Covariant quantization of Yang-Mills theory in the first order formalism
aa r X i v : . [ h e p - t h ] J a n Covariant quantization of Yang-Mills theory in the firstorder formalism
P.M. Lavrov ( a,b )1 ( a ) Tomsk State Pedagogical University,Kievskaya St. 60, 634061 Tomsk, Russia ( b ) National Research Tomsk State University,Lenin Av. 36, 634050 Tomsk, Russia
Abstract
In the present paper the Yang-Mills theory in the first order formalism is studied. As agauge theory it has more rich gauge algebra in comparison with the second order formal-ism. On classical level both formalisms are equivalent. It is proven that the covariantquantization of this theory leads to the statement about quantum non-equivalence withthe quantization based on the second order formalism.
Keywords:
Quantization of Yang-Mills fields, FP-method, BV-formalism.PACS numbers: 11.10.Ef, 11.15.Bt E-mail: [email protected]
Introduction
Yang-Mills theories have always attracted (see, for example, the famous textbook by Weinberg[1] for qualitative discussions and presentations of numerous aspects of classical and quantumproperties of Yang-Mills fields) and continue to attract [2, 3, 4, 5] the attention of researchers,since they play a key role in the mainstream of modern models of fundamental interactions. Incontrast with the electrodynamics the Yang-Mills theories belong to non-abelian gauge theories.It caused the problem indicated firstly by Feynman [6] and related to the S-matrix unitarityin Yang-Mills theories when simple modification of quantization rules adopted in quantumelectrodynamics is used. Later on the correct quantization of Yang-Mills theories has beenfound [7, 8].Recently, there has been activity in the study of quantum properties of the Yang-Millstheory formulated in the first order formalism [9, 10, 11, 12] instead of standard approachbased on the second order form [7]. On classical level both formulations are equivalent. Itwas really assumed in [11, 12] that all Green functions appeared in the standard quantizationcan be reproduced with the help of using more simple first order formulation. This proposal isclosely related to quantum equivalence of both presentations of the Yang-Mills theory. In ourknowledge the quantum equivalence of the Yang-Mills theory presented in the first and secondorders was not studied before in the scientific literature.In the present paper we study the Yang-Mills theory in the first order formulation. Thisformulation requires introduction additional antisymmetric second order tensor fields, F aµν . Inturn it leads to appearing additional gauge symmetry. Corresponding gauge algebra containsadditional gauge generators. Further quantization depends on properties of the gauge algebrato be closed/open and irreducible/reducible. Then the found structure of gauge algebra allowsto construct a quantum action using, in general, the BV-formalism [13, 14]. Fixing a gaugevia the standard BV procedure one obtains the full quantum action which is used to definethe generating functional of Green functions in the form of functional integral. Quantumequivalence or non-equivalence of the Yang-Mills theory in two formulations can be studied bycomparison of corresponding vacuum functionals. If the vacuum functional coincide then thequantum equivalence is realized. Otherwise one meets the quantum non-equivalence.The paper is organized as follows. In Sec. 2 gauge symmetries of the Yang-Mills theorywritten in the first order formulation are studied. In Sec. 3 the full quantum action foundas solution to the classical master equation of the BV-formalism with applying the standardgauge fixing procedure in linear Lorentz invariant gauges is constructed. In Sec. 4 the generatingfunctional of Green functions in the form of functional integral over only fields of initial classicalaction is found. In Sec. 5 non-equivalence of two approaches for quantization of the Yang-Millstheory is proven. Finally, in Sec. 6 the results obtained in the paper are discussed.In the paper the DeWitt’s condensed notations are used [17]. The right and left functional2erivatives with respect to fields and antifields are marked by special symbols ” ← ” and ” → ”respectively. Arguments of any functional are enclosed in square brackets [ ], and arguments ofany function are enclosed in parentheses, ( ). Standard formulation of the Yang-Mills fields, A aµ , operates with the second order action S (2) [ A ] = − F aµν ( A ) F aµν ( A ) , (2.1)where F aµν ( A ) is the field strength F aµν ( A ) = ∂ µ A aν − ∂ ν A αµ + f abc A bµ A cν , (2.2)and f abc are completely antisymmetric structure constants of the Lie algebra satisfying theJacobi identity f abc f cde + f aec f cbd + f adc f ced ≡ . (2.3)The equations of motion read δS (2) [ A ] δA aµ = D abν ( A ) F bνµ ( A ) = 0 , (2.4)where D abµ ( A ) is the covariant derivative D abµ ( A ) = δ ab ∂ µ + f acb A cµ . (2.5)The action (2.1) is invariant under the gauge transformations of A aµ , δ ξ S (2) [ A ] = 0 , δ ξ A aµ = D abµ ( A ) ξ b , (2.6)where ξ a are arbitrary functions of space-time coordinates. Notice that under the gauge trans-formations the field strength tensor transforms by the tensor law, δ ξ F aµν ( A ) = f abc F bµν ( A ) ξ c . (2.7)Algebra of gauge transformations[ δ ξ , δ ξ ] A aµ = D abµ ( A ) ξ b , ξ a = f abc ξ b ξ c , (2.8)is closed and irreducible.The first order formulation of Yang-Mills fields is based on the action S (1) [ A, F ] = 14 F aµν F aµν − F aµν ( A ) F aµν , (2.9)3here F aµν are new antisymmetric tensor fields, F aµν = −F aνµ . The equations of motion read δS (1) [ A, F ] δA aµ = D abν ( A ) F bνµ = 0 , δS (1) [ A, F ] δ F aµν = 12 (cid:0) F bµν − F bµν ( A ) (cid:1) = 0 . (2.10)From the second in (2.10) it follows F aµν = F aµν ( A ). Substituting this result in the first of (2.10)one obtains (2.4). On classical level we have two equivalent descriptions of Yang-Mills fields.The action (2.9) is invariant under the following gauge transformations δ ξ S (1) [ A, F ] = 0 , δ ξ A aµ = D abµ ( A ) ξ b , δ ξ F aµν = f abc F bµν ξ c , (2.11)which are considered as generalization of (2.6). Introduction of new fields leads to increasingthe degrees of freedom and as consequence to existence of additional gauge symmetry. Indeed,the action (2.9) is invariant under the additional gauge transformations δ ¯ ξ S (1) [ A, F ] = 0 , δ ¯ ξ A aµ = 0 , δ ¯ ξ F aµν = f abc (cid:0) F bµν − F bµν ( A ) (cid:1) ¯ ξ c , (2.12)where ¯ ξ a are arbitrary functions of space-time coordinates. The first order formulation ofYang-Mills fields is accompanying by the following gauge algebra[ δ ξ , δ ξ ] A aµ = D abµ ( A ) ξ b , [ δ ¯ ξ , δ ξ ] A aµ = 0 , [ δ ¯ ξ , δ ¯ ξ ] A aµ = 0 , [ δ ξ , δ ξ ] F aµν = f acb F cµν ξ b , [ δ ¯ ξ , δ ξ ] F aµν = f acb (cid:0) F cµν − F cµν ( A ) (cid:1)e ξ b , [ δ ¯ ξ , δ ¯ ξ ] F aµν = f acb F cµν ¯ ξ b , (2.13)where ξ a = f abc ξ b ξ c , e ξ a = f abc ¯ ξ b ξ c , ¯ ξ a = f abc ¯ ξ b ¯ ξ c . (2.14)Due to the second in (2.10) the commutator [ δ ¯ ξ , δ ξ ] F aµν can be written in the form[ δ ¯ ξ , δ ξ ] F aµν = 2 f acb δS (1) [ A, F ] δ F cµν e ξ b , (2.15)proportional to the equations of motion which is typical for open algebras. But final answer tobe a gauge algebra closed or open depends on fulfilment of the Jacobi identity without/withusing the equations of motion. Direct calculations show that the Jacobi identity for gaugetransformations (2.11), (2.12) is satisfied exactly. The gauge algebra (2.13) is closed and irre-ducible. In construction of quantum action for Yang-Mills theory in the first order form we will followprescriptions of the BV-formalism [13]. Minimal antisymplectic space is defined by the set offields φ Amin and antifields φ ∗ A min φ Amin = (cid:0) A aµ , F aµν , C a , C a (cid:1) , φ ∗ A min = (cid:0) A ∗ aµ , F ∗ aµν , C ∗ a , C ∗ a (cid:1) , (3.1)4here C a and C a are the ghost fields corresponding to the gauge symmetries (2.11) and (2.12)respectively. Due to closedness and irreducibility of gauge algebra the minimal quantum action, S min = S [ φ min , φ ∗ min ], being a solution to the classical master equation ( S min , S min ) = 0 , (3.2)satisfying the boundary condition S min (cid:12)(cid:12) φ ∗ min =0 = S (1) [ A, F ] (3.3)can be found in the form linear in antifields S min = S (1) [ A, F ] + A ∗ aµ X aµ ( φ ) + F ∗ aµν Y aµν ( φ ) + C ∗ a Z a ( φ ) + C ∗ a U a ( φ ) . (3.4)The result reads X aµ ( φ ) = D abµ ( A ) C b , Y aµν ( φ ) = f abc F bµν C c + f abc (cid:0) F bµν − F bµν ( A ) (cid:1) C c , (3.5) Z a ( φ ) = − f abc C b C c , U a ( φ ) = f abc (cid:16) C b C c + C b C c (cid:17) . (3.6)Now the extended action S = S [ φ, φ ∗ ] in full antisymplectic space φ A = (cid:0) A aµ , F aµν , C a , C a , ¯ C a , ¯ C a , B a , B a (cid:1) , φ ∗ A = (cid:0) A ∗ aµ , F ∗ aµν , C ∗ a , C ∗ a ¯ C ∗ a , ¯ C ∗ a , B ∗ a , B ∗ a (cid:1) , (3.7)has the form S = S (1) [ A, F ] + A ∗ aµ D abµ ( A ) C b + F ∗ aµν f abc (cid:16) F bµν C c + (cid:0) F bµν − F bµν ( A ) (cid:1) C c (cid:17) −− C ∗ a f abc C b C c + C ∗ a f abc (cid:16) C b C c + C b C c (cid:17) + ¯ C ∗ a B a + ¯ C ∗ a B a (3.8)and satisfies the classical master equation and the boundary condition( S, S ) = 0 , S (cid:12)(cid:12) φ ∗ =0 = S (1) [ A, F ] . (3.9)Here ¯ C a and ¯ C a are the antighost fields to C a and C a and B a , B a are auxiliary fields (Nakanishi-Lautrup fields). The gauge fixed action S eff [ φ ] in the BV-formalism is defined as S eff [ φ ] = S (cid:2) φ, φ ∗ = Ψ ←− ∂ φ (cid:3) , (3.10)where Ψ = Ψ[ φ ] is an odd gauge fixing functional. For the closed and irreducible gaugealgebra S eff [ φ ] is usually refereed as the Faddeev -Popov action S F P [ φ ]. For the model underconsideration the functional Ψ corresponding to the case of non-singular Lorentz invariant andlinear gauges can be chosen in the formΨ = ¯ C a χ a ( A, B ) + ¯ C a χ a ( F , B ) , (3.11) For any set of fields φ A and antifields φ ∗ A and any functionals F and G the antibracket is defined by therule ( F, G ) = F (cid:16) ←− ∂ φ A −→ ∂ φ ∗ A − ←− ∂ φ ∗ A −→ ∂ φ A (cid:17) G . χ a ( A, B ) and χ a ( F are gauge fixing functions, χ a ( A, B ) = ∂ µ A aµ + ξ B a , χ a ( F , B ) = ε µν F aµν + ξ B a . (3.12)In (3.12) ε µν is the Lorentz invariant antisymmetric tensor, ξ and ξ are constant gauge param-eters.The Faddeev-Popov action for the model (2.9) in the gauges (3.12) reads S (1) F P [ φ ] = S (1) [ A, F ] + ¯ C a ∂ µ D abµ C b + B a χ a ( A, B ) ++ ¯ C a f abc (cid:0) F b − F b ( A ) (cid:1) C c + ¯ C a f abc F b C c + B a χ a ( F , B ) , (3.13)where the following notations F a = ε µν F aµν , F a ( A ) = ε µν F aµν ( A ) (3.14)are used.The Faddeev-Popov action for the model (2.1) (3.12) has the form S (2) F P [ φ ] = S (2) [ A ] + ¯ C a ∂ µ D abµ ( A ) C b + B a χ a ( A, B ) , (3.15)where the gauge fixing function χ a ( A, B ) is the first in (3.12). The cases ξ = 0 for (3.15) and ξ = 0, ξ = 0 for (3.13) correspond to singular gauges being useful for theoretical treatmentsof quantum properties while for practical quantum calculations the cases ξ = 0 and ξ = 0 aremore preferred. The generating functional of Green functions of fields A aµ and F aµν for the Yang-Mills theory inthe first order form reads Z (1) [ j, J ] = Z Dφ exp n i ~ (cid:0) S (1) F P [ φ ] + jA + J F (cid:1)o , (4.1)where j = { j aµ } and J = { J aµν } are external sources to fields A aµ and F aµν respectively and thefollowing abbrevations jA = j aµ A aµ , J F = J aµν F aµν (4.2)are used. Integrating over fields B a and B a reproduces the following presentation for Z (1) [ j, J ], Z (1) [ j, J ] = Z DAD F DCD ¯ CDC D ¯ C exp n i ~ (cid:0) S (1) [ A, F , C, ¯ C, C , ¯ C ] + jA + J F (cid:1)o , (4.3)where S (1) [ A, F , C, ¯ C, C , ¯ C ] = S (1) [ A, F ] + ¯ C a ∂ µ D abµ ( A ) C b + ¯ C a f acb ( F c − F c ( A )) C b ++ ¯ C a f acb ( F c − F c ( A )) C b + 1 ξ ( ∂ µ A aµ ) + 1 ξ ( F a ) . (4.4)6urther integration over ghost and antighost fields allows to present the functional Z (1) [ j, J ] inthe form Z (1) [ j, J ] = Z DAD F exp n i ~ (cid:16) F aµν F aµν − F aµν ( A ) F aµν + 1 ξ ( ∂ µ A aµ ) + 1 ξ ( F a ) ++ jA + J F (cid:17)o ∆ F P [ A ]∆[ F − F ( A )] , (4.5)where ∆ F P [ A ] is the Faddeev-Popov determinant [7]∆ F P [ A ] = Det ∂ µ D abµ ( A ) , (4.6)and ∆[ F − F ( A )] is the functional determinant associated with the additional gauge symmetry,∆[ F − F ( A )] = Det f acb ( F c − F c ( A )) . (4.7)Making use the change of integration variables in the functional integral (4.5) F aµν = E aµν + F aµν ( A ) , (4.8)we obtain Z (1) [ j, J ] = Z DA exp n i ~ (cid:16) − F aµν F aµν + 1 ξ ( ∂ µ A aµ ) + jA + J F ( A ) (cid:17)o ∆ F P ( A )Σ[ A, J ] , (4.9)where the functional Σ[ A, J ] is defined by the following functional integralΣ[
A, J ] = Z DE exp n i ~ (cid:16) E aµν E aµν + 1 ξ ( E a ) + 2 ξ E a F a ( A ) + J E (cid:17)o ∆( E ) . (4.10)From (4.10) it follows that Σ[ A, J ] remains a non-trivial functional of A aµ , Σ[ A ], even when J aµν = 0,Σ[ A ] = Z DE exp n i ~ (cid:16) E aµν E aµν + 1 ξ ( E a ) + 2 ξ E a F a ( A ) (cid:17)o ∆( E ) = const . (4.11)The last fact plays a crucial role in solving the quantum equivalence problem for the Yang-Millstheory in the first and second order formulations. The generating functional of Green functions of fields A aµ for Yang-Mills theory in the secondorder form is defined as Z (2) [ j ] = Z Dφ exp n i ~ (cid:0) S (2) F P [ φ ] + jA (cid:1)o . (5.1)Integration over fields C, ¯ C, B leads to the well-known result [7] Z (2) [ j ] = Z DA exp n i ~ (cid:16) − F aµν ( A ) F aµν ( A ) + 1 ξ ( ∂ µ A aµ ) + jA (cid:17)o ∆ F P [ A ] . (5.2)7ow we are in position to study quantum (non)equivalence. To do this we have to compare vac-uum functionals for the Yang-Mills theory formulated in the first and second order formalisms.From (5.2) and (4.5) it follows Z (2) [0] = Z DA exp n i ~ (cid:16) − F aµν ( A ) F aµν ( A ) + 1 ξ ( ∂ µ A aµ ) (cid:17)o ∆ F P [ A ] , (5.3) Z (1) [0 ,
0] = Z DA exp n i ~ (cid:16) − F aµν ( A ) F aµν ( A ) + 1 ξ ( ∂ µ A aµ ) (cid:17)o ∆ F P ( A )Σ[ A ] . (5.4)Due to the relation (4.11) the vacuum functionals do not coincide, Z (1) [0 , = Z (2) [0] . (5.5)It means the quantum non-equivalence of two schemes of quantizations.The relation Z (2) [ j ] = Z (1) [ j,
0] (5.6)used in [11, 12] to support calculations of Green functions within the standard quantum ap-proach to the Yang-Mills theory with the help of the first order formulation, is broken Z (2) [ j ] = Z (1) [ j,
0] (5.7)as it follows from (5.2) and (4.9).
In the paper we have analyzed the Yang-Mills theory in the first order formulation. Thisformulation operates with two set of fields { A aµν } and {F aµν } instead of { A aµν } in the secondorder formalism. On classical level both formulations are equivalent. Equivalence on quantumlevel needs in special study. We have studied gauge symmetry of classical action in the first orderformulation and found two types of gauge transformations. One of them can be considered asnatural extension of gauge transformations of vector fields A aµ in the second order formulationbut the second type of gauge symmetry is specific for the first order formulation and is causedby the presence of second order tensor fields F aµν . Then it was calculated and proved that thefull gauge algebra is closed and irreducible. To construct full quantum action we have usedthe BV-formalism based on solution to classical master equation and gauge fixing procedure[13, 14]. With the help of full quantum action the generating functional of Green functionsin the first order formulation has been constructed and presented in the form of functionalintegral over fields A aµν and F aµν . It was found that the vacuum functionals constructed in thefirst and second order formulations do not coincide. It means quantum non-equivalence of theYang-Mills theory considered in two possible formulations. In its turn it means that the action(2.9) corresponds to the gauge model of vector fields A aµ and second order antisymmetric tensorfields F aµν which is differed the model of Yang-Mills fields (2.1).8 cknowledgments The author thanks I.L. Buchbinder and I.V. Tyutin for useful discussions. The work is sup-ported by Ministry of Education of Russian Federation, project FEWF-2020-0003.
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