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Faddeev–Jackiw quantization of four dimensional BF theory
Annals of Physics 374 (2016) 375–394
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Faddeev–Jackiw quantization of four dimensional BF theory Alberto Escalante ∗ , Prihel Cavildo Sánchez Instituto de Física, Universidad Autónoma de Puebla, Apartado Postal J-48 72570, Puebla Pue., Mexico
highlights • • • • •
We report a detailed symplectic analysis for BF theory. We report the Faddeev–Jackiw constraints. A pure Dirac’s analysis is performed. The complete structure of Dirac’s constraints is reported. We show that symplectic and Dirac’s brackets coincide to each other.
article
abstract
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Article history: Received 11 July 2016 Accepted 9 September 2016 Available online 13 September 2016 Keywords: Topological theories Symplectic quantization Hamiltonian dynamics
The symplectic analysis of a four dimensional BF theory in the context of the Faddeev–Jackiw symplectic approach is performed. It is shown that this method is more economical than Dirac’s formalism. In particular, the complete set of Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported. In addition, we show that the generalized Faddeev–Jackiw brackets and the Dirac ones coincide to each other. Finally, the similarities and advantages between Faddeev–Jackiw method and Dirac’s formalism are briefly discussed. © 2016 Elsevier Inc. All rights reserved.
1. Introduction It is well-known that the topological theories have a relevant role in the context of gravity. In fact, topological theories are good laboratories for testing classical and quantum ideas of generally covariant gauge systems. Topological theories are characterized by lacking of physical degrees of
∗
Corresponding author. E-mail addresses: [email protected] (A. Escalante), [email protected] (P.C. Sánchez).
http://dx.doi.org/10.1016/j.aop.2016.09.003 0003-4916/© 2016 Elsevier Inc. All rights reserved.
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freedom, either in three or four dimensions they have a close relation with General Relativity [GR] just as the background independence and the diffeomorphisms covariance, this is, all the dynamical variables characterizing the theory are dynamical ones. In the three dimensional case a relevant example of topological theory is the Chern–Simons theory. In fact, basically Chern–Simons theory describes GR, it has been showed that these theories are equivalent up to a total derivative [1,2], and also there exists a relation between these theories defined with (or without) an Immirizi-like parameter [3,4]. Furthermore, we can find a recent work where the Chern–Simons state describes a topological state with unbroken diffeomorphism invariance in Yang–Mills and GR [5]. In the Loop Quantum Gravity context, that state is called the Kodama state and has been studied in interesting works by Smolin, arguing that the Kodama state at least for the de Sitter spacetime, Loop Quantum Gravity does have a good low energy limit [6]. On the other hand, in four dimensions there exists the so-called BF theory. In fact, BF theories were introduced as generalizations of three dimensional Chern–Simons actions or in other cases, can also be considered as a zero coupling limit of Yang–Mills theories [7,8]. Moreover, we find in the literature several examples where BF theories with additional extra constraints describe gravity, for instance, the well-known formulations of Plebanski and MacDowell–Mansouri [9,10]. In addition, within the modern quantization scheme using tools developed in Loop Quantum Gravity, BF theories have been studied in the context of spin foams. In fact, in this approach is not considered the traditional Fock space formalism but holonomies along paths as the basic variables to be quantized [11]. With respect to the classical context, there are several works studying the canonical structure of a BF theory, see for instance Refs. [12–14]. However, in these works has been used the canonical formalism by using a reduced phase space, this means, it has been considered as dynamical variables only those that occur in the Lagrangian density with temporal derivative, however, in several cases this approach is not convenient, for instance in Palatini’s theory the price to pay for developing the standard approach is that we cannot know the full structure of the constraints and their algebra is not closed [15], thus, the better way to carry out the canonical formalism is by following all Dirac’s steps as it has been commented in [16–21]. In consideration with the commented above, either the classical or the quantum study of BF theories and their close relation with GR are at the present a frontier subject of study [22,23]. In this manner, with the ideas explained previously in this work the Faddeev–Jackiw [FJ] symplectic quantization of a four-dimensional BF theory is performed. In fact, the FJ method provides an alternative approach for studying constrained systems based on a first-order Lagrangian [24,25]. The FJ method is a symplectic study and the basic feature of this approach is to treat all the constraints at the same footing. In other words, in FJ method one avoids the classification of the constraints into firstclass and second-class ones as in Dirac’s framework is done. In addition, some essential elements of a physical theory such as the degrees of freedom, the gauge symmetry and the quantization brackets called the generalized FJ brackets can also be derived; Dirac’s and generalized FJ brackets coincide to each other. However, it is important to remark that in the canonical formalism we must to work by following all Dirac’s steps in order to compare with the FJ symplectic formalism. In fact, it has been showed that by following all Dirac’s steps, the Dirac results and the FJ ones coincide [26]. In this respect, in this paper we also develop a pure canonical analysis and we compare the obtained results with the FJ ones. We will start with a SO(3, 1) invariant four-dimensional BF theory, however, we will break down the Lorentz group in order to work with a compact group, the remaining group will be SO(3). It is important to comment that in [27] a pure canonical analysis of a SO(3, 1) invariant BF theory has been performed, however, in that paper the Dirac brackets were not reported. The reason is that by working with the SO(3, 1) group either the Dirac or FJ constraints of the theory have not a simple structure and this fact difficults the construction of such brackets. In this respect, we report the complete structure of the constraints of the theory, then the Dirac and the generalized FJ brackets are computed, we will show that the Dirac brackets and the FJ ones coincide to each other. In this manner, our results complete and extend those reported in the literature. The paper is organized as follows. In Section 2 the FJ analysis for a four-dimensional BF theory is performed; we report the complete set of FJ constraints. Moreover, in order to obtain a symplectic tensor we fix the gauge, then the generalized FJ brackets are found. In Section 3, we develop a pure canonical analysis of the theory under study. We report the complete structure of the first class and second class constraints and we show that the algebra between the constraints is closed
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in full agreement with the canonical rules of Dirac’s formulation. Then by introducing the Dirac brackets we eliminate the second class constraints. In Section 4 we present some remarks and conclusions. 2. Faddeev–Jackiw framework for BF theory In this section we shall perform the FJ analysis, our laboratory will be given by a four-dimensional BF theory described by the following action S [AIJµ , BKL αβ ] = Ξ
F IJ ∧ BIJ ,
(1)
M
∧ dxβ is a set of six SO(3, 1) valued two forms, the two-form curvature F of the Lorentz connection is defined as usual by Fµν IJ = ∂µ Aν IJ −∂ν AµIJ + AµIK AνKJ − AµJK AνKI . Here, I , J , K . . . = 0, 1, 2, 3 are internal Lorentz indices that can be raised and lowered by the internal metric ηIJ = (−1, 1, 1, 1), xµ are the coordinates that label the points of the four-dimensional manifold M, and α, β, µ, . . . = 0, 1, 2, 3 are space–time indices. By performing the 3 + 1 decomposition and breaking the Lorentz group down to SO(3) we obtain where Ξ is a constant, BIJ =
1 IJ B dxα 2 αβ
the following Lagrangian density L = Ξ
1 1 ηabc Bab 0i A˙ c0i + Bab ij A˙ cij + A0ij ∂c Bab ij + Bab il Acj l + 2Bab 0i Ac0 j 2
2
+ A00i ∂c Bab 0i + Bab 0j Aci j + Bab ij Ac0j + B0a 0i ∂b Ac0i − ∂c Ab0i + Ab0j Acj i + Ac0 j Abij 1 + B0a ij ∂b Acij − ∂c Abij + Abil Acl j + Abi0 Ac0j − Abjl Acl i − Abj0 Ac0i d3 x, 2
(2)
here, a, b, c , . . . = 1, 2, 3, ϵ 0abc = ηabc and i, j, k, l . . . = 1, 2, 3 are lowered and raised with the Euclidean metric ηij = (1, 1, 1). By introducing the following variables [28] Aaij ≡ −ϵijk Aa k , A0ij ≡ −ϵijk A0 k , Babij ≡ −ϵijk Bab k , B0aij ≡ −ϵijk B0a k , Aai ≡ Υai ,
(3)
the Lagrangian takes the following form
L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci
(4)
− −A0 i ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υc k − Ξ ηabc ϵjki Bab 0j Ac0 k − A00i ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υc k − Ξ ηabc ϵ ijk Babk Ac0j − Ξ ηabc B0a 0i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k − Ξ ηabc B0ai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k .
(5)
In this manner, we can identify the symplectic Lagrangian given by (0)
(0)
L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci − V ,
(6)
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(0)
where V is the symplectic potential expressed as (0)
V = −A0 i ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υc k − Ξ ηabc ϵjki Bab 0j Ac0 k − A00i ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υc k − Ξ ηabc ϵ ijk Babk Ac0j
− Ξ ηabc B0a 0i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k − Ξ ηabc B0ai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k .
(7) (0)
Lagrangian (6) we identify the following = From the0i symplectic symplectic variables ξ Aa0i , Bab , Υa i , Babi , A0 i , A00i , B0a 0i , B0ai and the 1-forms a(0) = Ξ ηabc Bab 0i , 0, Ξ ηabc Babi , 0, 0, 0, δ a (y) δ a (x) (0) 0, 0 . In this manner, the symplectic matrix defined as fij (x, y) = δξ ji (x) − δξ ji (y) , is given by
(0)
f
ij
0
Ξ ηabc δji 0 = 0 0 0 0 0
−Ξ ηabc δji 0 0 0 0 0 0 0
0 0 0
Ξη
0 0
δ
abc i j
0 0 0 0
−Ξ ηabc δji 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
(0)
0 0 0 0 δ 3 (x − y), 0 0 0 0
(0)
we observe that fij is singular and therefore, there will constraints. The modes of f the following 4 vectors
i v (0) 1 = 0, 0, 0, 0, V A0 , 0, 0, 0 , v (0) 2 = 0, 0, 0, 0, 0, V A00i , 0, 0 , 0i v (0) 3 = 0, 0, 0, 0, 0, 0, V B0a , 0 , v (0) 4 = 0, 0, 0, 0, 0, 0, 0, V B0ai , Ai0
B0i 0a
(8)
ij
are given by
(9) (10) (11) (12)
where V , V A00i , V and V B0ai are arbitrary functions. Hence, by using these modes we find the following constraints (0)
(0) δ d3 y V (ξ ) i δξ (0) 3 Ai0 δ = d xV d3 y V (ξ ) i δ A0 = ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υck − Ξ ηabc ϵjki Bab 0j Ac0 k , (0) 3 (0) i δ = d xv 2 i d3 y V (ξ ) δξ (0) δ 3 A00i = d xV d3 y V (ξ ) δ A00i = ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υck − Ξ ηabc ϵ ijk Babk Ac0j , (0) δ = d3 xv (0) i3 i d3 y V (ξ ) δξ (0) δ 3 B0a 0i = d xV d3 y V (ξ ) 0i δ B0a = Ξ ηabc ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k ,
Ωi =
(0)
Ω 00i
(0)
Ω 0a 0i
d3 xv (0) i1
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379
(0) δ d3 y V (ξ ) i δξ (0) δ 3 B0ai d3 y V (ξ ) = d xV δ B0ai = Ξ ηabc ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k ,
(0)
Ω 0ai =
d3 xv (0) i4
(13)
we can observe that these constraints are the secondary constraints obtained by using the Dirac method (see the following section). Now we shall observe if there are more constraints, for this aim, we calculate the following system [29] f¯kj ξ˙ (0)j = Zk (ξ ),
(14)
where
δ V (0) δξ (0)j and Zk = 0 . 0
(0)
fij
f¯kj = δ Ω (0) i
δξ (0)j
(15)
0
Thus, the symplectic matrix f¯ij is given by
−Ξ ηabc δji
0
Ξη δ 0 0 0 0 f¯ij = 0 0 −Ξ ηabc ϵjki Bab oj −Ξηabc ϵ ikj Babj 2Ξ ηabc δik ∂c − ϵi kj Υbj abc i j
0 0 0 0 0 0 0
−Ξηabc ϵkji Ac0 j Ξ ηabc δki ∂c − ϵ i jk Υcj 0
−2Ξ ηabc ϵ ijk Ab0j 0 0
−Ξ ηabc δji 0 0 0 0 0
Ξ ηabc δik ∂c + ϵ k ij Υcj −Ξ ηabc ϵ ijk Ac0j 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
Ξ ηabc δji 0 0 0 0
Ξ ηabc ϵ j ik Babj −Ξ ηabc ϵ i jk Bab 0j 2Ξηabc ϵi jk Ab0j
j
2Ξ ηabc δki ∂b + ϵ i jk Υb
0 0 0 0 0 0 3 δ (x − y). 0 0 0 0 0 0
(16)
The matrix fij is not a square matrix as expected, however it has null vectors. The null vectors are given by
ϵkji Ac0 j V i , −ϵjki Bab 0j V i , ∂c V k + ϵ k ij Υci V j , ϵ j ik Babj V i , 0, 0, 0, 0, V i , 0, 0, 0 , = ∂c Vk − ϵ i kj Υcj Vi , ϵ ikj Babj Vi , ϵ kij Ac0j Vi , ϵ k ji Bab 0j V i , 0, 0, 0, 0, 0, Vi , 0, 0 , = 0, 2 ∂b V k − ϵi kj Υbj V i , 0, 2ϵi jk Ab0j V i , 0, 0, 0, 0, 0, 0, V i , 0 ,
V⃗1 =
(17)
V⃗2
(18)
V⃗3
(19)
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j
V⃗4 = 0, 2ϵ ijk Ab0j Vi , 0, 2 ∂b V k + ϵ k ji Υb V i , 0, 0, 0, 0, 0, 0, 0, Vi .
(20)
On the other hand, Zk (ξ ) is given by
(0)
δ V (ξ ) δξ i
Zk (ξ ) =
0
=
Ξ ηabc ϵj ki A0 i Bab 0j + ηabc ϵ ikj A00i Babj + 2Ξ ηabc ∂b B0a 0k abc kj 0i abc ijk − 2Ξ η ϵi B0a Υcj + 2Ξ η ϵ B0ai Ab0j Ξ ηabc ϵikj A0 j Ac0 k + Ξ ηabc ∂c A00i + Ξ ηabc ϵ j ik A00j Υck − Ξ ηabc ϵ j ik A0 i Babj + Ξ ηabc ϵ i jk A00i Bab 0j − 2Ξ ηabc ϵi jk Ab0j B0a 0i −2Ξ ηabc ∂b B0ak − 2ϵ i jk Υbj B0ai Ξ ηabc ∂c A0 i − Ξ ηabc ϵ i jk A0 j Υck + Ξ ηabc ϵ jki A00j Ac0k (0) Ωi . (0) 00i Ω (0) 0a Ω 0i (0) 0ai Ω 0 0 0 0
µ The contraction of the null vectors with Zk , namely, V⃗i Zµ (ξ ) = 0, gives identities. For instance,
from the contraction of V⃗1 with Zk (ξ ) we obtain µ
V⃗1 Zµ (ξ ) = ϵ j ik A0 i V k ∂c Ξ ηabc Babj + Ξ ηϵ m jkl Babm Υcl + Ξ ηabc ϵplj Bab 0p Ac0 l
+ ϵ i kj A00i V k Ξ ηabc ∂c Bab 0j − Ξ ηabc ϵ ji m Babm Ac0 l − Ξ ηϵ j nl Bab 0n Υcl − ϵ i jk B0ai V k Ξ ηabc ∂b Υcj − ∂c Υbj + Ξ ηabc ϵ j pl Υbp Υcl − Ξ ηabc ϵ j pl Ab0 p Ac0 l − ϵijk B0a 0i V k Ξ ηabc ∂b Ac0 j − ∂c Ab0 j + ϵ j ml Υcl Ab0 m − ϵjml Υbl Ac0 m = 0,
(21)
where we can observe that the left hand side vanishes because there is a linear combination of constraints. Hence, there are no more FJ constraints. Furthermore, we will add the constraints given in (13) to the symplectic Lagrangian using the i ˙ i , B0a 0i = ς˙ a , B0ai = χ˙ ai , thus the following Lagrange multipliers, namely, A0 i = T˙ i , A00i = Λ 2 2 symplectic Lagrangian takes the form [29]
(1)
(0)
(0)
˙ i Ω 00i − L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci − T˙ i Ω i − Λ −
χ˙ ai 2
ς˙ ai 2
(0)
Ω 0a 0i
(1)
(0)
Ω 0ai − V ,
where V (1) = V (0) | (0)
(0)
Ω i, Ω
(22)
(0) (0) 00i , Ω 0a , Ω 0ai =0 0i
= 0, this result is expected because of the general covariance
of the theory just as it is present in General Relativity. (1)
From the symplectic Lagrangian (22) we identify the following symplectic variables ξ
Ac0i , Bab ,
0i
Υci
, Babi , T , Λi , ςa , χai and the 1 - forms a i
i
(1)
=
Ξη
abc
Bab , 0, Ξ η 0i
abc
Babi , 0, −Ω
= (0)
i
,
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
−Ω (0) 00i , −
(0)
Ω
0a 0i
2
( 0)
0ai
, − Ω2
(1)
f
ij
. Hence, the symplectic matrix has the following form
−Ξ ηabc δji
0
381
Ξ ηabc δji 0 0 = −Ξ ηabc ϵ k Bab 0j ji −Ξ ηabc ϵ ikj B abj Ξ ηabc d k
0 0 0
0 0 0
η δ η ϵ η ϵ η ϵ η
0 0
bi
Ξ ηabc ϵj ki Bab 0j Ξ ηabc ϵkji Ac0 j −Ξ ηabc ϵ j ik Babj −Ξ ηabc Dcki
Ξ ηabc ϵ ikj Babj −Ξ ηabc dci k Ξ ηabc ϵ i jk Bab 0i Ξ ηabc ϵ ijk Ac0j
0 0 0 0
0 0 0 0
−Ξ ηabc δji
Ξ abc ji Ξ abc j ik Babj Ξ abc i jk Bab 0i j Ξ abc i k Ab0j abc i Ξ db k
−Ξ ηabc ϵkji Ac0 j Ξ ηabc dci k
−Ξ ηabc ϵ ijk Ab0j
0 0
Ξ ηabc dai k
0
Ξ ηabc Dcki Ξ ηabc ϵ ijk Ac0j 0 0
Ξ ηabc ϵ ijk Ab0j
0
0 −Ξ ηabc Dbi j 3 0 δ (x − y), 0 0
−Ξ ηabc ϵi jk Ab0j 0 0 0 0 0
(23)
0 0
where we have used the notation Dal i = δli ∂a + ϵl ik Υak and dai l = δil ∂a − ϵi lk Υak . We can observe that this matrix is singular, however we have showed that there are not more constraints, therefore the theory under study has a gauge symmetry. In order to obtain a symplectic tensor, we fix the following temporal gauge A0 i = 0,
(24)
A00i = 0,
(25)
= 0,
(26)
B0ai = 0,
(27)
B0a
0i
˙ i = 0, ς˙ ai = 0 and χ˙ ai = 0. In this manner, we introduce more Lagrange this mean that T˙ i = 0, Λ multipliers enforcing the gauge fixing. The Lagrange multipliers introduced are βi , α i , ρia , σai , thus, the symplectic Lagrangian takes the form (2)
(0)
(0)
˙i L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci − Ω i − βi T˙ i − Ω 00i − α i Λ
−
(0) 0a Ω 0i 2
− ρia ς˙ ai −
(0) 0ai Ω 2
− σ ai χ˙ ai ,
(28)
(2)
from this symplectic Lagrangian we identify the following symplectic variables ξ = Aa0i , Bab 0i , Υai ,
Babi , T i , Λi , ςai , χai , βi , α i , ρia , σ a(2) =
ai
, and the 1-forms
(0) (0) Ξ ηabc Bab 0i , 0, Ξ ηabc Babi , 0, − Ω i − βi , − Ω 00i − αi , (0) (0) 0a 0ai Ω Ω 0i − − ρi , − − σi , 0, 0, 0, 0 .
2
2
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A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
Thus, the symplectic matrix is given by 0
(2)
f
ij
Ξ ηabc δji 0 0 −Ξ ηabc ϵ i Bab 0j jk = −Ξ ηabc ϵ kij Babj −Ξ ηabc d i bl −Ξ ηabc ϵ kji Ab0j 0 0
−Ξ ηabc δji
0
0
Ξ ηabc ϵj ik Bab oj
0
0
0
Ξ ηabc ϵijk Ac0 j
0
0
−Ξ η
0
Ξ ηabc δji
−Ξ ηabc ϵijk Ac0 j Ξη
abc
0 0
Ξ ηabc ϵ kij Babj
− η
Ξ dci l Ξ k ji Bab 0j
−Ξ ηabc ϵ i jk Υck
ϵ ji Bab
abc k
0j
Ξ ηabc ϵ i jk Υck
0
−Ξ ηabc ϵ kji Ac0j
0 0
0
−Ξ ηabc ϵi jk Ab0j
0
0
Ξ ηabc dbl i
0
0
0 0 0 0
0 0 0 0
0 0 0 0
δji 0 0 0
Ξ ηabc dbj i
Ξ ηabc ϵ kji Ab0j
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0 0
abc
ϵ
−Ξ η
−Ξ ηabc ϵ j ki Babj
0
Ξ ηabc ϵ j ki Babj
dci l
δ
abc i j
Ξη
abc
Ξ ηabc ϵ kji Ac0j
ϵi Ab0j jk
−Ξ η
0 0 0 0 0 0 0
0 0 0 0 0
δji
dbl i
0 0 0 0 0 0 0 0
δba δji
0 0
abc
δba δji
0
−δji
−δji
0 0 0 0 0 0 0
0 0 0 0 0 0
−δba δji 0 0 0 0 0
0 0 0 0 δ 3 (x − y), (29) 0 −δba δji 0 0 0 0
where dai l ≡ δil ∂a − ϵi lk Υak . We can observe that this matrix is not singular, after a long calculation, (2)
the inverse of f
(2)
1 f − ij
ij
is given by
0
1 i − 2Ξ ηbgc δj 0 0 0 = 0 0 0 ϵijl Ac0 j 0 0 0
1 2Ξ
ηbgc δji
0
0
0
0 0
0
−
1
1 2Ξ
ηabc δji
2Ξ 0 0 0 0
0
0
0
0
0
0
0
0
0
0
ηabc δji
0
0
0
0
0
0
0
0
0
0 0 0 0
−δji
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0
−δji
0
0
0 0 0 0 l −ϵj k Bab 0j
−ϵ l jk Υck
0 0 0 0 j −ϵ kl Bmnj
−ϵ klj Babj
ϵ kjl Ac0j
ab − δmn ϵkj l Ab0j
0
ab − δmn Dbkl
0
0
−δba δji
0
0
0
0
0
0
−δba δji
1
ab − δmn Dbi l
2
1
ab kjl − δmn ϵ Ab0j
2
1
2
1 2
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
−ϵijl Ac0 j
0
ϵj lk Bab 0j
ϵ klj Babj
ϵ l jk Υck 2
ab δmn ϵkj l Ab0j
δji
2
Ξ 2
Ξ 2
0 1 2
ab δmn Dakl
0
0 0
δba δji
0 0
Ξ
0
2 mi q
0
ab δmn Dai l
δji
0 0 0
Ξ
2
−ϵ kjl Ac0j 1
ϵ j kl Bmnj
−
0 1
0 mi q
Hjl
−
Ξ 2
Hjl
Ξ ηabc ϵ ijk Babk Ac0j
0
E ai j
0
0
l Fm i
−Gkjpi
0
E ai j
1 2
383
ab kjl δmn ϵ Aboj 0 0 0 0 3 0 δ (x − y), a i δb δj Ξ ml Fi 2 mjpi G 0
(30)
0
where we have defined Dal i ≡ δli ∂a + ϵl ik Υak , E ai j ≡ ηabc ϵ i jp ϵ plk Ab0l Ac0k + ηabc ϵ l pj ϵ i kl Υbk Υcp , l Fm ≡ ηmnc ϵijk ϵ kpl Ac0 j An0p , i jp
Gmjpi ≡ ηmnc ϵ k ϵj li An0l Υck , mi q
Hjl
mn i ≡ ηabc δab ϵ jk ϵ qp l An0p Υck .
Therefore, from the symplectic tensor (30) we can identify the generalized FJ brackets by means of
{ξi(2) (x), ξj(2) (y)}FD = [fij(2) (x, y)]−1 ,
(31)
thus, the following generalized brackets arise Ac0i (x), Bab 0j (y)
Υci (x), Babj (y)
FJ
FJ
= =
1 2Ξ 1 2Ξ
ηabc δij δ 3 (x − y),
(32)
ηabc δji δ 3 (x − y),
(33)
where we can observe that the FJ brackets and the Dirac ones coincide to each other (see the section below). Furthermore, in FJ framework there are less constraints than in Dirac’s framework, in this sense, the FJ is more economical to perform; we will see this point in more detail in the following section. Finally, we carry out the counting of physical degrees of freedom. As we have commented above, in FJ formalism there is not a classification of constraints, they are at the same level, thus, the counting of physical degrees of freedom is performed as [DF = dynamical variables − independent constraints]. In this manner, there are 18 canonical variables given by (Acoi , Υai ) and 18 independent first class constraints (Ω (0)i , Ω (0) 00i , Ω (0) 0ai , Ω (0) 0a 0i ); for BF theory it is well-known that the constraints are reducible, the reducibility between the constraints is given by ∂a Ω (0) 0ai = (0)
ϵij k Υak Ω 0aj + ϵij k Aa0k Ω (0) 0a 0j and ∂a Ω (0) 0a 0i = ϵ i jk Υak Ω (0) 0a 0j + ϵ i k j Aa0k Ω (0) 0aj . Therefore, the theory is devoid of physical degrees of freedom as expected. It is important to comment, that all results found in this section are not reported in the literature.
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3. Hamiltonian analysis In this section a pure Dirac’s canonical analysis for the four-dimensional BF theory will be performed, we will follow all Dirac’s steps in order to obtain the better canonical description of the theory [16]. For this aim, we start with the Lagrangian given in (2) L = Ξ
1 ηabc Bab 0i A˙ c0i + Bab ij A˙ cij 2
1
+ A0ij ∂c Bab ij + Bab il Acj l + 2Bab 0i Ac0 j 2 + A00i ∂c Bab 0i + Bab 0j Aci j + Bab ij Ac0j + B0a 0i ∂b Ac0i − ∂c Ab0i + Ab0j Acj i + Ac0 j Abij 1 + B0a ij ∂b Acij − ∂c Abij + Abil Acl j + Abi0 Ac0j − Abjl Acl i − Abj0 Ac0i d3 x, 2
(34)
by considering the following change of variables [28,30] Aaij ≡ −ϵijk Aa k , A0ij ≡ −ϵijk A0 k , Babij ≡ −ϵijk Bab k , B0aij ≡ −ϵijk B0a k , Aa0i ≡ Aa0i , Aai ≡ Υai , A0 i ≡ −T i , A00i ≡ −Λi , 1 B0a 0i ≡ − ςa i , 2 1 B0ai ≡ − χai , 2
(35)
the Lagrangian takes the following form L = L [Aa0i , Υai , Ti , Λi , ςai , χai , Bab0i , Babi ]
=
abc 0i Ξ η Bab A˙ c0i + Ξ ηabc Babi Υ˙ ci − T i ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υc k − Ξ ηabc ϵjki Bab 0j Ac0 k − Λi ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υc k − Ξ ηabc ϵ ijk Babk Ac0j 1 − Ξ ηabc ςa i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k 2 1 − Ξ ηabc χai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k d3 x. 2
In this manner, the canonically momenta of the dynamical variables are given by
∂L = Ξ ηabc Bbc 0i , ∂ A˙ a0i ∂L π ai ≡ = Ξ ηabc Bbc i , ∂ Υ˙ ai
pa0i ≡
(36)
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
Tˆ i ≡
ˆi ≡ Λ ςˆ ai ≡ χˆ ai ≡ pab0i ≡ pabi ≡
385
∂L = 0, ∂ T˙i ∂L = 0, ˙i ∂Λ ∂L = 0, ∂ ς˙ ai ∂L = 0, ∂ χ˙ ai ∂L = 0, ˙ ∂ Bab0i ∂L = 0, ∂ B˙ abi
(37)
with the following non-vanishing fundamental Poisson brackets between the fields
1 j Υai (x), π bj (y) = δab δi δ 3 (x − y), 2 1
(y) = δab δij δ 3 (x − y), 2 1 j j ˆ Ti (x), T (y) = δi δ 3 (x − y), 2 1 ˆ j (y) = δij δ 3 (x − y), Λi (x), Λ 2 ςai (x), ςˆ bj (y) = δab δij δ 3 (x − y), χai (x), χˆ bj (y) = δab δij δ 3 (x − y), 1 d e δa δb − δae δbd δij δ 3 (x − y), Bab0i (x), pde0j (y) =
Aa0i (x), p
b0j
Babi (x), p (y) = dej
4 1 4
δad δbe − δae δbd δij δ 3 (x − y).
(38)
Furthermore, from the definition of the momenta, we identify the following 60 primary constraints
φ1a0i ≡ pa0i − Ξ ηabc Bbc 0i ≈ 0, φ2ai ≡ π ai − Ξ ηabc Bbc i ≈ 0, φ3i ≡ Tˆ i ≈ 0, ˆ i ≈ 0, φ4i ≡ Λ φ5ai ≡ ςˆ ai ≈ 0, φ6ai ≡ χˆ ai ≈ 0, φ7ab0i ≡ pab0i ≈ 0, φ8abi ≡ pabi ≈ 0.
(39)
The canonical Hamiltonian of the theory is given by Hc =
=
˙ iΛ ˆ i + ς˙ ai ςˆ ai + B˙ ab0i pab0i + B˙ abi pabi − L d3 x A˙ a0i pa0i + Υ˙ ai π ai + T˙i Tˆ i + Λ
i T ∂a π a i − ϵi jk π a j Υak − ϵijk pa0j Aa0 k + Λi ∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j
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A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
1 + Ξ ηabc ςa i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck + ϵijk Ac0 j Υb k 2 3 1 abc i i i j k ijk + Ξ η χai ∂b Υc − ∂c Υb + ϵ jk Υb Υc − ϵ Ab0j Ac0k d x, 2
(40)
by adding the primary constraints we obtain the primary Hamiltonian H1 =
jk
Hc + T i ∂a π a i − ϵi π a j Υak − ϵijk pa0j Aa0 k
+ Λi ∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j 1 + Ξ ηabc ςa i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck + ϵijk Ac0 j Υb k 2 1
+ Ξ ηabc χai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k 2 ˆi + λa0i pa0i − Ξ ηabc Bbc 0i + λai π ai − Ξ ηabc Bbc i + αi Tˆ i + βi Λ + θai ςˆ ai + µai χˆ ai + λab0i pab0i + λabi pabi d3 x,
(41)
where λa0i , λai , αi , βi , θai , µai , λab0i , λabi are Lagrange multipliers enforcing the primary constraints. From consistency of the constraints, we identify the following 24 secondary constraints
φ˙ 3i ≈ 0 ⇒ ϕ3i ≡ − φ˙ 4i ≈ 0 ⇒ ϕ4i ≡ − φ˙ 5ai ≈ 0 ⇒ ϕ5ai ≡ −
1
∂a π ai − ϵ ijk π a j Υak − ϵ i jk pa0j Aa0 k ≈ 0,
(42)
∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j ≈ 0,
(43)
2 1 2
Ξ 2
ηabc ∂b Ac 0i − ∂c Ab 0i − ϵ ijk Ab0j Υck + ϵ ijk Ac0j Υbk
≈ 0,
(44)
Ξ φ˙ 6ai ≈ 0 ⇒ ϕ6ai ≡ − ηabc ∂b Υc i − ∂c Υb i + ϵ ijk Υbj Υck − ϵ ijk Ab0j Ac0k 2
≈ 0,
(45)
and the following 36 Lagrange multipliers
λab 0i ≈
λab i
1
ηabc ϵ i jk T j pc0k − ηabc ϵ ijk Λj π c k + Ξ ∂b ςa i − Ξ ∂a ςb i − Ξ ϵ ij k ςa j Υbk + Ξ ϵ ij k ςb j Υak + ϵ ijk χaj Ab0k − ϵ ijk χbj Aa0k , 1 ≈ ηabc ϵ ij k T j π c k + ηabc ϵ ij k pc0k Λj − Ξ ϵ ij k ςa j Ab0k + Ξ ϵ ij k ςb j Aa0k 2Ξ + Ξ ∂b χa i − Ξ ∂a χb i − Ξ ϵ ijk χaj Υbk + Ξ ϵ ijk χbj Υak , 2Ξ
λa0i ≈ 0, λai ≈ 0.
(46)
(47) (48) (49)
For this theory there are not tertiary constraints. Hence, in order to perform the classification of the constraints in first class and second class we proceed to calculate the following matrix whose entries are the Poisson brackets between all the constraints, it is
1 ϕ3i (x), ϕ3l (y) = − ϵ ilk ϕ3k δ 3 (x − y) ≈ 0,
(50)
1 ϕ4i (x), ϕ4l (y) = ϵ ilk ϕ3k δ 3 (x − y) ≈ 0.
(51)
4
4
Ξ φ1a0i (x), φ7de0l (y) = ηade ηil δ 3 (x − y), 2
(52)
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
1 φ1a0i (x), ϕ3l (y) = − ϵ il j pa0j δ 3 (x − y),
387
(53)
4
a0i 1 φ1 (x), ϕ4l (y) = ϵ ilj π a j δ 3 (x − y),
(54)
a0i Ξ il ade −η η ∂x,e + ηade ϵ ilj Υej δ 3 (x − y), φ1 (x), ϕ5dl (y) = −
(55)
a0i Ξ φ1 (x), ϕ6dl (y) = − ηade ϵ ilj Ae0j δ 3 (x − y),
(56)
ai Ξ φ2 (x), φ8del (y) = − ηil ηade δ 3 (x − y),
(57)
ai φ2 (x), ϕ3l (y) = − ϵ ilj π a j δ 3 (x − y),
(58)
ai φ2 (x), ϕ4l (y) = − ϵ il j pa0j δ 3 (x − y),
(59)
ai Ξ φ2 (x), ϕ5dl (y) = − ηade ϵ ilj Ae0j δ 3 (x − y),
(60)
ai Ξ φ2 (x), ϕ6dl (y) = ηade −ηil ∂x,e + ϵ ilj Υej δ 3 (x − y),
(61)
4
2 2
2 1 4 1 4
2
2
i 1 ϕ3 (x), ϕ4l (y) = − ϵ il j ϕ4j δ 3 (x − y) ≈ 0,
(62)
4 1
ϕ3i (x), ϕ5dl (y) = − ϵ il k ϕ5dk δ 3 (x − y) ≈ 0,
(63)
ϕ3i (x), ϕ6dl (y) = − ϵ il k ϕ6dk δ 3 (x − y) ≈ 0,
(64)
1 ϕ4i (x), ϕ5dl (y) = ϵ il k ϕ6k δ 3 (x − y) ≈ 0,
(65)
1 ϕ4i (x), ϕ6dl (y) = − ϵ il j ϕ5dj δ 3 (x − y) ≈ 0
(66)
4 1 4
4
4
we find that this matrix has rank = 36 and 48 null vectors. From the null vectors we find the following 48 first class constraints
γ1i ≡ Tˆ i ≈ 0, ˆ i ≈ 0, γ2i ≡ Λ γ3ai ≡ ςˆ ai ≈ 0, γ4ai ≡ χˆ ai ≈ 0, γ5 i ≡ ∂a π a i − ϵi jk π a j Υak − ϵijk pa0j Aa0 k + γ60i ≡ ∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j + γ7a 0i ≡
Ξ 2
1 2Ξ 1 2Ξ
ηabc ϵijk π aj pbck − pa0j pbc0k ≈ 0, ηabc ϵ i jk π aj pbc0k + pa0j pbck ≈ 0,
ηabc ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k
− ∂b pab 0i + ϵi jk Υbk pab 0j + ϵi jk Ab0k pab j ≈ 0, Ξ γ8ai ≡ ηabc ∂b Υc i − ∂c Υb i + ϵ ijk Υbj Υck − ϵ ijk Ab0j Ac0k 2
− ∂b pabi − ϵ ijk Ab0k pab 0j + ϵ ijk Υbk pab j ≈ 0
(67)
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and the rank allows us to identify the following 36 second class constraints
Γ1a0i ≡ pa0i − Ξ ηabc Bbc 0i ≈ 0, Γ2ai ≡ π ai − Ξ ηabc Bbc i ≈ 0, Γ3ab0i ≡ pab0i ≈ 0, Γ4abi ≡ pabi ≈ 0.
(68)
It is important to remark that the complete structure of the constraints (67) is not reported in the literature and this is a result of performing a pure Dirac’s formulation. In fact, by working with the standard form, it is not possible to obtain a full structure of the constraints and the algebra could not be closed just as is present in four-dimensional Palatini’s theory [15]. Furthermore, in order to compare the symplectic framework with the Dirac one, it is necessary to follow all steps of the Dirac formulation as has been developed in this paper. With all information obtained, we can carry out the counting of physical degrees of the theory in the following form; there are 120 dynamical variables, 48 first class constraints and 36 second class constraints, hence we obtain −6 degrees of freedom. However, it is well-known that BF theory is a reducible theory, this is, the constraints are not independent to each other. The reducibility of the constraints is given by 1
1
∂a γ7a 0i = ϵi jk Υak γ7a 0j + ϵi jk Aa0k γ8a j + ϵi jk Fabk Γ3ab0j + ϵi jk Fab0k Γ4abj ∂ γ =ϵ ai a 8
ijk
γ −ϵ
Υak 8a j
ijk
γ
Aa0k 7a 0j
2 1
− ϵ 2
ijk
Fab0k Γ3ab0j
2 1
+ ϵ ijk Fabk Γ4abj , 2
hence, there are 42 independent first class constraints. Therefore, by performing the counting of physical degrees of freedom we conclude that the theory is devoid of degrees of freedom, the theory is a topological one as expected. The algebra between the constraints is given by
Ξ Γ1a0i (x), Γ3de0l (y) = ηade ηil δ 3 (x − y), 2
ai Ξ Γ2 (x), Γ4del (y) = − ηade ηil δ 3 (x − y), 2
γ (x), γ (y) = i 5
l 5
γ5i (x), γ60l (y) =
γ5 i (x), γ7d 0l (y) =
1 2 1 2 1 2
ϵ il j γ5j δ 3 (x − y) ≈ 0, ϵ il j γ6j δ 3 (x − y) ≈ 0, ϵil k γ7d 0k −
1 4Ξ
ηabc Γ4bcl Γ3da0i +
1 4Ξ
da ηabc Γ3bc0 l Γ4 i
≈ 0, 1 1 1 l γ5 i (x), γ8dl (y) = ϵi lk γ8d k − ηabc Γ3bc0l Γ3da0i + δi ηabc Γ3bc0k Γ3da0k 2 4Ξ 4Ξ +
1 4Ξ
ηabc δil Γ4bck Γ4dak −
1 4Ξ
ηabc Γ4bcl Γ4dai
≈ 0, 0i 1 γ6 (x), γ60l (y) = − ϵ ilj γ5 j δ 3 (x − y) ≈ 0, 2
0i 1 1 l 1 da0k da0i d − δi ηabc Γ3bc0 + ηabc Γ3bc0 γ6 (x), γ7d 0l (y) = ϵ il k γ8k k Γ3 l Γ3 2 4Ξ 4Ξ
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
1
+
4Ξ
ηabc δli Γ4bck Γ4dak −
1 4Ξ
389
ηabc Γ4bcl Γ4dai
≈ 0, 0i 1 1 1 γ6 (x), γ8dl (y) = − ϵ iljγ7d 0j + ηabc g il Γ3bc0k Γ4dak − ηabc Γ3bc0l Γ4dai 2 4Ξ 4Ξ 1
−
4Ξ
ηabc Γ4bcl Γ3da0i +
1 4Ξ
g il ηabc Γ4bck Γ3da0 k
≈ 0, a 1 γ7 0i (x), Γ1d0l (y) = − ϵi lj pad j δ 3 (x − y) 2 1
= − ϵi lj Γ4adj δ 3 (x − y) ≈ 0, 2
γ
a 7 0i
(x),
Γ2dl
1 (y) = − ϵi lj pad0j δ 3 (x − y) 2 1
= − ϵi lj Γ3ad0j δ 3 (x − y) ≈ 0, 2
ai 1 γ8 (x), Γ1d0l (y) = ϵ ilj pad 0j = ai γ8 (x), Γ2dl (y) = =
2 1 2 1 2 1 2
ϵ ilj Γ3ad0j ≈ 0, ϵ ilj pad j ϵ ilj Γ4adj ≈ 0,
(69)
where we observe that the algebra is closed and it obeys the rules of the canonical formalism of the algebra between the constraints [16,17,21], namely, the result of the Poisson brackets between first class constraints with first class must be linear in first class constraints and square in second class; the result of the Poisson brackets between first class constraints with second class constraints must be linear in first class constraints and linear in second class constraints. We can observe that our results are in full agreement with these rules. Furthermore, the constraints γ5 i and γ60i are identified as generators of rotations and boost respectively whereas γ7 0i and γ8 ai are generators of translations, this can be seen from the algebra between these constraints. On the other hand, first class constraints are generators of gauge transformations. Hence, by defining the gauge generator in terms of first class constraints
G=
θ1,i γ1i + θ2,i γ2i + θ3,ai γ3ai + θ4,ai γ4ai + θ5i γ5,i + θ6,0i γ60i + θ70i,a γ7a,0i + θ8,ai γ8ai d3 x,
we find the following gauge transformations
δ Ad0l = {Ad0l , G} 1 1 1 = −∂d θ6,0l + θ5i ϵlik Ad0 k − ϵ i lk θ6,0i Υd k + ηdbc ϵlik θ5i pbc0k + ηdbc ϵ i lk pbck 2 2Ξ 2Ξ δ Υdl = {Υdl , G} 1 1 1 k i ij i bck i bc0k −∂d θ5,l + ϵli θ5 Υdk − ϵl θ6,0i Ad0l − = ηdbc ϵlik θ5 p + ηdbc ϵ lk θ6,0i p 2 2Ξ 2Ξ δ Tl = {Tl , G} = θ1,l δ Λi = {Λl , G} = θ2,l
(70)
(71) (72) (73)
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A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
δςdl = {ςdl , G} = θ3,dl δχdl = {χdl , G} = θ4,dl δ Bde0l = {Bde0l , G} 1 ∂e θ7,d0l − ∂d θ7,e0l − θl ij θ7,d0j Υek + ϵl ij θ7,e0i Υdj − ϵl ik θ8,di Ae0k = 4 1 1 ik i a0j i aj + ϵl θ8,ei Adok − ηdea ϵlij θ5 p + ηdea ϵl j θ6,0i π Ξ Ξ δ Bdel = {Bdel , G} 1 ∂e θ8,dl − ∂d θ8,el − ϵl ik θ8,di Υek + ϵl ik θ8,ei Υdk − ϵli k θ70i,d Ae0k + ϵli k θ70i,e Ad0k = 4 1 1 + ηdea ϵlij θ5i π aj + ηdea ϵl ij θ6,0i pa0j Ξ Ξ δ pd0l = pd0l , G 1 lj = Ξ ηdab ∂b θ70l,a + ϵ l ij θ5i pd0j − ϵ lij θ6,0i π d k − ϵ j θ7,a 0i pad j + ϵ lij θ8,ai pad 0j 2 − Ξ ηdac ϵ li k θ7,a 0i Υck + Ξ ηdac ϵ lik θ8,ai Ac0k δπ dl = π dl , G 1 lj lj Ξ ηdab ∂b θ8l ,a + ϵ i θ5i π d j + ϵ li j θ6,0i pd0j + ϵ i θ7,a 0i pda 0j + ϵ lij θ8,ai pda j = 2 − Ξ ηdab ϵ li j θ7,a 0i Ab0j − Ξ ηdac ϵ lik θ8,ai Υck δ Tˆ l = Tˆ l , G = 0, ˆ l , G = 0, ˆl = Λ δΛ δ ςˆ dl = ςˆ dl , G = 0, δ χˆ dl = χˆ l , G = 0, δ pde0l = pde0l , G = 0, δ pdel = pdel , G = 0.
(74) (75)
(76)
(77)
(78)
(79) (80) (81) (82) (83) (84) (85)
Now, with the second class constraints we can calculate the Dirac brackets. In order to perform this aim we calculate the matrix C αβ = Γ α , Γ β whose entries are given by the Poisson brackets of the second class constraints, namely
Cαβ
a0i d0l Γ1 , Γ1 Γ ai , Γ d0l 2 1 = ab0i Γ3 , Γ1d0l abi d0l Γ4 , Γ1
Γ1a0i , Γ2dl ai dl Γ2 , Γ2 ab0i Γ3 , Γ2dl abi dl Γ4 , Γ2
0
0 = Ξ abd il − η η 2 0
0
Ξ 2
Ξ 2
Γ1a0i , Γ3de0l ai de0l Γ2 , Γ3 ab0i Γ3 , Γ3de0l abi de0l Γ4 , Γ3
ηade ηil
0
0
0
0
ηabd ηil
0
0
Ξ
Γ1a0i , Γ4del ai del Γ2 , Γ4 ab0i Γ3 , Γ4del abi del Γ4 , Γ4
− η η 3 2 δ (x − y), 0 ade il
0
(86)
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
391
and the inverse of Cαβ is given by
−1 Cαβ
0
−
0
0 = 1 ηabd ηil Ξ −
0
1
Ξ
2
Ξ
ηade ηil
0
0
0
0
ηabd ηil
0
0
ηade ηil 3 Ξ δ (x − y). 0 2
(87)
0
Hence, the Dirac brackets between two functionals, namely F (q, p) and G(q, p), are defined by
{F , G}D ≡ {F , G} −
−1 dudv {F , Γ α (u)} Cαβ (u, v) Γ β (v), G ,
(88)
where Γ α represent the second class constraints. Therefore, the Dirac brackets between the fields are given by Aa0i (x), pd0j (y)
D
Υai (x), π dj (y) D Ti (x), Tˆ j (y) D j ˆ (y) Λi (x), Λ D ςai (x), ςˆ dj (y) D χai (x), χˆ dj (y) D Bab0i (x), pde0j (y) D Babi (x), pdej (y) D Aa0i (x), Bde0j (y) D
1 = Aa0i (x), pd0j (y) = δad δij δ 3 (x − y), 2 1 d
(89)
= Υai (x), π dj (y) = δa δij δ 3 (x − y), 2 1 j ˆ = Ti (x), T (y) = δij δ 3 (x − y), 2 1 d0j ˆ (y) = δij δ 3 (x − y), = Λa0i (x), Λ 2 = ςai (x), ςˆ dj (y) = δad δij δ 3 (x − y), = χai (x), χˆ dj (y) = δad δij δ 3 (x − y),
(90)
= 0,
(95)
= 0,
(96)
=−
Υai (x), Bdej (y) D =
1
2Ξ 1
2Ξ
ηade ηij δ 3 (x − y),
ηade ηij δ 3 (x − y).
(91) (92) (93) (94)
(97) (98)
In addition, we can also calculate the Dirac brackets by gauge fixing, in this case we will use the temporal gauge, namely, we take Ti ≈ 0,
(99)
Λi ≈ 0, ςai ≈ 0, χai ≈ 0.
(100) (101) (102)
These conditions are considered as a new set of second class constraints, hence, now there are the following 84 second class constraints
Γ1a0i ≡ pa0i − Ξ ηabc Bbc 0i ≈ 0, Γ2ai ≡ π ai − Ξ ηabc Bbc i ≈ 0, Γ3ab0i ≡ pab0i ≈ 0, Γ4abi ≡ pabi ≈ 0,
392
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≡ Ti ≈ 0, ≡ Λi ≈ 0, ≡ ςai ≈ 0, ≡ χai ≈ 0, Γ9i ≡ Tˆi ≈ 0, ˆ i ≈ 0, Γ10i ≡ Λ Γ5i Γ6i Γ7ai Γ8ai
Γ11ai ≡ ςˆ ai ≈ 0, Γ12ai ≡ χˆ ai ≈ 0.
(103)
β
In this manner, the matrix Cαβ = Γ α , Γ whose entries are given by the Poisson brackets between the second class constraints (103) is given by
0
0
Ξ 2
ηade ηil
0 0 Ξ − ηabd ηil 0 2 Ξ abd il η η 0 2 0 0 Cαβ = 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
ηade ηil
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 l δ 2 i
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 − δli 2
0
0
0
0
0
1 − δli 2
0
0
0
0
0
−δda δli
0
0
−
Ξ 2
0 0
0
0
0
0
0
0
0
−δda δli
0
0
0
0
0
3 1 l δi 0 0 δ (x − y), 2 0 δad δil 0 0 0 δad δil 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
(104) and its inverse
0
0
0 0 1 η η 0 Ξ abd il 1 0 − ηabd ηil Ξ −1 Cαβ = 0 0 0 0 0 0 0 0 0 0 0 0
−
2
Ξ
ηade ηil 0
0 2
Ξ
0
0
0
0
ηade ηil 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2δil
0
0
0
0
0
0
2δil
0
0 0
0
0
0
0
0
0
δad δil
0
0
0
0
0
0
0
δad δil
0
0
0
0
0 0 0 0 0 0 0 0 3 δ (x − y). i −2δl 0 0 0 i 0 −2δl 0 0 d i 0 0 −δa δl 0 0 0 0 −δad δli 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
(105) In this manner, by using (105) we construct the Dirac brackets given by Aa0i (x), pd0j (y)
D
1 = Aa0i (x), pd0j (y) = δad δij δ 3 (x − y), 2
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
Υai (x), π dj (y) D Ti (x), Tˆ j (y) D j ˆ Λi (x), Λ (y) D ςai (x), ςˆ dj (y) D χai (x), χˆ dj (y) D Bab0i (x), pde0j (y) D Babi (x), pdej (y) D Aa0i (x), Bde0j (y) D
393
1 = Υai (x), π dj (y) = δad δij δ 3 (x − y), 2
= 0, = 0, = 0, = 0, = 0, = 0, =−
Υai (x), Bdej (y) D =
1
2Ξ 1
2Ξ
ηade ηij δ 3 (x − y),
ηade ηij δ 3 (x − y),
(106)
where we can observe that the Dirac brackets and the generalized FJ brackets coincide to each other. 4. Conclusions and prospects In this paper, the symplectic analysis of a four-dimensional BF theory has been performed. We reported the complete set of FJ constraints and we observe that there are present less constraints than in Dirac’s method. Furthermore, we have carried out the counting of physical degrees of freedom concluding that the theory is a topological one as expected. In addition, we have used a temporal gauge in order to obtain a symplectic tensor, then the quantization brackets of FJ were obtained. On the other hand, a pure canonical analysis has been performed. Under a laborious work, we have reported the complete set of first class and second class constraints, the algebra between the constraints is in full agreement with the canonical rules; then using a temporal gauge the Dirac brackets were computed. The FJ and Dirac’s brackets coincide to each other, thus we can conclude that the FJ is more economical than Dirac framework. Of course, if in the Dirac approach are introduced the Dirac brackets and the second class constraints are considered as strong equations, then the FJ and Dirac’s constraints coincide to each other. Finally we would like to comment that in this work we provide the necessary tools for studying in alternative way the BF formulations of gravity such as that reported in [22]. In fact, in that work the canonical formulation of BF gravity has been performed, and will be interesting to study that theory by using the ideas of the symplectic formalism of FJ. All these ideas are in progress and will be the subject of forthcoming works. Acknowledgements 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. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Further reading [1] L. Liao, Y.C. Huang, Ann. Phys. 322 (2007) 2469–2484.
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Faddeev–Jackiw quantization of four dimensional BF theory Alberto Escalante ∗ , Prihel Cavildo Sánchez Instituto de Física, Universidad Autónoma de Puebla, Apartado Postal J-48 72570, Puebla Pue., Mexico
highlights • • • • •
We report a detailed symplectic analysis for BF theory. We report the Faddeev–Jackiw constraints. A pure Dirac’s analysis is performed. The complete structure of Dirac’s constraints is reported. We show that symplectic and Dirac’s brackets coincide to each other.
article
abstract
info
Article history: Received 11 July 2016 Accepted 9 September 2016 Available online 13 September 2016 Keywords: Topological theories Symplectic quantization Hamiltonian dynamics
The symplectic analysis of a four dimensional BF theory in the context of the Faddeev–Jackiw symplectic approach is performed. It is shown that this method is more economical than Dirac’s formalism. In particular, the complete set of Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported. In addition, we show that the generalized Faddeev–Jackiw brackets and the Dirac ones coincide to each other. Finally, the similarities and advantages between Faddeev–Jackiw method and Dirac’s formalism are briefly discussed. © 2016 Elsevier Inc. All rights reserved.
1. Introduction It is well-known that the topological theories have a relevant role in the context of gravity. In fact, topological theories are good laboratories for testing classical and quantum ideas of generally covariant gauge systems. Topological theories are characterized by lacking of physical degrees of
∗
Corresponding author. E-mail addresses: [email protected] (A. Escalante), [email protected] (P.C. Sánchez).
http://dx.doi.org/10.1016/j.aop.2016.09.003 0003-4916/© 2016 Elsevier Inc. All rights reserved.
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freedom, either in three or four dimensions they have a close relation with General Relativity [GR] just as the background independence and the diffeomorphisms covariance, this is, all the dynamical variables characterizing the theory are dynamical ones. In the three dimensional case a relevant example of topological theory is the Chern–Simons theory. In fact, basically Chern–Simons theory describes GR, it has been showed that these theories are equivalent up to a total derivative [1,2], and also there exists a relation between these theories defined with (or without) an Immirizi-like parameter [3,4]. Furthermore, we can find a recent work where the Chern–Simons state describes a topological state with unbroken diffeomorphism invariance in Yang–Mills and GR [5]. In the Loop Quantum Gravity context, that state is called the Kodama state and has been studied in interesting works by Smolin, arguing that the Kodama state at least for the de Sitter spacetime, Loop Quantum Gravity does have a good low energy limit [6]. On the other hand, in four dimensions there exists the so-called BF theory. In fact, BF theories were introduced as generalizations of three dimensional Chern–Simons actions or in other cases, can also be considered as a zero coupling limit of Yang–Mills theories [7,8]. Moreover, we find in the literature several examples where BF theories with additional extra constraints describe gravity, for instance, the well-known formulations of Plebanski and MacDowell–Mansouri [9,10]. In addition, within the modern quantization scheme using tools developed in Loop Quantum Gravity, BF theories have been studied in the context of spin foams. In fact, in this approach is not considered the traditional Fock space formalism but holonomies along paths as the basic variables to be quantized [11]. With respect to the classical context, there are several works studying the canonical structure of a BF theory, see for instance Refs. [12–14]. However, in these works has been used the canonical formalism by using a reduced phase space, this means, it has been considered as dynamical variables only those that occur in the Lagrangian density with temporal derivative, however, in several cases this approach is not convenient, for instance in Palatini’s theory the price to pay for developing the standard approach is that we cannot know the full structure of the constraints and their algebra is not closed [15], thus, the better way to carry out the canonical formalism is by following all Dirac’s steps as it has been commented in [16–21]. In consideration with the commented above, either the classical or the quantum study of BF theories and their close relation with GR are at the present a frontier subject of study [22,23]. In this manner, with the ideas explained previously in this work the Faddeev–Jackiw [FJ] symplectic quantization of a four-dimensional BF theory is performed. In fact, the FJ method provides an alternative approach for studying constrained systems based on a first-order Lagrangian [24,25]. The FJ method is a symplectic study and the basic feature of this approach is to treat all the constraints at the same footing. In other words, in FJ method one avoids the classification of the constraints into firstclass and second-class ones as in Dirac’s framework is done. In addition, some essential elements of a physical theory such as the degrees of freedom, the gauge symmetry and the quantization brackets called the generalized FJ brackets can also be derived; Dirac’s and generalized FJ brackets coincide to each other. However, it is important to remark that in the canonical formalism we must to work by following all Dirac’s steps in order to compare with the FJ symplectic formalism. In fact, it has been showed that by following all Dirac’s steps, the Dirac results and the FJ ones coincide [26]. In this respect, in this paper we also develop a pure canonical analysis and we compare the obtained results with the FJ ones. We will start with a SO(3, 1) invariant four-dimensional BF theory, however, we will break down the Lorentz group in order to work with a compact group, the remaining group will be SO(3). It is important to comment that in [27] a pure canonical analysis of a SO(3, 1) invariant BF theory has been performed, however, in that paper the Dirac brackets were not reported. The reason is that by working with the SO(3, 1) group either the Dirac or FJ constraints of the theory have not a simple structure and this fact difficults the construction of such brackets. In this respect, we report the complete structure of the constraints of the theory, then the Dirac and the generalized FJ brackets are computed, we will show that the Dirac brackets and the FJ ones coincide to each other. In this manner, our results complete and extend those reported in the literature. The paper is organized as follows. In Section 2 the FJ analysis for a four-dimensional BF theory is performed; we report the complete set of FJ constraints. Moreover, in order to obtain a symplectic tensor we fix the gauge, then the generalized FJ brackets are found. In Section 3, we develop a pure canonical analysis of the theory under study. We report the complete structure of the first class and second class constraints and we show that the algebra between the constraints is closed
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377
in full agreement with the canonical rules of Dirac’s formulation. Then by introducing the Dirac brackets we eliminate the second class constraints. In Section 4 we present some remarks and conclusions. 2. Faddeev–Jackiw framework for BF theory In this section we shall perform the FJ analysis, our laboratory will be given by a four-dimensional BF theory described by the following action S [AIJµ , BKL αβ ] = Ξ
F IJ ∧ BIJ ,
(1)
M
∧ dxβ is a set of six SO(3, 1) valued two forms, the two-form curvature F of the Lorentz connection is defined as usual by Fµν IJ = ∂µ Aν IJ −∂ν AµIJ + AµIK AνKJ − AµJK AνKI . Here, I , J , K . . . = 0, 1, 2, 3 are internal Lorentz indices that can be raised and lowered by the internal metric ηIJ = (−1, 1, 1, 1), xµ are the coordinates that label the points of the four-dimensional manifold M, and α, β, µ, . . . = 0, 1, 2, 3 are space–time indices. By performing the 3 + 1 decomposition and breaking the Lorentz group down to SO(3) we obtain where Ξ is a constant, BIJ =
1 IJ B dxα 2 αβ
the following Lagrangian density L = Ξ
1 1 ηabc Bab 0i A˙ c0i + Bab ij A˙ cij + A0ij ∂c Bab ij + Bab il Acj l + 2Bab 0i Ac0 j 2
2
+ A00i ∂c Bab 0i + Bab 0j Aci j + Bab ij Ac0j + B0a 0i ∂b Ac0i − ∂c Ab0i + Ab0j Acj i + Ac0 j Abij 1 + B0a ij ∂b Acij − ∂c Abij + Abil Acl j + Abi0 Ac0j − Abjl Acl i − Abj0 Ac0i d3 x, 2
(2)
here, a, b, c , . . . = 1, 2, 3, ϵ 0abc = ηabc and i, j, k, l . . . = 1, 2, 3 are lowered and raised with the Euclidean metric ηij = (1, 1, 1). By introducing the following variables [28] Aaij ≡ −ϵijk Aa k , A0ij ≡ −ϵijk A0 k , Babij ≡ −ϵijk Bab k , B0aij ≡ −ϵijk B0a k , Aai ≡ Υai ,
(3)
the Lagrangian takes the following form
L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci
(4)
− −A0 i ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υc k − Ξ ηabc ϵjki Bab 0j Ac0 k − A00i ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υc k − Ξ ηabc ϵ ijk Babk Ac0j − Ξ ηabc B0a 0i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k − Ξ ηabc B0ai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k .
(5)
In this manner, we can identify the symplectic Lagrangian given by (0)
(0)
L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci − V ,
(6)
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(0)
where V is the symplectic potential expressed as (0)
V = −A0 i ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υc k − Ξ ηabc ϵjki Bab 0j Ac0 k − A00i ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υc k − Ξ ηabc ϵ ijk Babk Ac0j
− Ξ ηabc B0a 0i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k − Ξ ηabc B0ai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k .
(7) (0)
Lagrangian (6) we identify the following = From the0i symplectic symplectic variables ξ Aa0i , Bab , Υa i , Babi , A0 i , A00i , B0a 0i , B0ai and the 1-forms a(0) = Ξ ηabc Bab 0i , 0, Ξ ηabc Babi , 0, 0, 0, δ a (y) δ a (x) (0) 0, 0 . In this manner, the symplectic matrix defined as fij (x, y) = δξ ji (x) − δξ ji (y) , is given by
(0)
f
ij
0
Ξ ηabc δji 0 = 0 0 0 0 0
−Ξ ηabc δji 0 0 0 0 0 0 0
0 0 0
Ξη
0 0
δ
abc i j
0 0 0 0
−Ξ ηabc δji 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
(0)
0 0 0 0 δ 3 (x − y), 0 0 0 0
(0)
we observe that fij is singular and therefore, there will constraints. The modes of f the following 4 vectors
i v (0) 1 = 0, 0, 0, 0, V A0 , 0, 0, 0 , v (0) 2 = 0, 0, 0, 0, 0, V A00i , 0, 0 , 0i v (0) 3 = 0, 0, 0, 0, 0, 0, V B0a , 0 , v (0) 4 = 0, 0, 0, 0, 0, 0, 0, V B0ai , Ai0
B0i 0a
(8)
ij
are given by
(9) (10) (11) (12)
where V , V A00i , V and V B0ai are arbitrary functions. Hence, by using these modes we find the following constraints (0)
(0) δ d3 y V (ξ ) i δξ (0) 3 Ai0 δ = d xV d3 y V (ξ ) i δ A0 = ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υck − Ξ ηabc ϵjki Bab 0j Ac0 k , (0) 3 (0) i δ = d xv 2 i d3 y V (ξ ) δξ (0) δ 3 A00i = d xV d3 y V (ξ ) δ A00i = ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υck − Ξ ηabc ϵ ijk Babk Ac0j , (0) δ = d3 xv (0) i3 i d3 y V (ξ ) δξ (0) δ 3 B0a 0i = d xV d3 y V (ξ ) 0i δ B0a = Ξ ηabc ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k ,
Ωi =
(0)
Ω 00i
(0)
Ω 0a 0i
d3 xv (0) i1
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379
(0) δ d3 y V (ξ ) i δξ (0) δ 3 B0ai d3 y V (ξ ) = d xV δ B0ai = Ξ ηabc ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k ,
(0)
Ω 0ai =
d3 xv (0) i4
(13)
we can observe that these constraints are the secondary constraints obtained by using the Dirac method (see the following section). Now we shall observe if there are more constraints, for this aim, we calculate the following system [29] f¯kj ξ˙ (0)j = Zk (ξ ),
(14)
where
δ V (0) δξ (0)j and Zk = 0 . 0
(0)
fij
f¯kj = δ Ω (0) i
δξ (0)j
(15)
0
Thus, the symplectic matrix f¯ij is given by
−Ξ ηabc δji
0
Ξη δ 0 0 0 0 f¯ij = 0 0 −Ξ ηabc ϵjki Bab oj −Ξηabc ϵ ikj Babj 2Ξ ηabc δik ∂c − ϵi kj Υbj abc i j
0 0 0 0 0 0 0
−Ξηabc ϵkji Ac0 j Ξ ηabc δki ∂c − ϵ i jk Υcj 0
−2Ξ ηabc ϵ ijk Ab0j 0 0
−Ξ ηabc δji 0 0 0 0 0
Ξ ηabc δik ∂c + ϵ k ij Υcj −Ξ ηabc ϵ ijk Ac0j 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
Ξ ηabc δji 0 0 0 0
Ξ ηabc ϵ j ik Babj −Ξ ηabc ϵ i jk Bab 0j 2Ξηabc ϵi jk Ab0j
j
2Ξ ηabc δki ∂b + ϵ i jk Υb
0 0 0 0 0 0 3 δ (x − y). 0 0 0 0 0 0
(16)
The matrix fij is not a square matrix as expected, however it has null vectors. The null vectors are given by
ϵkji Ac0 j V i , −ϵjki Bab 0j V i , ∂c V k + ϵ k ij Υci V j , ϵ j ik Babj V i , 0, 0, 0, 0, V i , 0, 0, 0 , = ∂c Vk − ϵ i kj Υcj Vi , ϵ ikj Babj Vi , ϵ kij Ac0j Vi , ϵ k ji Bab 0j V i , 0, 0, 0, 0, 0, Vi , 0, 0 , = 0, 2 ∂b V k − ϵi kj Υbj V i , 0, 2ϵi jk Ab0j V i , 0, 0, 0, 0, 0, 0, V i , 0 ,
V⃗1 =
(17)
V⃗2
(18)
V⃗3
(19)
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j
V⃗4 = 0, 2ϵ ijk Ab0j Vi , 0, 2 ∂b V k + ϵ k ji Υb V i , 0, 0, 0, 0, 0, 0, 0, Vi .
(20)
On the other hand, Zk (ξ ) is given by
(0)
δ V (ξ ) δξ i
Zk (ξ ) =
0
=
Ξ ηabc ϵj ki A0 i Bab 0j + ηabc ϵ ikj A00i Babj + 2Ξ ηabc ∂b B0a 0k abc kj 0i abc ijk − 2Ξ η ϵi B0a Υcj + 2Ξ η ϵ B0ai Ab0j Ξ ηabc ϵikj A0 j Ac0 k + Ξ ηabc ∂c A00i + Ξ ηabc ϵ j ik A00j Υck − Ξ ηabc ϵ j ik A0 i Babj + Ξ ηabc ϵ i jk A00i Bab 0j − 2Ξ ηabc ϵi jk Ab0j B0a 0i −2Ξ ηabc ∂b B0ak − 2ϵ i jk Υbj B0ai Ξ ηabc ∂c A0 i − Ξ ηabc ϵ i jk A0 j Υck + Ξ ηabc ϵ jki A00j Ac0k (0) Ωi . (0) 00i Ω (0) 0a Ω 0i (0) 0ai Ω 0 0 0 0
µ The contraction of the null vectors with Zk , namely, V⃗i Zµ (ξ ) = 0, gives identities. For instance,
from the contraction of V⃗1 with Zk (ξ ) we obtain µ
V⃗1 Zµ (ξ ) = ϵ j ik A0 i V k ∂c Ξ ηabc Babj + Ξ ηϵ m jkl Babm Υcl + Ξ ηabc ϵplj Bab 0p Ac0 l
+ ϵ i kj A00i V k Ξ ηabc ∂c Bab 0j − Ξ ηabc ϵ ji m Babm Ac0 l − Ξ ηϵ j nl Bab 0n Υcl − ϵ i jk B0ai V k Ξ ηabc ∂b Υcj − ∂c Υbj + Ξ ηabc ϵ j pl Υbp Υcl − Ξ ηabc ϵ j pl Ab0 p Ac0 l − ϵijk B0a 0i V k Ξ ηabc ∂b Ac0 j − ∂c Ab0 j + ϵ j ml Υcl Ab0 m − ϵjml Υbl Ac0 m = 0,
(21)
where we can observe that the left hand side vanishes because there is a linear combination of constraints. Hence, there are no more FJ constraints. Furthermore, we will add the constraints given in (13) to the symplectic Lagrangian using the i ˙ i , B0a 0i = ς˙ a , B0ai = χ˙ ai , thus the following Lagrange multipliers, namely, A0 i = T˙ i , A00i = Λ 2 2 symplectic Lagrangian takes the form [29]
(1)
(0)
(0)
˙ i Ω 00i − L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci − T˙ i Ω i − Λ −
χ˙ ai 2
ς˙ ai 2
(0)
Ω 0a 0i
(1)
(0)
Ω 0ai − V ,
where V (1) = V (0) | (0)
(0)
Ω i, Ω
(22)
(0) (0) 00i , Ω 0a , Ω 0ai =0 0i
= 0, this result is expected because of the general covariance
of the theory just as it is present in General Relativity. (1)
From the symplectic Lagrangian (22) we identify the following symplectic variables ξ
Ac0i , Bab ,
0i
Υci
, Babi , T , Λi , ςa , χai and the 1 - forms a i
i
(1)
=
Ξη
abc
Bab , 0, Ξ η 0i
abc
Babi , 0, −Ω
= (0)
i
,
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
−Ω (0) 00i , −
(0)
Ω
0a 0i
2
( 0)
0ai
, − Ω2
(1)
f
ij
. Hence, the symplectic matrix has the following form
−Ξ ηabc δji
0
381
Ξ ηabc δji 0 0 = −Ξ ηabc ϵ k Bab 0j ji −Ξ ηabc ϵ ikj B abj Ξ ηabc d k
0 0 0
0 0 0
η δ η ϵ η ϵ η ϵ η
0 0
bi
Ξ ηabc ϵj ki Bab 0j Ξ ηabc ϵkji Ac0 j −Ξ ηabc ϵ j ik Babj −Ξ ηabc Dcki
Ξ ηabc ϵ ikj Babj −Ξ ηabc dci k Ξ ηabc ϵ i jk Bab 0i Ξ ηabc ϵ ijk Ac0j
0 0 0 0
0 0 0 0
−Ξ ηabc δji
Ξ abc ji Ξ abc j ik Babj Ξ abc i jk Bab 0i j Ξ abc i k Ab0j abc i Ξ db k
−Ξ ηabc ϵkji Ac0 j Ξ ηabc dci k
−Ξ ηabc ϵ ijk Ab0j
0 0
Ξ ηabc dai k
0
Ξ ηabc Dcki Ξ ηabc ϵ ijk Ac0j 0 0
Ξ ηabc ϵ ijk Ab0j
0
0 −Ξ ηabc Dbi j 3 0 δ (x − y), 0 0
−Ξ ηabc ϵi jk Ab0j 0 0 0 0 0
(23)
0 0
where we have used the notation Dal i = δli ∂a + ϵl ik Υak and dai l = δil ∂a − ϵi lk Υak . We can observe that this matrix is singular, however we have showed that there are not more constraints, therefore the theory under study has a gauge symmetry. In order to obtain a symplectic tensor, we fix the following temporal gauge A0 i = 0,
(24)
A00i = 0,
(25)
= 0,
(26)
B0ai = 0,
(27)
B0a
0i
˙ i = 0, ς˙ ai = 0 and χ˙ ai = 0. In this manner, we introduce more Lagrange this mean that T˙ i = 0, Λ multipliers enforcing the gauge fixing. The Lagrange multipliers introduced are βi , α i , ρia , σai , thus, the symplectic Lagrangian takes the form (2)
(0)
(0)
˙i L = Ξ ηabc Bab 0i A˙ c0i + Ξ ηabc Babi Υ˙ ci − Ω i − βi T˙ i − Ω 00i − α i Λ
−
(0) 0a Ω 0i 2
− ρia ς˙ ai −
(0) 0ai Ω 2
− σ ai χ˙ ai ,
(28)
(2)
from this symplectic Lagrangian we identify the following symplectic variables ξ = Aa0i , Bab 0i , Υai ,
Babi , T i , Λi , ςai , χai , βi , α i , ρia , σ a(2) =
ai
, and the 1-forms
(0) (0) Ξ ηabc Bab 0i , 0, Ξ ηabc Babi , 0, − Ω i − βi , − Ω 00i − αi , (0) (0) 0a 0ai Ω Ω 0i − − ρi , − − σi , 0, 0, 0, 0 .
2
2
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A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
Thus, the symplectic matrix is given by 0
(2)
f
ij
Ξ ηabc δji 0 0 −Ξ ηabc ϵ i Bab 0j jk = −Ξ ηabc ϵ kij Babj −Ξ ηabc d i bl −Ξ ηabc ϵ kji Ab0j 0 0
−Ξ ηabc δji
0
0
Ξ ηabc ϵj ik Bab oj
0
0
0
Ξ ηabc ϵijk Ac0 j
0
0
−Ξ η
0
Ξ ηabc δji
−Ξ ηabc ϵijk Ac0 j Ξη
abc
0 0
Ξ ηabc ϵ kij Babj
− η
Ξ dci l Ξ k ji Bab 0j
−Ξ ηabc ϵ i jk Υck
ϵ ji Bab
abc k
0j
Ξ ηabc ϵ i jk Υck
0
−Ξ ηabc ϵ kji Ac0j
0 0
0
−Ξ ηabc ϵi jk Ab0j
0
0
Ξ ηabc dbl i
0
0
0 0 0 0
0 0 0 0
0 0 0 0
δji 0 0 0
Ξ ηabc dbj i
Ξ ηabc ϵ kji Ab0j
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0 0
abc
ϵ
−Ξ η
−Ξ ηabc ϵ j ki Babj
0
Ξ ηabc ϵ j ki Babj
dci l
δ
abc i j
Ξη
abc
Ξ ηabc ϵ kji Ac0j
ϵi Ab0j jk
−Ξ η
0 0 0 0 0 0 0
0 0 0 0 0
δji
dbl i
0 0 0 0 0 0 0 0
δba δji
0 0
abc
δba δji
0
−δji
−δji
0 0 0 0 0 0 0
0 0 0 0 0 0
−δba δji 0 0 0 0 0
0 0 0 0 δ 3 (x − y), (29) 0 −δba δji 0 0 0 0
where dai l ≡ δil ∂a − ϵi lk Υak . We can observe that this matrix is not singular, after a long calculation, (2)
the inverse of f
(2)
1 f − ij
ij
is given by
0
1 i − 2Ξ ηbgc δj 0 0 0 = 0 0 0 ϵijl Ac0 j 0 0 0
1 2Ξ
ηbgc δji
0
0
0
0 0
0
−
1
1 2Ξ
ηabc δji
2Ξ 0 0 0 0
0
0
0
0
0
0
0
0
0
0
ηabc δji
0
0
0
0
0
0
0
0
0
0 0 0 0
−δji
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0
−δji
0
0
0 0 0 0 l −ϵj k Bab 0j
−ϵ l jk Υck
0 0 0 0 j −ϵ kl Bmnj
−ϵ klj Babj
ϵ kjl Ac0j
ab − δmn ϵkj l Ab0j
0
ab − δmn Dbkl
0
0
−δba δji
0
0
0
0
0
0
−δba δji
1
ab − δmn Dbi l
2
1
ab kjl − δmn ϵ Ab0j
2
1
2
1 2
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
−ϵijl Ac0 j
0
ϵj lk Bab 0j
ϵ klj Babj
ϵ l jk Υck 2
ab δmn ϵkj l Ab0j
δji
2
Ξ 2
Ξ 2
0 1 2
ab δmn Dakl
0
0 0
δba δji
0 0
Ξ
0
2 mi q
0
ab δmn Dai l
δji
0 0 0
Ξ
2
−ϵ kjl Ac0j 1
ϵ j kl Bmnj
−
0 1
0 mi q
Hjl
−
Ξ 2
Hjl
Ξ ηabc ϵ ijk Babk Ac0j
0
E ai j
0
0
l Fm i
−Gkjpi
0
E ai j
1 2
383
ab kjl δmn ϵ Aboj 0 0 0 0 3 0 δ (x − y), a i δb δj Ξ ml Fi 2 mjpi G 0
(30)
0
where we have defined Dal i ≡ δli ∂a + ϵl ik Υak , E ai j ≡ ηabc ϵ i jp ϵ plk Ab0l Ac0k + ηabc ϵ l pj ϵ i kl Υbk Υcp , l Fm ≡ ηmnc ϵijk ϵ kpl Ac0 j An0p , i jp
Gmjpi ≡ ηmnc ϵ k ϵj li An0l Υck , mi q
Hjl
mn i ≡ ηabc δab ϵ jk ϵ qp l An0p Υck .
Therefore, from the symplectic tensor (30) we can identify the generalized FJ brackets by means of
{ξi(2) (x), ξj(2) (y)}FD = [fij(2) (x, y)]−1 ,
(31)
thus, the following generalized brackets arise Ac0i (x), Bab 0j (y)
Υci (x), Babj (y)
FJ
FJ
= =
1 2Ξ 1 2Ξ
ηabc δij δ 3 (x − y),
(32)
ηabc δji δ 3 (x − y),
(33)
where we can observe that the FJ brackets and the Dirac ones coincide to each other (see the section below). Furthermore, in FJ framework there are less constraints than in Dirac’s framework, in this sense, the FJ is more economical to perform; we will see this point in more detail in the following section. Finally, we carry out the counting of physical degrees of freedom. As we have commented above, in FJ formalism there is not a classification of constraints, they are at the same level, thus, the counting of physical degrees of freedom is performed as [DF = dynamical variables − independent constraints]. In this manner, there are 18 canonical variables given by (Acoi , Υai ) and 18 independent first class constraints (Ω (0)i , Ω (0) 00i , Ω (0) 0ai , Ω (0) 0a 0i ); for BF theory it is well-known that the constraints are reducible, the reducibility between the constraints is given by ∂a Ω (0) 0ai = (0)
ϵij k Υak Ω 0aj + ϵij k Aa0k Ω (0) 0a 0j and ∂a Ω (0) 0a 0i = ϵ i jk Υak Ω (0) 0a 0j + ϵ i k j Aa0k Ω (0) 0aj . Therefore, the theory is devoid of physical degrees of freedom as expected. It is important to comment, that all results found in this section are not reported in the literature.
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A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
3. Hamiltonian analysis In this section a pure Dirac’s canonical analysis for the four-dimensional BF theory will be performed, we will follow all Dirac’s steps in order to obtain the better canonical description of the theory [16]. For this aim, we start with the Lagrangian given in (2) L = Ξ
1 ηabc Bab 0i A˙ c0i + Bab ij A˙ cij 2
1
+ A0ij ∂c Bab ij + Bab il Acj l + 2Bab 0i Ac0 j 2 + A00i ∂c Bab 0i + Bab 0j Aci j + Bab ij Ac0j + B0a 0i ∂b Ac0i − ∂c Ab0i + Ab0j Acj i + Ac0 j Abij 1 + B0a ij ∂b Acij − ∂c Abij + Abil Acl j + Abi0 Ac0j − Abjl Acl i − Abj0 Ac0i d3 x, 2
(34)
by considering the following change of variables [28,30] Aaij ≡ −ϵijk Aa k , A0ij ≡ −ϵijk A0 k , Babij ≡ −ϵijk Bab k , B0aij ≡ −ϵijk B0a k , Aa0i ≡ Aa0i , Aai ≡ Υai , A0 i ≡ −T i , A00i ≡ −Λi , 1 B0a 0i ≡ − ςa i , 2 1 B0ai ≡ − χai , 2
(35)
the Lagrangian takes the following form L = L [Aa0i , Υai , Ti , Λi , ςai , χai , Bab0i , Babi ]
=
abc 0i Ξ η Bab A˙ c0i + Ξ ηabc Babi Υ˙ ci − T i ∂c Ξ ηabc Babi + Ξ ηabc ϵ j ik Babj Υc k − Ξ ηabc ϵjki Bab 0j Ac0 k − Λi ∂c Ξ ηabc Bab 0i − Ξ ηabc ϵ i jk Bab 0j Υc k − Ξ ηabc ϵ ijk Babk Ac0j 1 − Ξ ηabc ςa i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k 2 1 − Ξ ηabc χai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k d3 x. 2
In this manner, the canonically momenta of the dynamical variables are given by
∂L = Ξ ηabc Bbc 0i , ∂ A˙ a0i ∂L π ai ≡ = Ξ ηabc Bbc i , ∂ Υ˙ ai
pa0i ≡
(36)
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
Tˆ i ≡
ˆi ≡ Λ ςˆ ai ≡ χˆ ai ≡ pab0i ≡ pabi ≡
385
∂L = 0, ∂ T˙i ∂L = 0, ˙i ∂Λ ∂L = 0, ∂ ς˙ ai ∂L = 0, ∂ χ˙ ai ∂L = 0, ˙ ∂ Bab0i ∂L = 0, ∂ B˙ abi
(37)
with the following non-vanishing fundamental Poisson brackets between the fields
1 j Υai (x), π bj (y) = δab δi δ 3 (x − y), 2 1
(y) = δab δij δ 3 (x − y), 2 1 j j ˆ Ti (x), T (y) = δi δ 3 (x − y), 2 1 ˆ j (y) = δij δ 3 (x − y), Λi (x), Λ 2 ςai (x), ςˆ bj (y) = δab δij δ 3 (x − y), χai (x), χˆ bj (y) = δab δij δ 3 (x − y), 1 d e δa δb − δae δbd δij δ 3 (x − y), Bab0i (x), pde0j (y) =
Aa0i (x), p
b0j
Babi (x), p (y) = dej
4 1 4
δad δbe − δae δbd δij δ 3 (x − y).
(38)
Furthermore, from the definition of the momenta, we identify the following 60 primary constraints
φ1a0i ≡ pa0i − Ξ ηabc Bbc 0i ≈ 0, φ2ai ≡ π ai − Ξ ηabc Bbc i ≈ 0, φ3i ≡ Tˆ i ≈ 0, ˆ i ≈ 0, φ4i ≡ Λ φ5ai ≡ ςˆ ai ≈ 0, φ6ai ≡ χˆ ai ≈ 0, φ7ab0i ≡ pab0i ≈ 0, φ8abi ≡ pabi ≈ 0.
(39)
The canonical Hamiltonian of the theory is given by Hc =
=
˙ iΛ ˆ i + ς˙ ai ςˆ ai + B˙ ab0i pab0i + B˙ abi pabi − L d3 x A˙ a0i pa0i + Υ˙ ai π ai + T˙i Tˆ i + Λ
i T ∂a π a i − ϵi jk π a j Υak − ϵijk pa0j Aa0 k + Λi ∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j
386
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
1 + Ξ ηabc ςa i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck + ϵijk Ac0 j Υb k 2 3 1 abc i i i j k ijk + Ξ η χai ∂b Υc − ∂c Υb + ϵ jk Υb Υc − ϵ Ab0j Ac0k d x, 2
(40)
by adding the primary constraints we obtain the primary Hamiltonian H1 =
jk
Hc + T i ∂a π a i − ϵi π a j Υak − ϵijk pa0j Aa0 k
+ Λi ∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j 1 + Ξ ηabc ςa i ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck + ϵijk Ac0 j Υb k 2 1
+ Ξ ηabc χai ∂b Υc i − ∂c Υb i + ϵ i jk Υb j Υc k − ϵ ijk Ab0j Ac0k 2 ˆi + λa0i pa0i − Ξ ηabc Bbc 0i + λai π ai − Ξ ηabc Bbc i + αi Tˆ i + βi Λ + θai ςˆ ai + µai χˆ ai + λab0i pab0i + λabi pabi d3 x,
(41)
where λa0i , λai , αi , βi , θai , µai , λab0i , λabi are Lagrange multipliers enforcing the primary constraints. From consistency of the constraints, we identify the following 24 secondary constraints
φ˙ 3i ≈ 0 ⇒ ϕ3i ≡ − φ˙ 4i ≈ 0 ⇒ ϕ4i ≡ − φ˙ 5ai ≈ 0 ⇒ ϕ5ai ≡ −
1
∂a π ai − ϵ ijk π a j Υak − ϵ i jk pa0j Aa0 k ≈ 0,
(42)
∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j ≈ 0,
(43)
2 1 2
Ξ 2
ηabc ∂b Ac 0i − ∂c Ab 0i − ϵ ijk Ab0j Υck + ϵ ijk Ac0j Υbk
≈ 0,
(44)
Ξ φ˙ 6ai ≈ 0 ⇒ ϕ6ai ≡ − ηabc ∂b Υc i − ∂c Υb i + ϵ ijk Υbj Υck − ϵ ijk Ab0j Ac0k 2
≈ 0,
(45)
and the following 36 Lagrange multipliers
λab 0i ≈
λab i
1
ηabc ϵ i jk T j pc0k − ηabc ϵ ijk Λj π c k + Ξ ∂b ςa i − Ξ ∂a ςb i − Ξ ϵ ij k ςa j Υbk + Ξ ϵ ij k ςb j Υak + ϵ ijk χaj Ab0k − ϵ ijk χbj Aa0k , 1 ≈ ηabc ϵ ij k T j π c k + ηabc ϵ ij k pc0k Λj − Ξ ϵ ij k ςa j Ab0k + Ξ ϵ ij k ςb j Aa0k 2Ξ + Ξ ∂b χa i − Ξ ∂a χb i − Ξ ϵ ijk χaj Υbk + Ξ ϵ ijk χbj Υak , 2Ξ
λa0i ≈ 0, λai ≈ 0.
(46)
(47) (48) (49)
For this theory there are not tertiary constraints. Hence, in order to perform the classification of the constraints in first class and second class we proceed to calculate the following matrix whose entries are the Poisson brackets between all the constraints, it is
1 ϕ3i (x), ϕ3l (y) = − ϵ ilk ϕ3k δ 3 (x − y) ≈ 0,
(50)
1 ϕ4i (x), ϕ4l (y) = ϵ ilk ϕ3k δ 3 (x − y) ≈ 0.
(51)
4
4
Ξ φ1a0i (x), φ7de0l (y) = ηade ηil δ 3 (x − y), 2
(52)
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
1 φ1a0i (x), ϕ3l (y) = − ϵ il j pa0j δ 3 (x − y),
387
(53)
4
a0i 1 φ1 (x), ϕ4l (y) = ϵ ilj π a j δ 3 (x − y),
(54)
a0i Ξ il ade −η η ∂x,e + ηade ϵ ilj Υej δ 3 (x − y), φ1 (x), ϕ5dl (y) = −
(55)
a0i Ξ φ1 (x), ϕ6dl (y) = − ηade ϵ ilj Ae0j δ 3 (x − y),
(56)
ai Ξ φ2 (x), φ8del (y) = − ηil ηade δ 3 (x − y),
(57)
ai φ2 (x), ϕ3l (y) = − ϵ ilj π a j δ 3 (x − y),
(58)
ai φ2 (x), ϕ4l (y) = − ϵ il j pa0j δ 3 (x − y),
(59)
ai Ξ φ2 (x), ϕ5dl (y) = − ηade ϵ ilj Ae0j δ 3 (x − y),
(60)
ai Ξ φ2 (x), ϕ6dl (y) = ηade −ηil ∂x,e + ϵ ilj Υej δ 3 (x − y),
(61)
4
2 2
2 1 4 1 4
2
2
i 1 ϕ3 (x), ϕ4l (y) = − ϵ il j ϕ4j δ 3 (x − y) ≈ 0,
(62)
4 1
ϕ3i (x), ϕ5dl (y) = − ϵ il k ϕ5dk δ 3 (x − y) ≈ 0,
(63)
ϕ3i (x), ϕ6dl (y) = − ϵ il k ϕ6dk δ 3 (x − y) ≈ 0,
(64)
1 ϕ4i (x), ϕ5dl (y) = ϵ il k ϕ6k δ 3 (x − y) ≈ 0,
(65)
1 ϕ4i (x), ϕ6dl (y) = − ϵ il j ϕ5dj δ 3 (x − y) ≈ 0
(66)
4 1 4
4
4
we find that this matrix has rank = 36 and 48 null vectors. From the null vectors we find the following 48 first class constraints
γ1i ≡ Tˆ i ≈ 0, ˆ i ≈ 0, γ2i ≡ Λ γ3ai ≡ ςˆ ai ≈ 0, γ4ai ≡ χˆ ai ≈ 0, γ5 i ≡ ∂a π a i − ϵi jk π a j Υak − ϵijk pa0j Aa0 k + γ60i ≡ ∂a pa0i − ϵ i jk pa0j Υa k − ϵ ijk π a k Aa0j + γ7a 0i ≡
Ξ 2
1 2Ξ 1 2Ξ
ηabc ϵijk π aj pbck − pa0j pbc0k ≈ 0, ηabc ϵ i jk π aj pbc0k + pa0j pbck ≈ 0,
ηabc ∂b Ac0i − ∂c Ab0i + ϵi jk Ab0j Υck − ϵijk Ac0 j Υb k
− ∂b pab 0i + ϵi jk Υbk pab 0j + ϵi jk Ab0k pab j ≈ 0, Ξ γ8ai ≡ ηabc ∂b Υc i − ∂c Υb i + ϵ ijk Υbj Υck − ϵ ijk Ab0j Ac0k 2
− ∂b pabi − ϵ ijk Ab0k pab 0j + ϵ ijk Υbk pab j ≈ 0
(67)
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A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
and the rank allows us to identify the following 36 second class constraints
Γ1a0i ≡ pa0i − Ξ ηabc Bbc 0i ≈ 0, Γ2ai ≡ π ai − Ξ ηabc Bbc i ≈ 0, Γ3ab0i ≡ pab0i ≈ 0, Γ4abi ≡ pabi ≈ 0.
(68)
It is important to remark that the complete structure of the constraints (67) is not reported in the literature and this is a result of performing a pure Dirac’s formulation. In fact, by working with the standard form, it is not possible to obtain a full structure of the constraints and the algebra could not be closed just as is present in four-dimensional Palatini’s theory [15]. Furthermore, in order to compare the symplectic framework with the Dirac one, it is necessary to follow all steps of the Dirac formulation as has been developed in this paper. With all information obtained, we can carry out the counting of physical degrees of the theory in the following form; there are 120 dynamical variables, 48 first class constraints and 36 second class constraints, hence we obtain −6 degrees of freedom. However, it is well-known that BF theory is a reducible theory, this is, the constraints are not independent to each other. The reducibility of the constraints is given by 1
1
∂a γ7a 0i = ϵi jk Υak γ7a 0j + ϵi jk Aa0k γ8a j + ϵi jk Fabk Γ3ab0j + ϵi jk Fab0k Γ4abj ∂ γ =ϵ ai a 8
ijk
γ −ϵ
Υak 8a j
ijk
γ
Aa0k 7a 0j
2 1
− ϵ 2
ijk
Fab0k Γ3ab0j
2 1
+ ϵ ijk Fabk Γ4abj , 2
hence, there are 42 independent first class constraints. Therefore, by performing the counting of physical degrees of freedom we conclude that the theory is devoid of degrees of freedom, the theory is a topological one as expected. The algebra between the constraints is given by
Ξ Γ1a0i (x), Γ3de0l (y) = ηade ηil δ 3 (x − y), 2
ai Ξ Γ2 (x), Γ4del (y) = − ηade ηil δ 3 (x − y), 2
γ (x), γ (y) = i 5
l 5
γ5i (x), γ60l (y) =
γ5 i (x), γ7d 0l (y) =
1 2 1 2 1 2
ϵ il j γ5j δ 3 (x − y) ≈ 0, ϵ il j γ6j δ 3 (x − y) ≈ 0, ϵil k γ7d 0k −
1 4Ξ
ηabc Γ4bcl Γ3da0i +
1 4Ξ
da ηabc Γ3bc0 l Γ4 i
≈ 0, 1 1 1 l γ5 i (x), γ8dl (y) = ϵi lk γ8d k − ηabc Γ3bc0l Γ3da0i + δi ηabc Γ3bc0k Γ3da0k 2 4Ξ 4Ξ +
1 4Ξ
ηabc δil Γ4bck Γ4dak −
1 4Ξ
ηabc Γ4bcl Γ4dai
≈ 0, 0i 1 γ6 (x), γ60l (y) = − ϵ ilj γ5 j δ 3 (x − y) ≈ 0, 2
0i 1 1 l 1 da0k da0i d − δi ηabc Γ3bc0 + ηabc Γ3bc0 γ6 (x), γ7d 0l (y) = ϵ il k γ8k k Γ3 l Γ3 2 4Ξ 4Ξ
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
1
+
4Ξ
ηabc δli Γ4bck Γ4dak −
1 4Ξ
389
ηabc Γ4bcl Γ4dai
≈ 0, 0i 1 1 1 γ6 (x), γ8dl (y) = − ϵ iljγ7d 0j + ηabc g il Γ3bc0k Γ4dak − ηabc Γ3bc0l Γ4dai 2 4Ξ 4Ξ 1
−
4Ξ
ηabc Γ4bcl Γ3da0i +
1 4Ξ
g il ηabc Γ4bck Γ3da0 k
≈ 0, a 1 γ7 0i (x), Γ1d0l (y) = − ϵi lj pad j δ 3 (x − y) 2 1
= − ϵi lj Γ4adj δ 3 (x − y) ≈ 0, 2
γ
a 7 0i
(x),
Γ2dl
1 (y) = − ϵi lj pad0j δ 3 (x − y) 2 1
= − ϵi lj Γ3ad0j δ 3 (x − y) ≈ 0, 2
ai 1 γ8 (x), Γ1d0l (y) = ϵ ilj pad 0j = ai γ8 (x), Γ2dl (y) = =
2 1 2 1 2 1 2
ϵ ilj Γ3ad0j ≈ 0, ϵ ilj pad j ϵ ilj Γ4adj ≈ 0,
(69)
where we observe that the algebra is closed and it obeys the rules of the canonical formalism of the algebra between the constraints [16,17,21], namely, the result of the Poisson brackets between first class constraints with first class must be linear in first class constraints and square in second class; the result of the Poisson brackets between first class constraints with second class constraints must be linear in first class constraints and linear in second class constraints. We can observe that our results are in full agreement with these rules. Furthermore, the constraints γ5 i and γ60i are identified as generators of rotations and boost respectively whereas γ7 0i and γ8 ai are generators of translations, this can be seen from the algebra between these constraints. On the other hand, first class constraints are generators of gauge transformations. Hence, by defining the gauge generator in terms of first class constraints
G=
θ1,i γ1i + θ2,i γ2i + θ3,ai γ3ai + θ4,ai γ4ai + θ5i γ5,i + θ6,0i γ60i + θ70i,a γ7a,0i + θ8,ai γ8ai d3 x,
we find the following gauge transformations
δ Ad0l = {Ad0l , G} 1 1 1 = −∂d θ6,0l + θ5i ϵlik Ad0 k − ϵ i lk θ6,0i Υd k + ηdbc ϵlik θ5i pbc0k + ηdbc ϵ i lk pbck 2 2Ξ 2Ξ δ Υdl = {Υdl , G} 1 1 1 k i ij i bck i bc0k −∂d θ5,l + ϵli θ5 Υdk − ϵl θ6,0i Ad0l − = ηdbc ϵlik θ5 p + ηdbc ϵ lk θ6,0i p 2 2Ξ 2Ξ δ Tl = {Tl , G} = θ1,l δ Λi = {Λl , G} = θ2,l
(70)
(71) (72) (73)
390
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
δςdl = {ςdl , G} = θ3,dl δχdl = {χdl , G} = θ4,dl δ Bde0l = {Bde0l , G} 1 ∂e θ7,d0l − ∂d θ7,e0l − θl ij θ7,d0j Υek + ϵl ij θ7,e0i Υdj − ϵl ik θ8,di Ae0k = 4 1 1 ik i a0j i aj + ϵl θ8,ei Adok − ηdea ϵlij θ5 p + ηdea ϵl j θ6,0i π Ξ Ξ δ Bdel = {Bdel , G} 1 ∂e θ8,dl − ∂d θ8,el − ϵl ik θ8,di Υek + ϵl ik θ8,ei Υdk − ϵli k θ70i,d Ae0k + ϵli k θ70i,e Ad0k = 4 1 1 + ηdea ϵlij θ5i π aj + ηdea ϵl ij θ6,0i pa0j Ξ Ξ δ pd0l = pd0l , G 1 lj = Ξ ηdab ∂b θ70l,a + ϵ l ij θ5i pd0j − ϵ lij θ6,0i π d k − ϵ j θ7,a 0i pad j + ϵ lij θ8,ai pad 0j 2 − Ξ ηdac ϵ li k θ7,a 0i Υck + Ξ ηdac ϵ lik θ8,ai Ac0k δπ dl = π dl , G 1 lj lj Ξ ηdab ∂b θ8l ,a + ϵ i θ5i π d j + ϵ li j θ6,0i pd0j + ϵ i θ7,a 0i pda 0j + ϵ lij θ8,ai pda j = 2 − Ξ ηdab ϵ li j θ7,a 0i Ab0j − Ξ ηdac ϵ lik θ8,ai Υck δ Tˆ l = Tˆ l , G = 0, ˆ l , G = 0, ˆl = Λ δΛ δ ςˆ dl = ςˆ dl , G = 0, δ χˆ dl = χˆ l , G = 0, δ pde0l = pde0l , G = 0, δ pdel = pdel , G = 0.
(74) (75)
(76)
(77)
(78)
(79) (80) (81) (82) (83) (84) (85)
Now, with the second class constraints we can calculate the Dirac brackets. In order to perform this aim we calculate the matrix C αβ = Γ α , Γ β whose entries are given by the Poisson brackets of the second class constraints, namely
Cαβ
a0i d0l Γ1 , Γ1 Γ ai , Γ d0l 2 1 = ab0i Γ3 , Γ1d0l abi d0l Γ4 , Γ1
Γ1a0i , Γ2dl ai dl Γ2 , Γ2 ab0i Γ3 , Γ2dl abi dl Γ4 , Γ2
0
0 = Ξ abd il − η η 2 0
0
Ξ 2
Ξ 2
Γ1a0i , Γ3de0l ai de0l Γ2 , Γ3 ab0i Γ3 , Γ3de0l abi de0l Γ4 , Γ3
ηade ηil
0
0
0
0
ηabd ηil
0
0
Ξ
Γ1a0i , Γ4del ai del Γ2 , Γ4 ab0i Γ3 , Γ4del abi del Γ4 , Γ4
− η η 3 2 δ (x − y), 0 ade il
0
(86)
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
391
and the inverse of Cαβ is given by
−1 Cαβ
0
−
0
0 = 1 ηabd ηil Ξ −
0
1
Ξ
2
Ξ
ηade ηil
0
0
0
0
ηabd ηil
0
0
ηade ηil 3 Ξ δ (x − y). 0 2
(87)
0
Hence, the Dirac brackets between two functionals, namely F (q, p) and G(q, p), are defined by
{F , G}D ≡ {F , G} −
−1 dudv {F , Γ α (u)} Cαβ (u, v) Γ β (v), G ,
(88)
where Γ α represent the second class constraints. Therefore, the Dirac brackets between the fields are given by Aa0i (x), pd0j (y)
D
Υai (x), π dj (y) D Ti (x), Tˆ j (y) D j ˆ (y) Λi (x), Λ D ςai (x), ςˆ dj (y) D χai (x), χˆ dj (y) D Bab0i (x), pde0j (y) D Babi (x), pdej (y) D Aa0i (x), Bde0j (y) D
1 = Aa0i (x), pd0j (y) = δad δij δ 3 (x − y), 2 1 d
(89)
= Υai (x), π dj (y) = δa δij δ 3 (x − y), 2 1 j ˆ = Ti (x), T (y) = δij δ 3 (x − y), 2 1 d0j ˆ (y) = δij δ 3 (x − y), = Λa0i (x), Λ 2 = ςai (x), ςˆ dj (y) = δad δij δ 3 (x − y), = χai (x), χˆ dj (y) = δad δij δ 3 (x − y),
(90)
= 0,
(95)
= 0,
(96)
=−
Υai (x), Bdej (y) D =
1
2Ξ 1
2Ξ
ηade ηij δ 3 (x − y),
ηade ηij δ 3 (x − y).
(91) (92) (93) (94)
(97) (98)
In addition, we can also calculate the Dirac brackets by gauge fixing, in this case we will use the temporal gauge, namely, we take Ti ≈ 0,
(99)
Λi ≈ 0, ςai ≈ 0, χai ≈ 0.
(100) (101) (102)
These conditions are considered as a new set of second class constraints, hence, now there are the following 84 second class constraints
Γ1a0i ≡ pa0i − Ξ ηabc Bbc 0i ≈ 0, Γ2ai ≡ π ai − Ξ ηabc Bbc i ≈ 0, Γ3ab0i ≡ pab0i ≈ 0, Γ4abi ≡ pabi ≈ 0,
392
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
≡ Ti ≈ 0, ≡ Λi ≈ 0, ≡ ςai ≈ 0, ≡ χai ≈ 0, Γ9i ≡ Tˆi ≈ 0, ˆ i ≈ 0, Γ10i ≡ Λ Γ5i Γ6i Γ7ai Γ8ai
Γ11ai ≡ ςˆ ai ≈ 0, Γ12ai ≡ χˆ ai ≈ 0.
(103)
β
In this manner, the matrix Cαβ = Γ α , Γ whose entries are given by the Poisson brackets between the second class constraints (103) is given by
0
0
Ξ 2
ηade ηil
0 0 Ξ − ηabd ηil 0 2 Ξ abd il η η 0 2 0 0 Cαβ = 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
ηade ηil
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 l δ 2 i
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 − δli 2
0
0
0
0
0
1 − δli 2
0
0
0
0
0
−δda δli
0
0
−
Ξ 2
0 0
0
0
0
0
0
0
0
−δda δli
0
0
0
0
0
3 1 l δi 0 0 δ (x − y), 2 0 δad δil 0 0 0 δad δil 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
(104) and its inverse
0
0
0 0 1 η η 0 Ξ abd il 1 0 − ηabd ηil Ξ −1 Cαβ = 0 0 0 0 0 0 0 0 0 0 0 0
−
2
Ξ
ηade ηil 0
0 2
Ξ
0
0
0
0
ηade ηil 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2δil
0
0
0
0
0
0
2δil
0
0 0
0
0
0
0
0
0
δad δil
0
0
0
0
0
0
0
δad δil
0
0
0
0
0 0 0 0 0 0 0 0 3 δ (x − y). i −2δl 0 0 0 i 0 −2δl 0 0 d i 0 0 −δa δl 0 0 0 0 −δad δli 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
(105) In this manner, by using (105) we construct the Dirac brackets given by Aa0i (x), pd0j (y)
D
1 = Aa0i (x), pd0j (y) = δad δij δ 3 (x − y), 2
A. Escalante, P.C. Sánchez / Annals of Physics 374 (2016) 375–394
Υai (x), π dj (y) D Ti (x), Tˆ j (y) D j ˆ Λi (x), Λ (y) D ςai (x), ςˆ dj (y) D χai (x), χˆ dj (y) D Bab0i (x), pde0j (y) D Babi (x), pdej (y) D Aa0i (x), Bde0j (y) D
393
1 = Υai (x), π dj (y) = δad δij δ 3 (x − y), 2
= 0, = 0, = 0, = 0, = 0, = 0, =−
Υai (x), Bdej (y) D =
1
2Ξ 1
2Ξ
ηade ηij δ 3 (x − y),
ηade ηij δ 3 (x − y),
(106)
where we can observe that the Dirac brackets and the generalized FJ brackets coincide to each other. 4. Conclusions and prospects In this paper, the symplectic analysis of a four-dimensional BF theory has been performed. We reported the complete set of FJ constraints and we observe that there are present less constraints than in Dirac’s method. Furthermore, we have carried out the counting of physical degrees of freedom concluding that the theory is a topological one as expected. In addition, we have used a temporal gauge in order to obtain a symplectic tensor, then the quantization brackets of FJ were obtained. On the other hand, a pure canonical analysis has been performed. Under a laborious work, we have reported the complete set of first class and second class constraints, the algebra between the constraints is in full agreement with the canonical rules; then using a temporal gauge the Dirac brackets were computed. The FJ and Dirac’s brackets coincide to each other, thus we can conclude that the FJ is more economical than Dirac framework. Of course, if in the Dirac approach are introduced the Dirac brackets and the second class constraints are considered as strong equations, then the FJ and Dirac’s constraints coincide to each other. Finally we would like to comment that in this work we provide the necessary tools for studying in alternative way the BF formulations of gravity such as that reported in [22]. In fact, in that work the canonical formulation of BF gravity has been performed, and will be interesting to study that theory by using the ideas of the symplectic formalism of FJ. All these ideas are in progress and will be the subject of forthcoming works. Acknowledgements 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. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Further reading [1] L. Liao, Y.C. Huang, Ann. Phys. 322 (2007) 2469–2484.