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2 1 0 2 Complex Ashtekar Variables and Reality Con- n ditions for Holst’s Action a J 4 2 Wolfgang M. Wieland ] c q Abstract. From the Holst action in terms of complex valued Ashtekar - variables additional reality conditions mimicking the linear simplicity r g constraintsofspinfoamgravityarefound.Inquantumtheorywiththe [ results of Ding and Rovelli we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kine- 2 maticalHilbertspacematchestheoriginaloneofloopquantumgravity, v 8 thatisforrealvaluedAshtekarconnection.Ourresultperfectlyfitwith 3 recent developments of Rovelli and Speziale concerning Lorentz covari- 7 ance within spin-form gravity. 1 . 2 1 0 1 1. Introduction : v i Motivation and Overview. Loop quantum gravity comes in two technically X different versions. On the one hand there is the canonical framework [1, 2, 3] r a tryingtosolvetheWheeler–DeWittconstraintequationdirectly.Ontheother handthereisthepathintegralapproachofspin-foamgravity[4,5]searching tocomputetransitionamplitudes.Bothapproachesareinterconnected,however the precise mathematical relation between them two is still a matter of debate. In a couple of papers Ding You and Carlo Rovelli [6, 7] shed much light on this relation. They proved that within the spin-foam formalism one can actuallyrecoverthekinematicsofloopquantumgravity.Inthispaperweare interestedinsomethingsimilar.Thepresentspin-foammodelsareformulated in terms of SL(2,C) holonomies. The canonical program has however much abandoned this gauge group in favour of SU(2) instead. In order to build a mathematicalframeworkallowingustocomparebothformulationsproperly, it seems reasonable to repeat the program of canonical quantisation with respect to complex variables. To do this section 2 recalls the Holst action in termsofcomplexAshtekarvariables.(SeealsotheworkofAlexandrov[8]for a very similar approach towards this goal.) 2 Wolfgang M. Wieland Next,withinsection3bytheuseof“timegauge” (i.e.apartialgaugefix- ingaligningtheinternaltimedirectiontothesurfacenormalofthet=const. hyper-surfaces) we successfully introduce a Hamiltonian and perform the Dirac analysis of the constraint equations. This gauge condition manifestly breaks SL(2,C) invariance; residual gauge transformations are restricted to the SU(2) subgroup. But this does certainly not force us to write everything in terms of SU(2) variables (e.g. in terms of the real valued Ashtekar connection(β)A=Γ+βK,anditsmomentumconjugate).Infacttheconstraint equations can be split into those being SL(2,C) gauge invariant and those breaking this symmetry. The former (i.e. the collection of Gauß, vector and Hamiltonian constraints) happen to be of first class, whereas the latter are of second class. The additional second class constraints, absent in the real valued formulation restrict the gauge transformations allowed to those pre- servingtheinternaltimedirection,and,ipsofacto,breakSL(2,C)invariance. One of these “reality conditions” will match the linear simplicity constraints ! C ∝K+βL=0 of spin-foam [5] gravity. Being interested to perform the program of canonical quantisation we are then let to the crucial question whether or not the same kinematical Hilbert space as in the SU(2) case emerges. Remember that this question is highlynontrivial,alreadyinquantummechanics whenaskingwhathappens if quantisation is performed starting from some arbitrarily chosen canonical pair. At the end of section 4 this question can be answered in the affirma- tive. Starting from complex Ashtekar variables we are able to recover the kinematical Hilbert space of loop quantum gravity. Notation. Inthisworksmallindicesa,b,c,... refertoabstractindicesintan- gent space, internal indices in four dimensional Minkowski space are marked by capitals I,J,K,... their three dimensional counterparts are denoted by smallindicesi,j,k,... fromthemiddleoftheromanalphabet.Onsl(2,C)af- ter choosing some internal time direction any element X can be decomposed according to Xiτ into three complex components Xi ∈ C. Here τi := 1σi i 2i are the canonical generators of SU(2) built from the Pauli spin matrices σi. AccordingtothisdecompositiontheparitytransformedelementX¯ ∈sl(2,C) is nothing but the complex conjugate X¯ = X¯iτ of the component vector i Xi ∈C3. 2. Holst action revisited The kinematical Hilbert space [9, 1, 3] of loop quantum gravity can most naturally be obtained from the Hamiltonian formulation emerging from the Complex Variables & Reality Conditions for Holst’s Action 3 so-called1Holst[11]action.Intermsoftheco-tetradfieldηI,andtheso(1,3)- valued spin connection ωI it can be written according to J (cid:126) (cid:104)(cid:90) (cid:16) 2 (cid:17) S [η,ω]= (cid:15) ηI ∧ηJ − η ∧η ∧RKL[ω]+ Holst 4(cid:96) 2 IJKL β K L P M (1) −2(cid:90) (cid:15) ηI ∧ηJ ∧(cid:0)nKDωnL(cid:1)(cid:105). IJKL ∂M Here RI [ω] = dωI +ωI ∧ωM denotes the curvature corresponding to J J M J the spin connection, internal indices (I,J,K,··· ∈ {0,1,2,3}) are raised by the flat Minkowski metric η =diag(−1,1,1,1) and (cid:15) is the internal IJ IJ IJKL Levi-Civita tensor determined from (cid:15) =1. The last part of (1) absent in 0123 the original paper [11] is an integral over the three dimensional boundary of M. In the case of the Hilbert–Palatini Lagrangian this boundary term was already introduced by Obukhov [12]. It is built from the covariant derivative ω D = d + ω of the internal vector n = η an associated to the surface I I a normaln of∂M.ForanexhaustiveanalysisofsurfacetermswithintheHolst a setting we strongly recommend [13]. Later by passing to the Hamiltonian formulation we will implicitly assume that ∂M is formed by two Cauchy surfaces Σ and Σ glued at spatial infinity. Several constants appear in this 0 1 action; (cid:96) = (cid:112)8π(cid:126)G/c3 ≈ 8 · 10−35m equals the rescaled Planck length, P and the Barbero–Immirzi parameter β is a dimensionless parameter unique to all models of loop quantum gravity. Leaving classical dynamics of general relativity unaffected this number can affect quantum theory only. In order to derive the equations of motion for general relativity from the principle of least action appropriate boundary conditions have to be im- posed. Otherwise the variation principle remains obscure. In the case of the HolstactionwewillonlyallowforvariationsδηI andδωIJ oftheelementary configuration variables subject to the following restrictions on the boundary hI (cid:0)ϕ∗δηJ(cid:1)=! 0, (2) J hI hJ (cid:0)ϕ∗δωMN(cid:1)=! 0. (3) M N HeretheinternalprojectorhI annihilatingnI togetherwiththeembedding J ϕ:∂M →M and the corresponding pull-back ϕ∗ have been used. WearenowinterestedtodecomposetheHolstactiondefinedby(1)into its self- and antiselfdual parts (cid:126) β+i (cid:126) β−i SHolst[η,ω]=−2(cid:96) 2 iβ SC+ 2(cid:96) 2 iβ S¯C+I∂M. (4) P P Here we have introduced the C-valued action (cid:90) SC = PIJMNηI ∧ηJ ∧RMN[ω], (5) M 1In fact it is rather misleading to call it that way. Holst though proving this action to reproducetheSU(2)Ashtekarvariables,didactuallynotintroduceitfirst.Thiswasdone by Hojman et al. [10] already in the 1980ies. I’m grateful to Friedrich Hehl for pointing thisout. 4 Wolfgang M. Wieland together with the selfdual projector PIJ := 1(cid:0)δI δJ − i(cid:15)IJ (cid:1), (6) MN 2 [M N] 2 MN fulfilling the elementary properties: (i)PIJ PMN =PIJ , MN LK LK (ii)P =−P =−P =P , IJMN JIMN IJNM MNIJ (iii)P¯IJ PMN =0, MN LK (iv)(cid:15)IJ PMN =2iPIJ . MN LK LK See also [8] on that. Notice that the boundary term represented by I has ∂M not been decomposed into self- and antiselfdual parts. In order to construct a Hamiltonian formulation of the theory defined by(1)letusfirstdecomposeallfour-dimensionalquantitiesintotheirspatio- temporal components. This is achieved [14] by introducing a global time function t : M → R (thereby implicitly assuming global hyperbolicity, i.e. M =(0,1)×Σ) together with a future directed vector field ta transversal to all t = const. hyper-surfaces Σ (i.e. ta∂ t = 1). We can then introduce the t a spatial components of the elementary configuration variables, namely co-triad: ei =prηi, (7) so(3)-connection: Γi =prωi , (8) j j extrinsic curvature: Ki=prωi . (9) o Here we have used the spatial projection prϕ := ϕ−dt∧ι ϕ of any four- t dimensional p-form ϕ onto Σ . Still additional variables representing the dt t components are missing. Introducing lapse-function N, shift-vector Na to- getherwithadditional“Lagrangianmultiplierfields” φi andφi representing o j both infinitesimal boosts and rotations along the time axis, we are left with the following decomposition of the four-dimensional configuration variables ηo =Ndt, ηi =Naei dt+ei, (10) a ωi =dtφi +Ki, ωi =dtφi +Γi . (11) o o j j j NoticethatlocalLorentzinvarianceallowedustochoose“timegauge” thereby setting prηo globally to zero. Inserting this decomposition of variables into the selfdual action we recover the original expression of Ashtekar [15] (cid:90) (cid:90) SC = dt −Eia(LtAia−DaΛi)+NaFiabEib+ R Σ (12) i + N(cid:15) lmFi E aE b. 2 i ab l m (cid:101) Where we have just introduced the old i.e. complex valued Ashtekar [16] variables Ashtekar connection: Ai =Γi +iKi , (13) a a a 1 Densitised triad: E a = ηabc(cid:15) el em . (14) i 2(cid:101) ilm b c Complex Variables & Reality Conditions for Holst’s Action 5 Inequation(12)L equalstheLiederivativealongthetime-flowvectorfield, t ηabc is the spatial Levi-Civita tensor density, so(3) elements have been de- (cid:101) composed into the generators of SO(3) according to e.g. Γi = (cid:15)i Γm, the j mj AshtekarcovariantderivativeisdenotedbyD =∂ +[A ,·],Fi =[D ,D ]i a a a ab a b is the curvature associated, and Λi :=φi+iφi . Furthermore N :=µ−1N is o a density of weight minus one, where µ=e1∧e2∧e3 represent(cid:101)s the oriented volume element on Σ, and µ−1 =e µe νe ρ∂ ∧∂ ∧∂ equals its inverse. 1 2 3 µ ν ρ What about the surface contribution? In terms of the decomposition introduced above I takes the usual form of the Gibbons–Hawking–York ∂M [17, 18] boundary term (cid:126) (cid:90) (cid:126) (cid:90) (cid:126) (cid:90) I =− E aKi =− E aKi + E aKi . (15) ∂M (cid:96) 2 i a (cid:96) 2 i a (cid:96) 2 i a P ∂M P Σ1 P Σ0 By the use of time-gauge the boundary conditions (2, 3) on ∂M simplify δE a| =0, δΓi | =0. (16) i ∂M a ∂M However there are no restrictions on the variations of lapse N, shift Na and extrinsic curvature Ki on the boundary. a 3. Hamiltonian formulation 3.1. Phase space, constraints and time evolution As a matter of fact Hamiltonian mechanics is about time evolution. In particular one tries to split the equations of motion into two parts. First of all one has the evolution equations generated by the Hamiltonian vector field X = {H,·} describing dynamics of the theory (e.g. the evolution equation H for the electric field ∂ E(cid:126) = ∇(cid:126) ×(∇(cid:126) ×A(cid:126)) in terms of the vector potential). t Secondly one might have additional constraint equations discarding all un- physical degrees of freedom (e.g. the Gauß-law ∇(cid:126) ·E(cid:126) =0). Furthermore the Hamiltonian formulation is not necessarily about performing a Legendre transformation. In particular the decomposition of the complex action (12) strongly suggests not to do so. The action is already in Hamiltonian form, performing a singular Legendre transformation would introduce an unnaturally large phase space containing momenta associated todensitisedlapseN,shiftvector,toΛi andtoboththedensitisedtriadand the connection. But(cid:101)this is not needed at all. In order to turn the equations of motion into Hamiltonian form let us first introduce the “natural” phase space P of smooth SL(2,C) connections Ai and corresponding momenta Π a. The symplectic structure is defined by a i the only non vanishing Poisson brackets, namely (cid:8)Π a(p),Aj (q)(cid:9)=δjδaδ(3)(p,q), i b i b (17) (cid:8)Π¯ a(p),A¯j (q)(cid:9)=δjδaδ(3)(p,q). i b i b 6 Wolfgang M. Wieland Notice that Π a⊗τi has to be understood as sl(2,C) valued vector density, i δ(3)(p,q) is the Dirac distribution (a scalar density) and X¯ denotes complex conjugation of X. Look at the first part of (12), which actually tell us that Π a and Π¯ a i i are related according to Πia(cid:12)(cid:12)EOM =+2(cid:96)(cid:126)2βi+β iEia, (18) P Π¯ia(cid:12)(cid:12)EOM =−2(cid:96)(cid:126)2βi−β iEia. (19) P And the abbreviation EOM should remind us that these relations have to be fulfilled when the equations of motion hold. However we can put equations (18, 19) upside down, allowing us to define E a on the entire phase space i (cid:96) 2(cid:16) iβ iβ (cid:17) E a := P Π a− Π¯ a = i (cid:126) β+i i β−i i (20) (cid:96) 2 β (cid:16) (cid:17) = P (Π a+Π¯ a)+iβ(Π a−Π¯ a) . (cid:126) β2+1 i i i i Similarlyequations(18,19)tellusthatinordertorecovertheoriginalnumber of degrees of freedom the quantity (cid:96) 2(cid:16) iβ iβ (cid:17) C a := P Π a+ Π¯ a = i i(cid:126) β+i i β−i i (21) (cid:96) 2 β (cid:16) (cid:17) = P −i(Π a−Π¯ a)+β(Π a+Π¯ a) EO=M0 (cid:126) β2+1 i i i i is constrained to vanish. Due to the striking similarity with the constraints C a = E a−E¯ a = 0 found within the old Ashtekar approach we call them i i i reality conditions. Let us stop here for a moment. Equation (21) strongly suggests to introduce real and imaginary parts of Π corresponding to both internal rotations and boosts; i.e. we set L ∝ Π + Π¯, and iK ∝ Π − Π¯. Dropping all decorating indices the reality conditions turn into C ∝K+βL=0. (22) Hence C = 0 though formal at this level takes the basic form of the linear [5, 19] simplicity constraints of spin foam gravity [4]. Within section 4.1 this relation will become more explicit. Perez and Rezende [20] have recovered the very same constraints from a more general setting, i.e. the Holst action augmented with all possible topological invariants together with a term containing the cosmological constant. Complex Variables & Reality Conditions for Holst’s Action 7 In order to proceed let us define the following smeared quantities (cid:90) reality condition: C a[Vi ]:= C aVi EO=M0, (23) i a i a Σ (cid:90) (cid:16) (cid:17) Gaußconstraint: G [Λi]:= −ΛiD Π a+cc. EO=M0, (24) i a i Σ (cid:90) (cid:16) (cid:17) vector constraint: H [Na]:= NaFi Π b+cc. EO=M0, (25) a ab i Σ (cid:96) 2 (cid:90) (cid:16) β Hamiltonian constraint: H[N]:=− P N · (cid:126) β+i Σ (26) (cid:101) (cid:101) (cid:17) ·(cid:15) lmFi Π aΠ b+cc. EO=M0, i ab l m where the symbol cc. means complex conjugation of everything preceding. Variation of the original action (1, 12) with respect to lapse, shift-vector and Λi immediately reveals that all of these quantities are constrained to vanish provided the equations of motion hold. The physical meaning of both Gaußand vector constraint is immedi- ate. The (Hamiltonian vector field of the) Gaußconstraint G [Λi] generates i SL(2,C) gauge transformations g =exp(Λiτ ) on phase space: Λ i exp(cid:0)X (cid:1)Πa =g−1Πag , (27) Gi[Λi] Λ Λ exp(cid:0)X (cid:1)A =g−1∂ g +g−1A g . (28) Gi[Λi] a Λ a Λ Λ a Λ Thus the momentum Πa ≡ Π a⊗τi transforms under the adjoint represen- i tation of SL(2,C). Except for the reality condition C a = 0 all constraints i introduced above are actually invariant under these gauge transformations. The vector constraint on the other hand generates spatial diffeomorphisms modulo SL(2,C) gauge transformations. In both cases the proof follows the lines of the SU(2) case. The set of constraint equations gives us the first part of the equations of motion, what about the other part, what about time evolution on phase space?InordertostudytheevolutionequationsletusfirstintroduceDirac’s primary Hamiltonian (see for instance [21] and also [1] for the details of the following formalism) H(cid:48) :=C a[Vi ]+G [Λi]+H [Na]+H[N]. (29) i a i a (cid:101) Notice the appearance of a C a[Vi ] term proportional to the reality condi- i a tions (21). We will later comment on the necessity of this additional expression absent in the original action (12). For any functional X of the phase space variables, let us first define its time evolution according to L X ={H(cid:48),X}. (30) t Having introduced the symplectic structure (17) essentially by hand, it is still an open question whether or not Hamiltonian time evolution defined in thesenseof (30)iscompatiblewiththeEuler–Lagrangeequationsofmotion. 8 Wolfgang M. Wieland In order to check this, one first proves equivalence between the evolution equationsderivedfromthevariationprincipleandthefollowingHamiltonian equations (cid:16)β+i (cid:17)(cid:12) (cid:110) β+i (cid:111)(cid:12) L Ai +cc. (cid:12) = H(cid:48), Ai +cc. (cid:12) , (31) iβ t a (cid:12)C=0 iβ a (cid:12)C=0 (cid:0)LtEia(cid:1)(cid:12)(cid:12)C=0 =(cid:8)H(cid:48),Eia(cid:9)(cid:12)(cid:12)C=0. (32) And C = 0 is an abbreviation for C a(p) = 0. Notice that the Euler– i Lagrangian evolution equations do not determine L A and L A¯ indepen- t t dently, but only the linear combination appearing in (31). What about the other linearly independent combination of A and A¯? Calculating the corresponding Poisson bracket reveals (cid:16)β+i (cid:17)(cid:12) 2(cid:96) 2 L Ai +cc. (cid:12) = P Vi + β t a (cid:12)C=0 (cid:126) a (33) (cid:110) β+i (cid:111)(cid:12) + G [Λi]+H [Va]+H[N], Ai +cc. (cid:12) . i a β a (cid:12)C=0 (cid:101) Asamatteroffactthelefthandsideofthisequation,i.e.thetimederivative ofΓi −1Ki doesnotappearinthelistofevolutionequationsderivedfrom a β a the Euler–Lagrangian framework. But still, this quantity is fully determined by all the equations of motion: Variation of the action with respect to the so(1,3) spin connection immediately reveals that torsion is forced to vanish. Butifthereisnotorsiontheevolutionequations(31)and(32)fullydetermine the left hand side of (33). All of this happens in the case of the Lagrangian formulation, but in the Hamiltonian framework the situation is slightly different. The vanishing of torsion emerges only in a secondary step explicitly studied later. Unable to use this constraint in order to calculate the left hand side of (33) we fix it by an additional Lagrangian multiplier Vi . This multiplier determines the a time derivative (33) to some yet unspecified value V a+..., but leaves both i (31) and (32) unchanged. Its value, later found to be V a =0, is derived by i the requirement that all constraints are preserved under the time evolution generatedby(30).Allofthisisnaturallyachievedbythechoiceoftheprimary Hamiltonian (29) introduced above. We have already proven (32) that the Hamiltonian time evolution for E a is perfectly consistent with the equations of motion found from the vari- i ationprinciple,butwhataboutC a,i.e.theotherlinearlyindependentcom- i binationofΠ a andΠ¯ a?ComparisonwiththeLagrangianframeworkreveals i i that the quantity LtCia(cid:12)(cid:12)C=0 =(cid:8)H(cid:48),Cia(cid:9)(cid:12)(cid:12)C=0 EO=M0. (34) is actually constrained to vanish. Which of course is nothing but the state- ment that the reality conditions have to be preserved under time evolution on phase space. This finishes the proof of compatibility between time evolution generated by (30) and the corresponding Euler–Lagrangian evolution equations. Complex Variables & Reality Conditions for Holst’s Action 9 3.2. Dirac consistency of the constraints Inthissectionwecheckifalltheconstraints(23,24,25,25,26)arepreserved under the time evolution defined by (30). As a preliminary step to calculate this one first proves the following list of Poisson brackets: (cid:8)G [Λi],G [Mj](cid:9)=−G (cid:2)[Λ,M]i(cid:3) (35) i j i (cid:8)G [Λi],H [Va](cid:9)=0 (36) i a (cid:8)G [Λi],H[N](cid:9)=0 (37) i (cid:8)H [Ua],H(cid:101)[Vb](cid:9)=−H (cid:2)[U,V]a(cid:3)−G (cid:2)Fi(U,V)(cid:3) (38) a b a i δH[N] (cid:8)H [Va],H[N](cid:9)=−H(cid:2)L N]−G (cid:104) Va(cid:105) (39) a V i δΠ(cid:101)a i (cid:101) (cid:101) (cid:8)H[M],H[N](cid:9)= 4(cid:96)P4 (cid:90) β2 (M∂ N −N∂ M)Π cΠja· (cid:126)2 (β+i)2 c c j (40) Σ (cid:102) (cid:101) (cid:102) (cid:101) (cid:101) (cid:102) ·Fi Π b+cc. ab i Here [U,V] = L V is the Lie bracket between vector fields, [Λ,M]i := U (cid:15)i ΛlMm is the commutator on sl(2,C), and L N = −N2∂ (VaN−1) is lm V a the Lie derivative of the inverse density. Observe t(cid:101)hat the(cid:101)Poisson (cid:101)bracket between to smeared Hamiltonian constraints does not give a linear combination of smeared Gauß, vector and Hamiltonian constraints again. Therefore on the full phase space of complex valued connections and corresponding momenta the algebra generated by G [Λi], H [Na] and H[N] does not close. i a Instead the Poisson bracket between two smeared Hamilt(cid:101)onian constraints vanishes on-shell. This can be seen as follows. Restricting the result of (40) to those parts of phase space where C a =0 holds we find i (cid:12) (cid:12) (cid:8)H[M],H[N](cid:9)(cid:12) =−H (cid:2)E aEjb(M∂ N −N∂ M)(cid:3)(cid:12) . (41) (cid:12) a j b b (cid:12) (cid:102) (cid:101) C=0 (cid:102) (cid:101) (cid:101) (cid:102) C=0 And the right hand side of this equation being proportional to the vector constraint again vanishes on the constraint hyper-surface. Notice also that even though M∂ N is ill defined without choosing a preferred derivative b actingonthed(cid:102)ensi(cid:101)tyweight(notnecessarilythemetriccompatibleone),the antisymmetric part M∂ N −N∂ M perfectly is. This follows from the fact b b that derivatives “acti(cid:102)ng” o(cid:101)n th(cid:101)e de(cid:102)nsity weight cancel. Consistency of C a[Vi ] = 0. Calculating the Poisson bracket generating i a the corresponding time derivative reveals that L+tC1ia((cid:12)(cid:12)DC=0−=D¯(cid:8))H(cid:0)N(cid:48),CbEiaa(cid:9)(cid:12)(cid:12)−C=N0a=E−b(cid:1)21+i(cid:15)i1lm(cid:15)(lΛml(−DΛ¯+l)ED¯m)a(cid:0)+NE bE a(cid:1) (42) 2i b b i i 2 i b b l m =−(cid:15) m(cid:16)1(Λl−Λ¯l)−NbKl −elb∂ N(cid:17)E a+Nηba(cid:101)c∇ e .  il 2i b b m (cid:101) b ic Wherewehavereintroducedthescalar(i.e.the“undensitised” lapse)N =µN and ∇ = 1(D +D¯ )=∂ +[Γ ,·] defines another covariant derivative. I(cid:101)n a 2 a a a a 10 Wolfgang M. Wieland the last line of this equation both the expression within the big bracket and the ∇e term must vanish independently. This can be seen as follows. Notice that the covariant derivative ∇ equals the Levi-Civita derivative modulo a a difference tensor ∆i . Using the fact that ∇ E a = 0 provided both the a a j Gaußconstraint and the reality condition C a = 0 hold, it follows that ∆i i a mustbesymmetric(inthesenseof(cid:15) ∆l Emb =0).ButL C amustvanish, jlm b t i which reveals that 1 (cid:12) − (Λj −Λ¯j)+NaKj +eja∂ N(cid:12) =! 2i a a (cid:12)C,G=0 1 (cid:12) =− N(eicejb−eibejc)∇ e (cid:12) = 2 b ic(cid:12)C,G=0 1 (cid:12) 1 (cid:12) =− N(eicejb−eibejc)(cid:15) ∆l em (cid:12) = N(cid:15) j∆l eib(cid:12) =0. 2 ilm b c(cid:12)C,G=0 2 il b (cid:12)C,G=0 (43) ThereforetheimaginarypartofΛi (describingboostsalongtheflowoftime) is completely fixed by the dynamics of the theory to the value 1(cid:0)Λi−Λ¯i(cid:1)=+NaKi +eia∂ N. (44) 2i a a Since the condition C a = 0 is invariant under internal rotations but does i not remain valid if boosted, this restriction should not surprise us. Comparisonwith(43)revealsthat∇emustvanishtoo.Thereforethere is the additional secondary constraint forcing the difference tensor ∆i to a vanish: ∆i = 1(cid:0)Ai −A¯i (cid:1)−LΓCi [E]=! 0. (45) a 2 a a a LC Where Γi [E]denotestheLevi-Civitaconnectionfunctionallydependingon a E a. This equation is highly non polynomial [1] in E, the equivalent but i technically different version 2T :=De+D¯e=! 0 (46) is much simpler to handle. The latter just sets the spatial part of the four dimensional torsion 2-form to zero. Consistency of T = 0. Here we just mention the final result, i.e: LtTiab(cid:12)(cid:12)C,G,T=0 = 12(cid:8)H(cid:48),(De)iab+cc.(cid:9)(cid:12)(cid:12)C,G,T=0 = (47) = 1(cid:15)i (cid:0)Vl em −Vl em (cid:1)=! 0. 2 lm a b b a And C,G,T = 0 shall remind us that these equations hold provided torsion T vanishes and both reality condition and Gauß constraint are satisfied. We conclude by stating that equation (47) restricts the Lagrangian multiplier Vi to the value a Vi =0 (48) a being proven by an elementary algebraic manipulation of (47).