Lorentz invariant quantization of the Yang-Mills theory free of Gribov ambiguity
aa r X i v : . [ h e p - t h ] F e b Lorentz invariant quantization of the Yang-Millstheory free of Gribov ambiguity.
A.A.Slavnov ∗ Steklov Mathematical Institute, Russian Academy of SciencesGubkina st.8, GSP-1,119991, MoscowOctober 27, 2018
Abstract
A new formulation of the Yang-Mills theory which allows to avoid theproblem of Gribov ambiguity of the gauge fixing is proposed.
The standard formulation of the Yang-Mills theory does not allow a unique gaugefixing. It was shown by V.N.Gribov [1] that the Coulomb gauge condition ∂ i A i = 0does not choose a unique representative in the class of gauge equivalent configura-tions, as the condition ∂ i A Ω i = 0 (1)considered as the equation for the elements of the gauge group Ω at the surface ∂ i A i = 0 for sufficiently large A has nontrivial solutions fastly decreasing at thespatial infinity. This result was generalised by I.Singer [2] to arbitrary covariantgauge conditions.In the framework of perturbation theory, that is for sufficiently small A theequaton (1) has only trivial solutions. Hence the Gribov umbiguity in this caseis absent. However beyond the perturbation theory this ambiguity exists, whichmakes problematic the standard way of canonical quantization of nonabelian gaugetheories. This problem was studied by many authors (see e.g. [3]), but in my opinionthe problem is still far from being clear.Recently I proposed an explicitely Lorentz invariant formulation of the quantumYang-Mills theory in which the effective Lagrangian of ghost fields is gauge invariant[4]. In the present paper I will show that in this approach the Yang-Mills theoryallows a quantization procedure which is free of the Gribov ambiguity and hencemay serve as a starting point for nonperturbative constructions. ∗ E-mail: [email protected] Unambiguos quantization of the Yang-Mills field.
We consider the model described by the classical Lagrangian L = − F aµν F aµν + ( D µ ϕ ) ∗ ( D µ ϕ ) − ( D µ χ ) ∗ ( D µ χ ) + i [( D µ b ) ∗ ( D µ e ) − ( D µ e ) ∗ ( D µ b )] (2)To save the place we shall consider the model with the SU (2) gauge group. Gen-eralization to other groups makes no problem. Here F aµν is the standard curvaturetensor for the Yang-Mills field. The scalar fields ϕ, χ, b, e form the complex SU (2)doublets parametrized by the hermitean components as follows: c = ic + c √ , c − ic √ ! (3)where c denotes any of doublets. The fields ϕ and χ are commuting, and the fields e and b are anticommuting. In the eq.(2) D µ denotes the usual covariant derivative,hence the Lagrangian (2) is gauge invariant. Note that due to the negative sign ofthe χ field Lagrangian, this field posesses negative energy.Let us make the following shifts in the Lagrangian (2): ϕ → ϕ + g − ˆ m ; χ → χ − g − ˆ m ; ˆ m = (0 , m ) (4)where m is a constant parameter. Due to the negative sign of the Lagrangian ofthe field χ the terms quadratic in m arising due to the shifts of the fields ϕ and χ mutualy compensate and the Lagrangian acquires a form L = − F aµν F aµν + ( D µ ϕ ) ∗ ( D µ ϕ ) − ( D µ χ ) ∗ ( D µ χ )+ g − [( D µ ϕ ) ∗ + ( D µ χ ) ∗ ]( D µ ˆ m ) + g − ( D µ ˆ m ) ∗ [ D µ ϕ + D µ χ ]+ i [( D µ b ) ∗ ( D µ e ) − ( D µ e ) ∗ ( D µ b )] (5)As before this Lagrangian describes massless vector particles.The Lagrangian (5) is obviously invariant with respect to the ”‘shifted”’ gaugetransformations, which in terms of hermitean components look as follows δA aµ = ∂ µ η a − gǫ abc A bµ η c δϕ +0 = g ϕ + a η a δϕ − = g ϕ − a η a δϕ + a = − g ǫ abc ϕ + b η c − g ϕ +0 η a δϕ − a = − mη a − g ǫ abc ϕ − b η c − g ϕ − η a δb a = − g ǫ adc b d η c − g b η a δe a = − g ǫ adc e d η c − g e η a δb = g b a η a δe = g e a η a (6)2ere the obvious notations ϕ ± α = ϕ α ± χ α √ δϕ ( x ) = iǫb ( x ) δχ ( x ) = − iǫb ( x ) δe ( x ) = ǫ [ ϕ ( x ) + χ ( x )] δb ( x ) = 0 (7)where ǫ is a constant anticommuting parameter.In the future we shall see that invariance with respect to the supersymmetrytransformations provides unitarity of the theory in the space which includes onlyphysical exitations of the fields. An explicit form of interaction is not essential.Only the symmetry properties are important. In principle any counterterms whichpreserve gauge invariance and supersymmetry are allowed.To quantize the model one has to impose a gauge condition. One could choose theCoulomb gauge, as it was done in our paper [4]. The supersymmetry transformationsdo not involve the gauge fields, hence the Coulomb gauge Lagrangian preserves thesupersymmetry. However in this case we would confront again the problem of Gribovambiguity. Although in the framework of perturbation theory such a choice doesnot lead to any problems and allows to prove easily equivalence of the model (5) tothe usual Yang-Mills theory, beyond the peturbation theory the ambiguity problemmay be essential.The explicit form of the gauge transformations (6) shows that under these trans-formations not only the Yang-Mills fields are shifted by the gradients of arbitraryfunctions, but also the fields ϕ − a are shifted by arbitrary functions. It allows to addto the Lagrangian (5) the gauge fixing term of the form12 α ( ϕ − a ) (8)and in particular to consider the gauge ϕ − a = 0 (9)Note that the relation ( ϕ Ω ) − a = 0 (10)considered as the equation for the group elements Ω at the surface (9) has no notrivialsolutions fastly decreasing at spatial infinity. So imposing the gauge (9) is unam-bigous. 3n the gauge ϕ − a = 0 the action acquires a form˜ A = Z d x {− F aµν F aµν + ∂ µ ϕ +0 ∂ µ ϕ − + mϕ + a ∂ µ A aµ + i [( D µ b ) ∗ ( D µ e ) − ( D µ e ) ∗ ( D µ b )]+ mg A µ ϕ +0 + g A µ ϕ +0 ϕ − + g∂ µ ϕ − A aµ ϕ + a + g ϕ − ϕ + a ∂ µ A aµ } (11)The canonical momentum for the field A a is p a = mϕ + a (1 + g m ϕ − ) (12)The Hamiltonian action looks as follows˜ A H = Z d x { p ai ˙ A ai + p a ˙ A a + p ϕ ˙ ϕ + p χ ˙ χ − ( p ai ) p a ) g/ (2 m ) ϕ − ) − p ϕ p χ A a ∂ i p ai − p a ∂ i A ai g/ (2 m ) ϕ − − F aik F aik + . . . } (13)Here . . . denote the terms corresponding to the fields b and e and all the interactionterms. We wrote explicitely the term quadratic in momenta p a because it generatesa nontrivial Jacobian when one passes from the path integral over the phase spaceto the integral in the coordinate space. Due to the presence of this factor integrationover canonical momenta generates the additional term in the measure Y x (1 + g m ϕ − ) (14)and the scattering matrix in the gauge ϕ − a = 0 may be written as a path integral S = Z exp n i ˜ A o dµ (15)where dµ = Y x ( m + g ϕ − ) dA µ dϕ + α dϕ − db α de α (16)The scattering matrix (15) acts in the space which contains unobservable exita-tions: unphysical components of A µ , ghosts corresponding to the fields ϕ +0 , ϕ − , b α , e α .Below we shall show that the Lagrangian (5) in the gauge ϕ − a = 0 similar to theLagrangian in the Coulomb gauge is invariant under some supersymmetry transfor-mations which provide the unitarity of the scattering matrix in the space includingonly physical exitations of the Yang-Mills field. In classical theory the transition from the Coulomb gauge to the gauge ϕ − a = 0 inthe Lagrangian (5) may be done with the help of the gauge transformation (6). A4auge transformation may be considered as a change of variables which does notalter the action. In the Coulomb gauge the action was invariant with respect to thesupersymmetry transformation (7). Hence the action in the gauge ϕ − a = 0 also willbe ivariant under some supersymmetry transformation. However this transformationdiffers of (7). The explicit form of this transformation may be found as follows.Under a gauge transformation the action does not change, but the gauge fixingterm R d xλ a ( x ) ∂ i A ai ( x ) in new variables will be replaced by R d xλ a ( x ) ˜ ϕ − a ( x ). Inthis way we get the equation determining the gauge function Z d xλ a ( x ) ∂ i ( A Ω ) ai ( x ) = Z d xλ a ( x ) ϕ − a ( x ) (17)Both gauges under consideration are admissible, therefore the equation (17) has asolution. In practice one can find a solution of the eq.(17) in perturbation theory.At the lowest order we have η a ( x ) = − ϕ − a ( x ) + ∂ i A ai ( x ) m (18)where the functions η a ( x ) parametrize the gauge group elements Ω( x ). Under thesupersymmetry transfomations the function η changes η a ( x ) → η a ( x ) − i √ ǫb a ( x ) m (19)Therefore ˜ A aµ ( x ) = A aµ ( x ) − ∂ µ η a ( x ) → ˜ A aµ ( x ) + i √ ǫ∂ µ ˜ b a ( x ) m ˜ e α ( x ) → ˜ e α ( x ) + ǫ √ ϕ + α ( x )˜ ϕ − ( x ) → ˜ ϕ − ( x ) + i √ ǫ ˜ b ( x ) (20)The remaining fields do not change at zero order.Invariance with respect to the supersymmetry transformations which in the freetheory are described by the eqs.(20), according to the Noether theorem generates aconserved charge Q . For asymptotic fields this charge reduces to the free one Q ,which is determined by the transformations (20). Note that here we rely on thehypothesis about adiabatic switching of interaction for asymptotic states. Validityof this hypothesis for nonabelian gauge fields is not obvious. For this reason a formalproof given below should be applied either to infrared regularized theory [5] or tothe functional generating Green functions of gauge invariant composite operators.Physical states are separated by the condition Q | ψ > as = 0 (21)The explicit form of the charge Q may be obtained by considering variation of thefree Lagrangian under the transformations (20). It may be presented in the form Q = ˜ Q + Q (22)5here ˜ Q = √ Z d x { m − ( ∂ i A − ∂ A i ) a ( ∂ i b a ) − ϕ + a ∂ b a } (23)and Q = √ Z d x { ∂ ϕ +0 b + ∂ b ϕ +0 } (24)Here the operator ˜ Q coincides with the free BRST-operator for the Yang-Millstheory in the Lorentz gauge if one identifies b a with the Faddeev-Popov ghost c a and e a with ¯ c a . The operator Q in terms of creation and annihilation operatorslooks as follows Q ∼ Z d k { b ∗ ( k ) ϕ ( k ) + ( ϕ ) ∗ ( k ) b ( k ) } (25)where h ϕ − ( k ) , ϕ + ∗ ( p ) i = δ ( k − p ) (26)[ e ( k ) , b ∗ ( p )] + = δ ( k − p ) (27)The operators ˜ Q and Q are independent and anticommuting. So the equation(24) may be fulfilled only if the admissible vectors | ψ > as are annihilated by bothoperators ˜ Q , Q .As it is well known from the theory of BRST-quantization(see [6], any vectorannihilated by ˜ Q may be presented in the form | ψ > = | ˜ ψ > + | ˜ N > (28)where the vector | ˜ ψ > contains only the exitations corresponding to three dimension-ally transversal components of the Yang-Mills field and the particles correspondingto ϕ +0 , ϕ − , b , e o , and | ˜ N > is a zero norm vector.Further proof of nonnegativity of the norms of the vectors satisfying the condition(21) goes in a standard way [7]. We introduce the number operator for unphysicalparticles corresponding to the operators e , b , ϕ +0 , ϕ − ˆ N = Z d k { b ∗ e + e ∗ b + ϕ + ∗ ϕ − + ϕ −∗ ϕ +0 } (29)and notice that this operator may be presented as the anticommutatorˆ N = [ Q , K ] + (30)where K = Z d k { e ∗ ( k ) ϕ − ( k ) + ϕ −∗ ( k ) e ( k ) } (31)Any vector with N unphysical particles ϕ ± , b , e ; ( N = 0) looks as follows | ψ > = 1 N { Q K | ψ > + K Q | ψ > } (32)If this vector is annihilated by the operator Q | ψ > = 1 N Q | χ > (33)6ombining the equations (28, 33) one sees that any vector annihilated by the oper-ator Q may be presented as | ψ > as = | ψ > tr + | N > (34)where the vector | ψ > tr contains only exitations corresponding to the three dimen-sionally transversal components of the field A µ , and the vector | N > has zero norm.Factorizing this space with respect to zero norm vectors we see that it coincides withthe ”physical” space of the Yang-Mills theory. By construction the scattering matrixis unitary in this space, and expectation value of any gauge invariant operator overstates (21) coincides with the expectation value over transversal state vectors.The expression (15) for the scattering matrix in principle makes sense beyoundperturbation theory. Similar expression may be written for the functional generatingGreen functions of gauge invariant composite operators.For practical calculations of such Green functions in the framework of perturba-tion theory it is convenient to pass in the path integral to the Lorentz gauge. It canbe done if one notes that at the surface ϕ − a = 0 the factor Q x ( a + g ϕ − ) may bewritten in the gauge invariant form Y x ( a + g ϕ − ) − = Z d Ω δ ( ϕ − Ω a ) | ϕ − a =0 (35)where the integration at the r.h.s. goes over invariant measure at the gauge group.Using this observation one can write the expression for the gauge invariant gen-erating functional in the form Z ( J µ ) = Z exp { i ( A + Z d xJ µ F µ ) } δ ( ϕ − a )∆ − dA µ dϕ − α ϕ + α db α de α (36)where F µ denotes a gauge invariant functional, A is the gauge invariant action,corresponding to the Lagrangian (5) and∆ − Z d Ω δ ( ∂ µ A Ω µ ) = 1 (37)Multiplying the integral (36) by ”‘one”∆ L Z δ ( ∂ µ A Ω µ ) d Ω = 1 (38)and making the change of variables which is the gauge transformation (6) we obtainthe expression for the generating functional in the Lorentz gauge Z ( J µ = Z exp (cid:26) i ( A + Z d xJ µ F µ ) (cid:27) ∆ L δ ( ∂ µ A µ ) dA µ dϕ α db α de α (39)In the framework of perturbation theory the transformation to the Lorentz gauge iswell defined. The perturbation series in the Lorentz gauge is explicitely renormaliz-able.One can also develop the perturbation theory directly in the gauge ϕ − a = 0.However the propagator of the fields ϕ + a ( x ) , A bµ ( y ) at large momenta behaves as k − β -function atthe lowest order ([8]) confirms this hypothesis. The model under consideration isasymptotically free and the running charge is expressed by the usual formula. In the present paper we constructed the quantization procedure for nonabelian gaugetheories which allows to get a unique expression for the Green functions of gaugeinvariant composite operators and infrared regularized scattering matrix. Imposingthe gauge condition does not introduce the ambiguity indicated by Gribov. Lorentzinvariance of the theory is also manifest. In the framework of perturbation theorythis model leads to the expression for the gauge invariant Green functions, whichcoincides with the usual one.
Acknowledgements.
This work was partially done while the author was visiting University of Milan.I wish to thank R.Ferrari for hospitality and Cariplo Foundation for a generoussupport. My thanks to the members of Theoretical group for helpful discussion.This researsh was supported in part by Russian Basic Research Fund under grant08-01-00281a and by the RAS program ”‘Nonlinear dynamics”.
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