Hamilton-Jacobi Analysis of the Four Dimensional BF Model With

arXiv:1702.00863v1 [hep-th] 2 Feb 2017 rvt e [8]. see gravity l,i sipratt nlz h osritsrcue sal,th Usually, structure. constraint the analyze to important is it els, 2,[3]. [2], ffedm hsfc si codnewt t rvttoa interpr gravitational freedom. its of with degrees accordance no in have them is dimensions of fact lower all This feature: common freedom. a of have models Those gravity. quantum 4 n h rie-trdbsvmdl[] nbt ae hybegin they cases both in [5], BF model Freidel-Starodubtsev the and [4] he-iesoa rvt 7 ihaPolynomial can a Freidel-Starodubsetv with the [7] of gravity construction three-dimensional similar a that shown cintr hc rasteoiia ymtyo the of symmetry int by original shown the is Freidel-S breaks this The which [6], constraints. term gravity simplicity MacDowell-Mansouri action the imposed to is equivalent it is that such field tional spet fltrtr eae oteapiaino ehiussc a such techniques of application act the Einsten-Hilbert to to the related functions of literature correlation one of its plenty the is than with the easier along topological be metric would are quantization space-time theories the Those on [1]. depend gravity with relationship close h motneo h akrudFed( Field Background the of importance The Introduction 1 aitnJcb nlsso h ordimensional four the of analysis Hamilton-Jacobi 3 2 1 BF nodrt dniytetu ere ffedmaddnmclva dynamical and freedom of degrees true the identify to order In oe dimensional Lower nfu iesos hr r w important two are there dimensions, four In valcarcel.fl[email protected] [email protected] [email protected] oe lsacsooia em ntePeasiter ti introdu is it theory Plebanski the In term. cosmological a plus model qain fmto n bantegnrtr fteguean gauge the Keywords differ of generators fundamental the the obtain From and motion space. of phase equations int the reduced guarantee the that identify Hamiltonians involutive of set plete esoa akrudFed( Field Background mensional hoisa ehdt aesm nihsaotteqatmbeha quantum the about insights some have to method a as theories nti okw efr h aitnJcb osritanal constraint Hamilton-Jacobi the perform we work this In 1 nvriaeFdrld B,02010SnoAd´,S,Br André, SP, Santo 09210-170 ABC, do Federal Universidade , 2 nttt eFıiaT´rc,UEP-Sa al tt Univ State São Paulo F´ısica - Teórica, de UNESP Instituto osrie ytm,Hmlo-aoifraim Backgr formalism, Hamilton-Jacobi Systems, Constrained : .O o 03-,01690 õPuo P Brazil SP, São Paulo, 01156-970, 70532-2, Box O. P. .B eGracia de B. G. oe ihcsooia term cosmological with model 3 etod Matemática, Computa¸cão de Cogni¸cãoCentro e BF oesaeas odlbrtre oaayelwrdimensional lower analyze to laboratories good also are models BF 1 oe ihcsooia em eoti h com- the obtain We term. cosmological with model ) .M Pimentel M. B. , Abstract BF 1 BF hoisle ntefc htte aea have they that fact the on lies theories ) BF oeso rvt:TePeasitheory Plebanski The gravity: of models 2 .E Valcárcel E. C. , cin o eetrve on review recent a For action. BF gaiiyo h ytmand system the of egrability oe.Rcnl,i a been has it Recently, model. hf transformations. shift d nilw eoe the recover we ential spoeuei aewith made is procedure is pnfa quantization foam spin s sso h ordi- four the of ysis ihafour-dimensional a with o,freape There example. for ion, rpgtszr degrees zero propagates tto ic rvt in gravity since etation eapidfrtoand two for applied be ibe fthe of riables udFedmodel. Field ound re n hyd not do they and ories , n,a rnil,its principle, at and, 3 . aoute model tarodubtsev ouiga inter- an roducing ersity azil iro gravity of vior e naddi- an ced . , BF BF mod- BF the Hamiltonian Dirac formalism [9] (also see [10]). The Hamiltonian analysis of Plebanski theory has been made in [11], the Freidel-Starodubtsev in [12], the two-dimensions Poly- nomial BF in [13] and the BF with cosmological term in [14]. However, there are other methods of constraint analysis, as the Hamilton-Jacobi formalism. The Hamilton-Jacobi formalism presented here follows the approach of Güler [15], which is an extension of Caratheodory’s equivalent Lagrangian method in the calculus of variations [16]. This formalism is characterized by a set of Hamilton-Jacobi differential equations called Hamiltonians. The dynamical evolution of the system is given in terms of a fundamental differential which depends on the time and other linear independent arbitrary parameters related to the involutive Hamiltonians [17], [18], obtained from the Frobenius’ Integrability Condition. The canonical transformations are obtained immediately from this fundamental differential when just the dynamics described by those arbitrary parameters are considered. Furthermore, the gauge transformations are the subgroup of those transformations that leave the lagrangian invariant. On the other hand, although the Dirac’s approach is a very powerful tool to the constraint analysis, it deals with gauge symmetries by the conjecture that they are all generated by the first class constraints of the theory. Unfortunately, there are some examples that contradict it [19]. Therefore, we claim again that the Hamilton-Jacobi formalism is a way to illuminate the canonical origin of the gauge structure of the four dimensional BF theory. This approach was used to study several examples of gauge systems such as topologically massive theories [20], gravity models [21], and the two dimensional and three dimensional BF theories [22], [23]. This formalism were also extended to higher order Lagrangians and Berezinian systems [24]. In this work, we will deal with the constraint analysis of the four dimensional BF theory with a cosmological term. This model is a natural extension of the lower dimensional ones in 1 + 1 [22] and in 2 + 1 [23] dimensions studied under the Hamilton-Jacobi formalism. A careful analysis of its constraint structure and symmetry properties is an excellent laboratory to a future investigation of the gravitational BF models since it can give us insights about the implications of adding a symmetry breaking term in its highly symmetric structure. We argue that a Hamilton-Jacobi analysis can be enriching because it deals with gauge symmetries in a very natural way. The following section will be devoted to explain the main features of the Hamilton-Jacobi formalism. In section 3 we present some general characteristics of the four dimensional BF model. In section 4 we give its constraint analysis and build the generalized brackets. In the section 5 we compute its characteristic equations generated by the fundamental differential and analyze the equivalence with the Lagrangian formalism, as well as obtain the canonical and gauge transformations, and from them we find generators of those transformations. Finally, in section 6 we discuss the results.

2 The Hamilton-Jacobi formalism

In this section we develop the Hamilton-Jacobi formalism for constrained systems, which are defined as the ones whose Lagrangian do not satisfy the Hessian condition. Let us consider a physical system whose Lagrangian has the form L = L(xi, x˙ i, t) where the Latin indices run from 1 to n, which is the dimension of the configuration space. The system is called constrained or singular if it does not satisfy the Hessian condition ∂2L det Wij =6 0 with the matrix Wij given by Wij = ∂x˙ i∂x˙ j . If the Hessian condition is satisfied, the transformation that leads the configuration space to the phase space is invertible. If it ∂L is not, some of the conjugated momenta pi = ∂x˙ i are not invertible on velocities and we

2 are lead to equations of the form Φ(q,p) = 0 which constrains the phase space. Now, if we consider k non-invertible momenta and m = n − k invertible momenta, we have ∂L p − =0, (1) z ∂x˙ z where z = 1, ...k. Then, the above equation deﬁnes the primary constraints of the theory. ∂L By using the deﬁnition Hz ≡ − ∂x˙ z we can rewrite the above equation as ′ Hz ≡ pz + Hz =0. (2)

∂S These constraints are called Hamiltonians. If we deﬁne p0 ≡ ∂t , the Hamilton-Jacobi equation can be written as

′ H0 ≡ p0 + H0 =0. (3)

a z The canonical Hamiltonian function H0 = pax˙ +pzx˙ −L with a =1, ..., m, is independent of the non-invertible velocitiesx ˙ z if the constraints are implemented. The uniﬁed notation is given by

′ Hα ≡ pα + Hα, (4) where α =0, ..., k.. The Cauchy’s method [16] is employed to ﬁnd the characteristic equations related to the above ﬁrst order equations

′ ′ a ∂Hα α a ∂Hα α a α dx = dt , dp = − dt , dS =(padx − Hαdt ). (5) ∂pa ∂xa The differentials written above depend on tα = (t0, tz ≡ xz) independent variables or parameters. The name Hamiltonians used for the constraints is now justified, once that Hz generates flows parameterized by tz in analogy with the temporal evolution generated by H0. From the characteristic equations one can use the Poisson brackets defined on the ex- a α tended phase space (x , t ,pa,pα) to express in a concise form the evolution of any function a α f = f(x , t ,pa,pα):

′ α df = {f,Hα}dt . (6)

This is the fundamental diﬀerential, from where we identify the Hamiltonians as the generators of the dynamical evolution of the phase-space functions. Let us deﬁne the operator δf X [f]= {γI ,H′ } ; γI =(xa, tα,p ,p ), (7) α α δγI a α XI

I ′ where Xα[∗] can be interpreted as k vectors whose 2(n + 1) components are {γ ,Hα}. The fundamental diﬀerential can be expressed in terms of this operator as

α df = Xα[f]dt . (8)

The Frobenius’ Integrability Condition (IC) ensures the system of equations (8) is integrable. The IC can be expressed as

a a a a a a ′ ′ Xα,Xβ (x ,p ) ≡ Xα[x ]Xβ[p ] − Xβ[x ]Xα[p ]= −{Hα,Hβ} =0. (9) 3 The above condition can be generalized to

′ ′ γ ′ {Hα,Hβ} = CαβHγ, (10)

γ where Cαβ are structure coeﬃcients (See [25]). Therefore, the IC ensures that the Hamilto- nians close an involutive algebra. In terms of the fundamental diﬀerential (6), the IC () can be written as

′ ′ ′ β γ ′ β β dHα = {Hα,Hβ}dt = CαβHγ dt =0 . (11)

The Hamiltonians that satisfy the IC are called involutives. However, not all Hamiltonians from a physical systems satisfy this condition identically. Therefore, we must deﬁne new Hamiltonians. ′ Let us suppose we have a set of non-involutive Hamiltonians Ha¯. Then

′ ′ ′ ′ ′ ¯b dHa¯ = {Ha¯,H0}dt + {Ha¯,H¯b}dx . (12)

′ ′ ′ ¯ Once that we impose dHa¯ = 0, we can deﬁne a matrix with components Ma¯b ≡ {Ha¯,Ha¯}. b −1 ′ ′ If this matrix is invertible, we can write dx = −Ma¯¯b {Ha¯,H0}dt, i.e., there is a depen- dence between the parameters related to the non-involutive Hamiltonians. Replacing in the fundamental diﬀerential, we have

′ ′ −1 ′ ′ ¯ dF = [{F,H0}−{F,Ha¯}Ma¯¯b {Hb,H0}]dt. (13) Therefore, we can deﬁne Generalized Brackets (GB) as:

∗ ′ −1 ′ {A, B} ≡{A, B}−{A, Ha¯}(M )a¯¯b{Ha¯, B}, (14) which redefine the dynamic of the constrained system reducing its phase-space, once that ′ ∗ dF = {F,H0} dt. This procedure is the result of the Integrability Condition and, as shown in [17], it allows the possibility that the matrix Mab is non-invertible, or that the system has involutive and non-involutive Hamiltonians as well. The dynamical evolution described by the resulting arbitrary parameters can be understood as canonical transformations, with the involutive Hamiltonians as generators. To I α I understand this, we need to check that the variation δγ = δt Xα[γ ] is generated by α a α α g =1+ δt Xα, also preserves the symplectic structure dx ∧ dpa + dt ∧ dpα + dHα ∧ dt . with fixed dt0. In order to relate canonical transformations with the gauge ones, we need to restrict the study to fixed times dt0 = 0. Then, the transformation on any variable γI is

I I ′ ∗ z γ = {γ ,Hz} δt . (15)

′ ′ ∗ z ′ The Hamiltonians must be involutives, then {Hx,Hy} = CxyHz. However, the IC ensures ′ ′ ∗ 0 ′ z ′ that {Hx,Hy} = CxyH0 + CxyHz. To conciliate these equations we must consider whether 0 ′ 0 Cxy = 0 or H0 = 0. The condition Cxy = 0 is almost never satisﬁed. On the other hand, ′ the condition H0 = 0 constrains the phase-space. Under this assumption, we deﬁne the generator of gauge transformations as

can ′ z G ≡ Hzδt , (16) since δγI = {γI ,Gcan}∗. More details on the role of involutive Hamiltonians in the HJ formalism can be found in [18].

4 3 The four-dimensional BF model with Cosmological Term

Let us consider the gauge group SO(1, 3) acting on a background ﬁeld 2−form B and a connection 1−form A of a four-dimensional manifold M. The generators of the gauge group MIJ = −MJI obey the following algebra:

[MIJ , MKL]= ηILMJK − ηIKMJL + ηJK MIL − ηJLMIK, (17) where I,J = 0, 1, 2, 3 and ηIJ = diag(+, −, −, −). We can also define the 2−form strength field as F = dA + A ∧ A. The action for the BF model is characterized by tr(B ∧ F ), which is gauge invariant due to the properties of the trace and wedge product. In four dimensions we can also add the expression tr(B ∧ B), called cosmological term, which also maintains the invariance properties of the BF model. The action of the BF model with cosmological term is given by β S = tr B ∧ F − B ∧ B , (18) ZM 2 where β is a constant parameter. Once, the B field and connection A acts on the SO(1, 3), we can write 1 A = AIJ M dxµ, B = BIJ M dxµ ∧ dxν . (19) µ IJ 2 µν IJ Furthermore, from the definition of field strength, we obtain

IJ IJ IJ I KJ I KJ Fµν = ∂µAν − ∂ν Aµ + AµK Aν − AνKAµ . (20) Instead of using diﬀerential forms, we use the components of the B and F ﬁelds, we obtain

1 4 µναβ IJ β IJ S = d xε BIJµν F − B BαβIJ , (21) 2 Z αβ 2 µν where εµναβ is the Levi-Civita symbol in M. The Levi-Civita symbol is a totally antisym- 0123 metric quantity and, as a convention, we have ε = 1 and, in Minkowski space ε0123 = −1. The equations of motion (EOM) are

µναβ IJ IJ 0 = ε (Fαβ − βBαβ), (22) µναβ IJ 0 = ε DνBαβ, (23) where Dµ is the component of the covariant derivative Dθ = dθ + [A, θ]. For a 2−form θ, the explicit expression for the covariant derivative is

IJ IJ I KJ J KI Dµθαβ = ∂µθαβ + AµK θαβ − AµK θαβ . (24) The EOM have a direct interpretation: Equation (22) states that the field strength and the background fields are parallel, while equation (23) states that the covariant derivative on the B field, and as a consequence, the covariant derivative of the field strength is zero. IJ IJ Furthermore, on-shell, we can replace the background field Bαβ = Fαβ /β and replace it on the BF action (21), obtaining a Yang-Mills (YM) like four-dimensional action. In [26], the relation between the BF model and the Yang-Mills was studied, while in [27], the BF model with a cosmological term gives exactly the YM theory for the SU(N) group, showing that the BF model can also be understood as a first-order action for the YM field.

5 4 The Hamilton-Jacobi Analysis

In order to perform a Hamilton-Jacobi analysis, we foliate the space-time M = R × M3, being M3 the space at constant time. The Lagrangian density from (21) becomes

ijk IJ ijk IJ ijk IJ IJ L = ε BjkIJ ∂0Ai + ε A0 DiBjkIJ + ε B0iIJ (Fjk − βBjk ), (25) where the lowercase Latin indices go from 1, 2, 3 and denote the space coordinates, while the capital Latin indices represent the internal indices from the SO(1, 3) group. Furthermore, εijk ≡ ε0ijk is the three-dimensional Levi-Civita symbol. IJ IJ IJ IJ The canonical momenta πµ , Πµν conjugated to Aµ and Bµν respectively, are deﬁned by

µ ∂L µν ∂L πIJ ≡ IJ , ΠIJ ≡ IJ . (26) ∂(∂0Aµ ) ∂(∂0Bµν ) From the explicit expression of the Lagrangian, we notice that (25) does not depend on any IJ IJ velocities ∂0Aµ , ∂0Bµν . Therefore they are canonical constraints of the theory. For the other velocities, we have linear expressions that also represent constraints. Furthermore, the canonical Hamiltonian is given by

ijk IJ ijk IJ IJ H0 = −ε A0 DiBjkIJ − ε B0iIJ (Fjk − βBjk ), (27) and the canonical variables satisfy the following Poisson Brackets

IJ ν ν IJ 3 {Aµ (x), πKL(y)} = δµ∆KLδ (x − y), (28) IJ αβ µν IJ 3 {Bµν (x), ΠKL(y)} = δαβ∆KLδ (x − y), (29)

µν 1 µ ν µ ν IJ 1 I J I J where δαβ = 2 (δαδβ − δβ δα) and ∆KL ≡ 2 (δK δL − δLδK). The presence of these anti- symmetrised Kronecker deltas is the result of the antisymmetry of the indices of the gauge group. According to the HJ formalism, we can define π ≡ ∂0S where S is the action. This definition allows us to write all the HJ PDE, or Hamiltonian, as H′ ≡ π + H =0, (30) 0 0 AIJ ≡ πIJ =0, (31) i i ijk AIJ ≡ πIJ − ε BjkIJ =0, (32) µν µν BIJ ≡ ΠIJ =0. (33) The first Hamiltonian H′ is associated with the time parameter t ≡ x0. The Hamiltonians µ IJ AIJ are related to the momenta conjugated to the variable Aµ with correspondent param- IJ µ eters λµ . Finally, the Hamiltonians BIJ , related to the momenta conjugated to the variable IJ IJ Bµν , have correspondent parameters ωµ . These parameters play an important role in the definition of the fundamental differential

df = d3y {f(x), H′(y)}dt + {f(x), Aµ (y)}dλIJ(y)+ {f(x), Bµν(y)}dωIJ(y) , (34) Z IJ µ IJ µν and consequently, in the canonical structure of the theory. The Hamiltonians that have vanishing Poisson brackets with themselves and all the remaining ones are called involutives. Otherwise, we have non-involutive Hamiltonians. i ij From the set of HJ PDE above, we identity that AIJ and BIJ are non-involutive, since i jk ijk RS {AIJ (x), BKL(y)} = −ε ηIRηJS∆KLδ(x − y). (35)

6 with these Hamiltonians we can deﬁne generalized bracket. First, let us build the matrix between these constraints 0 −εijk M ijk (x, y)= η η ∆RS δ(x − y). (36) IJKL εijk 0 IR JS KL

This matrix has inverse, given by

−1 IJKL 1 0 −εijk IR JS KL (M )ijk (x, y)= η η ∆RS δ(x − y). (37) 2 εijk 0

Once that this inverse exists, we can deﬁne the GB. Following equation (14), we obtain the non-vanishing fundamental GB: 1 {A (x), BKL(y)}∗ = − ε ∆KLδ3(x − y), (38) iIJ kl 2 ikl IJ IJ ν ∗ ν IJ 3 {Aµ (x), πKL(y)} = δµ∆KLδ (x − y), (39)

IJ 0j ∗ 1 i IJ 3 {B0 (x), Π (y)} = δ ∆ δ (x − y). (40) i KL 2 j KL

IJ Note that the PB (28) remains unaltered. Furthermore, from (38), we notice that Bij is IJ now proportional to the canonical momenta of the variables Ai , in agreement with the Hamiltonian (32). The GB redeﬁne the dynamics of the system and the fundamental diﬀerential (34) now takes the form:

3 ′ ∗ 0 ∗ IJ 0k ∗ IJ df(x)= d y {f(x), H (y)} dt + {f(x), A (y)} dλ0 (y)+ {f(x), B (y)} dω0 (y) . (41) Z IJ IJ k

0 0k At this point we impose the IC for the remaining Hamiltonians: AIJ (y) and BIJ (y). The 0 0k condition dAIJ (y) = 0 and dBIJ (y) = 0 introduce new Hamiltonians

ijk CIJ ≡ ε DiBjkIJ, (42) iIJ ijk IJ IJ D ≡ ε (Fjk − βBjk ), (43) note that these constraints can be identiﬁed as the EOM (22) and (23). Furthermore, the canonical Hamiltonian (27) can now be written as the linear combination of the Hamiltonians:

IJ IJ k H0 = −A0 CIJ − B0k DIJ . (44)

0 0i IJ iIJ The IC are satisﬁed for the full set of Hamiltonians AIJ , BIJ , C , D . Moreover, they satisfy the following algebra:

{CIJ (x), CKL(y)}∗ = ηILCJK − ηIKCJL + ηJKCIL − ηJLCIK, (45) IJ KL ∗ IL JK IK JL JK IL JL IK {C (x), Dk (y)} = η Dk − η Dk + η Dk − η Dk , (46) and all other brackets between the Hamiltonians strictly zero. We conclude that the Hamil- tonians are involutive.

7 5 Characteristic equations

The fundamental diﬀerential allows us to deﬁne the evolution of any function of the phase- space as a function of the time and the local parameters and it is built with the complete set of involutive Hamiltonians. Let us renamed them as

0 IJ IJ HIJ ≡ AIJ → λ = λ0 , k 0k IJ IJ HIJ ≡ BIJ → ωk = ω0k , IJ GIJ ≡ CIJ → ζ , k k IJ GIJ ≡ DIJ → χk .

IJ IJ IJ IJ We have also introduced the parameters: (λ ,ωk ,ζ , χk ) related to each Hamiltonian. Therefore, the fundamental diﬀerential is given by the linear combination

3 ′ ∗ ∗ IJ k ∗ IJ df(x) = d y {f(x), H (y)} dt + {f(x), HIJ (y)} dλ + {f(x), H (y)} dω Z IJ k + {f(x), Gk }∗dχIJ + {f(x), G }∗dζIJ . (47) IJ k IJ

The characteristic equations are the ones that govern the evolution of the canonical IJ variables of the phase-space. In our case, for the variables Aµ we have

IJ IJ IJ IJ IJ dAi = DiA0 + βB0i dt − Didζ − βdχi , (48) IJ IJ dA0 = dλ , (49)

IJ and for the , Bµν ﬁeld we have

IJ 1 kmn I KJ J KI x IJ dB = ε ε A0 B − A0 B − 2D B0 dt ij 2 kij K mn K mn m n I KJ J KI IJ IJ + − BijKdζ − BijKdζ − Didχj − Djdχi , (50)

IJ 1 IJ dB0 = dω . (51) i 2 i

IJ IJ IJ IJ Notice that the variables A0 and B0i only depend on parameters λ and ωi respectively, enforcing that they only act as Lagrange multipliers. This can also be inferred from the form of the canonical Hamiltonian written in terms of the constraints. The ﬁnal form of the fundamental diﬀerential is given in terms of independent parameters, we can analyze the temporal evolution of the canonical variables independently. For the spatial components, we have

IJ IJ IJ ∂0Ai = DiA0 + βB0µ , (52)

IJ 1 kmn I KJ J KI x IJ ∂0B = ǫ ε A0 B − A0 B − 2D B0 . (53) ij 2 kij K mn K mn m n kij Equation (52) is in agreement with (22), while (53) becomes ε (D0Bµν − 2DjB0k) = 0, which is equivalent to the component µ = k of (23).

8 5.1 Generators of canonical and gauge transformations The canonical transformations of the theory are obtained by setting dt = 0 in the Charac- teristic Equations. Therefore, we have

IJ IJ IJ δAi = −Diδζ − βδχi , (54) IJ IJ δA0 = δλ , (55) IJ I KJ J KI IJ IJ δBij = − BijKδζ − BijKδζ − Diδχj − Djδχi , (56)

IJ 1 IJ δB0 = δω . (57) i 2 i Then, the generator of canonical transformations is the linear combination of the involutive Hamiltonians:

can 3 IJ k IJ k IJ IJ G = d y HIJ δλ + H δω + G δχ + GIJ δζ , (58) Z IJ k IJ k once that

IJ IJ can ∗ IJ IJ can ∗ δAµ = {Aµ ,G } , δBµν = {Bµν ,G } . (59) In order to obtain the gauge transformations, i.e, the set of canonical transformations that leaves the Lagrangian (quasi-)invariant, we need to compute its variation δL induced by the ﬁeld’s canonical transformations and then impose δL = 0. This procedure generates constraints between the local parameters. From (21), we have the variation of Lagrangian

ijk IJ IJ IJ IJ δL = ε [(Fjk − βBjk )δB0iIJ +(F0k − βB0k )δBijIJ ] ijk IJ IJ IJ + ε [2B0iIJ DjδAk + BijIJ (D0δAk − DkδA0 )] (60)

IJ IJ By replacing the expressions for δBµν and δAµ , we will have equations that will relate the four parameters: (λIJ ,ωIJ,ζIJ , χIJ ). A good approach to solve δL = 0 is to consider special cases where some parameters are set to zero. From (54) and (56) we see that educated IJ IJ guesses are choosing δχi = 0 and δζ = 0 independently. First, let us set δζIJ = 0. In this case, the equation for the variation δL = 0 becomes

ijk IJ 1 IJ 1 1 0= ε F δω + D0δχ − βB δω + D0δχ + D δλ , (61) ij 2 kIJ kIJ ij 2 kIJ kIJ β k IJ

1 1 IJ which can be solved for 2 δωkIJ = −D0δχkIJ − β DkδλIJ . The term proportional to Bij IJ becomes zero as well as the term proportional to Fij (up to a boundary term). Let us rename δλIJ = −βδχ0IJ . Therefore, we have

IJ IJ δAµ = −βδχµ , (62) IJ IJ IJ δBµν = − Dµδχν − Dνδχµ , (63) which is a shift translation for the A ﬁeld. Also notice that this translation only appears due to the cosmological term. Moreover, since we are dropping boundary terms this transformations leaves the Lagrangian quasi-invariant. IJ Now, as a second choice let us consider δχi = 0. Replacing in (60) we obtain

ijk 1 IJ I KJ J KI IJ IJ 0= ε (F − βB ) δω + B0 δζ − B0 δζ − B D D0δζ + δλ .(64) ijIJ ijIJ 2 k kK kK ijIJ k 9 IJ IJ IJ I KJ J KI This equation is zero for δλ = −D0δζ and δωk = −2 B0kKδζ − B0kKδζ . This corresponds to the gauge transformation

IJ IJ δAµ = −Dµδζ , (65) IJ I KJ J KI δBµν = − BµνK δζ − BµνK δζ . (66) This transformation does not depend on the parameter β, in contrast with (62). Both, shift and gauge transformations obtained through the Hamilton-Jacobi formalism are in perfect agreement with the results presented in [28].

6 Final remarks

In this work we have used the Hamilton-Jacobi formalism to analyze the BF model with cosmological term. This procedure consisted in finding the complete set of Hamiltonians that generates the dynamical evolution of the system. The Integrability Condition ensures that these hamiltonians are involutives. However, as we see from equation (35), there are Hamiltonians which do not satisfy the IC. These non-involutive Hamiltonians can be elim- inated if we define the generalized brackets which redefines the dynamic of the system, as shown in (38), (39) and (40). Related to each involutive Hamiltonians we have a independent parameter, one of them being the time. An interesting feature of the Hamilton-Jacobi formalism is the fact that all the dynamics of the theory is given in terms of the fundamental differential. Particularly, for the four-dimensional BF model the fundamental differential is given by (47). Since the local parameters are linearly independent we can study the system’s temporal evolution, from where we recover the equations of motion, as well as the canonical transformations (whenever we consider dt = 0). In order to obtain the symmetry transformations we vary the Lagrangian with respect to the fields, substituting its variations induced by the infinitesimal canonical transformations and set δL = 0. This procedure will generate relations between the local arbitrary parameters. Therefore, the generator of gauge transformations is equal to the generator of canonical transformation with dependent parameters. The Hamilton-Jacobi formalism pro- vides a simple method to find the symmetries from the fundamental differential. The gauge and shift generators are obtained directly from the inner structure of the theory. This is an important motivation to use the Hamilton-Jacobi approach in the study of BF theories. Our plan is to extend this work to the Freidel-Starodubtsev BF model, by adding a break invariance term. As shown in [29] and [30], the study of the BF theory plus cosmological term give some of the properties of the BF equivalents theories of gravity.

7 Acknowledgements

We would like to thank M. C. Bertin for reading the article and suggestions. GBG was supported by CNPq. BMP was partially supported by CNPq. CEV was supported by CNPq process 150407/2016-5.

10 References

[1] L. Freidel and S. Speziale, On the relations between gravity and BF theories, SIGMA 8 032 (2012). [2] J.C. Baez, An introduction to spin foam models of quantum gravity and BF theory, Lect. Notes Phys. 543 25 (2000 ). [3] A. Perez, The spin foam approach to quantum gravity, Living Rev. Rel. 16 3 (2013).

[4] J. F. Plebanski, On the separation of Einstenian substructures, J. Math. Phys. 18 2511 (1977). [5] L. Freidel and A. Starodubtsev, Quantum gravity in terms of topological observables ArXiv:hep-th/0501191 (2005). [6] S.W. MacDowell, F. Mansouri, Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38:739,(1977)., Erratum-ibid.38:1376,(1977). [7] R. Paszko, R. da Rocha, Quadratic gravity from BF theory in two and three dimensions, Gen. Rel. Grav. 47, 94 (2015). [8] M. Celada, D. González, M. Montesinos, BF Gravity, Class. Quant. Grav. 33, no 21, 213001 (2016). [9] P.A.M. Dirac, Generalized Hamiltonian dynamics, Can. J.Math. 2 129 (1950); P. A.M. Dirac, The Hamiltonian form of field dynamics, Can. J. Math. 3 1 (1951); P. A.M. Dirac, Lectures on quantum mechanics New York:Yeshiva University (1964). [10] K. Sundermeyer, Constrained Dynamics:With applications to Yang Mills Theory, Gen- eral Relativity, Classical Spin, Dual String model. Berlin; New York: Springer-Verlag (1982); H.J. Rothe and K.D. Rothe, Classical and Quantum Dynamics of Constrained Hamil- tonian Systems Singapore; World scientific (2010); D.M. Gitman, I.V. Tyutin, Quantization of fields with constraints Berlin-Heidelberg; Springer-Verlag (1990). [11] E. Buffenoir, M. Henneaux, K. Noui, Ph. Roche, Hamiltonian Analysis of Plebanski Theory, Class. Quant. Grav. 21 5203 (2004). [12] R. Durka, J. Kowalski-Glikman, Hamiltonian analysis of SO(4, 1) constrained BF theory, Class. Quant. Grav. 27 185008 (2010). [13] C. E. Valcárcel, Constraint Analysis of Two-Dimensional Quadratic Gravity from BF Theory, Gen. Rel. Grav. 49:11 (2017).

[14] A. Escalante, I. Rubalcava-Garc´ıa, A pure Dirac’s canonical analysis for four dimensional BF theories, Int. J. Geom. Methods Mod. Phys. 9 1250053 (2012). [15] Y. G¨uler, On the dynamics of singular, continuous systems, J. Math. Phys. 30 785 (1989). [16] C. Caratheodory, Calculus of Variations and Partial Diferential equations of the First Order 3rd edn (American Mathematical Society) (1999).

11 [17] M.C. Bertin, B.M. Pimentel, C.E. Valc´arcel, Non-involutive constrained systems and the Hamilton-Jacobi formalism, Ann. Phys. 323 3137 (2008).

[18] M.C. Bertin, B.M. Pimentel, C.E. Valc´arcel, Involutive constrained systems and the Hamilton-Jacobi formalism, J. Math. Phys. 55 112901 (2014).

[19] D.M. Gitman, I.V. Tyutin, Symmetries and physical functions in general gauge theory, Int. J. Mod. Phys. A 21 327 (2006).

[20] M.C. Bertin, B.M. Pimentel, C.E. Valc´arcel and G.R. Zambrano, Topologically massive Yang-Mills ﬁeld: A Hamilton-Jacobi approach, J. Math. Phys. 55 042902 (2014).

[21] B.M. Pimentel, P.J. Pompeia, J.F. da Rocha-Neto, The Hamilton-Jacobi approach to teleparalelism, Il Nuovo cimento B 120 981 (2005); M.C. Bertin, B.M. Pimentel, P.J. Pompeia General relativity in two dimensions: A Hamilton-Jacobi constraint analysis, Ann.Phys. 325 2499 (2010).

[22] M.C. Bertin, B.M. Pimentel, C.E. Valc´arcel, Two dimensional background ﬁeld gravity: A Hamilton-Jacobi analysis, J. Math. Phys. 53 102901 (2012).

[23] N.T. Maia, B.M. Pimentel, C.E. Valc´arcel, Three dimensional background ﬁeld gravity: A Hamilton-Jacobi analysis, Class. Quantum Grav. 32 185013 (2015).

[24] B.M. Pimentel, R.G. Teixeira, J.L. Tomazelli, Hamilton-Jacobi approach to Berezinian singular systems, Ann. Phys. 267 75 (1998); M.C. Bertin, B.M. Pimentel, P.J. Pompeia, First order actions: A new view, Mod. Phys. Lett. A 20 2873 (2005); M.C. Bertin, B.M. Pimentel, P.J. Pompeia, Hamilton-Jacobi approach for ﬁrst order actions and theories with higher derivatives, Ann. Phys. 323 527 (2008).

[25] A. S. Mishchenko, A. T. Fomenko, Generalized Liouville method of integration of hamiltonian systems, Funct. Anal. Appl. 12, 113 (1978); A. T. Fomenko, V. V. Troﬁmov, Integrable Systems on Lie Algebras and Symmetric Spaces. Advanced Studies in Contemporary Mathematics Vol. 2 (Gordon and Breach Science Publishers, 1988).

[26] A. S. Cattaneo, P. Cotta-Ramusino, F. Fucito, M.Martellini, M. Rinaldi, A. Tanzini, M. Zeni, Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory, Commun. Math. Phys. 197 571 (1998).

[27] A. Escalante, I. Garc´ıa, Covariant Canonical Method for Yang-Mills Theory Expressed as a Constrained BF -Like Theory, Adv. in Math. Phys. (2012), 852329.

[28] M. Mondragon, M. Montesinos, Covariant canonical formalism for four dimensional BF theory, J. Math. Phys. 47 022301 (2006).

[29] G. T. Horowitz,Exactly Soluble Diﬀeomorphism Invariant Theories, Commun. Math. Phys. 125, 417 (1989).

[30] I. Oda, A Relation between topological quantum ﬁeld theory and the Kodama state, ArXiv: hep-th/0311149.