Hamiltonian Dynamics for Proca's Theories in Five Dimensions with A

RESEARCH Revista Mexicana de F´ısica 61 (2015) 188–195 MAY-JUNE 2015

Hamiltonian dynamics for Proca’s theories in ﬁve dimensions with a compact dimension

A. Escalante and C.L. Pando Lambruschini Instituto de F´ısica Luis Rivera Terrazas, Benemerita´ Universidad Autonoma´ de Puebla, Apartado postal J-48 72570 Puebla. Pue., Mexico,´ e-mail: [email protected] P. Cavildo Instituto de F´ısica Luis Rivera Terrazas, Benemerita´ Universidad Autonoma´ de Puebla, Apartado postal J-48 72570 Puebla. Pue., Mexico,´ Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad Autonoma´ de Puebla, Apartado postal 1152, 72001 Puebla, Pue., Mexico.´ Received 5 September 2014; accepted 19 February 2015

The canonical analysis of Proca’s theory in five dimensions with a compact dimension is performed. From the Proca five dimensional action, 1 we perform the compactification process on a S /Z2 orbifold, then, we analyze the four dimensional effective action that emerges from the compactification process. We report the extended action, the extended Hamiltonian and we carry out the counting of physical degrees of freedom of the theory. We show that the theory with the compact dimension continues laking of first class constraints. In fact, the final theory is not a gauge theory and describes the propagation of a massive vector field plus a tower of massive KK-excitations and one massive scalar field. Finally, we develop the analysis of a 5D BF -like theory with a Proca mass term, we perform the compactification process on a 1 S /Z2 orbifold and we find all the constraints of the effective theory, we also carry out the counting of physical degrees of freedom; with these results, we show that the theory is not topological but reducible in the first class constraints.

Keywords: Proca theory; extra dimensions; Hamiltonian dynamics.

PACS: 98.80.-k; 98.80.Cq

1. Introduction compactiﬁcation process is a crucial step in the construction of models with extra dimensions [9, 10].

Nowadays, the introduction of extra dimensions in field theo- By taking into account the ideas explained above, in this ries have allowed a new way of looking at several problems in paper we perform the Hamiltonian analysis of Proca’s theory theoretical physics. It is well-know that the first proposal in- in 5D with a compact dimension. It is well-known that four troducing extra dimensions beyond the fourth dimension was dimensional Proca’s theory is not a gauge theory, the the- considered around 1920’s, when Kaluza and Klein (KK) tried ory describes a massive vector field and the physical degrees to unify electromagnetism with Einstein’s gravity by propos- of freedom are three, this is, the addition of a mass term to ing a theory in 5D where the fifth dimension is compactified Maxwell theory breaks the gauge invariance of the theory and on a circle S1 of radius R, and the electromagnetic field is adds one physical degree of freedom to electromagnetic de- contained as a component of the metric tensor [1]. The study grees of freedom [11, 12]. Hence, in the present work, we of models involving extra dimensions has an important ac- study the effects of the compact extra dimension on a 5D tivity in order to explain and solve some fundamental issues Proca’s theory. Our study is based on a pure Dirac’s analysis, found in theoretical physics, such as, the problem of mass this means that we will develop all Dirac’s steps in order to hierarchy, the explanation of dark energy, dark matter and obtain a complete canonical analysis of the theory [13–16]. inflation etc., [2]. Moreover, extra dimensions become also We shall find the full constraints of the theory; we need to re- important in theories of grand unification trying of incorpo- member that the correct identification of the constraints will rating gravity and gauge interactions in a theory of everyt- play a key role to make progress in the study of the quanti- ing. In this respect, it is well known that extra dimensions zation aspects. We also report the extended Hamiltonian and have a fundamental role in the developing of string theory, we will determine the full Lagrange multipliers in order to since all versions of the theory are formulated in a space- construct the extended action. It is important to comment, time of more than four dimensions [3, 4]. For some time, that usually from consistency of the constraints it is not pos- however, it was conventional to assume that in string theory sible to determine the complete set of Lagrange multipliers, such extra dimensions were compactified to complex man- so a pure Dirac’s analysis becomes useful for determining all ifolds of small sizes about the order of the Planck length, them. Finally, we study a 5D BF -like theory with a massive −33 1 `P ∼ 10 cm [4, 5], or they could be even of lower size term. We perform the compactification process on a S /Z2 independently of the Plank Length [6–8]; in this respect, the orbifold and we obtain a 4D effective action, then we study HAMILTONIAN DYNAMICS FOR PROCA’S THEORIES IN FIVE DIMENSIONS WITH A COMPACT DIMENSION 189 the action developing the Hamiltonian analysis, we report the thus, the fields can be expanded in terms of Fourier series as full constraints program and we show that the 4D effective follows theory is a reducible system in the first class constraints. All 1 X∞ 1 ³ny ´ these ideas will be clarified along the paper. A (x, y) = √ A(0)(x) + √ A(n)(x) cos , The paper is organized as follows: In Sec. 1, we ana- µ µ µ R 2πR n=1 πR lyze a Proca’s theory in 5D, after performing the compacti- X∞ ³ ´ fication process on a S1/Z orbifold we obtain a 4D effec- 1 (n) ny 2 A (x, y) = √ A (x) sin . (3) tive Lagrangian. We perform the Hamiltonian analysis and 5 5 R n=1 πR we obtain the complete constraints of the theory, the full La- grange multipliers associated to the second class constraints We shall suppose that the number of KK-modes is k, and we and we construct the extended action. In addition, we carry- will take the limit k → ∞ at the end of the calculations, thus, out the counting of physical degrees of freedom. Addition- n = 1, 2, 3...k − 1. Moreover, by expanding the five dimen- ally, in Sec. 2, we perform the Hamiltonian analysis for a sional Lagrangian L5p, takes the following form 5D BF -like theory with a Proca mass term; we also perform the compactification process on a S1/Z orbifold, we find 1 m2 2 L (x, y)=− F (x, y)F µν (x, y)+ A (x, y)Aµ(x, y) the complete set of constraints and then the effective action is 5p 4 µν 2 µ obtained. We show that for this theory there exist reducibility 1 m2 conditions among the first class constraints associated with − F (x, y)F µ5(x, y) + A (x, y)A5(x, y). (4) 2 µ5 2 5 the zero mode and the excited modes. Finally we carry out the counting of physical degrees of freedom. In Sec. 3, we Now, by inserting (3) into (4), and after performing the in- present some remarks and prospects. tegration on the y coordinate, we obtain the following 4D effective Lagrangian 2. Hamiltonian Dynamics for Proca theory in Z ( 1 m2 five dimensions with a compact dimension L (x) = − F (0)(x)F µν (x) + A(0)(x)Aµ (x) p 4 µν (0) 2 µ (0) In this section, we shall perform the canonical analysis for · X∞ 1 m2 Proca’s theory in five dimensions, then we will perform the + − F (n)(x)F µν (x) + A(n)(x)Aµ (x) S1/Z 4 µν (n) 2 µ (n) compactfication process on a 2 orbifold. For this aim, n=1 the notation that we will use along the paper is the fol- 2 ³ ´ lowing: the capital latin indices M,N run over 0, 1, 2, 3, 5, m (n) 5 1 (n) m (n) + A5 (x)A(n)(x) − ∂µA5 (x) + Aµ (x) here as usual, 5 label the compact dimension. The M,N 2 2 R ) indices can be raised and lowered by the five-dimensional ³ ´ ¸ (n) m Minkowski metric η = (−1, 1, 1, 1, 1); y will represent × ∂µA (x) + Aµ(n)(x) dx4. (5) MN 5 R the coordinate in the compact dimension, xµ the coordinates that label the points of the four-dimensional manifold M 4 The terms given by and µ, ν = 0, 1, 2, 3 are spacetime indices; furthermore we 1 µ ¶ µ ¶ will suppose that the compact dimension is a S /Z2 orbifold 1 m2 whose radius is R. − F (0)(x)F µν (x) + A(0)(x)Aµ (x) 4 µν (0) 2 µ (0) The Proca Lagrangian in five dimensions without sources is given by are called the zero mode of the Proca theory [11,12], and the 1 following terms are identified as a tower of KK-modes [17]. L = − F (x, y)F MN (x, y) 5p 4 MN In order to perform the Hamiltonian analysis, we observe that the theory is singular. In fact, it is straightforward to m2 + A (x, y)AM (x, y), (1) observe that the Hessian for the zero mode given by 2 M ∂2L where FMN (x, y) = ∂M AN (x, y) − ∂N AM (x, y). W ρλ(0) = p Because of the compactification of the fifth dimension (0) (0) ∂(∂0Aρ )∂(∂0Aλ ) will be carry out on a S1/Z orbifold of radius R, such a 2 " (0) choose imposes parity and periodic conditions on the gauge gµαgνβ ∂F = − (δ0 δλ − δ0δλ) αβ fields given by µ ν ν µ (0) 4 ∂(∂0Aρ ) # (0) AM (x, y) = AM (x, y + 2πR), ¡ ¢ ∂F + δ0 δλ − δ0 δλ µν α β β α (0) Aµ(x, y) = Aµ(x, −y), ∂(∂0Aρ ) ρ0 λ0 ρλ 00 ρ0 λ0 ρλ A5(x, y) = −A5(x, −y), (2) = g g − g g = g g + g ,

Rev. Mex. Fis. 61 (2015) 188–195 190 A. ESCALANTE, C.L. PANDO LAMBRUSCHINI, AND P. CAVILDO

(0) (0) (n) has detW = 0, rank= 3 and one null vector. Furthermore, where λ1 y λ1 are Lagrange multipliers enforcing the con- the Hessian of the KK-modes has the following form straints. ∂2L Therefore, from the consistency of the constraints we find W HL(l) = p (l) (l) that ∂(∂0AH )∂(∂0AL ) H0 L0 HL 00 00 L H = g g − g g − g δ5 δ5 ˙1 1 φ(0)(x) = {φ(0),H1} H0 L0 HL L H = g g + g + δ5 δ5 , i 2 0 = ∂iπ(0)(x) + m A(0)(x) ≈ 0, and has detW (l) = 0, rank= 4k − 4 and k − 1 null vectors. Thus, a pure Dirac’s method calls the definition of the and i 5 canonical momenta (π(0), π(n), π(n)) to the dynamical vari- (0) (n) (n) n o ables (Aµ ,Aµ ,A5 ) given by ˙1 1 φ(n)(x) = φ(n),H1 i i (0) i(0) π = −∂ A + ∂0A , (6) (0) 0 i 2 0 n 5 = ∂iπ(n)(x) + m A(n)(x) + π(n)(x) ≈ 0. i i (n) i(n) R π(n) = −∂ A0 + ∂0A , (7) (n) n (n) In this manner, there are the following secondary constraints π5 = ∂ A + A . (8) (n) 0 5 R 0 From the null vectors we identify the following k primary 2 i 2 0 φ (x) = ∂iπ (x) + m A (x) ≈ 0, (13) constraints (0) (0) (0) 2 i n 5 2 0 1 0 φ (x) = ∂iπ (x) + π (x) + m A (x) ≈ 0. (14) φ(0) = π(0) ≈ 0, (9) (n) (n) R (n) (n) φ1 = π0 ≈ 0. (10) (n) (n) On the other hand, concistency of secondary constraints im- Hence, the canonical Hamiltonian is given by plies that Z " (0) i Hc = − A0 (x)∂iπ(0)(x) ˙2 2 i 2 (0) φ(0)(x) = m ∂iA(0)(x) − m λ1 (x) ≈ 0, 1 1 + πi (x)π(0)(x) + F (0)(x)F ij (x) hence, 2 (0) i 4 ij (0) ∞ µ 2 X (0) m (0) µ (n) i i − A (x)A (x) + − A (x)∂ π (x) λ1 (x) ≈ ∂iA(0)(x), (15) 2 µ (0) 0 i (n) n=1 1 1 and + πi (x)π(n)(x) + π5 (x)π5 (x) 2 (n) i 2 (n) (n) 2n ³ n ´ n 1 φ˙2 (x)=m2∂ Ai (x)− ∂ ∂iA(n)(x) + Ai(n)(x) − π5 (x)A(n)(x) + F (n)(x)F ij (x) (n) i (n) R i 5 R R (n) 0 4 ij (n) m2n m2 m2 − m2λ(n)(x) + A5 (x) ≈ 0, − A(n)(x)Aµ (x) − A(n)(x)A5 (x) 1 R (n) 2 µ (n) 2 5 (n) 1 ³ n ´ + ∂ A(n)(x) + A(n)(x) thus 2 i 5 R i # ¡ ³ ´ ¶ (n) i(n) 2n i (n) i (n) n i(n) 3 λ1 (x) ≈ ∂iA (x) − 2 ∂i ∂ A5 (x) × ∂ A5 (x) + A (x) d x, (11) m R R n ¢ n + Ai(n)(x) + A5(n)(x). (16) by using the primary constraints, we identify the primary R R Hamiltonian Z (0) 1 3 For this theory there are not third contraints. Hp ≡ Hc + λ (x)φ (x)d x 1 (0) By following with the method, we need to identify the Z X∞ first class and second class constraints. For this step we cal- (n) 1 3 + λ1 (x)φ(n)(x)d x, (12) culate the Poisson brackets among the primary and secondary n=1 constraints for the zero mode, obtaining

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Ã ! ³ ´ {φ1 (x), φ1 (z)}{φ1 (x), φ2 (z)} 0αβ(0) (0) (0) (0) (0) W = 2 1 2 2 {φ(0)(x), φ(0)(z)}{φ(0)(x), φ(0)(z)} Ã ! 0 i 2 0 0 {π(0)(x), ∂iπ(0)(z) + m A(0)(z)} = i 2 0 0 {∂iπ(0)(x) + m A(0)(x), π(0)(z)} 0 Ã ! Ã ! 0 m2δ3(x − z) 0 1 = = m2 δ3(x − z). −m2δ3(x − z) 0 −1 0

0 The matrix W αβ (0) has a rank= 2, therefore the constraints found for the zero mode are of second class. In the Therefore the Lagrange multipliers are given by [16] same way, the Poisson brackets among the constraints related for the KK-modes, we find −1 ρ Ã ! λβ = −Cβρ h . (19) ³ ´ 1 1 1 2 0 {φ (x), φ (z)}{φ (x), φ (z)} W αβ(n) = (n) (n) (n) (n) 2 1 2 2 In this manner, for the zero mode we obtain {φ(n)(x), φ(n)(z)}{φ(n)(x), φ(n)(z)} Ã ! n o n o 0 2 0   0 {π(n)(x), m A(n)(z)} ³ ´ φ1 (x), φ1 (z) φ1 (x), φ2 (z) = αβ(0)  (0) (0) (0) (0)  {m2A0 (x), π0 (z)} 0 C = n o n o (n) (n) φ2 (x), φ1 (z) φ2 (x), φ2 (z) Ã ! (0) (0) (0) (0) 0 m2δ3(x − z) µ ¶ = 2 0 1 3 2 3 = m δ (x − z), −m δ (x − z) 0 −1 0 Ã ! 0 1 = m2 δ3(x − z), and its inverse is given by −1 0 µ ¶ 0 1 0 −1 the matrix W αβ(n) has a rank= 2(k−1), thus, the constraints C(0)−1 = δ3(x − z), αβ 2 1 0 associated to the KK-modes are of second class as well. In m this manner, the counting of physical degrees of freedom is (0) given in the following way: there are 10k − 2 dynamical so, for the constraints associated with the zero modes, h is variables, and 2k − 2 + 2 = 2k second class constraints, obtained from there are not first class constraints. Therefore, the number  n o  1 of physical degrees of freedom is 4k − 1. It is important to φ (x),Hc h(0) =  n (0) o  note that for k = 1 we obtain the three degrees of freedom 2 φ (x),Hc of a four dimensional Proca’s theory identified with the zero (0) Ã ! mode [11, 12]. i 2 0 ∂iπ (x) + m A (x) = (0) (0) , Furthermore, we found 2k second class constraints, m2∂ Ai (x) which implies that 2k Lagrange multipliers must be fixed; i (0) however, we have found only k given in the expressions (15) and (16). Hence, let us to find the full Lagrange multipliers; therefore, by using (19) we can determine the Lagrange mul- it is important to comment that usually the Lagrange multi- tipliers associated with the zero modes, pliers can be determined by means consistency conditions, Ã ! µ ¶ λ(0)(x) 1 0 −1 however, for the theory under study this is not possible be- 1 ≈ − (0) 2 1 0 cause some of them did not emerge from the consistency of λ2 (x) m the constraints. In order to construct the extended action and Ã ! i 2 0 the extended Hamiltonian, we need identify all the Lagrange ∂iπ(0)(x) + m A(0)(x) 3 × 2 i δ (x − z), multipliers, hence, for this important step, we can find the m ∂iA(0)(x) Lagrange multipliers by means of © ª ˙α α α β φ (x) = {φ ,Hc} + λβ φ , φ ≈ 0, (17) thus β where φ© are allª the constrains found. In fact, by calling (0) i(0) αβ α β α α λ1 (x) ≈ ∂iA (x), (20) C = φ , φ and h = {φ ,Hc}, we rewrite (17) as α αβ (0) 1 i(0) 0(0) h + C λβ ≈ 0. (18) λ (x) ≈ − ∂ π (x) − A (x). (21) 2 m2 i

Rev. Mex. Fis. 61 (2015) 188–195 192 A. ESCALANTE, C.L. PANDO LAMBRUSCHINI, AND P. CAVILDO

In the same way, for the KK-modes we observe that n o n o ³ ´ 1 1 1 2 µ ¶ φ(n)(x), φ(n)(z) φ(n)(x), φ(n)(z) 0 1 Cαβ(n) = n o n o = m2 δ3(x − z), 2 1 2 2 −1 0 φ(n)(x), φ(n)(z) φ(n)(x), φ(n)(z) where the inverse is µ ¶ 1 0 −1 C(n)−1 = δ3(x − z), αβ m2 1 0 so, for the constraints associated with the excited modes, h(n) are given by  n o  Ã ! 1 i 2 0 n 5 φ (x),H ∂iπ (x) + m A (x) + π (x) (n)  n (n) co  (n) ³ (0) R (n)´ h = = 2 i(n) 2n i (n) n i(n) nm2 5(n) , 2 m ∂iA (x) − ∂i ∂ A (x) + A (x) + A (x) φ(n)(x),Hc R 5 R R additionally by using (19) we obtain

Ã ! µ ¶ Ã i 2 0(0) n 5 ! (n) ∂iπ (x) + m A (x) + π (x) 3 λ1 (x) 0 −1 (n) ³ R (n)´ δ (x − z) = − 2 , (n) 1 0 2 i(n) 2n i (n) n i(n) nm 5(n) 2 λ2 (x) m ∂iA (x) − R ∂i ∂ A5 (x) + R A (x) + R A (x) m thus, the 2k Lagrange multipliers associated for the second class constraints of the KK-modes read

(n) i(n) 2n 1 0 λ (x) = ∂ A (x) − ∂ χ(n)(x) ≡ π(n)(x) ≈ 0, 1 i m2R i ³ ´ n n n χ2 (x) ≡ ∂ πi (x) + m2A0 (x) + π5 (x) ≈ 0, × ∂iA5(n)(x) + Ai(n)(x) + A5(n)(x), (22) (n) i (n) (0) R (n) R R (n) 1 i now, by using the second class constraints and the Lagrange λ2 (x) = − ∂iπ(n)(x) m2 multipliers found for the zero mode and the excited modes, n + A(n)(x) − π5 (x). (23) the extended action has the following expression 0 m2R (n) Z Therefore, we have seen that by using a pure Dirac’s method n S [A, π, v¯] = A˙ (0)πµ + A(0)(x)∂ πi (x) we were able to identify all Lagrange multipliers of the the- E µ (0) 0 i (0) ory. We have commented above that Lagrange multipliers are 1 (0) 1 (0) essential in order to construct the extended action and then − πi (x)π (x) − F (x)F ij (x) 2 (0) i 4 ij (0) identify from it the extended Hamiltonian. In fact, the im- m2 portance for finding the extended action is that it is defined + A(0)(x)Aµ (x) − v¯(0)χj on the full phase space, therefore, if there are first class con- 2 µ (0) j (0) straints, then the extended action is full gauge invariant. On X∞ h ˙ (n) µ ˙ (n) 5 (n) i the other hand, from the extended Hamiltonian, we are able to + Aµ π(n) + A5 πn + A0 (x)∂iπ(n)(x) obtain the equations of motion which are mathematical dif- n=1 ferent from Euler-Lagrange equations, but the difference is 1 1 − πi (x)π(n)(x) − π5 (x)π5 (x) unphysical. It is interesting to point out that in [11, 12] the 2 (n) i 2 (n) (n) complete Lagrange multipliers were not reported, thus, our n 5 (n) 1 (n) ij approach extend the results reported in those works. + π(n)(x)A0 (x) − Fij (x)F(n)(x) (0)−1 (n)−1 R 4 Furtheremore, by using the matrix Cαβ and Cαβ it 2 2 is straightforward to calculate the Dirac brackets of the the- m (n) µ m (n) 5 + Aµ (x)A(n)(x) + A5 (x)A(n)(x) ory, thus, with the results of this paper we have a complete 2 2 ³ ´ hamiltonian description of the system. By using all our re- 1 (n) n (n) − ∂ A (x) + A (x) sults, we will identify the extended action, for this aim, we 2 i 5 R i ³ ´ io write the second class constraints as i (n) n i(n) (n) j 4 ∂ A5 (x) + A (x) − v¯j χ(n) d x, (24) 1 0 R χ(0)(x) ≡ π(0)(x) ≈ 0, 2 i 2 0 (0) (n) χ(0)(x) ≡ ∂iπ(0)(x) + m A(0)(x) ≈ 0, where v¯j and v¯j are Lagrange multipliers enforcing the second class constraints. From the extended action, we are able to identify the extended Hamiltonian given by

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MN NM here B = −B is an antisymmetric field, and AM Z ½ 1 is the connexion. The Hamiltonian analysis of the BF - H = A(0)(x)∂ πi (x) − πi (x)π(0)(x) like term without a compact dimension has been developed E 0 i (0) 2 (0) i in [18], the theory is devoid of physical degrees of freedom, 1 m2 the first class constraints present reducibility conditions and − F (0)(x)F ij (x) + A(0)(x)Aµ (x) 4 ij (0) 2 µ (0) the extended Hamiltonian is a linear combination of first class constrains. Hence, it is an interesting exercise to perform the X∞ h 1 + A(n)(x)∂ πi (x) − πi (x)π(n)(x) analysis of the action (26) in the context of extra dimensions. 0 i (n) 2 (n) i n=1 We expect that the massive term gives physical degrees of 1 n freedom to the full action. − π5 (x)π5 (x) + π5 (x)A(n)(x) 2 (n) (n) R (n) 0 We shall resume the complete Hamiltonian analysis 2 of (26); for this aim, we perform the 4 + 1 decomposition, 1 (n) ij m (n) µ − Fij (x)F(n)(x) + Aµ (x)A(n)(x) and then we will carry out the compactification process on 4 2 1 a S /Z2 orbifold in order to obtain the following effective m2 + A(n)(x)A5 (x) Lagrangian, 2 5 (n) ³ ´ " 1 (n) n (n) 2 X∞ − ∂ A (x) + A (x) µν m µ µν 2 i 5 R i L = B F (0) − A(0)A + B F (n) (0) µν 4 µ (0) (n) µν ³ ´ io n=1 i (n) n i(n) 3 × ∂ A5 (x) + A (x) d x. (25) # R m2 ³ n ´ − A(n)Aµ + 2Bµ5 ∂ A(n) + An . (27) It is worth to comment that there are not first class con- 4 µ (n) µ 5 R µ straints, therefore there is not gauge symmetry; the system under study is not a gauge theory and we can observe from (5) (n) By performing the Hamiltonian analysis of the action (27) we that the field A is a massive vector field with a mass term µ obtain the following results: there are 6 first class constraints given by (m2 + (n2/R2)) and A5 is a massive scalar field (n) for the zero mode with a mass term given by m2. 1 h i γ(0) = F (0) − ∂ Π(0) − ∂ Π(0) ≈ 0, (28) 3. Hamiltonian Dynamics for a Bf-like theory ij ij 2 i 0j j 0i plus a Proca term in five dimensions with a 0 (0) (0) γ ij = Π ≈ 0, (29) compact dimension ij ³ ´ The study of topological field theories is a topic of great inter- here, Π(n) , ΠM are canonically conjugate to ³ MN´ (n) est in physics. The importance to study these theories arises MN (n) because they have a close relation- ship with field theories as B(n) ,AM respectively. Furthermore, these constraints Yang-Mills and General Relativity [13–15, 18]. In fact, for are not independent because there exist the reducibility con- ijk (0) the former we can cite to Martellini’s model; this model conditions given by ∂i² γjk = 0; thus, there are [6 − 1] = 5 sists in expressing Yang Mills theory as a BF -like theory, and independent first class constraints for the zero mode. More- the BF first-order formulation is equivalent (on shell) to the over, there are 8 second class constraints usual (second-order) formulation. In fact, both formulations of the theory possess the same perturbative quantum proper- m2 χ = ∂ Πi − A ≈ 0, (30) ties [22]. On the other hand, with respect General Relativity (0) i (0) 2 0(0) we can cite the Mcdowell-Mansouri formulation; this formu- χi = Πi − 2B0i ≈ 0, (31) lation consist in breaking down the group symmetry of a BF (0) (0) (0) theory from SO(5) to SO(4), hence, it is obtained the Pala- 0 0 χ(0) = Π(0) ≈ 0, (32) tini action plus the sum of the second Chern and Euler topo- 0i 0i logical invariants [23], and since these topological classes χ(0) = Π(0) ≈ 0, (33) have trivial local variations that do not contribute classically to the dynamics, one obtains essentially General Relativity. thus, with that information we carry out the counting the In this manner, the study of BF formulations becomes rele- physical degrees of freedom for the zero mode, we find that vant, and we will study in the context of extra dimensions to there is one physical degree of freedom. In fact, the mas- an abelian BF -like theory with a Proca mass term. In this sive term adds that degree of freedom to the theory, just like manner, we shall analyze the following action Proca’s term to Maxwell theory. Z µ 2 ¶ MN m M 5 On the other hand, for the exited modes there are 12k−12 S[A, B] = B FMN − AM A dx , (26) first class constraints given by M 4

Rev. Mex. Fis. 61 (2015) 188–195 194 A. ESCALANTE, C.L. PANDO LAMBRUSCHINI, AND P. CAVILDO

describes the propagation of a massive vector field associated n h n i with the zero mode plus a tower of excited massive vector γ(n)=∂ A(n)+ A(n)− ∂ Π(n)+ Π(n) ≈ 0, (34) i i 5 R i i 05 2R 0i fields and a massive scalar field. Furthermore, we carry out 1 h i the counting of physical degrees of freedom, in particular, our γ(n) = F (n) − ∂ Π(n) − ∂ Π(n) ≈ 0, (35) ij ij 2 i 0j j 0i results reproduce those ones known for Proca’s theory without a compact dimension. Finally, in order to construct the 0 (n) (n) γ ij = Πij ≈ 0, (36) extended action, we have identified the complete set of La-

(n) (n) grange multipliers; we observed that usually Lagrange multi- γi5 = Πi5 ≈ 0, (37) pliers emerge from consistency conditions of the constraints. however, also these constraints are not independent because However, if the Lagrange multipliers are mixed, then it is dif- there exist the following reducibility conditions; there are ficult identify them. In those cases, it is necessary to perform ijk (n) a pure Dirac’s analysis as was developed in this paper, thus, k − 1 conditions given by ² ∂iγjk = 0, and 3(k − 1) con- (n) (n) n (n) all Lagrange multipliers can be determined. ditions given by ∂iγj − ∂jγi − R γij = 0. Hence, there are [(12k − 12) − (4k − 4)] = 8k − 8 independent first class On the other hand, we developed the Hamiltonian analy- BF constraints. Furthermore, there are 10k − 10 second class sis of a 5D -like theory with a massive Proca term. From constraints our analysis, we conclude that the theory is not topological anymore. In fact, the massive term breakdown the topolog- i i 0i χ(n) = π(n) − 2B(n) ≈ 0, (38) ical structure of the BF -like term. The theory is reducible, 0 0 it present first and second class constraints and we used this χ(n) = π(n) ≈ 0 (39) fact in order to carry out the counting of physical degrees of (n) (n) freedom. The physical effect of the massive term, is that it χ = Π ≈ 0, (40) 0i 0i adds degrees of freedom to the topological BF -like term just (n) (n) like Proca’s term adds degrees of freedom to Maxwell theory, χ05 = Π05 ≈ 0, (41) in addition, the massive term did not allowed the presence of 5 5 05 χ(n) = Π(n) − B(0) ≈ 0, (42) pseudo-Goldstone bosons just like is present in Maxwell or 2 Stueckelberg theory [24]. Hence, we have in this work all the i n 5 m 0 χ(n) = ∂iπ(n) + π(n) + A(n) ≈ 0. (43) necessary tools for studying the quantization aspects of the R 2 theories annalized along this paper. It is worth to comment In this manner, by performing the counting of physical de- that our results can be extended to models that generalize the grees of freedom we find that there are 2k − 2 physical de- dynamics of Yang-Mills theory, as for instance the following grees of freedom for the excited modes. So, for the full the- Lagrangian [19] ory, zero modes plus KK-modes, there are 2k − 1 physical degrees of freedom. Therefore, the theory present reducibil- 1 L = ²µνρσB F a ity conditions among the first class constraints of the zero 4 aµν ρσ mode and there are reducibility in the first class constraints of 2 e aµν N µνρσ a the KK-modes. We observe from the first class constraints − Baµν B + ² Faµν Fρσ. (44) (n) 2 4 4 that the field Aµ has a mass term given by m and is not (n) a gauge field. On the other hand, Bij is a massless gauge here, a are indices of SU(N) group. Such generalization pro- field. vide a generalized QCD theory. In fact, it is claimed in [19] that the analysis of this kind of Lagrangians is mandatory for 4. Conclusions and Prospects studying the dynamics of the gluons with further interactions, in addition, that QCD generalization could be amenable to In this paper, the Hamiltonian analysis for a 5D Proca’s the- experimental test. In this respect, our work can be useful for ory in the context of extra dimensions has been developed. In studying that model within the extra dimensions context. oder to obtain a 4D effective theory, we performed the com- 1 pactification process on a S /Z2 orbifold. From the analysis of the effective action, we obtained the complete set of con- Acknowledgments straints, the full Lagrange multipliers and the extended action was found. From our results we conclude that the theory is This work was supported by Sistema Nacional de Inves- not a gauge theory, namely, there are only second class con- tigadores Mexico.´ The authors want to thank R. Cartas- straints. Thus, 5D Proca’s theory with a compact dimension, Fuentevilla for reading the manuscript.

Rev. Mex. Fis. 61 (2015) 188–195 HAMILTONIAN DYNAMICS FOR PROCA’S THEORIES IN FIVE DIMENSIONS WITH A COMPACT DIMENSION 195

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