Generators of local gauge transformations in the covariant canonical formalism of fields
GGenerators of local gauge transformations in the covariantcanonical formalism of fields
Satoshi Nakajima ∗ Abstract
We investigate generators of local gauge transformations in the covariant canonical for-malism (CCF) for matter fields, gauge fields and the second order formalism of gravity. TheCCF treats space and time on an equal footing regarding the differential forms as the ba-sic variables. The conjugate forms π A are defined as derivatives of the Lagrangian D -form L ( ψ A , dψ A ) with respect to dψ A , namely π A := ∂L/∂dψ A , where ψ A are p -form dynamicalfields. The form-canonical equations are derived from the form-Legendre transformation ofthe Lagrangian form H := dψ A ∧ π A − L . We show that the generator of the local gaugetransformation in the CCF is given by ε r G r + dε r ∧ F r where ε r are infinitesimal parametersand G r are the Noether currents which are ( D − { G r , G s } = f trs G t holds where {• , •} is the Poisson bracket of the CCF and f trs are the structure constants of the gaugegroup. For the gauge fields and the gravity, G r = −{ F r , H } holds. For the matter fields, F r = 0 holds. Additionally, we apply the CCF to the second order formalism of gravity withDirac fields for the arbitrary dimension ( D ≥ Contents D ≥
44 Generators of local gauge transformations 115 Summary 15A Formulas 16B Noether currents 16C Covariant canonical formalism of gauge fields 17 ∗ [email protected] a r X i v : . [ g r- q c ] J u l Introduction
In the traditional analytical mechanics of fields, the canonical formalism gives especial weight totime. The covariant canonical formalism (CCF) [1–12] is a covariant extension of the traditionalcanonical formalism. The form-Legendre transformation and the form-canonical equations arederived from a Lagrangian D -form with p -form dynamical fields ψ A . The conjugate forms aredefined as derivatives of the Lagrangian form with respect to dψ A . One can obtain the form-canonical equations of gauge theories or those of the second order formalism of gravity withoutfixing a gauge nor introducing Dirac bracket nor any other artificial tricks. Although the secondorder formalism of gravity (of which the dynamical variable is only the frame form (vielbein)) isa non-constrained system in the CCF, the first order formalism (of which the dynamical variablesare both the frame form and the connection form) is a constrained system even in the CCF. InRefs. [1–4], the CCFs of the first order formalism of gravity and super-gravity have been studied.Only for D = 4, the CCF of the second order formalism of gravity without Dirac field [7] and withDirac field [9] have been studied .Poisson brackets of the CCF are proposed in Refs. [1, 12] and in Ref. [11] independently. Theseare equivalent. Although the form-canonical equations of the CCF are equivalent to modified DeDonder-Weyl equations [10], the Poisson bracket of the CCF is not equivalent to it of the DeDonder-Weyl theory proposed in Ref. [13]. In Ref. [12], the generators of the CCF for the localLorentz transformations of gravity have been studied in the first order formalism.In this paper, first we apply the CCF to the second order formalism of gravity with Diracfields for the arbitrary dimension ( D ≥
3) in Sec.3. Next, we investigate generators of local gaugetransformations in CCF for matter fields, gauge fields and the second order formalism of gravity(Sec.4). The total generator which is a ( D − G = ε r G r + dε r ∧ F r where ε r are infinitesimal parameters and G r are the Noether currents. The Noether currents satisfy { G r , G s } = f trs G t where {• , •} is the Poisson bracket of the CCF and f trs are the structureconstants of the gauge group. F r = 0 holds for the matter fields. For the gauge fields and thegravity, G r = −{ F r , H } holds. Here, H is the form-Legendre transformation of the Lagrangianform.There are three appendices in this paper. In Appendix A, several formulas are listed. InAppendix B, we review the Noether currents. In Appendix C, we review the CCF of gauge fields. In this section, we review the covariant canonical formalism.Let us consider D dimension space-time. Suppose a p -form β is described by forms { α i } ki =1 . Ifthere exists the form ω i such that β behaves under variations δα i as δβ = δα i ∧ ω i , (2.1) In Refs. [7, 9], the method to derive the form-canonical equations used special characteristics of D = 4. In thispaper, we derive the form-canonical equations without using special characteristics of D = 4.
2e call ω i the derivative of β by α i and denote ∂β∂α i := ω i , (2.2)namely, δβ = δα i ∧ ∂β∂α i .The Lagrangian D -form L is given by L = L η where L is the Lagrangian density and η = ∗ and described by ψ and dψ , L = L ( ψ, dψ ), where ψ is a set the forms of thedynamical fields. For simplicity, we treat ψ as single p -form in this section. The Euler-Lagrangeequation is given by ∂L∂ψ − ( − p d ∂L∂dψ = 0 . (2.3)The above Euler-Lagrange equation has been used since the 1970’s [14–16].The conjugate form π is defined by π := ∂L∂dψ . (2.4)This is a q -form where q := D − p −
1. The
Hamilton D -form is defined by H = H ( ψ, π ) := dψ ∧ π − L (2.5)and described by ψ and π . The variation of H is given by δH = ( − ( p +1) q δπ ∧ dψ − δψ ∧ ∂L∂ψ . (2.6)Then, we obtain ∂H∂ψ = − ∂L∂ψ , ∂H∂π = ( − ( p +1) q dψ. (2.7)By substituting the Euler-Lagrange equation (2.3), we obtain the canonical equations dψ = ( − ( p +1) q ∂H∂π , dπ = − ( − p ∂H∂ψ . (2.8)The Poisson bracket proposed in Ref. [11] is given by { F, G } = ( − p ( f + D +1) ∂F∂ψ ∧ ∂G∂π − ( − ( D + p − f +1) ∂F∂π ∧ ∂G∂ψ . (2.9) In this case, β is differentiable by α i . The Hodge operator ∗ maps an arbitrary p -form ω = ω µ ··· µ p dx µ ∧ · · · ∧ dx µ p ( p = 0 , , · · · , D ) to a D − p = r -form as ∗ ω = 1 r ! E µ ··· µ p ν ··· ν r ω µ ··· µ p dx ν ∧ · · · ∧ dx ν r . Here, E µ ··· µ D is the complete anti-symmetric tensor such that E ··· D − = √− g ( g is the determinant of the metric g µν ). And ∗ ∗ ω = − ( − p ( D − p ) ω holds. F and G are differentiable by ψ and π , and F is a f -form. The Poisson bracket proposed inRef. [12], denoted by { F, G } F , is given by { F, G } F = −{ G, F } . If F , G and H are f -form, g -formand h -form respectively and differentiable by ψ and π , { G, F } = − ( − ( f + D +1)( g + D +1) { F, G } , (2.10) { F, G ∧ H } = { F, G } ∧ H + ( − ( f + D +1) g G ∧ { F, H } , (2.11)and ( − ( f + D +1)( h + D +1) { F, { G, H }} + ( − ( g + D +1)( f + D +1) { G, { H, F }} + ( − ( h + D +1)( g + D +1) { H, { F, G }} = 0 (2.12)hold. The canonical equations can be written as dψ = −{ H, ψ } , dπ = −{ H, π } . (2.13)The fundamental brackets are { ψ, π } = ( − Dp , { π, ψ } = − , { ψ, ψ } = 0 = { π, π } . (2.14)If a form F is differentiable by ψ and π , and F does not depend positively on space-time points, dF = dψ ∧ ∂F∂ψ + dπ ∧ ∂F∂π = ( − ( p +1) q ∂H∂π ∧ ∂F∂ψ − ( − p ∂H∂ψ ∧ ∂F∂π = −{ H, F } (2.15)holds. D ≥ In this section, we apply the CCF to the second order formalism of gravity with Dirac fields forthe arbitrary dimension ( D ≥ We explain the notations used in this paper. Let g be the metric of which has signature ( − + · · · +), and let { θ a } D − a =0 denote an orthonormal frame. We have g = g ◦ ab θ a ⊗ θ b with g ◦ ab :=diag( − , , · · · , g ◦ ab or its inverse g ab ◦ . The first structureequation is dθ a + ω ab ∧ θ b = Θ a , (3.1)where ω ab is the connection form and Θ a = 12 C abc θ b ∧ θ c (3.2)4s the torsion 2-form. In the following of this paper, we suppose ω ba = − ω ab . We put η a = ∗ θ a , η ab = ∗ ( θ a ∧ θ b ) , η abc = ∗ ( θ a ∧ θ b ∧ θ c ) , η abcd = ∗ ( θ a ∧ θ b ∧ θ c ∧ θ d ) . (3.3)In Appendix A, several identities about θ a ∧ η a ··· a r ( r = 1 , , ,
4) , δη a ··· a r ( r = 0 , , ,
3) and dη a ··· a r ( r = 1 , ,
3) are listed. The curvature 2-form Ω ab is given byΩ ab = dω ab + ω ac ∧ ω cb . (3.4)Expanding the curvature form as Ω ab = 12 R abcd θ c ∧ θ d , (3.5)we define R ab := R cacb , R := R aa . (3.6)Because of (A.5), ∗ R can be written as ∗ R = Ω ab ∧ η ab . (3.7) The Lagrangian form of the gravity in the second order formalism is given by L ( θ, dθ ) = L G ( θ, dθ ) + L mat ( θ, dθ ) . (3.8)Here, L G is the Lagrangian form for the pure gravity given by L G ( θ, dθ ) = 12 κ N (cid:48) , N (cid:48) := ∗ R − d ( ω ab ∧ η ab ) , (3.9)and L mat ( θ, dθ ) = L mat ( θ, ω ( θ, dθ )) is the Lagrangian form of “matters” which are scaler fields,Dirac fields and gauge fields. Here, κ is the Einstein constant. Only the Dirac fields couple to ω ab . We derive the Euler-Lagrange equation of the gravity. The variation of L is given by δL ( θ, dθ ) = δθ c ∧ (cid:16) κ [Ω ab ∧ η abc − d ( ω ab ∧ η abc )] + T c (cid:17) + δdθ c ∧ κ ω ab ∧ η abc + δω ab ( θ, dθ ) ∧ (cid:16) κ [ dη ab − ω ca ∧ η cb − ω cb ∧ η ac ] + ∂L mat ∂ω ab (cid:17) , (3.10) δN (cid:48) = δ ( ω ac ∧ ω cb ∧ η ab + ω ab ∧ dη ab ) and ω ab ∧ δdη ab = δdθ c ∧ ω ab ∧ η abc + δθ c ∧ ω ab ∧ dη abc hold. In the second equation, we used the following formulas: dη ab = dθ c ∧ η abc , δη abc = δθ d ∧ η abcd , dη abc = dθ d ∧ η abcd . T a := ∂L mat ( θ, ω ) ∂θ a (3.11)is the energy-momentum form. We suppose that12 κ [ dη ab − ω ca ∧ η cb − ω cb ∧ η ac ] + ∂L mat ∂ω ab = 0 , (3.12)which are the same as the Euler-Lagrange of the first order formalism for the connection. Underthis supposition, (3.10) leads ∂L∂θ c = 12 κ [Ω ab ∧ η abc − d ( ω ab ∧ η abc )] + T c , ∂L∂dθ c = 12 κ ω ab ∧ η abc . (3.13)The Euler-Lagrange equation ∂L∂θ c + d ∂L∂dθ c = 0 becomes the Einstein equation − κ Ω ab ∧ η abc = T c . (3.14)If we expand T c as T c = T bc η b , the above Einstein equation leads R ab − Rδ ab = κT ab . (3.15) We determine the torsion C abc . The Lagrangian form of the “matters” can be written as L mat = L ( θ ) + ω ab ∧ S ab ( θ ) , (3.16)where S ab = η c S c,ab and S c,ab is independent from θ a and described by the Dirac fields. Using(3.12) and (A.12), we have 12 κ Θ c ∧ η abc = − ∂L mat ∂ω ab = − η c S c,ab . (3.17)This equation leads κ [ − C a g ◦ cb + C cab + C b g ◦ ca ] = − S c,ab , (3.18)where C a := C bab . Then, we have12 κ C cab = − S c,ab + 1 D − S a g ◦ cb − S b g ◦ ca ] (3.19)with S a := S bab . Here, D is the space-time dimension. Ω ab ∧ η abc = ( Rδ bc − R bc ) η b holds. Θ c ∧ η abc = η c ( g ◦ ac C b − g ◦ cb C a + C cab ) holds. .3 Covariant canonical formalism Next, we consider the covariant canonical formalism.In (3.9), N (cid:48) can be rewritten as N (cid:48) = ω ac ∧ ω cb ∧ η ab + ω ab ∧ dη ab = N + Θ a ∧ ω bc ∧ η abc (3.20)with N := ω ac ∧ ω cb ∧ η ba . (3.21)In the second line of (3.20), we used (A.12). Using (3.1) and (A.2), N can be rewritten as N = dθ a ∧ ω bc ∧ η abc − Θ a ∧ ω bc ∧ η abc . (3.22)The conjugate form of θ a is given by π a = 12 κ ω bc ∧ η abc . (3.23)The Hamilton form is given by H ( θ, π ) = dθ a ∧ π a − L = H G ( θ, π ) − L mat ( θ, π ) (3.24)with H G ( θ, π ) = N κ . (3.25)Here, we used (3.20) and (3.22). In Sec.3.4, we represent N by θ a and π a and take derivativesby these. Because C abc is described by the Dirac fields, it is independent from θ a and π a . Then,Θ a = C abc θ b ∧ θ c is independent from π a .The canonical equations are given by dθ a = ∂H G ∂π a − ∂L mat ∂π a , (3.26) dπ a = ∂H G ∂θ a − ∂L mat ( θ, π ) ∂θ a . (3.27)In the RHS of (3.26), the second term can be rewritten as − ∂L mat ∂π a = − ∂∂π a (cid:104) ω bc ∧ S bc (cid:105) = ∂∂π a [Θ b ∧ π b ] = Θ a . (3.28)In the second equality, we used ω ab ∧ S ab = − Θ a ∧ π a , (3.29)which is derived form (3.17). Then, (3.26) becomes dθ a = ∂H G ∂π a + Θ a . (3.30)In Sec.3.4, we calculate ∂H G ∂π a , ∂H G ∂θ a and ∂L mat ( θ,π ) ∂θ a . 7 .4 Variation of the Hamilton form π a We represent N by θ a and π a . Using (A.5), N can be rewritten as N = ( ω abc ω bca + ω a ω a ) η. (3.31)Here, we expand ω ab as ω ab = ω abc θ c and put ω a := ω bab . We can represent ω abc by θ a and π a as ω abc = κ (cid:104) v c,ab + 1 D − g ◦ ac v b − g ◦ bc v a ) (cid:105) , (3.32) v c,ab := − ∗ V c,ab , V c,ab := π c ∧ θ a ∧ θ b (3.33)with v a := v bab .Next, we calculate δv c,ab η . For an arbitrary D -form ξ , δv c,ab ξ = − δv c,ab ξ ∗ η = − δv c,ab η ∗ ξ = ( − δ [ v c,ab η ] + v c,ab δη ) ∗ ξ = ( − δV c,ab + v c,ab δη ) ∗ ξ (3.34)holds. Then, we have δv c,ab η = δV c,ab − v c,ab δη. (3.35)The variation by only π a is given by δv c,ab η = δπ c ∧ θ a ∧ θ b . (3.36)We take the variation of H G by π a : δH G = ψ abc ω c [ ab ] + ψ a ω a , (3.37)where ψ abc := 1 κ δω abc η = δπ c ∧ θ a ∧ θ b + δπ d ∧ D − g ◦ ac θ b ∧ θ d − g ◦ bc θ a ∧ θ d ) , (3.38) ψ a := ψ bab and ω c [ ab ] := ( ω cab − ω cba ). (3.37) leads δH G = δπ c ∧ θ a ∧ θ b ω c [ ab ] . (3.39)Then, we have ∂H G ∂π c = − ω ca ∧ θ a . (3.40)8 .4.2 Variation by θ a We take the variation of H G by θ a using (3.35). From (3.35), we have δv c,ab η = δ ( π c ∧ θ a ∧ θ b ) − v c,ab δη = ( δθ a ∧ θ b − δθ b ∧ θ a ) ∧ π c − v c,ab δθ d ∧ η d (3.41)for the variation by only θ a . Here, we used (A.10). By the way, H G can be written as H G = H G η , H G := 12 κ ( ω abc ω bca + ω a ω a ) . (3.42)The variation of H G by θ a is given by δH G = Y abc ω c [ ab ] + Y a ω a + H G δθ d ∧ η d , (3.43)where Y abc := κ δω abc η and Y a := Y bab . Using (3.41) and (3.32), we have Y abc = 2 δθ [ a ∧ θ b ] ∧ π c + 2 D − g ◦ c [ a ( δθ b ] ∧ θ d − δθ d ∧ θ b ] ) ∧ π d − ω abc δθ d ∧ η d . (3.44)Then, we obtain δH G = 2 g ◦ ad δθ d ∧ θ b ∧ π c ω c [ ab ] − H G δθ d ∧ η d . (3.45)This leads ∂H G ∂θ d = ω cd ∧ π c − θ b ∧ π c ω cbd − e d (cid:99) H G . (3.46)We used (A.14). Here, (cid:99) is the interior product and { e a } is the dual basis of { θ a } ( e a (cid:99) θ b = δ ba ).Next, we derive ∂H G ∂θ d = 12 κ ( ω cd ∧ ω ab ∧ η abc + ω ac ∧ ω bc ∧ η bad ) (3.47)from (3.46). Substituting (3.23) to (3.46), we have ∂H G ∂θ d = 12 κ ( ω cd ∧ ω ab ∧ η cab − θ b ∧ ω ae ∧ η cae ω cbd ) − e d (cid:99) H G . (3.48)Using (A.2), − θ b ∧ ω ae ∧ η cae ω cbd = − ω ae ∧ η ce ω cad + ω ab ∧ η ca ω cbd (3.49)holds. On the other hand, using (A.16), e d (cid:99) H G = 12 κ ( ω acd ω cb ∧ η ba − ω cbd ω ac ∧ η ba + ω ac ∧ ω cb ∧ η bad ) (3.50)holds. Then, we obtain (3.47). The RHS of (3.47) is same with Sparling’s form except for acoefficient and relates to the gravitational energy-momentum pseudo-tensor [6, 8].9 .4.3 Variation of L mat We calculate ∂L mat ( θ, π ) /∂θ c . Using (3.16) and (3.29), we have ∂L mat ( θ, ω ) ∂θ c = ∂L ∂θ c − ω ab ∧ η dc S dab , (3.51) ∂L mat ( θ, π ) ∂θ c = ∂L ∂θ c − ∂ Θ a ∂θ c ∧ π a = ∂L ∂θ c − C acb θ b ∧ π a , (3.52)and t c := ∂L mat ( θ, π ) ∂θ c − T c = ∂L mat ( θ, π ) ∂θ c − ∂L mat ( θ, ω ) ∂θ c = − C acb θ b ∧ κ ω de ∧ η ade + ω ab ∧ η dc S dab =: ω ab ∧ b c,ab . (3.53)We can show that B c,ab := 12 κ Θ d ∧ η abcd = b c,ab (3.54)using (3.18) . Then, we have t c = ω ab ∧ B c,ab and − ∂L mat ( θ, π ) ∂θ c = − T c − κ ω ab ∧ Θ d ∧ η abcd . (3.55) We derive the canonical equations. Substituting (3.40) to (3.26), we have dθ a = − ω ab ∧ θ b + Θ a , (3.56)which is equivalent to the first structure equation (3.1). Using (3.47) and (3.55), (3.27) becomes dπ c = 12 κ ( ω db ∧ ω ab ∧ η adc + ω dc ∧ ω ab ∧ η abd − ω ab ∧ Θ d ∧ η abcd ) − T c . (3.57)We show that (3.57) is equivalent to the Einstein equation. (3.47) can be rewritten as ∂H G ∂θ c = 12 κ (cid:104) − ω ad ∧ ω bd ∧ η abc − ω ab ∧ ( ω da ∧ η dbc + ω db ∧ η adc + ω dc ∧ η abd ) (cid:105) = 12 κ (cid:104) − Ω ab ∧ η abc + d ( ω ab ∧ η abc ) (cid:105) + ω ab ∧ B c,ab . (3.58) B c,ab = b c,ab = κ (cid:104) C c η ab + 2 C dc [ a η b ] d + 2 C [ a η b ] c + C dab η cd (cid:105) holds. B c,ab can be rewritten as B c,ab = 12 κ (cid:104) dη abc − ( ω da ∧ η dbc + ω db ∧ η adc + ω dc ∧ η abd ) (cid:105) (3.59)using (A.11). Then, (3.57) becomes dπ c = 12 κ [ − Ω ab ∧ η abc + d ( ω ab ∧ η abc )] − T c = − κ Ω ab ∧ η abc + dπ c − T c . (3.60)This leads the Einstein equation (3.14). Let us consider that an infinitesimal transformation of dynamical fields ψ A and its conjugate forms π A : ψ A → ψ A + δψ A , π A → π A + δπ A . (4.1)Here, A is the label of the fields. If there exists ( D − G such that δψ A = { ψ A , G } , δπ A = { π A , G } , (4.2)we call G the generator of the transformation. If a form F is differentiable by ψ A and π A , thetransformation of F is given by δF = { F, G } . (4.3)In this section, we find the generators of the gauge transformations for matter fields (Sec.4.1),gauge fields (Sec.4.2) and the gravitational field within the second order formalism (Sec.4.3). Let us consider that an infinitesimal global gauge transformation of matter fields: δψ A = ε r ( G r ) AB ψ B , δL = 0 . (4.4)Here, ε r are infinitesimal parameters, G r are representations of the generators of a linear Lie group G and L ( ψ A , dψ A ) is the Lagrangian form of the matter fields ψ A . The matrices G r satisfy[ G r , G s ] = f trs G t , (4.5)where [ A, B ] := AB − BA and f trs are the structure constants of G . Under the transformation(4.4), the conjugate forms π A behave as δπ A = − ε r ( G r ) BA π B . (4.6)11he Noether currents (B.4) are given by G (0) r = ( G r ) AB ψ B ∧ π A . (4.7)The Noether currents G (0) r satisfy { ψ A , G (0) r } = ( G r ) AB ψ B , (4.8) { π A , G (0) r } = − ( G r ) BA π B , (4.9)and { G (0) r , G (0) s } = f trs G (0) t . (4.10)The generator of the transformation (4.4) is given by ε r G (0) r .To generalize (4.4) to the local gauge transformation, L ( ψ A , dψ A ) should be replaced by L ( ψ A , ( Dψ ) A ) where ( Dψ ) A := dψ A + A r ( G r ) AB ∧ ψ B and A r are the gauge fields. The forms ψ A and π A are independent from A r and π r where π r is the conjugate forms of A r . Let us consider that the infinitesimal local gauge transformation of the gauge fields: δA r = ε s f rst A t − dε r , δL = 0 . (4.11)Here, L is the Lagrangian form of the gauge fields. Under the transformation (4.11), π r behaveas δπ r = − ε s f tsr π t . (4.12)The Noether currents (B.4) are given by G (1) s = f rst A t ∧ π r . (4.13)The Noether currents G (1) r satisfy { A s , G (1) r } = f srt A t , (4.14) { π s , G (1) r } = − f trs π t , (4.15)and { G (1) r , G (1) s } = f trs G (1) t . (4.16)We put G r := G (0) r + G (1) r . The Noether currents G r satisfy { G r , G s } = f trs G t . (4.17)12he generator of the transformation without dε r is given by ˜ G := ε r G r . We assume that thegenerator of the local gauge transformation denoted by G is given by G = ˜ G + dε r ∧ F r , (4.18)where F r are unknown ( D − π r . Because { A s , dε r ∧ F r } = dε r ∧ ∂F r ∂π s (4.19)holds, we obtain F r = − π r . (4.20)Then, G is given by G = ε r G r − dε r ∧ π r . (4.21)The ( D − F r does not affect to ψ A , π A and π r : δψ A = { ψ A , G } = ε r ( G r ) AB ψ B , (4.22) δA r = { A r , G } = ε s f rst A t − dε r , (4.23) δπ A = { π A , G } = − ε r ( G r ) BA π B , (4.24) δπ r = { π r , G } = − ε s f tsr π t . (4.25) Let us consider that an infinitesimal local Lorentz transformation δθ a = ε ab θ b . (4.26)Here, ε ab are infinitesimal parameters which satisfy ε ab = − ε ba . Under the transformation, ω ab behave as δω ab = ε ac ω cb + ε bc ω ac − dε ab (4.27)and π a given by (3.23) behave as δπ a = 12 κ δω bc ∧ η abc + 12 κ ω bc ∧ δθ d ∧ η abcd = ε bc ( − g ◦ a [ c π b ] ) − dε bc ∧ κ η abc . (4.28)The Noether currents (B.4) are given by G cd = 2 θ [ d ∧ π c ] . (4.29)13he Noether currents G cd satisfy { θ a , G cd } = 2 θ [ d δ ac ] , (4.30) { π a , G cd } = − g ◦ a [ d π c ] , (4.31)and { G ab , G cd } = g ◦ bc G ad − g ◦ ac G bd + g ◦ ad G bc − g ◦ bd G ac . (4.32)The above equation corresponds to the commutation relations of the generators of the Lorentzgroup: [ G ab , G cd ] = g ◦ bc G ad − g ◦ ac G bd + g ◦ ad G bc − g ◦ bd G ac . (4.33)The generator of the transformation without dε ab is given by ˜ G := ε ab G ab . We assume that thegenerator of the local Lorentz transformation (4.26) denoted by G is given by G = ˜ G + 12 dε ab ∧ F ab , (4.34)where F ab are ( D − θ a . Because { π a , dε bc ∧ F bc } = 12 dε bc ∧ ∂F bc ∂θ a (4.35)holds and the RHS of the above equation should be − dε bc ∧ κ η abc , we obtain F bc = − κ η bc . (4.36)Then, G is given by G = 12 ε ab G ab − κ dε ab ∧ η ab . (4.37)The ( D − F ab does not affect to θ a : δθ a = { θ a , G } = ε ab θ b , (4.38) δπ a = { π a , G } = − ε ba π b − dε bc ∧ κ η abc . (4.39) F r and G r According to Ref. [12], if a generator is given by G = ε r G r + dε r ∧ F r with nonzero F r , G r = −{ F r , H } (4.40)14olds. We check this relationship for the gauge fields and the gravitational field. For the gaugefields, −{ F r , H } = { π r , H } = − ∂H ∂A r = f crb A b ∧ π c = G (1) r (4.41)holds. Here, H is the Hamilton form of the gauge fields (C.10) and we used (C.11) in the thirdline. For the gravitational field, −{ F ab , H G } = 1 κ { η ab , H G } = − κ ∂η ab ∂θ c ∧ ∂H G ∂π c = − κ η abc ∧ ( − ω cd ∧ θ d )= 1 κ ω cd ∧ θ d ∧ η abc (4.42)holds. In the third line, we used (3.40). Because κG ab = − ω cd ∧ θ [ b ∧ η a ] cd = − ω c [ b ∧ η a ] c , (4.43) − κ { F ab , H G } = ω cd ∧ θ d ∧ η abc = − ω c [ b ∧ η a ] c (4.44)hold, we have G ab = −{ F ab , H G } . (4.45) We investigated generators of local gauge transformations in the covariant canonical formalism(CCF) for matter fields, gauge fields and the second order formalism of gravity. The total generator G is given by G = ε r G r + dε r ∧ F r where ε r are infinitesimal parameters and G r are the Noethercurrents. { G r , G s } = f trs G t holds. Here, {• , •} is the Poisson bracket of the CCF and f trs arethe structure constants of the gauge group. For the matter fields, F r = 0 holds. For the gaugefields and the gravitational field, G r = −{ F r , H } holds. Here, H is the Hamilton form which isthe form-Legendre transformation of the Lagrangian form. Additionally, we applied the CCF tothe second order formalism of gravity with Dirac fields for the arbitrary dimension ( D ≥ Acknowledgment
We acknowledge helpful discussions with M. Matsuo.15
Formulas
Several useful formulas are listed. For θ a ∧ η a ··· a r ( r = 1 , , , θ a ∧ η bcde = − δ ab η cde + δ ac η bde − δ ad η bce + δ ae η bcd , (A.1) θ a ∧ η bcd = δ ab η cd − δ ac η bd + δ ad η bc , (A.2) θ a ∧ η bc = − δ ab η c + δ ac η b , (A.3) θ a ∧ η b = δ ab η (A.4)hold [16]. Using (A.3) and (A.4), we have θ a ∧ θ b ∧ η cd = ( δ ac δ bd − δ ad δ bc ) η. (A.5)Using (A.2) and (A.3), we have θ a ∧ θ b ∧ η cde = ( δ ad δ be − δ ae δ bd ) η c − ( δ ac δ be − δ ae δ bc ) η d + ( δ ac δ bd − δ ad δ bc ) η e . (A.6)For δη a ··· a r ( r = 0 , , , δη abc = δθ d ∧ η abcd , (A.7) δη ab = δθ c ∧ η abc , (A.8) δη a = δθ b ∧ η ab , (A.9) δη = δθ a ∧ η a (A.10)hold. For dη a ··· a r ( r = 1 , , dη abc = ω da ∧ η dbc + ω db ∧ η adc + ω dc ∧ η abd + Θ d ∧ η abcd , (A.11) dη ab = ω ca ∧ η cb + ω cb ∧ η ac + Θ c ∧ η abc , (A.12) dη a = ω ba ∧ η b + Θ b ∧ η ab (A.13)hold. Forms η a ··· a r ( r = 1 , , ,
4) can be written as [16] η a = e a (cid:99) η, (A.14) η ab = e b (cid:99) η a , (A.15) η abc = e c (cid:99) η ab , (A.16) η abcd = e d (cid:99) η abc . (A.17)Here, (cid:99) is the interior product and { e a } is the dual basis of { θ a } ( e a (cid:99) θ b = δ ba ). B Noether currents
We explain the Noether currents. For an infinitesimal transformation of p -form dynamical fields ψ A → ψ A + δψ A , an identical equation δL ≡ δψ A ∧ (cid:16) ∂L∂ψ A − ( − p d ∂L∂dψ A (cid:17) + d (cid:16) δψ A ∧ ∂L∂dψ A (cid:17) (B.1)16olds. Here, ≡ denotes identical equation which holds without using the Euler-Lagrange equations.For a global transformation δψ A = ε r ∆ Ar , δL = ε r dl r , (B.2)we have dl r ≡ ∆ Ar ∧ [ L ] A + d (cid:16) ∆ Ar ∧ ∂L∂dψ A (cid:17) (B.3)from (B.1). Here, [ L ] A := ∂L∂ψ A − ( − p d ∂L∂dψ A . Under the Euler-Lagrange equations [ L ] A = 0, theNoether currents N r := ∆ Ar ∧ ∂L∂dψ A − l r (B.4)are conserved: dN r = 0. C Covariant canonical formalism of gauge fields
In this section, we review that the covariant canonical formalism of gauge fields.Let us consider that a global transformation ψ (cid:48) A = [ T ( ε )] AB ψ B (C.1)of matter fields { ψ A } . Here, T ( ε ) is a representation of a linear Lie group G and ε = { ε r } nr =1 isthe set of the continuous parameters. For infinitesimal transformation, (C.1) becomes δψ A = ε r ( G r ) AB ψ B , (C.2)where G r are the representation of generators of the group G . The matrices G r satisfy [ G r , G s ] = f ars G a where f ars are the structure constants of G . Suppose that the Lagrangian form of thematter fields L ( ψ A , dψ A ) is invariant under the transformation. Let us replace L ( ψ A , dψ A ) by L ( ψ A , ( Dψ ) A ) where ( Dψ ) A := dψ A + A r ( G r ) AB ∧ ψ B is the covariant derivative and A r are thegauge fields. L ( ψ A , ( Dψ ) A ) is invariant under the local transformation ψ (cid:48) A = [ T ( ε ( x ))] AB ψ B . Forthe infinitesimal local transformation, A r behave as δA r = ε s f rst A t − dε r and ( Dψ ) A behave as δ ( Dψ ) A = ε r ( x )( G r ) AB ( Dψ ) B . The latter is the same form with (C.2).The curvature of the gauge fields is defined by F r := dA r + 12 f rbc A b ∧ A c . (C.3)Under the local transformation (C.2), F r behave as δF r = ε a f ras F s . We put F r := κ rs F s withthe Killing form κ rs := − f arb f bsa (= κ sr ). Under the transformation (C.2), F r behave as δF r = − ε a f sar F s . The Lagrangian form of the gauge fields L is given by L = − k F r ∧ ∗ F r , (C.4)17here k is a positive constant. L is gauge invariant.We consider the Euler-Lagrange equation of the gauge fields. We put L := L ( ψ A , ( Dψ ) A ) + L .The derivatives of L are given by ∂L∂A a = − k f cab A b ∧ ∗ F c + J a , ∂L∂dA a = − k ∗ F a (C.5)with J a := ∂L ( ψ A , ( Dψ ) A ) ∂A a . (C.6)The Euler-Lagrange equation ∂L/∂A a + d ( ∂L/∂dA a ) = 0 is given by D ∗ F a := d ∗ F a + f cab A b ∧ ∗ F c = kJ a . (C.7) D ∗ F a is the covariant derivative. The above equation is the Yang-Mills-Utiyama equation.The conjugate form of A a is given by π a = − k ∗ F a . (C.8)The Hamilton form is given by H = H − L ( ψ A , ( Dψ ) A ) , (C.9) H = − f abc A b ∧ A c ∧ π a + k π a ∧ ∗ π a , (C.10)with π a := ( κ − ) ab π b = − k ∗ F a . The derivatives of H are given by ∂H∂A a = − f cab A b ∧ π c − J a , (C.11) ∂H∂π a = k ∗ π a − f abc A b ∧ A c . (C.12)The canonical equations dA a = ∂H∂π a and dπ a = ∂H∂A a became dA a = k ∗ π a − f abc A b ∧ A c , (C.13) dπ a = − f cab A b ∧ π c − J a . (C.14)The above two equations can be rewritten as F a = k ∗ π a , Dπ a := dπ a + f cab A b ∧ π c = − J a . (C.15)The former is equivalent with the definition of π a . The latter is equivalent with the Yang-Mills-Utiyama equation. The covariant canonical formalism does not need the gauge fixing.18 eferences [1] A. D’Adda, J. E. Nelson and T. Regge, “Covariant canonical formalism for the group mani-fold”, Annals of Physics , 384 (1985).[2] J. E. Nelson and T. Regge, “Covariant Canonical Formalism for Gravity”, Annals of Physics , 234 (1986).[3] A. Lerda, J. E. Nelson and T. Regge, “Covariant Canonical Formalism for Supergravity”,Phys. Lett. , 294 (1985).[4] A. Lerda, J. E. Nelson and T. Regge, “The Group Manifold Hamiltonian for Supergravity”,Phys. Lett. , 297 (1985).[5] T. Nakamura, Bussei Kenkyu , 2 (2002) (in Japanese).[6] J. M. Nester, “General pseudotensors and quasilocal quantities”, Classical and QuantumGravity , S261 (2004).[7] Y. Kaminaga, “Covariant Analytic Mechanics with Differential Forms and Its Application toGravity”, Electron. J. Theor. Phys. , 199 (2012).[8] Chiang-Mei Chen, J. M. Nester and Roh-Suan Tung, “Gravitational energy for GR andPoincar´e gauge theories: a covariant Hamiltonian approach”, Int. J. Mod. Phys. D , 1530026(2015).[9] S. Nakajima, “Application of covariant analytic mechanics with differential forms to gravitywith Dirac field”, Electron. J. Theor. Phys. , 95 (2016).[10] S. Nakajima, “Reconsideration of De Donder-Weyl theory by covariant analytic mechanics”,arXiv:1602.04849v2.[11] Y. Kaminaga, “Poisson Bracket and Symplectic Structure of Covariant Canonical Formalismof Fields”, Electron. J. Theor. Phys. , 55 (2018).[12] L. Castellani and A. D’Adda, “Covariant Hamiltonian for gravity coupled to p -forms”, Phys.Rev. D , 025015 (2020).[13] I. V. Kanatchikov, “Canonical structure of classical field theory in the polymomentum phasespace”, Rept. Math. Phys. , 49 (1998).[14] A. Trautman, “On the Einstein-Cartan equations”, Bull. Acad. Pol. Sci , 185 (1972).[15] W. Thirring, “A Course in Mathematical Physics 2”, Springer (second edition, 1978).[16] F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’eman, “Metric-affine gauge theory ofgravity: field equations, Noether identities, world spinors, and breaking of dilation invariance”,Phys. Rep.258