Redalyc.Hamiltonian Dynamics: Four Dimensional BF-Like Theories with A

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Escalante, A.; Zarate Reyes, M. Hamiltonian dynamics: four dimensional BF-like theories with a compact dimension Revista Mexicana de Física, vol. 62, núm. 1, enero-febrero, 2016, pp. 31-44 Sociedad Mexicana de Física A.C. Distrito Federal, México

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Hamiltonian dynamics: four dimensional BF-like theories with a compact dimension

A. Escalante and M. Zarate Reyes Instituto de F´ısica Luis Rivera Terrazas, Benemerita´ Universidad Autonoma´ de Puebla, Apartado postal J-48 72570 Puebla. Pue., Mexico,´ e-mail: [email protected]; [email protected] Received 29 July 2015; accepted 23 October 2015

A detailed Dirac’s canonical analysis for a topological four dimensional BF -like theory with a compact dimension is developed. By performing the compactification process we find out the relevant symmetries of the theory, namely, the full structure of the constraints and the extended action. We show that the extended Hamiltonian is a linear combination of first class constraints, which means that the general covariance of the theory is not affected by the compactification process. Furthermore, in order to carry out the correct counting of physical degrees of freedom, we show that must be taken into account reducibility conditions among the first class constraints associated with the excited KK modes. Moreover, we perform the Hamiltonian analysis of Maxwell theory written as a BF -like theory with a compact dimension, we analyze the constraints of the theory and we calculate the fundamental Dirac’s brackets, finally the results obtained are compared with those found in the literature.

Keywords: Topological theories; extra dimensions; Hamiltonian dynamics.

PACS: 98.80.-k;98.80.Cq

1. Introduction effects could be tested in the actual LHC collider, and in the International Linear Collider [10]. We can find several works involving extra dimensions, for Models that involve extra dimensions have introduced com- instance, in Refs. 4, 5, and 9 is developed the canonical anal- pletely new ways of looking up on old problems in theoretical ysis of Maxwell theory in five dimensions with a compact physics; the possible existence of a dimension extra beyond dimension, after performing the compactification and fixing the fourth dimension was considered around 1920’s, when the gauge parameters, the final theory describes to Maxwell Kaluza and Klein (KK) tried to unify electromagnetism with theory plus a tower of KK excitations corresponding to mas- Einstein’s gravity by proposing a theory in 5D, where the fifth sive Proca fields. Furthermore, in the context of Yang-Mills dimension is a circle S1 of radius R, and the gauge field is (YM) theories, in Ref. 11 it has been carry out the canonical contained in the extra component of the metric tensor (see analysis of a 5D YM theory with a compact dimension; in Ref. 1 and references therein). Nowadays, the study of mod- that work were obtained different scenarios for the 4D effec- els involving extra dimensions have an important activity in tive action obtained after the compactification; if the gauge order to explain and solve some fundamental problems found parameters propagate in the bulk, then the excited KK modes in theoretical physics, such as, the problem of mass hierar- are gauge fields, and they are matter vector fields provided chy, the explanation of dark energy, dark matter and infla- that those parameters are confined in the 3-brane. tion [2]. Moreover, extra dimensions become also impor- On the other side, the study of alternative models describ- tant in theories of grand unification trying of incorporating ing Maxwell and YM theories expressed as the coupling of gravity and gauge interactions consistently. In this respect, topological theories have attracted attention recently because it is well known that extra dimensions have a fundamental of its close relation with gravity. In fact, the study of topo- role in the developing of string theory, since all versions of logical actions has been motived in several contexts of the- the theory are natural and consistently formulated only in oretical physics given their interesting relation with physical a spacetime of more than four dimensions [3,4]. For some theories. One example of this is the well-known MacDowell- time, however, it was conventional to assume that in string Maunsouri formulation of gravity (see Ref. 12 and references theory such extra dimensions were compactified to complex therein). In this formulation, breaking the SO(5) symmetry manifolds of small sizes about the order of the Planck length, of a BF -theory for SO(5) group down to SO(4) we can ob- −33 `P ∼ 10 cm [4,5], or they could be even of lower size tain the Palatini action plus the sum of second Chern and Eu- independently of the Plank Length [6-8]; in this respect, the ler topological invariants. Due to these topological classes compactification process is a crucial step in the construction have trivial local variations that do not contribute classically of models with extra dimensions [9]. to the dynamics, we thus obtain essentially general relativ- On the other hand, there are phenomenological and theo- ity [13,14]. Furthermore, in Refs. 15 and 16 , an analysis of retical motivations to quantize a gauge theory in extra dimen- specific limits in the gauge coupling of topological theories sions, for instance, if there exist extra dimensions, then their yielding a pure YM dynamics in four and three dimensions 32 ALBERTO ESCALANTE Y MOISES ZARATE REYES has been reported. In this respect, in the four-dimensional of the Dirac algorithm and also the Dirac brackets of the the- case, nonperturbative topological configurations of the gauge ory were constructed by using the field redefinition method. fields are defined as having an important role in realistic the- Moreover, a pure Dirac’s analysis of the action (1) has been ories, e.g. quantum chromodynamics. Moreover, the 3D reported in Ref. 19, where it was showed that the action can case is analyzed at the Lagrangian level, and the action be- be split in two terms lacking physical degrees of freedom, comes the coupling of BF -like terms in order to generalize the complete action, however, does have physical degrees of the quantum dynamics of YM [15]. freedom, the Maxwellian degrees of freedom. The studio of Because of the ideas expressed above, in this paper we action (1) with a compact dimension becomes important be- analyze a four dimensional BF -like theory and the Maxwell cause could expose some information among the topological theory written as a BF -like theory (it is also called first or- sector given in the second term on the right hand side of (1) der Maxwell action [17]) with a compact dimension. First, (the BF term), and the dynamical sector given in the full ac- we perform the analysis for the BF term; in this case we are tion. Furthermore, our study could be useful for extending interested in knowing the symmetries of a topological the- the work reported in Ref. 17 in order to find the dual theory ory defined in four dimensions with a compact dimension. of Maxwell with extra dimensions. We shall show that in order to obtain the correct counting of Hence, we first analyze the following action physical degrees of freedom, we must take into account re- Z © ª ducibility conditions among the first class constraints of the 4 MN S2 [A, B] = d x B (∂M AN − ∂N AM ) . (4) KK excitations; hence, in this paper we present the study of a model with reducibility conditions in the KK modes. Finally, The action S is a topological theory, and its study in the we perform the Hamiltonian analysis of first order Maxwell 2 context of extra dimensions become relevant. We need to re- action with a compact dimension, and we compare our results member that topological field theories are characterized by with those found in the literature. In addition, we have added being devoid of local degrees of freedom. That is, the theo- as appendix the fundamental Dirac’s brackets of the theories ries are susceptible only to global degrees of freedom asso- under study, thus we develop the first steps for studying the ciated with non-trivial topologies of the manifold in which quantization aspects. they are defined and topologies of the gauge bundle, thus the next question arises; it is affected the topological nature of S2 2. Hamiltonian dynamics of a BF-like topolog- because of the compactification process?. Moreover, in or- ical theory with a compact dimension der to carry out the counting of physical degrees of freedom of (4) without a compact dimension, we must take into ac- In the following lines, we shall study the Hamiltonian dynam- count reducibility conditions among the constraints [19,20], ics for a four dimensional BF -like topological theory with a hence, it is interesting to investigate if reducible constraints compact dimension; then we develop the canonical analysis are still present after performing the compactification pro- of a four dimensional Maxwell theory written as a BF -like cess. In fact, the Hamiltonian analysis of theories with re- theory with a compact dimension. ducibility conditions among the constraints in the context of Let us start with the following action reported in Sun- extra dimensions has not been performed, and we shall an- dermeyer’s book [18] (also Yang-Mills theory is written as a swer these questions along this paper. BF -like theory in that book) defined in four dimensions For simplicity, we shall work with a four dimensional ac- Z ½ tion. Then we will perform the compactification process in 4 1 MN S1 [A, B] = d x B BMN order to obtain a three dimensional effective Lagrangian. It 4 is straightforward perform the extension of our results to di- ¾ 1 mensions higher than four. The notation that we will use − BMN (∂ A − ∂ A ) , (1) 2 M N N M along the paper is the following: the capital latin indices M,N run over 0, 1, 2, 3 here 3 label the compact dimen- where BMN = −BNM . The equations of motion obtained sion and these indices can be raised and lowered by the from (1) are given by four-dimensional Minkowski metric ηMN = (−1, 1, 1, 1); z will represent the coordinate in the compact dimension and M µ ∂ BMN = 0, (2) µ, ν = 0, 1, 2 are spacetime indices, x the coordinates that label the points for the three-dimensional manifold M3; furthermore we will suppose that the compact dimension is a BMN = ∂M AN − ∂N AM . (3) 1 S /Z2 orbifold whose radius is R; then any dynamical vari- able defined on M ×S1/Z can be expanded in terms of the By taking into account (3) in (2), we obtain the Maxwell’s 3 2 complete set of harmonics [4,5,11,21] free field equations. The action (1) is written in the first order form, it have been analysed without a compacta dimension in 1 X∞ ³nz ´ Ref. 17. In that paper was worked the context of S-duality B3µ(x, z) =√ B3µ (x) sin , πR (n) R transformation by taking into account the general covariance n=1

Rev. Mex. Fits. 62 (2016) 31–44 HAMILTONIAN DYNAMICS: FOUR DIMENSIONAL BF-LIKE THEORIES WITH A COMPACT DIMENSION 33

here, n = 1, 2, 3, .., k − 1. We also can observe that the rank of the Hessian is zero, so we expect 10k − 4 primary con- µν 1 µν B (x, z) = √ B (x) straints; from the definition of the momenta (8). We identify 2πR (0) the following primary constraints: 1 X∞ ³nz ´ + √ Bµν (x) cos , (n) R zero-modes k-modes πR n=1 (0) (0) (n) (n) ∞ φ ≡Π ≈0, φ ≡ Π ≈0, 1 X ³nz ´ 0j 0j 03 03 A (x, z) = √ A(n)(x) sin , 3 3 R φ(0)≡Π(0)≈0, φ(n)≡Π(n)≈0, πR n=1 ij ij i3 i3 φi ≡Πi −2B0i ≈0, φ(n)≡Π(n)≈0, 1 (0) (0) (0) (0) 0i 0i (9) Aµ(x, z) = √ Aµ (x) 2πR 0 0 (n) (n) φ(0)≡Π(0)≈0, φ0i ≡Π0i ≈0, 1 X∞ ³nz ´ 3 3 03 (n) φ(n)≡Π(n)−2B(n)≈0, + √ Aµ (x) cos . (5) πR R i i 0i n=1 φ(n)≡Π(n)−2B(n)≈0, (0) 0 0 φ(n)≡Π(n)≈0. The dynamical variables of the theory are given by Ai , A(0), B0i , Bij , A(n), A(n), A(n), B03 , Bi3 , B0i , Bij , 0 (0) (0) 3 i 0 (n) (n) (n) (n) Furthermore, the canonical Hamiltonian is given by with i, j = 1, 2. Ã Let us perform the Hamiltonian analysis of the topologi- Z 2 (0) i ij (0) cal term given by S2 Hc = d x − A0 ∂iΠ(0) − B(0)Fij Z 2ZπR © ª ∞ 3 MN X h ³ n ´ S2 [A, B] = d x dz B (∂M AN −∂N AM ) . (6) + − 2Bi3 ∂ A(n) + A(n) (n) i 3 R i 0 n=1 ! First, we start the analysis by performing the 3+1 decompo- ³ ´ i (n) i n 3 ij (n) sition and we use explicitly the expansions given in (5); then − A0 ∂iΠ(n) + Π(n) B(n)Fij . 1 R we perform the compactification process on a S /Z2 orbifold, we obtain the following effective Lagrangian Thus by using the primary constraints (9), we define the pri- 0i ˙ (0) (0) 0i ij (0) mary Hamiltonian given by L2 = 2B(0)Ai + 2A0 ∂iB(0) + B(0)Fij " Z " ∞ X 2 0j (0) ij (0) (0) i (n) 0i 0i ˙ (n) HP = Hc+ dx λ φ +λ φ +λ φ + 2A0 ∂iB(n) + 2B(n)Ai (0) 0j (0) ij i (0) n=1 Ã ³ ´ ∞ (n) n (n) X i3 (0) 0 03 (n) j3 (n) (n) 3 + 2B(n) ∂iA3 + Ai +λ φ + λ φ +λ φ +λ φ R 0 (0) (n) 03 (n) j3 3 (n) # n=1 ³ n ´ !# + 2B03 ∂ A(n) + A(n) + Bij F (n) , (7) (n) 0 3 R 0 (n) ij 0j (n) ij (n) (n) i (n) 0 + λ(n)φ0j +λ(n)φij +λi φ(n)+λ0 φ(n) , (10)

(m) (m) (m) where Fij = ∂iAj − ∂jAi . The ﬁrst three terms on 03 j3 (n) 0j ij (n) (n) 0j ij the left hand side are called the zero modes and the theory de- where λ(n), λ(n), λ3 , λ(n), λ(n), λi , λ0 and λ(0), λ(0), scribes a topological theory [19,20,22], the following terms λ(0), λ(0) are Lagrange multipliers enforcing the constraints. αβ (n) i 0 correspond to a KK tower; in fact, both B(n) and Aα are The non-vanishing fundamental Poisson brackets for the the- called Kaluza-Klein (KK) modes. In the following, we shall ory under study are given by suppose that the number of KK modes is given by k, taking the limit k → ∞ at the end of the calculations. (m) 0 N 0 M m 2 {AM (x , x), Π(n)(x , y)} = δ N δ nδ (x − y), The theory under study is a singular system, it is easy to 1 ¡ observe that the Hessian is a 10k − 4 × 10k − 4 matrix, it {BMN (x0, x), Π(n)(x0, y)} = δm δM δN (m) IJ 2 n I J has a determinant equal to zero. Hence, the Hamiltonian¡ for-¢ ¢ malism calls for the deﬁnition of the momenta Π(n) , ΠM − δN δM δ2(x − y). (11) ¡ ¢ MN (n) I J canonically conjugate to A(n),BMN , M (n) Let us now analyze if secondary constraints arise from the δL δL consistency conditions over the primary constraints. For this ΠM = ³ 2 ´ , Π(n) = ³ 2 ´ , (8) (n) (n) MN aim, we construct the (10k − 4) × (10k − 4) matrix formed δ ∂ A δ ∂ BMN 0 M 0 (n) by the Poisson brackets among the primary constraints; the

Rev. Mex. Fits. 62 (2016) 31–44 34 ALBERTO ESCALANTE Y MOISES ZARATE REYES non-vanishing Poisson brackets between primary constraints are given by

(0) j j 2 {φ0i (x), φ(0)(y)} = δ iδ (x − y),

(m) 3 m 2 {φ03 (x), φ(n)(y)} = δ nδ (x − y), (12)

(m) j m j 2 {φ0i (x), φ(n)(y)} = δ nδ iδ (x − y).

If we write to (12) in matrix form, it is straightforward to observe that has rank=6k−2 and 4k−2 null vectors. From consistency and by using the null vectors, we ﬁnd the following 4k − 2 secondary constraints

˙0 0 k φ(0)(x) = {φ(0)(x),HP } ≈ 0 ⇒ ψ(0) = ∂kΠ(0), ≈ 0. ˙(0) (0) (0) (0) φij (x) = {φij (x),HP } ≈ 0 ⇒ ψij = Fij ≈ 0, m φ˙(m)(x) = {φ(m)(x),H } ≈ 0 ⇒ ψ(m) = ∂ A(m) + A(m) ≈ 0, k3 k3 P k3 k 3 R k m φ˙0 (x) = {φ0 (x),H } ≈ 0 ⇒ ψ3 = ∂ Πk + Π ≈ 0, (m) (m) P (m) k (m) R (m) ˙(m) (m) (m) (m) φij (x) = {φij (x),HP } ≈ 0 ⇒ ψij = Fij ≈ 0; (13) and the rank allows us to ﬁx the following 6k − 2 Lagrange multipliers

zero-modes k-modes

0j ij 03 i3 λ(0) = −2∂iB(0), λ(n) = −2∂iB(n), (0) (n) (14) λi = 0, λ3 = 0, 0k ik 2m k3 λ(m) = −2∂iB(m) + R B(m), (m) λi = 0.

For this theory there are not third constraints. Hence, this completes Dirac’s consistency procedure for ﬁnding the complete set of constraints. Explicitly the set of constraints primary and secondary obtained are given by

zero-modes k-modes

(0) (0) (n) (n) φ0j ≡ Π0j ≈ 0, φ03 ≡ Π03 ≈ 0, (0) (0) (n) (n) φij ≡ Πij ≈ 0, φi3 ≡ Πi3 ≈ 0,

i i 0i (n) (n) φ(0) ≡ Π(0) − 2B(0) ≈ 0, φ0i ≡ Π0i ≈ 0, 0 0 (n) (n) φ(0) ≡ Π(0) ≈ 0, φij ≡ Πij ≈ 0, 0 k 3 3 03 (15) ψ(0) ≡ ∂kΠ(0) ≈ 0, φ(n) ≡ Π(n) − 2B(n) ≈ 0, (0) (0) i i 0i ψij ≡ Fij ≈ 0, φ(n) ≡ Π(n) − 2B(n) ≈ 0, 0 0 φ(n) ≡ Π(n) ≈ 0, (n) (n) n (n) ψk3 ≡ ∂kA3 + R Ak ≈ 0, 3 k n 3 ψ(n) ≡ ∂kΠ(n) + R Π(n) ≈ 0, (n) (n) ψij ≡ Fij ≈ 0.

Once identified all the constraints as primary, secondary etc., we may verify which ones correspond to first and second class. For this purpose we will construct the matrix formed by the Poisson brackets among the primary and secondary constraints; in order to achieve this aim, we find that the non-zero Poisson brackets among primary and secondary constraints are given by

Rev. Mex. Fits. 62 (2016) 31–44 HAMILTONIAN DYNAMICS: FOUR DIMENSIONAL BF-LIKE THEORIES WITH A COMPACT DIMENSION 35

(0) (0)j j 2 {φ0i (x), φ (y)} = δ iδ (x − y),

j (0) ¡ j y j y¢ 2 {φ(0)(x), ψls (y)} = − δ s∂l − δ l∂s δ (x − y),

(m) 3 m 2 {φ03 (x), φ(n)(y)} = δ nδ (x − y),

(m) j m i 2 {φ0i (x), φ(n)(y)} = δ nδ jδ (x − y), (16)

3 (n) m y 2 {φ(m)(x), ψk3 (y)} = −δ n∂k δ (x − y), n {φj (x), ψ(n)(y)} = − δm δk δ2(x − y), (m) k3 R n j j (n) m ¡ j y j y¢ 2 {φ(m)(x), ψls (y)} = −δ n δ s∂l − δ l∂s δ (x − y).

Again, if we write to (16) in matrix form, we find that has a rank = 6k − 2 and 8k − 4 null vectors. Thus, by using the variables, 8k − 4 first class constraints and 6k − 2 second rank and the null vectors, we find the following 4 first class class constraints, therefore the number of degrees of freedom constraints for the zero modes is given by γ˜(0) =F (0) − ∂ Π(0) + ∂ Π(0) ≈ 0, 1 ij ij i 0j j 0i G= (20k−8− (2(8k−4)+6k−2)) = − (k − 1). (21) i 2 γ(0) =∂iΠ(0) ≈ 0, This is an interesting fact, the counting of degrees of free- (0) (0) γij =Πij ≈ 0, dom is negative and this can not be correct. It is important to comment, that in a four dimensional BF theory without a 0 0 γ(0) =Π(0) ≈ 0, (17) compact dimension, in order to carry out the correct counting and the following 4 second class constraints for the zero of physical degrees of freedom, we must take into account the modes reducibility conditions among first class constraints [20,22] Hence, if we observe the constraints found above, we can (0) (0) χ0i =Π0i ≈ 0, see that the reducibility among the constraints is also present; however, there exist reducibility conditions in the first class χi =Πi − 2B0i ≈ 0. (18) (0) (0) (0) constraints of the KK excitations and there are not in the zero Furthermore, we identifying the following 8k − 8 first class mode. In fact, it can be showed that the reducibility condi- constraints for the KK-modes tions are identified by the following k − 1 relations

(m) (m) m (m) (m) m (m) (m) (m) m (m) γ˜i3 =∂iA3 + Ai − ∂iΠ03 − Π0i ≈ 0, ∂ γ˜ − ∂ γ˜ − γ˜ = 0. (22) R R i j3 j i3 R ij (m) (m) (m) (m) γ˜ij =Fij − ∂iΠ0j + ∂jΠ0i ≈ 0, In this manner, the number of independent first class con- (m) (m) straints are (8k − 4 − k + 1 = 7k − 3); then, this implies that γi3 =Πi3 ≈ 0, the number of physical degrees of freedom is (m) (m) γ =Π ≈ 0, 1 ij ij G = (20k − 8 − (2(7k − 3) + 6k − 2)) = 0. (23) 0 0 2 γ(m) =Π(m) ≈ 0, m Therefore, the BF -like theory with a compact dimension is γ =∂ Πi + Π3 ≈ 0, (19) still topological one. It is important to comment that if we (m) i (m) R (m) perform the counting of physical degrees of freedom for the and 6k − 6 second class constraints zero mode, then we find that it is devoid of local degrees (m) (m) of freedom as expected; for the zero mode defined in three χ03 =Π03 ≈ 0, dimensions there are not reducibility conditions. All this 3 3 03 χ(m) =Π(m) − 2B(m) ≈ 0, information become relevant, because after performing the compactification process there are already reducibility condi- χi =Πi − 2B0i ≈ 0, (m) (m) (m) tions; we need to remember that the correct identification of χ(m) =Π(m) ≈ 0. (20) the constraints is a relevant step because they allows us iden- 0i 0i tify observables and constraints are the best guideline to per- With all this information at hand, the counting of degrees of form the quantization; similarly the reducibility conditions in freedom is carry out as follows: there are 20k − 8 dynamical the KK modes must be taken into account in that process.

Rev. Mex. Fits. 62 (2016) 31–44 36 ALBERTO ESCALANTE Y MOISES ZARATE REYES

With all this information, we can identify the extended action; thus, we use the ﬁrst class constraints (20), the second class constraints (18), the Lagrange multipliers (14) and we ﬁnd that the extended action takes the form ³ ´ Z h 3 ˙ (0) ν ˙ νµ (0) (0) ij (0) (0) ij (0) (0) 0 0i (0) SE QK ,PK , λK = d x Aν Π(0) + B(0)Πνµ − H − α˜(0)γ˜ij − α γ(0) − α(0)γij − α0 γ(0) − λ(0)χ0i

Xk n (0) i ˙ (n) N ˙ MN (n) (n) i3 (n) ij (n) i3 (n) − λi χ(0) + AN Π(n) + B(n) ΠMN − H − α(n)γ˜i3 − α(n)γ˜ij − λ(n)γi3 n=1 oi ij (n) (n) 0 (n) (n) i 0i (n) 03 (n) (n) 3 − λ(n)γij − λ0 γ(n) − α γ(n) − λi χ(n) − λ(n)χ0i − λ(n)χ03 − λ3 χ(n) , (24) where we abbreviate with QK y PK all the dynamical variables and the generalized momenta; λK stand for all Lagrange multipliers associated with the ﬁrst and second class constraints. From the extended action, it is possible to identify the extended Hamiltonian which is given by

Z h Xk h i 2 (0) (0) ij (0) (n) 3 i3 (n) ij (n) ij (0) (0) ij (0) Hext = d x − A0 γ − B(0)γ˜ij + − A0 γ(n) − 2B(n)γ˜i3 − B(n)γ˜ij + α(0)γ˜ij + α γ(0) + λ(0)γij n=1 Xk n oi (0) 0 i3 (n) ij (n) i3 (n) ij (n) (n) 0 (n) + α0 γ(0) + α γ˜i3 + α(n)γ˜ij + λ(n)γi3 + λ(n)γij + λ0 γ(n) + α γ(n) . (25) n=1 We can observe that this expression is a linear combination of constraints. In fact, they are first class constraints of the zero mode and first class constrains of the KK-modes. It is well-known, that for the action (6) without compact dimensions, its extended Hamiltonian is a linear combination of first class constraints [20,22]. Thus, we can notice that the general covariance of the theory is not affected by the compactification process. Hence, in order to perform a quantization of the theory, it is not possible to construct the Schrodinger equation because the action of the Hamiltonian on physical states is annihilation. In Dirac’s quantization of systems with general covariance, the restriction on physical states is archived by demanding that the first class constraints in their quantum form must be satisfied; thus in this paper we have all tools for studying the quantization of the theory by means a canonical framework. By following with our analysis, we need to know the gauge transformations on the phase space. For this important step, we shall define the following gauge generator in terms of the first class constraints (20) Z " # 2 i3 (n) ij (n) (n) i3 (n) ij (n) (n) 0 ij (0) ij (0) (0) 0 (0) G= d x ε(n)γ˜i3 +ε(n)γ˜ij +ε0 γ(n)+ε ˙(n)γi3 +ε ˙(n)γij +ε ˙0 γ(n)+ε(0)γ˜ij +ε ˙(0)γij +ε0 γ(0)+ε ˙0 γ(0) . (26) Σ

Thus we obtain that the gauge transformations on the phase space are given by gauge parameters. In the following lines, we shall perform the Hamiltonian analysis of the action (1) and we will find zero-mode k-mode that the field BMN is not a gauge field anymore. There are (0) (0) (n) (n) not reducibility conditions among the constraints, moreover, δAµ = −∂µε0 , δAµ = −∂µε0 , there exist physical degrees of freedom and the fixing of the 0i ki (n) n (n) δB(0) = ∂kε(0), δA3 = R ε0 , gauge parameters will allow us to find massive Proca fields ij ij 03 1 i3 and pseudo-Goldston bosons as expected. Furthermore, we δB = ∂0ε , δB = ∂iε , (0) (0) (n) 2 (n) have added in the Appendix B the Dirac brackets of the the- δΠi = ∂ εki , δB0i = −∂ εik − n εi3 , (0) k (0) (n) k (n) 2R (n) ory which are important for studying the quantization. 0 i3 1 i3 (27) δΠ(0) = 0, δB(n) = 2 ∂0ε(n), MN ij ij δΠ(0) = 0, δB(n) = ∂0ε(n), 3 i3 3. Hamiltonian analysis of the four- δΠ = −∂iε , (n) (n) dimensional Maxwell theory written as a δΠi = ∂ εki − n εi3 , (n) k (n) R (n) BF-like theory with a compact dimension 0 δΠ(n) = 0, MN δΠ(n) = 0. By following the steps developed in previous section, we can perform the Hamiltonian analysis of (1). In this section we MN We notice that the fields B and AM are gauge fields; there shall resume the complete analysis; thus by performing the are not degrees of freedom, thus, it is not relevant to fix the 3 + 1 decomposition. Using the expansion of the fields (5),

Rev. Mex. Fits. 62 (2016) 31–44 HAMILTONIAN DYNAMICS: FOUR DIMENSIONAL BF-LIKE THEORIES WITH A COMPACT DIMENSION 37

1 and developing the compactification process on a S /Z2 orb- and the following 12k − 6 second class constraints ifold, we obtain the following effective Lagrangian written as " ∞ (0) (0) 1 1 X 1 χ0i = Π0i ≈ 0, L = Bµν B(0) − Bµν F (0) + Bν3 B(n) 4 (0) µν 2 (0) µν 2 (n) ν3 (0) (0) n=1 χij = Πij ≈ 0, ³ n ´ − Bµ3 ∂ A(n) + A(n) χj = Πj + B0j ≈ 0, (n) µ 3 R µ (0) (0) (0) # ³ ´ (0) 1 (0) (0) 1 µν (n) 1 µν (n) + B B − B F . (28) χ˜ij = Bij − Fij ≈ 0, 4 (n) µν 2 (n) µν 2 We identify the zero mode given by (n) (n) χ03 = Π03 ≈ 0, 1 µν (0) 1 µν (0) B(0)Bµν − B(0)Fµν (n) (n) 4 2 χi3 = Πi3 ≈ 0, and the following terms are identified as the KK excitations. (n) (n) We have commented above, that the action (28) describes χ0j = Π0j ≈ 0, Maxwell theory in three dimensions (zero mode) plus a tower (n) (n) of KK-modes. The theory is singular, there exists the same χij = Πij ≈ 0, number of dynamical variables defined above. Hence, after 3 3 03 developing a pure Dirac’s analysis, we find a set of 2k first χ(n) = Π(n) + B(n) ≈ 0, class constraints given by i i 0i χ(n) = Π(n) + B(n) ≈ 0, γ0 = Π0 ≈ 0, (0) (0) ³ ³ ´´ (n) 1 (n) (n) n (n) i χ˜i3 = Bi3 − ∂iA3 + Ai ≈ 0, γ(0) = ∂iΠ(0) ≈ 0, 2 R ³ ´ 0 0 (n) 1 (n) (n) γ(n) = Π(n) ≈ 0, χ˜ = B − F ≈ 0. (30) ij 2 ij ij i n 3 γ(n) = ∂iΠ + Π ≈ 0, (29) (n) R (n) The identification of second class constraints, allows us to fix the following 12k − 6 Lagrange multipliers

zero-modes k-modes ³ ´ 0k ik ik 03 i3 (n) n (n) λ(0) = −4∂iB(0) + 2∂iF(0), λ(n) = −4∂iB(n) + 2∂i ∂iA3 + R Ai , ¡ ¢ ij i j j i i3 i 3 n i λ(0) = ∂ Π(0) − ∂ Π(0), λ(n) = 2 ∂ Π(n) + R Π(n) , ³ ´ (0) 0i ki ki 2n (n) n (n) λi = 0, λ(n) = −4∂kB(n) + 2∂kF(n) + R ∂iA3 + R Ai ,

ij ij ij ij i j j i β(0) = B(0) − F(0), λ(n) = ∂ Π(n) − ∂ Π(n), (31)

(n) λ3 = 0,

(n) λi = 0, ³ ´ i3 i3 i3 β(n) = 2 B(n) − F(n) ,

ij ij ij β(n) = B(n) − F(n).

By using all this information it is possible to carry out the counting of degrees of freedom as follows; there are 10k − 4 dynamical variables, 2k ﬁrst class constraints and 12k − 6 second class constraints, thus

1 G = (20k − 8 − (2(2k) + 12k − 6)) = 2k − 1. 2 We observe that if k = 1 we obtain one degree of freedom as expected for Maxwell theory in three dimensions. By using the ﬁrst class constraints (29), the second class constraints (30), and the Lagrange multipliers we ﬁnd that the extended action takes the form

Rev. Mex. Fits. 62 (2016) 31–44 38 ALBERTO ESCALANTE Y MOISES ZARATE REYES

³ ´ Z h ˙ (0) ν ˙ (0) νµ (0) (0) (0) 0 (0) i ij (0) ij (0) SE QK ,PK , λK = Aν Π(0) + Bνµ Π(0) − H − β γ(0) − λ0 γ(0) − λi χ(0) − λ(0)χij − β(0)χ˜ij

XN n ˙ (n) N ˙ (n) MN (n) (n) 0 (n) (n) i (n) 3 (32) + AN Π(n) + BMN Π(n) − H − λ0 γ(n) − β γ(n) − λi χ(n) − λ3 χ(n) n=1 oi 03 (n) 0i (n) i3 (n) ij (n) i3 (n) ij (n) 3 − λ(n)χ03 − λ(n)χ0i − λ(n)χi3 − λ(n)χij − β(n)χ˜i3 − β(n)χ˜ij dx , where the corresponding extended Hamiltonian is given by

Z h XN n oi (0) (0) 0 (n) 0 (n) 3 Hext = H + β(0)γ + λ0 γ(0) + λ0 γ(n) + β γ(n) dx . (33) n=1 Here we deﬁne Z Ã 1 1 ³ ´ H = d2x Πi Π(0) + Bij B(0) − A(0)γ + −4∂ Bji + 2∂ F ji χ(0) + 2χ(0)∂ Π(0) − χ˜(0)F ij 2 (0) i 4 (0) ij 0 (0) j (0) j (0) 0i ij j i ij (0)

" XN 1 1 1 1 + Πi Π(n) + Π3 Π(n) + Bij B(n) + Bi3 B(n) − A(n)γ − F ij χ˜(n) + 2χ(n)∂ Π(n) 2 (n) i 2 (n) 3 4 (n) ij 2 (n) i3 0 (n) (n) ij ij i j n=1

³ n ´ ³ n ´³ n ´ ³ n ´ − Bi3 ∂ A(n) + A(n) + ∂ A(n) + A(n) ∂ A(n) + A(n) + 2 ∂ Π(n) + Π(n) χ(n) (34) (n) i 3 R i i 3 R i i 3 R i i 3 R i i3 µ ¶ #! 2n³ n ´ ³ ³ n ´´ + −4∂ Bji + 2∂ F ji + ∂ A(n) + A(n) χ(n) + −4∂ Bi3 + 2∂ ∂ A(n) + A(n) χ(n) j (n) j (n) R i 3 R i 0i i (n) i i 3 R i 03

Ã ! Z XN = d2x H(0) + H(n) . n=1

MN Note, that the extended Hamiltonian is not a linear combination of constraints anymore, the term B BMN of the action (1) breaks down the general covariance of the theory and eliminates the reducibility relations present in the BF -like term.

Now, the first class constraints allows us to know the fundamental gauge transformations. For this important step, we and the gauge transformation for the KK modes use the Castellani’s procedure [23,24] to construct the gauge (n) generators δAµ = −∂µε(n), (37) Z h i n (n) 0 (n) (0) 0 (0) 2 δA(n) = ε , G= ε0 γ(n)+ε γ(n)+ε0 γ(0)+ε γ(0) dx . (35) 3 (n) (38) Σ R (n) Thus, we find that the gauge transformations on the phase δBµν = 0, (39) space are given for µ δΠ = 0, (40) zero modes (n) µν (0) δΠ(n) = 0, (41) δAµ = − ∂µε(0), (0) we can observe that the gauge transformations for the zero δBµν =0, mode are the same given for Maxwell theory written in the µ δΠ(0) =0, standard form [24], and we also observe that the B filed is µν not a gauge field anymore. Finally, the transformations of δΠ(0) =0, (36) n n the fields Aµ, A3 corresponding for the k-th mode are the

Rev. Mex. Fits. 62 (2016) 31–44 HAMILTONIAN DYNAMICS: FOUR DIMENSIONAL BF-LIKE THEORIES WITH A COMPACT DIMENSION 39 same to those reported in the literature (see [4,24] and the could provide generalized QCD theories as it is claimed in cites therein). Hence, by fixing the gauge parameters by Ref. 15. In this manner, our results can be used for studying (n) ε(n) = −(R/n)A3 and considering the second class con- those generalizations in the context of extra dimensions. Fur- straints as strong identities, the effective action (28) is re- thermore, we have commented that our results can be used for duced to that reported in Ref. 4 and 25, namely extending those reported in Ref. 17. In fact, in this paper we have at hand all the tools for study S-Duality of Maxwell the- 1 ory with a compact dimension. Moreover, our results also can L = − F µν F (0) 4 (0) µν be used for studying models that are present in string theory " µ ¶ # such as those models described by Kalb-Ramond fields. Fi- X∞ 1 1 2n 2 + − F νµ F (n) + A(n)Aµ(n) , (42) nally, all the results presented in this work will be useful in or- 4 (n) νµ 2 R µ der to compare the Dirac quantization with other schemes, for n=1 instance, with the Faddeev-Jackiw quantization (see the Ref. where we able to observe that the KK-modes are massive (n) 26 and cites therein). In fact, in Faddeev-Jackiw approach it Proca fields, A3 has been absorbed and it is identified as a is possible to obtain all the relevant Dirac’s results; the gener- pseudo-Goldstone boson [4,25]. Furthermore, we have added alized Faddeev-Jackiw brackets coincide with the Dirac ones, in the appendix A the Dirac brackets, thus we have developed basically in this formulation we only choose the symplectic a full Hamiltonian analysis of the theory under study. variables either the configuration space or the phase space and by fixing the appropriated gauge, we can invert the symplectic matrix in order to obtain a complete analysis. In this 4. Conclusions respect, in order to make progress in the quantization, we will work with the Faddeev-Jackiw formulation and we will con- In this paper, the Hamiltonian analysis for a topological BF - firm the results obtained along this paper. All these ideas are like theory and for Maxwell theory expressed as a BF -like in progress and will be the subject of future works. theory with a compact dimension has been performed. For the former, we performed the compactification process on a 1 S /Z2 orbifold, then we analysed the effective theory and all the constraints, gauge transformations and the extended Appendix Hamiltonian have been obtained. We also found that the extended Hamiltonian is given by a linear combination of A. first class constraints of the zero mode and first class constraints of the KK modes, this indicates that the compact- In this section we will compute the Dirac brackets for the BF ification process does not break the general covariance of theory with a compact dimension given by the action (7). By the theory. Moreover, we observed that reducibility relations using the constraints given in (17), (18) and the fixed gauge among the constraints are preserved before and after perform- (0) (0) i 0j ij ∂iAi ≈ 0, A0 ≈ 0, 2ηij∂ B(0) ≈ 0 and 2B(0) ≈ 0 we ing the compatification process, however, the reducibility is obtain the following set of second class constraints given among the first class constraints of the excited modes, there is not reducibility in the zero modes. This important (0) (0) χˆ = ∂iAi ≈ 0, fact allowed us to conclude that the theory is a topological (0) (0) one. χ0 = A0 ≈ 0, Finally, for Maxwell theory written in the first order for- χ = ∂ Πi ≈ 0, malism with a compact dimension, we found the constraints, (0) i (0) the gauge transformations and the extended action. We ob- 0 0 χ¯(0) = Π(0) ≈ 0, served that the theory do not present reducibility conditions among the constraints, and the theory is not topological any- (0) 1 ij (0) ij (0) χ˜ = η F − η ∂iΠ ≈ 0, more. In fact, the theory has the same symmetries and de- 2 ij 0j grees of freedom than Maxwell theory with a compact di- ij 0j χˆ(0) = 2η ∂iB ≈ 0, mension [4,9]. Finally by fixing the gauge parameters we (0) ij ij noted that the theory is reduced to Maxwell theory in three χ(0) = 2B(0) ≈ 0, dimensions described by the zero mode plus a tower of mas- (0) (0) sive Proca fields excitations. χ˜ij = Πij ≈ 0, We would to comment that our results are generic and can χi = Πi − 2B0i ≈ 0, be extended to a 5D theory and models with a close relation (0) (0) (0) to YM and general relativity. In fact, we have commented χ(0) = Π(0) ≈ 0. above that there are topological generalizations of Maxwell 0i 0i and Yang-Mills theories in three and four dimensions, that (A.1)

Rev. Mex. Fits. 62 (2016) 31–44 40 ALBERTO ESCALANTE Y MOISES ZARATE REYES

Thus, the matrix whose entries are given by the Poisson brackets among the constraints (A.1) takes the form χˆ(0) χ(0) χ χ¯0 χ˜(0) χˆ χkl χ˜(0) χk χ(0)  0 (0) (0) (0) (0) kl (0) 0k 

(0)  2  χˆ  0 0 −∇ 0 0 0 0 0 ∂k 0      (0)   χ0  0 0 0 1 0 0 0 0 0 0     2  χ(0)  ∇ 0 0 0 0 0 0 0 0 0      χ¯0  0 −1 0 0 0 0 0 0 0 0  (0)       χ˜(0)  0 0 0 0 0 −∇2 0 0 0 0  (0)   2 Gαν =  δ (x − y)  2 ik  χˆ(0)  0 0 0 0 ∇ 0 0 0 0 η ∂i     ¡ ¢  ij  i j i j  χ(0)  0 0 0 0 0 0 0 δ kδ l − δ lδ k 0 0     ¡ ¢  (0)  i j i j  χ˜  0 0 0 0 0 0 − δ kδ l − δ lδ k 0 0 0  ij     i  k  χ(0)  −∂i 0 0 0 0 0 0 0 0 −δ i    (0) ji k χ0i 0 0 0 0 0 −η ∂j 0 0 δ i 0. (A.2) Hence, the inverse is given χˆ(0) χ(0) χ χ¯0 χ˜(0) χˆ χkl χ˜(0) χk χ(0)  0 (0) (0) (0) (0) kl (0) 0k  (0)  1  χˆ  0 0 2 0 0 0 0 0 0 0   ∇    (0)   χ0  0 0 0 −1 0 0 0 0 0 0      χ  − 1 0 0 0 0 0 0 0 0 ∂k  (0)  ∇2 ∇2      χ¯0  0 1 0 0 0 0 0 0 0 0  (0)      ik  χ˜(0)  0 0 0 0 0 1 0 0 η ∂i 0  (0)−1  ∇2 ∇2  2 Gαν =  δ (x − y). (A.3)   χˆ  0 0 0 0 − 1 0 0 0 0 0  (0)  ∇2   ¡ ¢  ij  i j i j  χ  0 0 0 0 0 0 0 − δ kδ l − δ lδ k 0 0  (0)     (0)  ¡ ¢  χ˜  0 0 0 0 0 0 δi δj − δi δj 0 0 0  ij  k l l k     ηji∂  χi  0 0 0 0 − j 0 0 0 0 δk  (0)  ∇2 i    χ(0) 0 0 − ∂i 0 0 0 0 0 −δk 0 0i ∇2 i In this manner, the nonzero Dirac’s brackets are given by 1 ¡ ¢ {A(0)(x), Πj (y)} =δj δ2(x − y) − ∂ ∂ − ηkiηlj∂ ∂ δ2(x − y), i (0) D i ∇2 i j k l 1 1 ¡ ¢ {B0i (x),A(0)(y)} = − δj δ2(x − y) − ∂ ∂ − ηkiηlj∂ ∂ δ2(x − y), (0) j D 2 i 2∇2 i j k l 0i (0) {B(0)(x), Π0j (y)}D =0,

ij (n) {B(n)(x), Πkl (y)}D =0. (A.4) We can obtain similar results for the excited modes. Therefore, we have in this work all the elements for studying the quantization of the theories under study. It is important to comment that all these results are not reported in the literature.

In this appendix, we calculate the Dirac brackets for Maxwell theory written as a BF-like theory. For our aims we will calculate the Dirac brackets for the zero mode, then we will calculate the brackets for the excited modes. Hence, by using the following i (0) (0) ﬁxed gauge ∂ Ai ≈ 0 and A0 ≈ 0, we obtain the following set of second class constraints

Rev. Mex. Fits. 62 (2016) 31–44 HAMILTONIAN DYNAMICS: FOUR DIMENSIONAL BF-LIKE THEORIES WITH A COMPACT DIMENSION 41

(0) (0) χ¯ =A0 ≈ 0,

0 χˆ(0) =Π(0) ≈ 0,

(0) (0) χ˜ =∂iAi ≈ 0,

i χ(0) =∂iΠ(0) ≈ 0,

(0) (0) χ0i =Π0i ≈ 0,

(0) (0) χij =Πij ≈ 0,

i i 0i χ(0) =Π(0) + B(0) ≈ 0,

1 ³ ´ χ˜(0) = B(0) − F (0) ≈ 0. (B.1) ij 2 ij ij

Thus, we can calculate the following matrix whose entries are given by the Poisson brackets between these constraints, obtaining

χ¯(0) χˆ χ˜(0) χ χ(0) χ(0) χk χ˜(0)  (0) (0) 0j kl (0) kl  (0)   χ¯  0 1 0 0 0 0 0 0      χˆ  −1 0 0 0 0 0 0 0  (0)     (0)  2  χ˜  0 0 0 −∇ 0 0 ∂k 0      χ  0 0 ∇2 0 0 0 0 0  (0) (0)   2 Cαν (x, y) =  δ (x − y). (0)  1 i  χ  0 0 0 0 0 0 − δ k 0  0i  2   ¡ ¢  (0)  1 i j i j  χ  0 0 0 0 0 0 0 − δ kδ l − δ lδ k  ij  4   ¡ ¢  i  1 i 1 i i  χ(0)  0 0 −∂i 0 2 δ j 0 0 2 δ l∂k − δ k∂l    ¡ ¢ ¡ ¢ (0) 1 i j i j 1 k k χ˜ij 0 0 0 0 0 4 δ kδ l − δ lδ k − 2 δ j ∂i − δ i∂j 0 (B.2)

The inverse of this matrix is given by

χ¯(0) χˆ χ˜(0) χ χ(0) χ(0) χk χ˜(0)  (0) (0) 0k kl (0) kl 

(0)   χ¯  0 −1 0 0 0 0 0 0      χˆ  1 0 0 0 0 0 0 0  (0)     (0)  1  χ˜  0 0 0 2 0 0 0 0   ∇    χ  0 0 − 1 0 − 2∂k 0 0 0  (0)−1 (0)  ∇2 ∇2  2 Cαν (x, y)=  δ (x − y).  ¡ ¢  (0)  2∂i i i i  χ0i  0 0 0 2 0 4 δ l∂k − δ k∂l 2δ k 0   ∇   ¡ ¢ ¡ ¢  (0)  k k i j i j  χ  0 0 0 0 −4 δ j ∂i − δ i∂j 0 0 4 δ kδ l − δ lδ k  ij     χi  0 0 0 0 −2δi 0 0 0  (0)  k    ¡ ¢ (0) i j i j χ˜ij 0 0 0 0 0 −4 δ kδ l − δ lδ k 0 0 (B.3)

Rev. Mex. Fits. 62 (2016) 31–44 42 ALBERTO ESCALANTE Y MOISES ZARATE REYES

Thus, we obtain the following Dirac’s brackets for the zero mode µ ¶ ∂ ∂ {A(0)(x), Πj (y)} = δj − j i δ2(x − y), i (0) D i ∇2

ij (0) {B(0)(x), Πkl (y)}D = 0

(0) (0) {Ai (x),Aj (y))}D = 0 i j {Π(0)(x), Π(0)(y)}D = 0

ij k ¡ k k ¢ 2 {B(0)(x), Π(0)(y)}D = 2 δ j∂i − δ i∂j δ (x − y),

ij 0k ¡ k k ¢ 2 {B(0)(x),B(0)(x)}D = −2 δ j∂i − δ i∂j δ (x − y), µ ¶ ∂ ∂ {A(0)(x),B0j (x)} = − δj − j i δ2(x − y). (B.4) i (0) D i ∇2

(0) j Observe that the Dirac brackets among the ﬁelds Ai , Π(0) are those knew for Maxwell theory [24]. Now we calculate Dirac’s brackets for the excited modes of the Maxwell BF -like theory. By working with the following ﬁxed (n) 3 n (n) gauge ∂iAi ≈ 0 and Π(n) + R A0 ≈ 0, we obtain the following set of second class constraints

(n) (n) χ˜ = ∂iAi ≈ 0, 0 0 χ(n) = Π(n) ≈ 0, n χ˜3 = Π3 + A(n) ≈ 0, (n) (n) R 0 n χ = ∂ Πi + Π3 ≈ 0, (n) i (n) R (n) (n) (n) χ0j = Π0j ≈ 0, (n) (n) χij = Πij ≈ 0, i i 0i χ(n) = Π(n) + B(n) ≈ 0, 1 ³ ´ χ˜(n) = B(n) − F (n) ≈ 0, ij 2 ij ij (n) (n) χ03 = Π03 ≈ 0, (n) (n) χi3 = Πi3 ≈ 0, 3 3 03 χ(n) = Π(n) + B(n) ≈ 0, 1 ³ ³ n ´´ χ˜(n) = B(n) − ∂ A(n) + A(n) ≈ 0. (B.5) i3 2 i3 i 3 R i Thus, we obtain the following matrix whose entries are given by the Poisson brackets between these second class constraints, obtaining

(n) (n) (n) (n) (n) (n) χ˜(n) χ0 χ˜3 χ χ χ χk χ˜ χ χ χ3 χ˜  (n) (n) (n) 0k kl (n) kl 03 k3 (n) k3  χ˜(n)  0 0 0 −∇2 0 0 ∂ 0 0 0 0 0   k  0  n  χ  0 0 − 0 0 0 0 0 0 0 0 0  (n)  R  3  n 1  χ˜  0 0 0 0 0 0 0 0 0 0 ∂k  (n)  R 2   2  χ  ∇ 0000 0 0 0 0000  (n)   (n)   χ  0 0 0 0 0 0 − 1 δi 0 0 0 0 0  0i  2 k ³ ´  (n)   χ  0 0 0 0 0 0 0 − 1 δi δj − δi δj 0 0 0 0  (n) ij  4 k l l k  2 Gαν =  ³ ´ δ (x − y). χi  −∂ 0 0 0 1 δi 0 0 1 δi ∂ − δi ∂ 0 0 0 n δi  (n)  i 2 k 2 l k k l 2R k   ³ ´ ³ ´  (n)  1 i j i j 1 k k  χ˜  0 0 0 0 0 δ δ − δ δ − δ ∂ − δ ∂ 0 0 0 0 0  ij  4 k l l k 2 j i i j  (n)   χ  0 0 0 0 0 0 0 0 0 0 − 1 0  03  2  (n)   χ  00000 0 0 0 000 − 1 δi  i3  4 k    χ3  0 0 0 0 0 0 0 0 1 0 0 1 ∂  (n)  2 2 k  (n) χ˜ 0 0 − 1 ∂ 0 0 0 − n δi 0 0 1 δi − 1 ∂ 0 i3 2 i 2R k 4 k 2 i (B.6)

Rev. Mex. Fits. 62 (2016) 31–44 HAMILTONIAN DYNAMICS: FOUR DIMENSIONAL BF-LIKE THEORIES WITH A COMPACT DIMENSION 43

Hence, the inverse matrix is given by

(n) (n) (n) (n) (n) (n) χ˜(n) χ0 χ˜3 χ χ χ χk χ˜ χ χ χ3 χ˜  (n) (n) (n) 0k kl (n) kl 03 k3 (n) k3  (n) 1 χ˜  0 0 0 0 0 0 0 0 0 0 0   ∇2  0  R 2R  χ  0 0 0 0 0 0 0 0 ∂ 0 0  (n)  n n k  3   χ˜  0 − R 0 0 0 0 0 0 0 0 0 0  (n)  n     1 ∂k  χ(n)  − 0 0 0 −2 0 0 0 0 0 0 0   ∇2 ∇2   ³ ´ i  (n)  ∂k i i i 4nδ k  χ  0 0 0 2 2 0 4 δ l∂k − δ k∂l 2δ k 0 0 0 0  0i  ∇ ³ ´ ³ ´ R  (n)   (n)−1 χ  0 0 0 0 −4 δk ∂ − δk ∂ 0 0 4 δi δj − δi δj 0 0 0 0  2 Gαν = ij  j i i j k l l k δ (x − y).   χi  0 0 0 0 −2δi 0 0 0 0 0 0 0  (n)  k   ³ ´  (n)  i j i j  χ˜  0 0 0 0 0 −4 δ δ − δ δ 0 0 0 0 0 0  ij  k l l k  (n)   χ  0 0 0 0 0 0 0 0 0 4∂ 2 0  03  k   i  (n)  2R 4nδ i  χ  0 − ∂ 0 0 − k 0 0 0 −4∂ 0 0 4δ  i3  n i R i k  χ3  0 0 0 0 0 0 0 0 −2 0 0 0  (n)   (n) χ˜ 0 0 0 0 0 0 0 0 0 −4δi 0 0 i3 k (B.7) In this manner, the Dirac brackets for the excited modes are given by µ ¶ ∂ ∂ {A(n)(x), Πj (y)} = δj − i j δ2(x − y), i (n) D i ∇2 µ ¶ n δ2(x − y) {A(n)(x), Πi (y)} = ∂ 3 (n) D R i ∇2

(n) 3 2 {A3 (x), Π(n)(y)}D = δ (x − y),

ij (n) {B(n)(x), Πkl (y)}D = 0,

ij k ¡ k k ¢ 2 {B(n)(x), Π(n)(y)}D = 2 δ j∂i − δ i∂j δ (x − y),

ij 0k ¡ k k ¢ 2 {B(n)(x),B(n)(x)}D = −2 δ j∂i − δ i∂j δ (x − y), i3 03 2 {B(n)(x),B(n)(x)}D = −∂iδ (x − y), n {Bi3 (x), Πj (y)} = δi δ2(x − y), (n) (n) D R j i3 3 2 {B(n)(x), Π(n)(y)}D = ∂iδ (x − y), µ ¶ ∂ ∂ {A(n)(x),B0j (x)} = − δj − j i δ2(x − y), i (n) D i ∇2

(n) 03 2 {A3 (x),B(n)(x)}D = δ (x − y). (B.8)

Acknowledgments

This work was supported by CONACyT under Grant No. CB-2014-01/ 240781. We would like to thank R. Cartas-Fuentevilla for discussion on the subject and reading of the manuscript.

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