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arXiv:0710.4953v2 [gr-qc] 29 Nov 2007
Different canonical formulations
of Einstein’s theory of gravity
V.A. Franke∗
St.-Petersburg State University, Russia
Abstract
We describe the four most famous versions of the classical canonical formalism in
the Einstein theory of gravity: the Arnovitt-Deser-Misner formalism, the Faddeev-Popov
formalism, the tetrad formalism in the usual form, and the tetrad formalism in the form
best suited for constructing the loop theory of gravity, which is now being developed. We
present the canonical transformations relating these formalisms. The paper is written
mainly for pedagogical purposes.
∗E-mail: franke@pobox.spbu.ru
1
1. Introduction
The most direct method for constructing a quantum theory is to quantize the corresponding
classical theory written in canonical form. Different equivalent canonical formulations of the
classical theory may then lead to not completely equivalent versions of the quantum theory. In
complicated cases, it is therefore beneficial to use different methods to represent the classical
theory in canonical form before quantization. In particular, this concerns the theory of gravity,
whose final quantum form has not yet been found. It is not improbable that choosing an
appropriate classical canonical formulation, we can here approach a satisfactory solution of the
quantization problem. Precisely this approach underlies the so-called loop theory of gravity
(see [1] and the references therein), which is currently being developed.
In this paper written mainly for pedagogical purposes, we describe several well-known equiv-
alent classical canonical formulations of the Einstein theory of gravity and relations between
these formulations. We first consider the Arnovitt-Deser-Misner (ADM) formalism [2]. We
then use the canonical transformation to pass to the Faddeev-Popov (FP) formalism [3]. We
next use a change of variables to introduce the frame (tetrad) formalism in the usual form.
Finally, we use the canonical transformation to reduce this formalism to the form underlying
the loop theory of gravity [1].
We do not consider the problem of quantizing gravity here and restrict ourself to only
several remarks on this subject, but the information presented here can be useful in studying
this problem.
2. The ADM formalism
First, we consider the classical ADM formalism [2]. Let xµbe coordinates in the Riemannian
space-time (µ, ν, . . . = 0,1,2,3). The coordinate x0=tis called time (we set c= 1, where c
is the speed of light). We assume that all hypersurfaces x0=const are spacelike. The space
coordinates are denoted by xi(i, k, . . . = 1,2,3). We use the metric signature (−,+,+,+).
We fix a hypersurface x0=const and let Σ denote it. In the coordinates xithe three-
dimensional metric induced on Σ coincides with the three-dimensional part of the four-dimensio-
nal metric. We let βik denote this three-dimensional metric and introduce βik by the condition
βikβkl =δl
i.(1)
Then
βik =gik,(2)
βik =gik −g0ig0k
g00 ,(3)
where gµν is the four-dimensional metric and gµνgν λ =δµ
λ. We introduce the notation
g= det gµν , β = det βik .(4)
As usual,
Rαβ,γδ =∂γΓα
δβ −∂δΓα
γβ + Γα
γρΓρ
δβ −Γα
δρΓρ
γβ ,(5)
Rβδ =Rαβ,αδ,(6)
2
R=gβδ Rβδ ,(7)
where Γα
βγ are the Christoffel symbols constructed from the metric gµν by a known method.
We determine the quantities
(3)
Γi
kl,
(3)
Rik,lm,
(3)
Rlm,
(3)
R, formed from βik,∂lβik,∂m∂lβik precisely as
the quantities Γα
βγ ,Rαβ,γδ ,Rβ δ ,Rare constructed from gµν ,∂αgµν ,∂α∂βgµν . We introduce the
covariant derivative
(3)
∇iacting on Σ by the connection
(3)
Γi
kl just as the derivative ∇µacts on
the entire space-time by the connection Γα
βγ . We determine the second fundamental tensor Kik
of the hypersurface Σ:
Kik =Kki =−∇ink
Σ=n0Γ0
ik
Σ=−1
p−g00 Γ0
ik
Σ
,(8)
where the field nµ(x) of unit normals to the surfaces x0=const is determined by the relations
nµnµ=−1, nµ=−δ0
µ
1
p−g00 , nµ=−g0µ
p−g00 .(9)
We have the identity
√−g R =√−g(3)
R+Ki
lKl
i−(Ki
i)2+ 2∂γ√−g(nγ∇δnδ−nδ∇δnγ),(10)
where Ki
l=βikKkl. The simplest derivation of this identity is based on the well-known Gauss
formula relating the curvature tensor of the hypersurface to the curvature tensor of the ambient
Riemannian space.
We consider only the gravitational field not interacting with other fields because all specific
features of the problem can be clearly seen in this case. We start from the action of the
gravitational field
S=Zd4xL(11)
where
L=1
2κ
√−ggαβ Γρ
αγ Γγ
ρβ −Γρ
αβΓγ
ργ −2Λ.(12)
Here κ= 8πγ,γis the Newtonian gravitational constant, and Λ is the cosmological constant.
Otherwise,
L=1
2κ
√−g(R−2Λ) + 1
2κ∂γ√−ggγαΓβ
αβ −gαβΓγ
αβ.(13)
whence we use identity (10) to obtain
L=1
2κ
√−g(3)
R+Ki
lKl
i−(Ki
i)2−2Λ+
+1
2κ∂γ√−ggγαΓβ
αβ −gαβΓγ
αβ + 2√−g(nγ∇δnδ−nδ∇δnγ).(14)
In the case of a closed universe, we can here omit the divergence, and in the case of an island
position of masses in an asymptotically three-dimensionally flat space-time, it suffices to only
take into account the essential part of the divergence equal to
1
2κ
(∂k∂kβll −∂i∂kβik).(15)
3
In the last case Λ = 0. We often omit the divergence and assume that the universe is closed
for simplicity; we also often omit the Λ term.
We choose the quantities
βik ≡gik, N ≡1
p−g00 , Ni=g0i.(16)
as independent ADM field variables. In what follows, the subscripts i, k, . . . are raised and
lowered by the three-dimensional tensors βik and βik. The following relations hold:
gik =βik, gik =βik −NiNk
N2, g0k=Nk, g0k=Nk
N2,(17)
g00 =−N2+NkNk, g00 =−1
N2,√−g=Npβ, (18)
nµ=−δ0
µN, n0=1
N, ni=−Ni
N,(19)
Kik =1
2N(3)
∇iNk+
(3)
∇kNi−∂0βik.(20)
In these variables, Lagrangian density (14) with the divergence omitted becomes
L(ADM )=N2Jij,kl Kij Kkl +√β
2κ(3)
R−2Λ,(21)
where
Jij,kl =1
4√β
2κβikβj l +βilβjk −2βij βkl.(22)
It is convenient to introduce the symbols
δik
lm ≡1
2δi
lδk
m+δi
mδk
l(23)
and to determine the quantity Jij,kl using the condition
Jij,klJkl,mn =δij
nm.(24)
In this case, we have
Jij,kl =2κ
√β(βikβj l +βil βjk −βij βkl),Jij,kl =Jkl,ij =Jji,kl =Jij,lk (25)
and similar relations hold for Jij,kl . We again omit the inessential part of the divergence arising
in the expression for L.
We set
L=Z
x0=const
d3xL(ADM )(26)
4
and determine the conjugate momenta,
P(N)(x)≡δL
δ(∂0N(x)),(27)
P(Ni)(x)≡δL
δ(∂0Ni(x)),(28)
Pik(x)≡δL
δ(∂0βik(x)) =∂L(ADM)
∂(∂0βik(x)) ,(29)
where x≡(x1, x2, x3) and δ/δ( ) is the three-dimensional variational derivative. We immedi-
ately obtain the primary constraints
P(N)(x) = 0, P (Ni)(x) = 0.(30)
We solve these constraints explicitly, i. e., we set P(N)and P(Ni)equal to zero everywhere as
they are encountered. Next,
Pik(x) = ∂L(ADM)
∂Klm
∂Klm
∂(∂0βik)=−2Jik,lmKlm,(31)
hence
Kik =−1
2Jik,lmPlm,(32)
and according to (20), we have
∂0βik =
(3)
∇iNk+
(3)
∇kNi−2NKik .(33)
The density of the generalized Hamiltonian is equal to
H(gen)=Pik∂0βik −L(ADM ).(34)
We again omit the inessential addition to the divergence and obtain
H(gen)=NH0+NiHi,(35)
where
H0=1
2Jik,lmPikPlm +√β
2κ−
(3)
R+ 2Λ,(36)
Hi=−2βispβ
(3)
∇lPls
√β,(37)
and we take into account that Pls√βis a tensor.
The density of the first-order Lagrangian is equal to
L(ADM )
(1) =Pik∂0βik −H(gen)=Pik ∂0βik −NH0−NiHi.(38)
We add the essential part of the divergence and obtain the relation for the island position of
masses in asymptotically three-dimensionally flat space:
L(ADM )
(1) =Pik∂0βik −H(gen)=Pik∂0βik −NH0−NiHi−1
2κ
(∂i∂kβik −∂k∂kβii).(39)
5
We vary L(ADM )
(1) in Nand Niand obtain the secondary constraints
H0(x) = 0,Hi(x) = 0.(40)
In the case of the island position of masses, the total energy reduces to the surface integral
Htotal =1
2κZd3x(∂i∂kβik −∂k∂kβii),(41)
and we now obtain
H(gen)=NH0+NiHi+1
2κ(∂i∂kβik −∂k∂kβii).(42)
In the case of a closed universe, the total energy is zero.
We introduce the Poisson brackets. If F1and F2are two three-dimensional functionals of
βik and Pik, then
{F1, F2}=Zd3xδF1
δβik (x)
δF2
δP ik (x)−δF2
δβik (x)
δF1
δP ik (x),(43)
where δ/δ() is the three-dimensional variational derivative. Obviously,
{F1, F2}=−{F2, F1},(44)
{F1,{F2, F3}} +{F2,{F3, F1}} +{F3,{F1, F2}} = 0,(45)
{F1, F2F3}={F1, F2}F3+F2{F1, F3}.(46)
We next use the notation
f≡f(x), f
∼≡f(˜x).(47)
In this notation, we obtain
nβik, P
∼
lmo=δlm
ik δ3(x−˜x),(48)
βik, β
∼
lm= 0,nPik , P
∼
lmo= 0.(49)
The following relations hold:
nHi,H
∼
ko=Hk∂iδ3(x−˜x)−H
∼
i∂
∼
kδ3(x−˜x),(50)
nHi,H
∼
0o=H0∂iδ3(x−˜x),(51)
nH0,H
∼
0o=βikHk∂iδ3(x−˜x)−β
∼
ikH
∼
k∂
∼
iδ3(x−˜x),(52)
where ∂
∼
i=∂
∂˜xi. Clearly, all the constraints in the classical theory are of the first kind. No new
constraints arise.
6
The constraints Hiare generators of three-dimensional transformations of coordinates on
the surface Σ. Indeed, after the change of coordinates
xi→x′i+ξi(x),(53)
where ξi(x) are infinitely small, we have
δβik ≡β′
ik(x)−βik (x) = −
(3)
∇iξk−
(3)
∇kξi,(54)
δP ik ≡P′ik (x)−Pik (x) = (∂lξi)Plk +Pil∂lξk−∂l(Pik ξl).(55)
It can be verified directly that
Zd3xHiξi, β
∼
kl=δβ
∼
kl,Zd3xHiξi, P
∼
kl=δP
∼
kl.(56)
Correspondingly, the constraint H0generates displacements of points of the surface Σ along
the normal to Σ. In this case, the variations in βik and Pik correspond to the solutions of the
Einstein equations.
We make several remarks about quantizing the described theory. Under quantization, the
variables βik and Pik are replaced with operators satisfying the conditions
[βik, P
∼
lm] = iδlm
ik δ3(x−˜x),(57)
[βik, β
∼
lm] = [Pik, P
∼
lm] = 0.(58)
Because constraints (36) and (37) are too complicated to be solved explicitly, these constraints
are usually imposed on the state vector. The theory thus obtained is consistent only under the
condition that the commutators of the constraints are equal to linear combinations of these
constraints with coefficients placed to the left of them. After quantization, the constraints sat-
isfy commutation relations of form (50)-(52) with the bracket { } replaced with −i[ ]. But the
order of the factors βik and Pik chosen in the expressions for the constraints is now important.
It may happen that the result of commuting the constraints contains these factors not in the
order originally accepted in the constraints and the coefficients of the constraints may arise
not only to the left of them. It is easy to see that this does not occur in quantum analogues
of relations (50) and (51), and these relations preserve the form after quantization (up to the
change { } → −i[ ]). In particular, the latter is due to the abovementioned geometric sense
of the constraints Hias generators of transformations of three-dimensional coordinates. This
sense is completely preserved under quantization.
The situation with the quantum analogue of relation (52) is quite different. If the operators
in the constraints H0and Hiare located such that these constraints are Hermitian, then the
quantity −i[H0,H
∼
0] obtained from the quantity {H0,H
∼
0}is also Hermitian. This means that
the non-Hermitian expressions βikHkcannot appear on the right in an analogue of relation (52)
(we take into account that βik and Hkdo not commute). The most that can be obtained for the
commutator −i[H0,H0
∼
] by choosing the order of the factors in H0and Hkwithout violating
7
the Hermitian property is an expression of the form
1
2βikHk+Hkβki∂iδ3(x−˜x)−1
2β
∼
ikH
∼
k+H
∼
kβ
∼
ki∂
∼
iδ3(x−˜x) =
=βikHk∂iδ3(x−˜x)−β
∼
ikH
∼
k∂
∼
iδ3(x−˜x) +
+δ3(0)(...)∂iδ3(x−˜x)−δ3(0)(...)
∼
∂
∼
iδ3(x−˜x),(59)
where (...) and (...)
∼
are some nonzero operator-valued functions. The symbols δ3(0) arise
from commuting the operators βik and Hkor β
∼
ik and H
∼
ktaken at the same point.
Clearly, an expression of form (59) containing the product δ3(0)∂iδ3(x−˜x) does not make
sense. A meaningful expression can be obtained from it only by regularization. This raises the
question of the possibility of choosing a regularization such that the extra terms in expression
(59) become zero and the general covariance of the theory is reestablished after the regulariza-
tion is removed. But a unique answer to this question has not yet been obtained. In several
published works, the problem of regularization and its removal was studied insufficiently rig-
orously. An explanation for this is that the regularization methods were studied in detail only
in the framework of the perturbation theory. But the problem is posed beyond this framework
here.
Although there is still a certain ambiguity in this problem, the theory of gravity was quan-
tized by the path-integral method by analogy with quantizing non-Abelian gauge theories (see
[3] and the references therein and also [4]). If a satisfactory perturbation theory were thus
obtained, then its consistency could be verified directly in the framework of the Feynman di-
agram formalism, and this would suffice. But it turned out that the constructed perturbation
theory is unrenormalizable. Under these circumstances, different approaches for constructing
the quantum theory of gravity are now being developed; the most well-known approaches are
superstring theory (see, e.g., [5]) and the so-called loop theory of gravity [1].
We also note that the above difficulties in closing the constraint algebra after quantization
are also typical of other versions of the canonical formalism in the theory of gravity, which are
described below.
3. The FP formalism
We now consider the classical canonical FP formalism [3]. We first introduce the quantities
hµν =√−g gµν ,(60)
in terms of which the subsidiary harmonic coordinate condition can be simply written as
∂µhµν = 0. For the original variables, we take the functions
qik ≡h0ihk0−h00hik ,(61)
and we write q≡det qik in what follows. Moreover, we preserve the functions Nand Ni
contained in the ADM formalism.
8
The ADM and FP formalisms are related by the canonical transformation
qik =ββik =1
2εilmεknpβlnβmp,
πik =−1
(2κ)2√βJik,lmPlm =β−11
2βikβlm −βilβkmPlm ,
(62)
βik =1
2√qεilmεknpqlnqmp,
Plm =−1
√qqikqlm −qliqmk πik ,
(63)
where πik are the momenta conjugate to the generalized coordinates qik ,εikl is a completely
antisymmetric symbol, and ε123 = 1. In this case
πik∂0qik =Pik ∂0βik ,(64)
nqik, π
∼
lmo=δik
lmδ3(x−˜x),(65)
qik, q
∼
lm= 0,nπik , π
∼
lmo= 0.(66)
In the FP formalism, the density of the first-order Lagrangian has the form
L(FP)
1=πik∂0qik −NH0−NiHi+1
2κ∂i∂kqik,(67)
where we write the part of the divergence that is essential in the case of the island position of
masses in an asymptotically three-dimensionally flat space-time. Now
H0=2κ
q1/4qlpqmq −qlmqpq πlmπpq −q1/4
2κ(3)
R−2Λ,(68)
Hi=2
q1/4qkl (3)
∇kq1/4πil−
(3)
∇iq1/4πkl,(69)
where we must express βik in
(3)
Rin terms of qlm according to (63).
The quantities H0and Hicontinue to satisfy relations (50)-(52), and the geometric meaning
of these quantities is preserved.
4. The usual frame formalism
We now consider the frame formalism. At each point of space-time, we introduce four
mutually pseudo-orthogonal normalized vectors eµ
A(x), where the subscript Anumbers the
vectors (A= 0,1,2,3), the superscript µnumbers their components in the coordinate basis
(µ= 0,1,2,3), and x≡ {x0, x1, x2, x3}. We assume that
eµ
A(x)gµν eν
B(x) = ηAB,(70)
9
where ηAB = diag(−1,1,1,1). We also introduce the variables eA
µ(x) by the relation
eA
µeν
A=δν
µ.(71)
It follows from relations (70) and (71) that
gµν (x) = eA
µηABeB
ν.(72)
We substitute this expression in expressions (11) and (12) above for the action of the gravita-
tional field and regard eA
µ(x) as functions describing this field in what follows. We thus obtain a
theory invariant under two groups of transformations: general coordinate transformations and
local Lorentz transformations of the variables eA
µ(x). The last transformations have the form
e′A
µ(x) = ωAB(x)eB
µ(x),(73)
under the condition
ηABωAD(x)ωBE(x) = ηDE .(74)
Relation (74) ensures the invariance of the metric gµν (x) under such transformations. The
variables eA
µand eµ
Aare called frame parameters.
Each vector referred to the coordinate basis is assigned a vector referred to the frame basis
according to the rule
aA=eA
µaµ, aA=eµ
Aaµ.(75)
The tensors Tµ,...,A,...
ν,...,B,... , which vary in the indices µ,...,ν,... as usual under the change of coor-
dinates and which are transformed by the Lorentz matrices in the indices A,...,B,... under
the change of parameters eA
µaccording to (73), can be introduced similarly. The tensor indices
A,...,B,... are raised and lowered using the symbols ηAB and ηAB.
The covariant derivatives are introduced as usual,
∇µaν=∂µaν+ Γν
µαaα,∇µaν=∂µaν−aαΓα
µν ,(76)
∇µaA=∂µaA+AµABaB,∇µaA=∂µaA−aBAµBA(77)
(and similarly in the case of tensors). Under the assumption that
∇µeA
ν= 0,∇µeν
A= 0,∇µηAB = 0,(78)
we establish the relation between Γα
µν and AµAB:
Γµ
αν =eµ
AAαABeB
ν+eµ
A∂αeA
ν,(79)
AαAB=eA
µΓµ
αν eν
B+eA
µ∂αeµ
B,(80)
where AαABis called a frame connection and Γµ
αν is called a coordinate connection.
The frame connection is similar to the gauge field with a Lorentz structure group. Therefore,
AαAB=AαADηDB ,(81)
10
where AαAD =−AαDA. If Aαis understood as the matrix AαAB, then we can construct the
analogue of the field strength
Fµν =∂µAν−∂νAµ+AµAν−AνAµ,(82)
and, moreover,
Rαβ,µν =eα
AFµν,ABeB
β.(83)
It is necessary to use the frame formalism to describe spinors in the Riemannian space-time
because the spinor representations of the Lorentz group cannot be extended to the representa-
tions of the total linear group. Therefore, the spinors cannot be referred to the local coordinate
basis; they can only be referred to the pseudo-orthogonal frame basis. But we use the frame
formalism for a different purpose in what follows. As before, we assume that the gravitational
field does not interact with other fields.
To remove the gauge arbitrariness completely, we must additionally impose four coordinate
and six frame subsidiary conditions. We use this possibility and remove only part of the frame
arbitrariness using the three conditions
eµ
(0)(x) = nµ,(84)
where nµ(x) is the normalized normal to the surface x0=const at the point x(see relations
(9)). Hereafter, we enclose the frame indices in parentheses if they are written as numbers
and write the coordinate indices without parentheses in this case. Relation (84) contains only
three conditions because the vectors nµand eµ
(0) are normalized. After conditions (84) are
introduced, the theory remains invariant under the semidirect product of the group of gen-
eral coordinate transformations and the group of three-dimensional orthogonal transformations
(i. e., the Lorentz transformations under which the vector eµ
(0)(x) = nµ(x) remains unchanged).
On each surface x0=const, we introduce the ADM variables βik ≡gik,Nand Ni. Next,
i, k, . . . are three-dimensional coordinate indices (i, k, . . . = 1,2,3), and a, b, . . . are three-
dimensional frame indices (a, b, . . . = 1,2,3). By conditions (84), the frame parameters eµ
Aand
eA
µcan be expressed in terms of ei
a,ea
i,Nand Nias
e0
(0) =1
N, e0
a= 0, ei
(0) =−Ni
N, e(0)
0=N, e(0)
i= 0, ea
0=ea
iNi.(85)
In this case, not only
eA
µeν
A=δν
µ,(86)
but also
ea
iek
a=δi
k, gik =βik =ea
iea
k, βik =ei
aek
a.(87)
We set
e≡det ea
i,(88)
and then
β=e2.(89)
Having in mind a possible application to the loop theory of gravity, for the main variables,
we take the functions
Qi
a≡eei
a=pβei
a, N, Ni.(90)
11
We set
Q≡det Qi
a,(91)
and
Q=β. (92)
We define the quantities Qa
iby the relations
Qa
iQk
a=δk
i.(93)
The vectors Qi
a(as well as ei
a) are tangent to the surface x0=const. It follows from relations
(87) that the indices a, b, . . . can be raised and lowered using the symbols δab and δab. Therefore,
there is no difference between the superscripts and subscripts a,b, and they can be written for
convenience.
We develop the canonical formalism on the hypersurface x0=const in terms of the three-
dimensional variables Qi
a,N,Ni, preserving the notation Σ for this hypersurface. The simplest
way to the goal is to start from the FP formalism (see Sec. 3). According to formulas (62), (87)
and (90), we have
qik =ββik =βei
aek
a=Qi
aQk
a.(94)
We substitute this expression in FP Lagrangian (67) and first assume that πik and Qi
aare
independent. We see that
πik∂0qik = 2πik Qk
a∂0Qi
a,(95)
and the variables Qi
aare thus assigned the conjugate momenta
Pa
i= 2πikQk
a.(96)
We hence have
πik =1
2Qa
kPa
i.(97)
But πik =πki, and the new constraints
Qa
kPa
i−Qa
iPa
k= 0 (98)
therefore appear. In view of relations (93), this is equivalent to the three constraints
Φa≡εabcQibPc
i= 0,(99)
where εabc is completely antisymmetric and ε123 = 1.
The action of the frame formalism can now be written in the canonical form
S(frame)
(1) =Zd4xL(frame)
(1) ,(100)
L(frame)
(1) =Pa
i∂0Qi
a−NH0−NiHi−λaΦa,(101)
where we take the new constraints into account using the Lagrange multipliers λa. We assume
that in FP formulas (68) and (69) for H0and Hi, the variables qik and πik are expressed in
12
terms of Pa
iand Qi
aa according to (94) and (97). Hereafter, for simplicity, we do not write the
divergence in L(frame)
(1) and consider the case of a closed universe. We have
H0=1
42κ
√QQk
bQl
bPc
kPc
l−(Qk
bPb
k)2−√Q
2κ(3)
R−2Λ,(102)
Hi=Qk
a(3)
∇kPa
i−
(3)
∇iPa
k,(103)
where
(3)
∇kas before is the covariant derivative on the hypersurface x0=const containing the
connection coefficients
(3)
Γi
kl and
(3)
Aiabexpressed in terms of Qk
aand Qa
k.
The coefficients
(3)
Aiabare determined by analogy with formula (80) by the relations
(3)
Akab=ea
i
(3)
Γi
klel
b−ea
i∂kei
b.(104)
We can verify that by condition (84)
(3)
Akab=Akab,(105)
where Akabis the three-dimensional part of the four-dimensional connection AµAB. The quan-
tities Akabform an SO(3) connection. Therefore,
Akab=Akab =−Akba =εabcAc
k,(106)
where
Ac
k=1
2εcabAkab.(107)
It follows from formulas (90) and (92) that
ei
a=Q−1/2Qi
a, ea
i=Q1/2Qa
i.(108)
Substituting this in relation (104) and taking expressions (105) and (107) into account, we
obtain
Ac
i=1
2εcab (∂kQa
i−∂iQa
k)Qk
b−Ql
a(∂lQd
m)Qm
bQd
i+Qb
iQk
aQl
d∂kQd
l=
=1
2εcab Qa
k∂iQkb +Qd
iQka(Qb
l∂kQld +Qd
l∂kQlb) + Qa
iQkbQd
l∂kQld.(109)
Letting Akdenote the matrix Akab, we can determine the three-dimensional field strength
(3)
Fik =∂iAk−∂kAi+AiAk−AkAi.(110)
In this case, the relations
(3)
Fikab =εabc (3)
Fikc,
(3)
Ri
k,lm =ei
a
(3)
Flmab eb
k(111)
13
hold.
The canonical variables now satisfy the relations
nQi
a,P
∼
b
ko=δi
kδa
bδ3(x−˜x),Qi
a, Q
∼
k
b= 0,nPb
i,P
∼
a
ko= 0.(112)
The symmetry condition πik =πki in the framework of the formalism considered in this section
is satisfied because of constraints (99), while this condition holds in Sec. 3 by definition. The
Poisson brackets relating πik and π
∼
lm differ from those introduced in Sec. 3. We now have
qik, q
∼
lm= 0,nqik , π
∼
lmo=δik
lmδ3(x−˜x),(113)
but nπik, π
∼
lmo=1
4Qb
kQb
mQd
iQc
lεcdaΦaδ3(x−˜x).(114)
New terms therefore appear in the right-hand sides of relations with Poisson brackets (50)-
(52), but all these terms are proportional to the constraints Φa. Moreover,
nΦa,Φ
∼
bo=εabcΦcδ3(x−˜x),nΦa,H
∼
0o= 0,nΦa,H
∼
io= 0.(115)
Therefore, the classical algebra of constraints is closed, and all the constraints H0,Hiand Φa
are constraints of the first kind.
Instead of the constraints Hiit is convenient to introduce the constraints
H′
i≡Hi+AicΦc≡Qbk(∂kPb
i−∂iPb
k) + (∂kQkb)Pb
i= 0.(116)
We can verify that the quantities H′
igenerate transformations of three-dimensional coordinates
without changing the frames as geometric objects and the quantities Φagenerate rotations of
frames without changing the coordinates.
The algebra of constraints in terms of the quantities H0,H′
iand Φahas the form
nH′
i,H
∼
′
ko=H′
k∂iδ3(x−˜x)−H
∼
′
i∂
∼
kδ3(x−˜x),
nH′
i,H
∼
0o=H0∂iδ3(x−˜x),
nH0,H
∼
0o=Qi
aQk
aH′
k∂iδ3(x−˜x)−Q
∼
i
aQ
∼
k
aH
∼
′
i∂
∼
kδ3(x−˜x) + (...)aΦa−(...)
∼
aΦ
∼
a,
nΦa,Φ
∼
bo=εabcΦcδ3(x−˜x),
nΦb,H
∼
′
io=−Φb∂iδ3(x−˜x),
nΦa,H
∼
0o= 0,(117)
where (...)aare certain expressions composed of Qi
aand Pa
i.
14
5. The formalism used in the loop theory of gravity
We now turn to the canonical formalism underlying the loop theory of gravity. We can
readily verify that three-dimensional frame connection (109) admits the representation
Aa
i=δF
δQi
a(x),(118)
where
F=1
2Z
x0=const
d3x εabcQl
aQb
k∂lQkc.(119)
Here, x≡(x1, x2, x3), and δ/δQi
aa is the three-dimensional variational derivative. Because
Pa
i(x), f[Qk
b]=−δf [Qk
b]
δQi
a(x),(120)
where f[Qk
b] is an arbitrary functional of the functions Qk
a(x), we have
nPf
m, A
∼
c
io=nP
∼
c
i, Af
mo.(121)
This permits performing the canonical transformation
Qi
a−→ Qi
a,
Pa
i−→ P′a
i=Pa
i+bAa
i,(122)
where bis a number called the Barbero-Immirzi parameter. This parameter can be assigned
any value. By (121), the condition
P′a
i,P′b
k= 0.(123)
holds. Because Aa
idepends only on Qa
i, the relation
nQi
a,P
∼
′b
ko=δi
kδb
aδ3(x−˜x) (124)
is also satisfied.
After (122), we can perform one more canonical change of variables:
Qi
a−→ Ba
i=1
bP′a
i=Aa
i+1
bPa
i,
P′a
i−→ Πi
a=−bQi
a.(125)
Under the change of the three-dimensional coordinates on the hypersurface x0=const and
of the frames satisfying condition (84) all the time, the quantity Ba
itransforms as the frame
connection Aa
i. Therefore, the path integral
tr W(C) = trPexp
−I
C
dxiBi
,(126)
15
where Biis the matrix with the entries Bab
i=εabcBc
iand the integration is over a closed path
on the hypersurface x0=const, is invariant under the SO(3) frame transformations generated
by the constraints Φa. This fact underlies the loop theory of gravity in which the quantities Ba
i
and Πi
aare used as canonical variables.
It follows from formulas (125) that
Qi
a=−1
bΠi
a,Pa
i=b(Ba
i−Aa
i),(127)
where
Aa
i≡Aa
i
Qi
a=−b−1Πi
a
.(128)
We substitute this expression in action (101), take (102) and (103) into account, and write the
result in the two forms
S1=Zd4xL1,(129)
L1= Πi
a∂0Ba
i−N′H′
0−N′iH′
i−λ′aΦa=
= Πi
a∂0Ba
i−N′′H′′
0−N′′iH′′
i−λ′′aΦa,(130)
where
Φa=−∂iΠi
a+εabcBc
iΠi
b=−√−Π
(3B)
∇iΠi
a
√−Π≡ −DiΠi
a,(131)
H′
k=Hk+Aa
kΦa=−Πi
c
(3)
Fc
ik(B) + Bb
kΦb,
H′
0= Πi
aΠk
bεabc(3)
Fc
ik(B) + 4b−31
(2κ)2Π(1 + κ2b2)
(3)
R−2Λ,(132)
H′′
k=−Πi
c
(3)
Fc
ik(B) + Bb
kΦb,
H′′
0= Πi
aΠk
bεabc(3)
Fc
ik(B)−
−(1 + κ2b2)Πk
bΠl
b(Bc
k−Ac
k)(Bc
l−Ac
l)−Πk
b(Bb
k−Ab
k)2−2b−1ΛΠ.(133)
Here,
N′=N
42κ
√Q, N′k=Nk,
λ′a=λa+N
42κ
√QQkdPb
kεdba −NkAb
k+bκei
a∂iN,
N′′ =−N
(2κ)b2√Q, N′′
k=Nk,
λ′′a=λa−NkAb
k−N(2κ)b2pQ−1Qk
dPf
kεdf a −2 ((2κ)b)−1ei
a∂iN,
Π = det Πi
a, Aa
k=Aa
k(Qi
a)
Qi
a=−b−1Πi
a
,
(3)
R=
(3)
R(Qi
b)
Qi
b=−b−1Πi
b
,
F(3)
ik (B) = ∂iBk−∂kBi+BiBk−BkBi,(134)
16
Bi,
(3)
Fik are matrices with the entries
Bab
i=εabcBc
i,
(3)
Fab
ik =εabc(3)
Fc
ik,(135)
and
(3B)
∇iis the three-dimensional covariant derivative with the connection Bab
i.
We can see that the expressions for H′
0and H′′
0are drastically simplified and become equal
to each other if we continue the theory to the complex domain and set
b=∓iκ.(136)
According to (125), we then have
Bc
i=Ac
i±iκPc
i.(137)
But in the frame basis where eµ
(0) =nµ, the relations
A(0)c
i=NΓ0
ikek
c=−Kikek
c,(138)
hold, and then
πik =1
(2κ)√βKik,Pa
i= 2πikQk
a=1
κKikek
a=−1
κA(0)c
i,(139)
which implies that relation (137) becomes
Bc
i=Ac
i∓iA(0)c
i,(140)
i. e.,
Bab
i=Aab
i∓iεabcA(0)c
i.(141)
We can introduce the fields
AAB(±)
µ=AAB
µ∓i
2ηADηBE εDE F H AF H
µ,(142)
whence
Aab(±)
i=Aab
i∓iεabcA(0)c
i.(143)
The fields AAB(±)
µsatisfy the self-duality (anti-self-duality) conditions
AAB(±)
µ=∓i
2ηADηB E εD EF H AF H(±)
µ,(144)
and
AAB
µ=AAB(+)
µ+AAB(−)
µ.(145)
Therefore, the expressions for H′
0and H′′
0are simplified if
Bab
µ=Aab(+)
µ(146)
or
Bab
µ=Aab(−)
µ.(147)
17
In this case, all the constraints depend polynomially on the variables Ba
iand Πi
a. But to return
to the real domain, the complicated condition
Ba
i+Ba
i
∗= 2A(Q)
Qi
a=−b−1Πi
a
(148)
must be satisfied, where ∗denotes complex conjugation in the classical case and Hermitian
conjugation after quantization. Because condition (148) is complicated, it is currently preferred
to construct the loop theory of gravity for a real value of the parameter bfor the case in which
the constraint H′
0(or H′′
0) is very complicated.
The above classical canonical formulations of the theory of gravity have found and continue
to find application in studying the problem of quantizing this theory.
Acknowledgments. The author thanks the UNESCO Regional Bureau for Science and
Culture in Europe for the support of the V.A. Fock International School of Physics.
References
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an introduction to current research”. Ed. Louis Witten. Wiley, 1962. Chapter 7, P. 227.
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[3] V.N. Popov, L.D. Faddeev. Sov. Phys. Usp. 1973. V. 111. P. 427.
[4] N.P. Konopleva, V.N. Popov. Gauge Fields [in Russian]. URSS, Moscow, 2000; English
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