3D gravity with propagating torsion: the AdS sector

0.22 MB·English

by  Milutin Blagojević

3D gravity with propagating torsion: the AdS sector
M. Blagojevi´c and B. Cvetkovi´c∗
Institute of Physics, P. O. Box 57, 11001 Belgrade, Serbia
2
1 January 20, 2012
0
2

n
a
J Abstract

0
2 We study the general parity-preserving model of three-dimensional gravity with

propagating torsion, with a focus on its nonlinear dynamics in the AdS sector. The
]
c model is shown to posses the black hole with torsion as a vacuum solution, provided
q
- we adopt certain restrictions on its coupling constants. The canonical analysis in the
r
g AdS asymptotic region yields the conserved charges of the black hole and the central
[ charges of the asymptotic conformal symmetry.

1
v
7 1 Introduction
7
2
4 In our attempts to properly understand basic aspects of the gravitational dynamics at both
.
1 classical and quantum level, we are naturally led to consider three-dimensional (3D) gravity
0
as a technically simpler model with the same conceptual features. Following a traditional
2
1 approach based on general relativity (GR), 3D gravity has been studied mainly in the realm
v: of Riemannian geometry, leading to a number of outstanding results [1]. In the early 1990s,
i a new approach to 3D gravity has been initiated by Mielke and Baekler [2]. The approach is
X
based on a modern gauge-field theoretic conception of gravity characterized by a Riemann–
r
a
Cartan geometry of spacetime; it is known as Poincar´e gauge theory (PGT), the theory in
which both the torsion and the curvature carry the dynamics of gravity, see [3, 4, 5, 6].
The Mielke–Baekler (MB) model is introduced as a topological 3D gravity with torsion,
with an idea to explore the influence of geometry on the dynamics of gravity. Recent
investigations along these lines led to remarkable results: (i) the MB model possesses the
black hole solution, (ii) it can be formulated as a Chern–Simons gauge theory, (iii) in the
AdS sector, asymptotic symmetry is described by two independent Virasoro algebras with
different central charges, (iv) the black hole entropy is found to depend on torsion, and (v)
the geometric idea of torsion is compatible with supersymmetry; see [7, 8, 9, 10, 11].
Einstein’sGRin3D,withorwithoutacosmologicalconstant, isalsoatopologicaltheory,
which has no propagating degrees of freedom. Such a degenerate situation is not quite a
realistic feature of the gravitational dynamics. Thus, one is naturally motivated to study
gravitational models with propagating degrees of freedom. In the context of Riemannian
∗Email addresses: [email protected], [email protected]
1
geometry, there are two well-known models of this type: topologically massive gravity [12],
and the Bergshoeeff–Hohm–Townsend (BHT) massive gravity [13]. In 3D gravity with
torsion, an extension that includes propagating modes is even more natural—it corresponds
toLagrangianswhich arequadratic inthefield strengths, asinthestandard gaugeapproach.
In the present paper, we begin an investigation of the parity-preserving 3D gravity with
propagating torsion. Compared to the topological MB model [2], here we have a Lagrangian
witharatherlargenumber ofparameters, andoneisfacedwiththeproblemofchoiceofaset
of parameters which defines an acceptable gravitational model. Some aspect of this problem
have beendiscussed intheliterature. Motivatedby theactual importanceof massive gravity
inhigh-energyphysics andcosmology, Hernaski et al. [14] used the spinprojection operators
to investigate how the existence of propagating torsion can be used to build up a unitary
massive gravity model of the BHT-type, in any dimension. After that, Helay¨el-Neto et al.
[15] studied the parity-preserving 3D gravity with propagating torsion combined with the
Chern-Simons term; using the requirements of no ghosts and no tachions, they found certain
restrictions on the parameters. Although these arguments are commonly accepted in the
literature, one should note that they essentially rely on the weak-field approximation of the
theory, in which inherently nonlinear properties of gravity remain untachable.
Our approach to 3D gravity with propagating torsion is aimed at studying essential
aspects of its nonlinear dynamics. In this paper, we start by introducing basic elements of
the Lagrangian formalism, whereupon we focus our attention on the AdS sector, examining
the existence of black holes and the nature of asymptotic symmetries. In the next paper
[16], we will use the criterion of stability of the canonical structure under linearization [17]
to find out the restrictions on parameters that define viable PGT models.
The paper is organized as follows. In section 2, we describe Lagrangian dynamics of the
general parity-preserving PGT in 3D. In section 3, we give a brief account of the linearized
theory, showing that masses of the propagating modes coincide with those found in [15]. In
section 4, we find the restrictions on parameters that allow the existence of the black hole
with torsion. In section 5, we apply the Hamiltonian formalism to construct the canoni-
cal gauge generator of the theory. Then, in section 6, we introduce the AdS asymptotic
conditions and calculate the improved form of the canonical generator, defined by suitable
surface terms. The improved generator is used to find the conserved charges of the black
hole with torsion. Moreover, the asymptotic symmetry, defined by the canonical algebra
of the improved generators, is described by two independent Virasoro algebras with central
charges, the values of which depend on the quadratic piece of the Lagrangian.
Our conventions are as follows: the Latin indices (i,j,k,...) refer to the local Lorentz
frame, the Greek indices (µ,ν,λ,...) refer to the coordinate frame, and both run over 0,1,2;
the metric components in the local Lorentz frame are η = (+,−,−); totally antisymmetric
ij
tensor εijk is normalized to ε012 = 1.
2 General Lagrangian formalism
Theory of gravity with torsion can be naturally described as a Poincar´e gauge theory, with
an underlying Riemann-Cartan (RC) geometry of spacetime [3, 4, 5, 6]. Basic gravitational
variables in PGT are the triad field bi and the Lorentz connection Aij = −Aji (1-forms),
and the corresponding field strengths are Ti := ∇bi and Rij := dAij +Ai ∧Akj (2-forms).
k
2
The covariant derivative ∇ = d+ 1AijΣ (1-form) acts on a tangent-frame spinor/tensor in
2 ij
accordance with its spinorial/tensorial structure, reflected in the form of the representation
of the spin matrix Σ .
ij
The antisymmetry of the Lorentz connection Aij implies that the geometric structure of
PGT corresponds to a RC geometry, in which bi is an orthonormal coframe, g := η bi ⊗bj
ij
is the metric of spacetime, Aij is the metric-compatible connection defined by ∇g = 0, and
Ti and Ri are the torsion and the RC curvature, respectively.
In local coordinates xµ, we can write bi = bi dxµ, the frame dual to bi reads h = h µ∂ ,
µ i i µ
and we have h ⌋bj = h µbj = δi, where ⌋ is the interior product.
i i µ j
2.1 Lagrangian and the field equations
General dynamics of 3D gravity with propagating torsion is defined by the Lagrangian
3-form
L = L (bi,Ti,Rij)+L (bi,ψ,∇ψ) (2.1a)
G M
where L denotes matter contribution, and the gravitational piece L is at most quadratic
M G
in torsion and curvature. Assuming that L preserves parity, we have
G
1
L = −aε bi ∧Rjk − Λ ε bi ∧bj ∧bk +L +L ,
G ijk 0 ijk T2 R2
3
L = Ti ∧⋆ a (1)T +a (2)T +a (3)T ,
T2 1 i 2 i 3 i
1 (cid:0) (cid:1)
L = Rij ∧⋆ b (4)R +b (5)R +b (6)R , (2.1b)
R2 4 ij 5 ij 6 ij
2
(cid:0) (cid:1)
where (a)T and (a)R are irreducible components of the torsion and the RC curvature, see
i ij
Appendix A. In what follows, we will omit the wedge product sign ∧ for simplicity.
Let us now introduce the covariant gravitational momenta (1-forms)
∂L ∂L
G G
H := , H := . (2.2)
i ∂Ti ij ∂Rij
In addition to that, we define the dynamical energy-momentum and spin currents (2-forms)
for the gravitational field:
∂L ∂L
G G
t := , s := , (2.3)
i ∂bi ij ∂Aij
as well as the corresponding matter currents (2-forms):
∂L ∂L ∂L
M M M
τ := , σ := = Σ ψ . (2.4)
i ∂bi ij ∂Aij ij ∂∇ψ
Then, using the relations [6]
δTi = ∇δbi +δAij ∧b , δRij = ∇δAij,
j
one finds that the variation of the Lagrangian (2.1a) with respect to bi and Aij produces
the following gravitational field equations:
∇H +t = −τ , (2.5a)
i i i
∇H +s = −σ . (2.5b)
ij ij ij
3
Explicit calculation based on the gravitational Lagrangian (2.1b) yields
H = 2⋆ a (1)T +a (2)T +a (3)T ,
i 1 i 2 i 3 i
H = −(cid:0)2aε bk +H′ , (cid:1)
ij ijk ij
H′ := 2⋆ b (4)R +b (5)R +b (6)R , (2.6)
ij 4 ij 5 ij 6 ij
(cid:0) (cid:1)
and
1
t = e ⌋L −(e ⌋Tm)∧H − (e ⌋Rmn)∧H ,
i i G i m i mn
2
s = −(b ∧H −b ∧H ) . (2.7)
ij i j j i
The second field equation can be now rewritten in an equivalent form as:
−2aε Tk +∇H′ +s = −σ . (2.5b′)
ijk ij ij ij
Using the above expressions for the gravitational field momenta, the gravitational La-
grangian can be written in a more compact form as:
1 1 1 1
L = TiH + Rij(−2aε bk)+ RijH′ − Λ ε bibjbk. (2.8)
2 i 2 ijk 4 ij 3 0 ijk
Bianchi identities for PGT read:
∇Ti = Ri bj , ∇Rij = 0. (2.9)
j
For the form of Noether identities, we refer the reader to Ref. [6].
In 3D gravity, the Weyl curvature vanishes:
1
W = R −(b Rˆ −b Rˆ )+ Rb b = 0. (2.10)
ij ij i j j i i j
2
As a consequence, the RC curvature R can be expressed in terms of the Ricci 1-form
ij
Rˆ = Rˆ bj (or more compactly, in terms of the Schouten 1-form L = Rˆ − 1Rb ). This
i ij i i 4 i
property can be easily accommodated in the general Lagrangian formalism.
2.2 The tensor formalism
For later convenience and an easier comparison with the results in the literature, we present
here the tensor form of Lagrangian and the field equations, see Appendix A and [3].
Using the notation L˜= bL, the gravitational Lagrangian reads
L = −aR−2Λ +L +L , (2.11a)
G 0 T2 R2
where
1
L := Tijk a (1)T +a (2)T +a (3)T
T2 1 ijk 2 ijk 3 ijk
2
(cid:0) (cid:1)
1
L := Rijkl b (4)R +b (5)R +b (6)R . (2.11b)
R2 4 ijkl 5 ijkl 6 ijkl
4
(cid:0) (cid:1)
4
The expansions H = 1H ⋆(bjbk) and H = 1H ⋆(bjbk) define the components of the
i 2 ijk i 2 ijk
gravitational momenta as:
H = 2 a (1)T +a (2)T +a (3)T ,
ijk 1 ijk 2 ijk 3 ijk
H = =(cid:0)−2a(η η −η η )+H′ , (cid:1)
ijkl ik jl jk il ijkl
H′ = 2 b (4)R +b (5)R +b (6)R .
ijkl 4 ijkl 5 ijkl 6 ijkl
(cid:0) (cid:1)
Note that H = ∂L /∂Tijk,H = ∂L /∂Rijkl. Then, the field equations read
ijk G ijkl G
∇ H µν −t ν = τ ν , (2.12a)
µ i i i
∇ H µν −s ν = σ ν , (2.12b)
µ ij ij ij
where H = bH , H = bH , and
ijk ijk ijkl ijkl
1
t ν = h νL˜ −H νTmn − H νRmnr ,
i i G mn i mnr i
2
s ν = −(H ν −H ν) .
ij ij ji
Isolating the contribution of the linear curvature term in H , the field equations read:
ijkl
1
∇ H µν −h νL˜ +H νTmn −2abRν + H′ νRmnk = τ ν , (2.13a)
µ i i G mn i i 2 mnk i i
−aεµνρTk +∇ H′ µν +2H ν = σ ν. (2.13b)
ijk µρ µ ij [ij] ij
We find it useful to give here an equivalent description of the T2 Lagrangian:
1
L = TijkH = Tijk(α T +α T +α η v ) , (2.14)
T2 ijk 1 ijk 2 kji 3 ij k
4
1 1 1
α = (2a +a ), α = (a −a ), α = (a −a ).
1 1 3 2 1 3 3 2 1
6 3 2
Moreover, thevanishing oftheWeyl curvature implies that bothH′ andL = 1RijklH′
ijkl R2 8 ijkl
ˆ
can be expressed in terms of the Ricci tensor R :
ij
H′ = 2(η γ −η γ )−(k ↔ l),
ijkl ik jl jk il
L = Rˆij β Rˆ +β Rˆ +β η R =: Rˆijγ , (2.15)
R2 1 ij 2 ji 3 ij ij
(cid:16) (cid:17)
1 1 1
β = (b +b ), β = (b −b ), β = (b −4b ).
1 4 5 2 4 5 3 6 4
2 2 12
2.3 Lie dual forms
Let us note that in 3D, for any antisymmetric form Xij = −Xji, one can define its Lie dual
form X by Xij = −εijkX . It is often convenient to replace (Aij,Rij) by the corresponding
k k
Lie duals (ωi,Ri), so that
1
Ti = dbi +εi ωjbk, Ri = dωi + εi ωjωk.
jk jk
2
We will switch to this Lie dual notation in section 4.
5
3 Particle spectrum on Minkowski background
The weak-field approximation of 3D gravity with propagating torsion around the Minkowski
background M yields an approximate picture of the gauge structure and dynamical content
3
of the theory. To what extent this picture reflects essential features of the full, nonlinear
theory is an issue that will be examined in [16].
The particle spectrum of 3D gravity with propagating torsion has been studied by
Helay¨el-Neto et. al. [15] in a model based on the Lagrangian (2.1) plus a parity-violating
Chern-Simons term. Using an extended basis of the spin projection operators, they were
able to examine the conditions for the theory to have well-behaved propagating modes (no
ghosts and no tachions). Here, we shall re-derive these results for the theory (2.1), using an
approach based on the covariant field equations, see [19].
The only propagating modes of the theory (2.1) are those associated to the Lorentz
connection Aij . In the weak field approximation, Aij has 9 independent modes. For mas-
µ µ
sive modes, the spin content can be determined by looking at the corresponding irreducible
representations of the little group SO(2). Since SO(2) is Abelian, all its representations
are one-dimensional. By subtracting 3 gauge degrees of freedom, corresponding to 3 local
Lorentz rotations, one finds that at most 6 degrees of freedom can be physical.
In two spatial dimensions, parity is defined as the inversion of one axis, the inversion
of both would be a rotation [20]; thus, for instance, we can define it as the inversion of y
axis. Since parity is a symmetry of the Lagrangian (2.1), the structure of the corresponding
irreducible representations is changed: they contain two states with the same value of spin
J, which transform into each other under P.
The irreducible representations of SO(2)×P provide a foundation for understanding the
particle content of any Lorentz-covariant field theory in 3D. However, it is usually simpler to
start with finite-dimensional representations of the full Lorentz group SO(1,2). Since these
covariant fields are not, in general, irreducible, they are combined with the field equations
and subsidiary conditions so as to remove the unphysical degrees of freedom [21].
For Λ = 0, the Minkowski configuration (bi ,Aij ) = (δi,0) is a solution of the field
0 µ µ µ
equations (2.13) in vacuum. The weak-field approximation around M takes the form
3
bi = δi +˜bi , Aij = A˜ij , (3.1a)
µ µ µ µ µ
where the tilde sign denotes small field excitations. Then:
Ti = T˜i +O , T˜i = ∂ ˜bi −∂ ˜bi +2A˜i ,
µν µν 2 µν µ ν ν µ [νµ]
Rij = R˜ij +O , R˜ij = ∂ A˜ij −∂ A˜ij . (3.1b)
µν µν 2 µν µ ν ν µ
The linearized field equations take the form:
∂ H˜ µν −2aG˜ν = τ˜ν , (3.2a)
µ i i i
−aεµνρT˜k +∂ H˜′ +2H˜ ν = σ˜ ν. (3.2b)
ijk µρ µ ij [ij] ij
Being basically interested in spins and masses of the tordion modes, we restrict our at-
tentionto thevacuum field equations. Moreover, inwhat followswe omit tilde forsimplicity.
Using (3.2a) we can express the Ricci curvature interms ofthe first derivatives of the torsion
tensor, whereupon (3.2b) can be transformed into a set of equations containing the torsion
6
tensor and its second derivatives. A suitable “diagonalization” of these equations leads to
several Klein–Gordon equations for the torsion modes. Here, we consider the case when
these modes are massive.
(a) The field a = 1ε Tijk satisfies the Klein–Gordon equation ((cid:3)+m2 )a = 0 with
6 ijk 0−
3(a−a )(a+2a )
m2 = 1 3 . (3.3)
0− (a +2a )b
1 3 5
Thus, a is a massive pseudoscalar state, JP = 0−.
(b) Similarly, the field σ = ∂iv is a massive scalar with
i
3a(a+a )
m2 = 2 , (3.4)
0+ a (b +2b )
2 4 6
and JP = 0+.
1
(c) For v¯i := vi + ∂iσ (with ∂kv¯ = 0), we find:
m2 k
0+
4(a−a )(a+a )
m2 = 1 2 . (3.5)
1 (a +a )(b +b )
1 2 4 5
The transverse field v¯ describes two massive states with spin J = 1; these two states
i
transform into each other under parity (like the helicity states in 4D).
(d) The field
a (a+a )2
χ = ∂kt + 1 2 ∂ v¯ ,
ij k(ij) (i j)
2(a−a )[a(a −a )−2a a ]
1 2 1 1 2
is symmetric, traceless and divergenceless, and it satisfies the Klein–Gordon equation with
a(a−a )
m2 = − 1 . (3.6)
2 a b
1 4
Hence, χ describes two massive degrees of freedom with spin J = 2. Again, these two
ij
states form a parity invariant multiplet.
Using the field equations and the identities εijkt = ti = 0, one can show that t and
ijk ik ijk
its first and second derivatives can be expressed in terms of (a, σ, v¯, χ ) and their first and
i ij
second derivatives. Thus, the spectrum of excitations around M consists of 6 independent
3
torsion modes: two spin-0∓ states (a,σ), two spin-1 states v¯, and two spin-2 states χ , in
i ij
agreement with the little group analysis. Our results coincide with those obtained in [15],
in the limit when the Chern–Simons coupling constant vanishes.
4 The black hole with torsion
Asafirst stepinourstudyofthenonlineardynamics of3Dgravitywithpropagatingtorsion,
we wish to examine whether the PGT model (2.1) admits the AdS black hole solution.
7
In the MB model of 3D gravity with torsion [2], there exists an interesting vacuum
solution, the black hole with torsion [7, 8, 9]. In the Schwarzschild-like coordinates xµ =
(t,r,ϕ), this solution is defined by the pair (bi,ωi), where (a) the triad field has the form
b0 = Ndt, b1 = N−1dr, b2 = r(dϕ+N dt), (4.1a)
ϕ
where N and N are the lapse and the shift functions of the Ban˜ados-Teitelboim-Zanelli
ϕ
(BTZ) metric [22] (m and J are the integration constants):
r2 16G2J2 4GJ
N2 = −8mG+ + , N = ,
ϕ
ℓ2 r2 r2
and (b) the connection ωi can be found as the solution of the MB vacuum field equations:
p
ωi = ω˜i + bi, (4.1b)
2
where ω˜i is the Riemannian connection, and p is a parameter that measures torsion. For
the solution (4.1), the field strengths have the following form:
2T = pε bjbk, 2R = qε bjbk, (4.2)
i ijk i ijk
where p and q are parameters, and we assume that the effective cosmological constant is
negative [8, 9]:
1 1
Λ = q − p2 = − < 0.
eff 4 ℓ2
In the context of RC geometry, the form-invariance of a given field configuration is
defined by the requirements δ bi = δ ωi = 0, which differ from the Killing equation in
0 µ 0 µ
GR (δ is the PGT analogue of the Lie derivative). Symmetry properties of the black hole
0
(4.1) are expressed by its form-invariance under the action of two Killing vectors, ∂/∂t
and ∂/∂ϕ. In a RC spacetime with Λ < 0, there exists another, maximally symmetric
eff
solution, known as the AdS solution (AdS ), which can be formally obtained from (4.1)
3
by the replacement J = 0, 2m = −1. The form-invariance of AdS is described by the
3
six-dimensional AdS group SO(2,2). The black hole and AdS are locally isometric, but
3
globally distinct solutions [9].
By combining (4.2) with the field equations (2.13) in vacuum, we can obtain certain
restrictions on p and q, under which the BTZ-like black hole, as well as AdS , is an exact
3
solution of the theory. To find these these restrictions, we use (4.2) to obtain
H = 2pa b ,
i 3 i
1 1
t = aq −Λ − p2a − q2b ε bjbk,
i 0 3 6 ijk
(cid:18) 2 2 (cid:19)
H = −2(a+qb )ε bk,
ij 6 ijk
s = −4pa b b ,
ij 3 i j
whereupon the field equations (2.13) in vacuum lead to:
1 1
aq −Λ + p2a − q2b = 0,
0 3 6
2 2
p(a+qb +2a ) = 0. (4.3)
6 3
8
These conditions guarantee that the black hole with torsion (4.1) is a solution of the PGT
model (2.1). The second equation naturally leads to the following two cases:
a) p = 0 ⇒
For b 6= 0, we have
6
qb = a± a2 −2b Λ .
6 6 0
p
If, additionally, a2 −2b Λ = 0, the value of qb is unique: qb = a.
6 0 6 6
For b = 0, the value of q is q = Λ /a.
6 0
b) a+qb +2a = 0 ⇒
6 3
1 1 1
a p2 = Λ + q(qb −2a) = Λ + (2a +a)(2a +3a).
3 0 6 0 3 3
2 2 2b
6
For a = 0, p remains undetermined, which is physically not acceptable.
3
5 Canonical gauge generator
As an important step in our examination of the asymptotic structure of spacetime, we
are going to construct the canonical gauge generator, which is our basic tool for studying
asymptotic symmetries and conserved charges of 3D gravity with propagating torsion.
5.1 First-order formalulation
Following Nester’s ideas [23], based on the analogy with the first-order formulation of elec-
trodynamics, L = dA∧⋆F + 1F ∧⋆F, we introduce the first-order formulation1 of the
ED 2
gravitational theory (2.1) by
1
L = Tiτ +Riρ −V(bi,τ ,ρ′)− Λ ε bibjbk +L , (5.1a)
i i i i 3 0 ijk M
where τ and ρ := 2ab +ρ′ are the covariant field momenta (1-forms), conjugate to bi and
i i i i
ωi. Here, not only bi and ωi, but also τ and ρ are independent dynamical variables. The
i i
piece 2ab in ρ is a correction stemming from the term linear in curvature. The potential
i i
V is quadratic in τ and ρ′, and its form is chosen so as to ensure the on-shell equivalence
of the new formulation (5.1a) with (2.1). Indeed, the variation with respect to τ and ρ
i i
produces the field equations τ = H and ρ′ := −ε kρ′ = H′ (Appendix B). The first-order
i i ij ij k ij
formulation leads to a particularly simple construction of the gauge generator [24].
Our approach is based on the canonical structure of the theory, which we investigate
in the standard tensor calculus. In local coordinates xµ, the first-order Lagrangian density,
corresponding to (5.1a), has the form:
1
L˜= εµνρ Ti τ +Ri ρ −V(b,τ,ρ′)+L˜ , (5.1b)
µν iρ µν iρ M
2
(cid:0) (cid:1)
1
The term “first-order formulation” used here differs from the common usage in gravity, where it refers
to the case in which bi and ωi are independent Lagrangianvariables (as in the standard form of PGT).
9
where V stands for the sum of V and the cosmological term. We are concerned here with
finite gravitational sources, characterized by matter fields which decrease sufficiently fast at
largedistances, sothattheircontributiontosurfaceintegralsvanishes. Thus, theasymptotic
dynamics can be studied by ignoring matter fields.
For reference, we display here the vacuum field equations obtained by varying L˜ with
respect to bi ,ωi ,τi , and ρi :
µ µ µ µ
∂V
εµνρ∇ τ − = 0, εµνρ ∇ ρ +ε bj τk = 0, (5.2a)
ν iρ ∂bi ν iρ ijk ν ρ
µ (cid:0) (cid:1)
1 ∂V 1 ∂V
εµνρT − = 0, εµνρR − = 0. (5.2b)
2 iνρ ∂τi 2 iνρ ∂ρi
µ µ
As we mentioned above, equations (5.2b) determine the values of τ and ρ′ .
iµ iµ
5.2 Hamiltonian and constraints
Primary constraints. By introducing the momenta π := (π µ,Π µ,p µ,P µ), conjugate
A i i i i
to the respective Lagrangian variables ϕA := (bi ,ωi ,τi ,ρi ), we obtain the following
µ µ µ µ
primary constraints:
φ 0 := π 0 ≈ 0, φ α := π α −ε0αβτ ≈ 0,
i i i i iβ
Φ 0 := Π 0 ≈ 0, Φ α := Π α −ε0αβρ ≈ 0.
i i i i iβ
p µ ≈ 0, P µ ≈ 0. (5.3)
i i
The canonical Hamiltonian is conveniently represented as
H = bi H +ωi K +τi T +ρi R +V +∂ Dα,
c 0 i 0 i 0 i 0 i α
where
H = −ε0αβ∇ τ ,
i α iβ
K = −ε0αβ ∇ ρ +ε bj τk ,
i α iβ ijk α β
1 (cid:0) (cid:1)
T = − ε0αβT ,
i iαβ
2
1
R = − ε0αβR ,
i iαβ
2
and Dα = ε0αβ(ωi ρ +bi τ ).
0 iβ 0 iβ
Secondary constraints. Going over to the total Hamiltonian,
H = H +ui φ µ +vi Φ µ +wi p µ +zi P µ,
T c µ i µ i µ i µ i
where (u,v,w,z) are canonical multipliers, we find that the consistency conditions of the
10