–negative self–adjoint operator h on L ( M, ψe − φ ) with a domain D ( h ) , so that ( f, f ′ ) h = ( hf, f ′ ) ψe − φ for any f ∈ D ( h ) , f ′ ∈ D .Moreover, if the constant function belongs to L ( M, ψe − φ ) then is an eigenfunction of theoperator h with the lowest eigenvalue zero.For any smooth non–vanishing function ψ on M , ψ ∈ C ∞ ( M ) , the operator h is a quantizationof Hamiltonian (26) in the sense of canonical quantization. The second part of the previous theorem ensures the existence of the lowest energy ground statefor the operator h .3.2. Application to the Yang-Mills Hamiltonian.
Now we are going to apply the idea of theprevious section to quantize the reduced Yang–Mills Hamiltonian defined by formula (25) on thereduced phase space µ − (0) / K ≃ T D / K . Note that according to this formula H red is of the sametype as the Hamiltonian h ( x, p ) considered in the previous section with φ = CS ( A ). So informally,according to Proposition 4, we should take D / K as M in the previous section. But the fact that D / K is infinite–dimensional now brings further difficulties.According to the philosophy of Section 3.1, firstly we should try to find a measure with “density”which resembles ψe − φ with φ = CS ( A ) and an appropriate ψ . The peculiarity of the infinite–dimensional case is that the existence of such measures is a very strong condition. In particular, all known measures of this kind are probability measures, so that the entire space has a finite volumeusually normalized to one. Therefore ψe − φ should rapidly decrease at infinity. As it can be easilyseen this condition is not fulfilled if we choose ψ = 1. It is natural to use ψ = exp( − < G, G > )and then(30) ψe − φ = exp( − < G, G > − CS ( A )) . This functional is explicitly gauge invariant, so it is well defined on D / K . But it turns out that ameasure with a “density” which resembles exp( − < G, G > − CS ( A )) still does not exist evenin the Abelian case, and a certain “renormalization” is required to define it. This phenomenon isrelated to the fact that there are no translation invariant measures on infinite–dimensional spaces,and one should not expect that a measure on an infinite–dimensional quotient space by an actionof an infinite–dimensional group is induced by a measure on the original space invariant under thegroup action. Therefore firstly we have to fix a model for D / K and then define a measure on it. Weshall do it now in the Abelian case when K = U (1).So from now on we assume that K = U (1). We identify the corresponding Lie algebra with R .Choose a model D for D / K being the space of the elements A of D which satisfy the conditiondiv A = 0, where div = div .In the Abelian case we have(31) exp( − < G, G > − CS ( A )) = exp( − < ∗ dA, ∗ dA > − < ∗ dA, A > ) , and this function is the exponent of an expression which is quadratic in A which means that themeasure that we are going to construct is likely to be Gaussian. To define such a measure we haveto ensure that the expression in the exponent is negative definite which is not true for (31). In orderto fulfill this condition we choose ψ = exp( − c < G, G > − ( c + m ) < A, A > ), A ∈ D , where c, m ∈ R are constants, c = 0, and m >
0. Then(32) ψe − φ = exp( − c < G, G > − CS ( A ) −
12 ( c + m ) < A, A > ) == exp( − c < ∗ dA, ∗ dA > − < ∗ dA, A > −
12 ( c + m ) < A, A > ) = exp( −
12 (Λ
A, A )) , where Λ = T + mId , and T = c curl + cId , curl = ∗ d are symmetric operators on D with respectto the scalar product < · , · > .Recall that Gaussian measures are actually defined on spaces dual to nuclear spaces (see e.g. [9]).This forces us to enlarge D and to replace it with the nuclear space S which consists of elements A of Ω ( R , k ) = Ω ( R ) the components of which with respect to the fixed Cartesian coordinatesystem belong to the Schwartz space and which satisfy the condition div A = 0, the topology on S being induced by that of the Schwartz space. Let S ∗ be the dual space.According to the Bochner–Minlos theorem (Theorem 1.5.2 in [9]) Gaussian measures on S ∗ are Fouries transforms of characteristic functionals on S , and the Gaussian measure with “den-sity” which resembles exp( − (Λ A, A )) should have the characteristic functional C ( A ) = exp( − < Λ − A, A > ). We claim that C ( A ), A ∈ S is a characteristic functional.To justify this claim we shall need some facts about the spectral decomposition for the operatorcurl (see [3], § H i , i = 0 , ic ( R , k ) = Ω ic ( R ) with respect to scalar product(6). Here we assume that k is identified with R and the Killing form is just minus the product of realnumbers. According to Lemma 8 (i) in [12] div : Ω c ( R ) → H is a closable operator. We denoteits closure by the same symbol, div : H → H . OWARDS NON–PERTURBATIVE QUANTIZATION OF THE YANG–MILLS FIELD 13
Ker div ⊂ H is naturally a Hilbert space with the scalar product inherited from H , and, infact, this Hilbert space is rigged. Namely,(33) S ⊂ Ker div ⊂ S ∗ is the corresponding Gelfand-Graev triple.Let H i C , i = 0 ,
1, Ker div C and S C be the complexifications of H i , i = 0 ,
1, Ker div and S ,respectively. We can identify H C with the Lebesgue space L ( R , C ) of square integrable functionswith values in C equipped with the scalar product induced from H C . The componentwise Fouriertransform F provides an isomorphism of L ( R , C ) onto itself under which Ker div C is mapped ontothe subspace F (Ker div C ) in L ( R , C ) which consists of C –valued functions f ( k ) ∈ L ( R , C ), k ∈ R satisfying the condition k · f ( k ) = 0, where · is the standard scalar product in C induced bythe Cartesian product in R fixed above. Also the Fourier transform maps S C ⊂ H C isomorphicallyonto the subspace F ( S C ) of C –valued functions f ( k ) ∈ L ( R , C ), k ∈ R with components fromthe complex Schwartz space and satisfying the condition k · f ( k ) = 0. Ker div C is an invariantsubspace for the natural extension of the operator curl to H C . Note that the action of curl on H C preserves H and Ker div, i.e. curl is a real operator and Ker div is an invariant subspace for it.curl : Ker div C → Ker div C is a self–adjoint operator with the natural domain { v ∈ Ker div C :curl v ∈ Ker div C } .Under the isomorphism F the operator curl acting on H C ≃ L ( R , C ) becomes the operator F curl F − acting by the cross vector product by ik on elements of f ( k ) ∈ L ( R , C ). For each k ∈ R this operator acts of f ( k ) ∈ C by the matrix(34) − ik ik ik − ik − ik ik which is nothing but the symbol of the operator curl. F (Ker div C ) is an invariant subspace for the operator F curl F − . For each fixed k the eigenvaluesof matrix (34) restricted to the subspace in C which consists of elements v ∈ C satisfying thecondition k · v = 0 are ±| k | . According to [3], § σ (curl) = σ ac (curl) = R , and hence curl has no eigenvectorsin the usual sense. But it has a complete basis of generalized eigenvectors (see, for instance, [6]).Namely, this operator can be easily diagonalized by means of the Fourier transform (see [3], Ch.8, § ±| k | , k ∈ R \ { } , can be chosen, for instance, in the form(35) 1 √ π ) e ik · x ( θ ( k ) ± iθ ( k )) , where θ , ( k ) are 1-forms on R dual to orthonormal vectors e , ( k ), with respect to the fixedCartesian scalar product, such that for every k = 0 k | k | , e ( k ) , e ( k ) is an orthonormal basis in R , k | k | × e ( k ) = e ( k ) (vector product) and e , ( k ) smoothly depend on k ∈ R \ { } .Since the operator curl sends real–valued functions to real valued functions one can also find realgeneralized eigenvectors e ± ( k ) ∈ S ∗ corresponding to ±| k | , k ∈ R \ { } .The vectors e ± ( k ) are generalized eigenvectors for the operator curl in the sense that(36) < e ± ( k ) , curl ω > = ±| k | < e ± ( k ) , ω > for any ω ∈ S . Note also that S is dense in Ker div.Now we show that Λ − : S → S is a continuous operator. The easiest way to see this is to observethat according to the results on the eigenvalues of matrix (34) mentioned above the eigenvalues ofthe symbol of the operator Λ acting for each fixed k on the subspace in C which consists of elements v ∈ C satisfying the condition k · v = 0 are ( ± c | k | + c ) + m ≥ m >
0. Therefore the inverse to thesymbol is well–defined for each k , the entries of the inverse to the symbol are smooth and bounded(in fact they are rational functions of the components of k with respect to the orthonormal basis in R ), and hence it gives rise to a bounded operator on F ( S C ). Applying the inverse to the Fouriertransform and recalling that Λ, and hence Λ − , preserve S ⊂ S C we deduce that Λ − : S → S isa continuous operator.Recall also that < · , · > is a continuous bilinear form on S . Therefore the functional C ( A ) =exp( − < Λ − A, A > ) is continuous. Obviously, C (0) = 1.Note that < Λ · , · > = < T · , T · > + m < · , · > . Therefore < Λ · , · > is a positive definite bilinearform on S , and Λ is a positive operator on Ker div. Thus Λ − is a positive operator on Ker div aswell, and < Λ − · , · > is a positive definite bilinear form on S .Note that the results on the spectrum of the operator curl above imply that the spectrum ofthe real operator Λ acting on the complexification of Ker div is continuous and coincides with theset [ m, ∞ ), and therefore the spectrum of the real operator Λ − acting on the complexification ofKer div is continuous and coincides with the set [0 , m ]. We deduce that the bilinear form < Λ − · , · > on S is non–degenerate and defines the structure of a pre–Hilbert space on S .Now by Lemma 2.1.1 in [9] C ( A ) is positive definite, i.e. for any α , . . . , α n ∈ C , ξ , . . . , ξ n ∈ S we have P nij =1 α i ¯ α j C ( ξ i − ξ j ) ≥ C ( A ), A ∈ S is a positive definite continuous functional on S satisfying C (0) = 1, i.e. itis a characteristic functional. We conclude that by the Bochner–Minlos theorem (Theorem 1.5.2 in[9]) there is a probability measure µ on S ∗ such that C ( A ) is the Fourier transform of µ , i.e. C ( A ) = Z S ∗ e i
Recall that the generalized Fourier transformΦ : Ker div → L ( R ) · + L − ( R ) , where L ± ( R ) are copies of the usual L ( R ), associated to the basis e ± ( k ) of generalized eigenvectorsis given in terms of components by(38) Φ( ω ) ± ( k ) = − l . i . m . R →∞ Z | x |≤ R ∗ ( ω ∧ ∗ e ± ( k )) d x, Φ( ω ) ± ∈ L ± ( R ) , Φ( ω ) = Φ( ω ) + · + Φ( ω ) − . Here l . i . m . stands for the limit with respect to the L ( R )-norm.For ω ∈ S we can also write(39) Φ( ω ) ± ( k ) = − lim R →∞ R | x |≤ R ∗ ( ω ∧ ∗ e ± ( k )) d x == − lim R →∞ R R ∗ ( ω ∧ ∗ χ R ( x ) e ± ( k )) d x = < ω, e ± ( k ) >, where χ R ( x ) is the characteristic function of the ball of radius R .Since the usual Fourier transform is unitary, one can normalize the generalized eigenvectors e ± ( k )in such a way that Φ is also a unitary map. We shall always assume that such normalization is fixed.Using the generalized eigenvectors e ± ( k ) and unitarity of Φ operator (37) can be rewritten in thefollowing form(40) H = 12 X ε = ± Z R d kD ∗ e ε ( k ) D e ε ( k ) . Note that by Proposition 4.3.11 in [9] for ξ, η ∈ S ⊂ S ∗ the operators D ξ , D ∗ η satisfy the followingcommutation relations(41) [ D ξ , D ∗ η ] = (Λ ξ, η ) , [ D ξ , D η ] = [ D ∗ ξ , D ∗ η ] = 0 . By definition the operator Λ acts on the generalized eigenvectors e ε ( k ) as follows Λ e ε ( k ) =(( c ε | k | + c ) + m ) e ε ( k ).The last two observations imply that one can establish a Hilbert space isomorphism between H and a Fock space as follows (see [2], Ch. 1, [9], Theorem 2.3.5).Let H ε ,...,ε n be the space of complex–valued symmetric functions f ( k , . . . , k n ) of n variables k i ∈ R , i = 1 , . . . , n such that k f k ε ,...,ε n = Z ( R ) n | f ( k , . . . , k n ) | n Y i =1 (( ε i c | k i | + c ) + m ) dk . . . dk n < ∞ .H ε ,...,ε n is a Hilbert space with the scalar product( f, g ) ε ,...,ε n = Z ( R ) n f ( k , . . . , k n ) g ( k , . . . , k n ) n Y i =1 (( ε i c | k i | + c ) + m ) dk . . . dk n . Let F be the space of all sequences ( f ε ,...,ε n ) ∞ n =0 , f ε ,...,ε n ∈ H ε ,...,ε n such that ∞ X n =0 X ε ,...,ε n = ± k f ε ,...,ε n k ε ,...,ε n < ∞ , where we assume that for n = 0 H ε ,...,ε n = C with the usual complex number scalar product andnorm. F is a Hilbert space with the scalar product indiced from ∞ M n =0 M ε ,...,ε n = ± H ε ,...,ε n . The following result is standard.
Theorem 9.
The map
F → H , ( f ε ,...,ε n ) ∞ n =0 ∞ X n =0 X ε ,...,ε n = ± Z ( R ) n dk . . . dk n f ε ,...,ε n D ∗ e ε ( k ) . . . D ∗ e εn ( k n ) is a well–defined Hilbert space isomorphism. D ∗ e ε ( k ) . . . D ∗ e εn ( k n ) S ∗ . Weare not going to define this space here (see e.g. [7], Ch. 3, 4).Commutation relations (41), formula (40) for H and the unitarity of the generalized Fouriertransform Φ imply that the elements D ∗ e ε ( k ) . . . D ∗ e εn ( k n ) H . Namely, at least formally, we have(42) HD ∗ e ε ( k ) . . . D ∗ e εn ( k n ) n X i =1 (( ε i c | k i | + c ) + m ) D ∗ e ε ( k ) . . . D ∗ e εn ( k n ) ,H . By Theorem 9 the set of the generalized eigenvectors is complete. Therefore using the formulas forthe generalized eigenvalues in (42) we deduce the following statement.
Theorem 10.
The spectrum of the operator H is { } ∪ [ m, ∞ ) . The eigenspace corresponding tothe eigenvalue is one–dimensional and is generated by the constant function ∈ H = L ( S ∗ , µ ) which can be regarded as the ground state. The other points of the spectrum belong to the absolutelycontinuous spectrum which is of Lebesgue type. The spectral multiplicity function takes the constantvalue N on the absolutely continuous spectrum. Thus σ pt ( H ) = { } , σ ac ( H ) = [ m, ∞ ) , σ ( H ) = σ pt ( H ) ∪ σ ac ( H ) , and the spectrum of H has a gap. In conclusion we remark that in the non–abelian case a properly quantizated Hamiltonian H red should act as a self–adjoint operator in an L –space associated to a measure with a “density” whichresembles functional (30) with an appropriate “renormalization”. If this measure was constructedthe quantized Hamiltonian would be immediately defined. References [1] Arnold, V. I., Mathematical methods of classical mechanics, Springer-Verlag, Berlin (1980).[2] Berezin, F., The method of secondary quantization, Academic Press, London (1966).[3] Birman, M. Sh., Solomyak, M. Z., Spectral theory of self–adjoint operators in Hilbert space, Reidel, Dordrecht(1987).[4] Dirac, P.A.M., Generalized Hamiltonian dynamics,
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