arXiv:0704.0367v6  [gr-qc]  18 Mar 2011
Instanton representation of Plebanski gravity.
Consistency of the initital value constraints under
time evolution
Eyo Eyo Ita III
October 24, 2018
Department of Applied Mathematics and Theoretical Physics
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road
Cambridge CB3 0WA, United Kingdom
eei20@cam.ac.uk
Abstract
The instanton representation of Plebanski gravity provides as equa-
tions of motion a Hodge self-duality condition and a set of ‘general-
ized’ Maxwell’s equations, subject to gravitational degrees of freedom
encoded in the initial value constraints of general relativity. The main
result of the present paper will be to prove that this constraint sur-
face is preserved under time evolution. We carry this out not using
the usual Dirac procedure, but rather the Lagrangian equations of mo-
tion themsleves. Finally, we provide a comparison with the Ashtekar
formulation to place these results into overall context.
http://arxiv.org/abs/0704.0367v6
1 Introduction
In [1] a new formulation of general relativity was presented, named the
instanton representation of Plebanski gravity. The basic dynamical variables
are an SO(3, C) gauge connection Aaµ and a matrix Ψae taking its values in
two copies of SO(3, C).1 The consequences of the associated action IInst
were determined via its equations of motion, which hinge crucially on weak
equalities implied by the the initial value constraints. For these consequences
to be self-consistent, the constraint surface must be preserved for all time
by the evolution equations. The present paper will demonstrate that this
is indeed the case. We will not use the usual Hamiltonian formulation for
totally constrained systems [2], since we will not make use of any canonical
structure implied by IInst. Rather, we will deduce the time evolution of the
dynamical variables directly from the equations of motion of IInst.
Sections 2 and 3 of this paper present the instanton representation ac-
tion and derive the time evolution of the basic variables. Sections 4, 5 and 6
demonstrate that the nondynamical equations, referred to as the diffeomor-
phism, Gauss’ law and Hamiltonian constraints, evolve into combinations of
the same constraint set. The result is that the time derivatives of these con-
straints are weakly equal to zero with no additional constraints generated on
the system. While we do not use the usual Dirac method in this paper, the
result is still that the instanton representation is in a sense Dirac consistent.
We will make this inference clearer by comparison with the Ashtekar vari-
ables in the discussion section. On a final note, the terms ‘diffeomorphism’
and ‘Gauss’ law’ constraints are used loosely in this paper, in that we have
not specified what transformations of the basic variables these constraints
generate. The use of these terms is mainly for notational purposes, due to
their counterparts which appear in the Ashtekar variables.
2 Instanton representation of Plebanski gravity
The starting action for the instanton representation of Plebanski gravity is
given by [1]
IInst =
d3xΨaeB
F a0i + ǫkjmB
−iN(detB)1/2
Λ+ trΨ−1
, (1)
1Index labelling conventions for this paper are that symbols a, b, . . . from the begini-
ing of the Latin alphabet denote internal SO(3, C) indices while those from the middle
i, j, k, . . . denote spatial indices. Both of these sets of indices take takes 1, 2 and 3. The
Greek symbols µ, ν, . . . refer to spacetime indices which take values 0, 1, 2, 3.
where Nµ = (N,N i) are the lapse function and shift vector from metric
general relativity, and Λ is the cosmological constant. The basic fields are
Ψae and A
i , and we action (1) is defined only on configurations restricted
to (detB) 6= 0 and (detΨ) 6= 0.2 In the Dirac procedure one refers to Nµ
as nondynamical fields, since their velocities do not appear in the action.
While the velocity Ψ̇ae also does not appear, we will distinguish this field
from Nµ since the action (1), unlike for the latter, is nonlinear in Ψae.
The equation of motion for the shift vector N i, the analogue of the
Hamilton equation for its conjugate momentum Π ~N , is given by
δIInst
= ǫmjkB
eΨae = (detB)(B
−1)diψd ∼ 0, (2)
where ψd = ǫdaeΨae is the antisymmetric part of Ψae. This is equivalent to
the diffeomorphism constraint Hi owing to the nondegeneracy of B
a, and
we will often use Hi and ψd interchangeably in this paper. The equation of
motion for the lapse function N , the analogue of the Hamilton equation for
its conjugte momentum ΠN , is given by
δIInst
= (detB)1/2
Λ+ trΨ−1
= 0. (3)
Nondegeneracy of Ψae and the magnetic field B
e implies that on-shell, the
following relation must be satisfied
Λ + trΨ−1 = 0, (4)
which we will similarly take as synonymous with the Hamiltonian constraint.
The equation of motion for Ψae is
δIInst
= BkeF
0k + ǫkjmB
m + iN
detΨ(Ψ−1Ψ−1)ea ∼ 0, (5)
up to a term proportional to (4) which we have set weakly equal to zero.
One could attempt to define a momentum conjugate to Ψae, for which (5)
would be the associated Hamilton’s equation of motion. But since Ψae forms
part of the canonical structure of (1), then our interpretation is that this is
not technically correct.3
The equation of motion for the connection Aaµ is given by
δIInst
∼ ǫµσνρDσ(ΨaeF eνρ)−
4ǫmjkN
mBkeΨ[de]
+N(B−1)dj
Λ+ trΨ−1
, (6)
2The latter case limits the application of our results to spacetimes of Petroc Types I,
D and O (See e.g. [3] and [4].
3This is because (5) contains a velocity Ȧa
within F a
and will therefore be regarded
as an evolution equation rather than a constraint. This is in stark contrast with (2) and
(3), which are genuine constraint equations due to the absence of any velocities.
where we have defined
ea(x, y) ≡
δAai (x)
Bje(y) = ǫ
−δae∂k + fedaAdk
δ(3)(x, y); D
ea ≡ 0. (7)
The terms in large round brackets in (6) vanish weakly, since they are pro-
portional to the constraints (2) and (4) and their spatial derivatives. For
the purposes of this paper we will regard (6) as synonymous with
ǫµσνρDσ(ΨaeF
νρ) ∼ 0. (8)
In an abuse of notation, we will treat (5) and (8) as strong equalities in
this paper. This will be justified once we have completed the demonstration
that the constraint surface defined collectively by (2), (3) and the Gauss’
constraint from (8) is indeed preserved under time evolution. As a note
prior to proceeding we will often make the identification
N(detB)1/2
detΨ ≡
−g (9)
as a shorthand notation, to avoid cluttering many of the derivations which
follow in this paper.
2.1 Internal consistency of the equations of motion
Prior to embarking upon the issue of consistency of time evolution of the
initial value constraints, we will check for internal consistency of IInst, which
entails probing of the physical content implied by (8) and (5). First, equation
(8) can be decomposed into its spatial and temporal parts as
Di(ΨbfB
f ) = 0; D0(ΨbfB
f ) = ǫ
ijkDj(ΨbfF
0k). (10)
The first equation of (10) is the Gauss’ law constraint of a SO(3) Yang–
Mills theory, when one makes the identification of ΨbfB
f ∼ Eib with the
Yang–Mills electric field. The Maxwell equations for U(1) gauge theory
with sources (ρ, ~J), in units where c = 1, are given by
~∇ · ~B = 0; Ḃ = −~∇× ~E = 0; ~∇ · ~E = ρ; ~̇E = − ~J + ~∇× ~B. (11)
Equations (10) can be seen as a generalization of the first two equations of
(11) to SO(3) nonabelian gauge theory in flat space when one: (i) identifies
with the SO(3) generalization of the electric field ~E, and (ii) one
chooses Ψae = kδae for some numerical constant k.
When ρ = 0 and ~J = 0, then one has the vacuum theory and equations
(11) are invariant under the transformation
( ~E, ~B) −→ (− ~B, ~E). (12)
Then the second pair of equations of (11) become implied by the first pair.
This is the condition that the Abelian curvature Fµν , where F0i = Ei and
ǫijkFjk = Bi, is Hodge self-dual with respect to the metric of a conformally
flat spacetime. But equations (10) for more general Ψae encode gravitational
degrees of freedom, which as shown in [1] generalizes the concept of self-
duality to more general spacetimes solving the Einstein equations. Let us
first attempt to derive the analogue for (10) of the second pair of (11) in the
vacuum case. Acting on the first equation of (10) with D0 yields
D0Di(ΨbfB
f ) = DiD0(ΨbfB
f ) + [D0,Di](ΨbfB
f ) = 0. (13)
Substituting the second equation of (10) into the first term on the right hand
side of (13) and using the definition of temporal curvature as the commutator
of covariant derivatives on the second term we have
ijkDj(ΨbfF
0k)) + fbcdF
0iΨdfB
f = fbcd
0k +B
Ψdf = 0 (14)
where we have also used the spatial part of the commutator ǫijkDiDjva =
fabcB
b vc. Note that the term in brackets in (14) is symmetric in f and c,
and also forms the symmetric part of the left hand side of (5)
0i + i
−g(Ψ−1Ψ−1)fb + ǫijkBifB
k = 0, (15)
re-written here for completeness. To make progress from (14), we will sub-
stitute (15) into (14). This causes the last term of (15) to drop out due to
antisymmetry, which leaves us with
−gfbcd
Ψdf (Ψ
−1Ψ−1)fc +Ψdf (Ψ
−1Ψ−1)fc
= −2i
−gfbcdΨ−1dc . (16)
The equations are consistent only if (16) vanishes, which is the requirement
that Ψae = Ψea be symmetric. This of course is the requirement that the
diffeomorphism constraint (2) be satisfied. So the analogue of the second
pair of (11) in the vacuum case must be encoded in the requirement that
Ψae = Ψea be symmetric.
3 The time evolution equations
We must now verify that the initial value constraints are preserved under
time evolution defined by the equations of motion (5) and (6). These equa-
tions are respectively the Hodge duality condition
0k + i
−g(Ψ−1Ψ−1)fb + ǫijkN iBjbB
f = 0, (17)
and one of the Bianchi identity-like equations
ǫijkDj(ΨaeF
ok) = D0(ΨaeB
e). (18)
Since the initial value constraints were used to obtain the second line of (17)
from (1), then we must verify that these constraints are preserved under
time evolution as a requirement of consistency. Using F b0i = Ȧ
i −DiAb0 and
defining
−g(B−1)fi (Ψ
−1Ψ−1)fb + ǫmnkN
mBnb ≡ iHbk, (19)
Then equation (17) can be written as a time evolution equation for the
connection, which is not the same as a constraint equation as noted earlier
F b0i = −iHbi −→ Ȧbi = DiAb0 − iHbi . (20)
From equation (20) we can obtain the following equation governing time
evolution equation for the magnetic field
Ḃie = ǫ
ijkDjȦ
k = ǫ
ijkDj
0 − iHek
= febcB
0 − iǫijkDjHek = −δ~θB
e − iǫijkDjHek, (21)
which will be useful. On the first term on the right hand side of (21) we
have used the definition of the curvature as the commutator of covariant
derivatives. The notation δ~θ in (21) suggests that that B
e transforms as
a SO(3, C) vector under gauge transformations parametrized by θb ≡ Ab0.4
Since we have not specified anything about the canonical structure of IInst,
then δ~θ as used in (21) and in (24) should at this stage simply be regarded
as a definition useful for shorthand notation.
We will now apply the Liebnitz rule in conjunction with the definition
of the temporal covariant derivatives to (18) to determine the equation gov-
erning the time evolution of Ψae. This is given by
D0(ΨaeB
e) = B
eΨ̇ae +ΨaeḂ
e + fabcA
0(ΨceB
e) = ǫ
ijkDj(ΨaeF
0k). (22)
Substituting (21) and (20) into the left and right hand sides of (22), we have
BieΨ̇ae +Ψae
febcB
0 − iǫijkDjHek
+ fabcA
0(ΨceB
e) = −iǫijkDj(ΨaeHek).(23)
In what follows, it will be convenient to use the following transformation
properties for Ψae as A
i under SO(3, C) gauge transformations
δ~θΨae =
fabcΨce + febcΨac
Ab0; δ~θA
i = −DiAa0; δ~θB
e = −febcBibAc0. (24)
4We will make the identification with SO(3, C) gauge transformations later in this
paper when we bring in the relation of IInst with the Ashtekar variables.
Then using (24), the time evolution equations for the phase space variables
ΩInst can be written in the following compact form
Ȧbi = −δ~θA
i − iHbi ; Ψ̇ae = −δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
k . (25)
We have found evolution equations for Ψae and A
i from the covariant equa-
tions of Aaµ and the Hodge-duality condition We have obtained these without
using Poisson brackets, and by assuming that the Hamiltonian and diffeo-
morphism constraints are satisfied. Therefore the first order of business is
then to check for the preservation of the initial value constraints under the
time evolution generated by (25). This means that we must check that
the time evolution of the diffeomorphism, Gauss’ law and Hamiltonian con-
straints are combinations of terms proportional to the same constraints and
their spatial derivatives, and terms which vanish when the constraints hold.5
These constraints are given by
we{Ψae} = 0; (detB)(B−1)diψd = 0; (detB)1/2
Λ+ trΨ−1
= 0(26)
where (detB) 6= 0 and (detΨ) 6= 0. We will occasionally make the identifi-
cation
N(detB)1/2(detΨ)1/2 ≡
−g (27)
for a shorthand notation. Additionally, the following definitions are provided
for the vector fields appearing in the Gauss’ constraint
we = B
eDi; ve = B
e∂i (28)
where Di is the SO(3, C) covariant derivative with respect to the connection
Aai . Equations (26) are the equations of motion for the auxilliary fields A
N i and N .
4 Consistency of the diffeomorphism constraint un-
der time evolution
The diffeomorphism constraint is directly proportional to ψd = ǫdaeΨae, the
antisymmetric part of Ψae. So to establish the consistency condition for
this constraint, it suffices to show that the antisymmetric part of the second
equation of (25) weakly vanishes. This is given by
ǫdaeΨ̇ae = −δ~θ(ǫdaeΨae)− iǫdaeǫ
ijk(B−1)ei (DjΨaf )H
k , (29)
5This includes any nonlinear function of linear order or higher in the constraints, a
situation which involves the diffeomorphism constraint.
which splits into two terms. Using (24), one finds that the first term of (29)
is given by
−ǫdaeδ~θΨae = −ǫdae
fabcΨce +Ψacfebc
δebδdc − δecδbd
Ψce +
δdbδac − δdcδab
Ψdb − δbdtrΨ + δdbtrΨ−Ψbd
Ab0 = 2Ψ[bd]A
0 = −ǫdbhAb0ψh, (30)
which is proportional to the diffeomorphism constraint. The second term of
(30) has two contributions due to H
k as defined in (19). The first contribu-
tion reduces to
−iǫdaeǫijk(B−1)ei (DjΨaf )(H(1))
= −iǫdaeǫijk(B−1)ei (DjΨaf )
−g(B−1)gk(Ψ
−1Ψ−1)gf
= iǫdae(detB)
−1ǫegh(Ψ−1Ψ−1)gfB
hDjΨaf
= i(detB)−1(Ψ−1Ψ−1)gf
a − δgaδhd
vv{Ψaf}
= i(detB)−1(Ψ−1Ψ−1)gf
dva{Ψaf} − vd{Ψgf}
= i(detB)−1
(Ψ−1Ψ−1)dfGf + vd{Λ+ trΨ−1}
. (31)
The first term on the final right hand side of (31) is the Gauss’ constraint
and the second term is the derivative of a term direction proportional to
the Hamiltonian constraint.6 The second contribution to the second term
of (29) is given by
ǫdaeǫ
ijk(B−1)ei (DjΨaf )(H(2))
k = ǫdaeǫ
ijk(B−1)ei (DjΨaf )ǫmnkN
= ǫdac
n − δinδjm
(B−1)ei (DjΨaf )N
= ǫdaeN
i(B−1)eivf{Ψaf} −N jDj(ǫdaeΨae) = ǫdaeN i(B−1)efGa −N jDjψd.(32)
The result is that the time evolution of the diffeomorphism constraint is
directly proportional to
ψ̇d =
i(detB)−1(Ψ−1Ψ−1)da + ǫdaeN
i(B−1)ei
Ab0ǫbdh − δdhN jDj
ψh + i(detB)
vd{(−g)−1/2H}, (33)
which is a linear combination of terms proportional to the constraints (26)
and their spatial derivatives. The result is that the diffeomorphism con-
straint Hi = 0 is consistent with respect to the Hamiltonian evolution gen-
erated by the equations (25). So it remains to verify consistency of Gauss’
law and the Hamiltonian constraints Ga and H.
6We have added in a term Λ, which can be regarded as a constant of integration with
respect to the spatial derivatives from vd.
5 Consistency of the Gauss’ constraint under time
evolution
Having verified the consistency of the diffeomorphism constraint under time
evolution, we now move on to the Gauss’ constraint. Application of the
Liebnitz rule to the first equation of (26) yields
Ġa = Ḃ
eDiΨae +B
eDiΨ̇ae +B
fabfΨfe + febgΨag
Ȧai . (34)
Upon substituion of (21) and (25) into (34), we have
Ġa =
−δ~θB
e − iǫijkDjHek
DiΨae +B
−δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
fabfΨfe + febgΨag
−δ~θA
i − iHbi
.(35)
Using the Liebniz rule to combine the δ~θ terms of (35), we have
Ġa = −δ~θGa − iǫ
k)DiΨae +B
e Dm((B
−1)ei (DjΨaf )H
fabfΨfe + febgΨag
i . (36)
The requirement of consistency is that we must show that the right hand
side of (36) vanishes weakly. First, we will show that the third term on the
right hand side of (36) vanishes up to terms of linear order and higher in
the diffeomorphism constraint. This term, up to an insignificant numerical
factor, has two contributions. The first contribution is
fabfΨfe + febgΨag
Bie(H(1))
fabfΨfe + febgΨag
(Ψ−1Ψ−1)eb
fabf (Ψ
−1)fb + febg(Ψ
−1Ψ−1)ebΨag
∼ δ(1)a (~ψ) ∼ 0, (37)
which is directly proportional to a nonlinear function of first order in ψd
which is proportional to the diffeomorphism constraint. The second contri-
bution to the third term on the right hand side of (36) is
fabfΨfe + febgΨag
Bie(H(2))
fabfΨfe + febgΨag
ǫkmnN
kBme B
fabfΨfe + febgΨag
(detB)Nk(B−1)dkǫdeb
= (detB)Nk(B−1)dk
δfdδae − δfeδad
Ψfe + 2δdgΨag
= (detB)Nk(B−1)dk
Ψda − δadtrΨ + 2Ψad
≡ δ(2)a ( ~N) (38)
which does not vanish, and neither is it expressible as a constraint. For
the Gauss’ law constraint to be consistent under time evolution, a necessary
condition is that this δ
a ( ~N) term must be exactly cancelled by another
term arising from the variation.
Let us expand the terms in square brackets in (36). This is given, using
the Liebniz rule on the second term, by
ǫijk(DjH
k)(DiΨae) + ǫ
ijkBme Dm((B
−1)ei (DjΨae)H
= ǫijk(DjH
k)(DiΨae)− ǫijkBme (B−1)en(DmBng )(B−1)
i (DjΨaf )H
+ǫmjk(DmDjΨaf )H
+ ǫmjk(DjΨaf )(DmH
). (39)
The first and last terms on the right hand side of (39) cancel, which can be
seen by relabelling of indices. Upon application of the definition of curvature
as the commutator of covariant derivatives to the third term, then (39)
reduces to
−ǫijk(DnBng )(B−1)
i (DjΨaf )H
fabcΨcf + ffbcΨac
. (40)
The first term of (40) vanishes on account of the Bianchi identity and the
second term contains two contributions which we must evaluate. The first
contribution is given by
(H(2))
fabcΨcf + ffbcΨac
= (detB)Nk(B−1)dkǫdbf
fabcΨcf + ffbcΨac
= (detB)Nk(B−1)dk
δdaδfc − δdcδfa
Ψcf − 2δdcΨac
= (detB)Nk(B−1)dk
δdatrΨ−Ψda − 2Ψad
= −δ(2)a ( ~N ),(41)
with δ
a ( ~N) as given in (37). So putting the results of (39), (40) and (41)
into (36), we have
Ġa = −δ~θGa + δ
~N) + δ(1)a (
~ψ) + δ(1)a (
~ψ)− δ(2)a ( ~N) = −δ~θGa + 2δ
(1)(~ψ).(42)
The velocity of the Gauss’ law constraint is a linear combination of the
Gauss’ constraint with terms of the diffeomorphism constraint of linear or-
der and higher. Hence the time evolution of the Gauss’ law constraint is con-
sistent in the sense that we have defined, since δ(1)(~ψ) vanishes for ψd = 0.
6 Consistency of the Hamiltonian constraint un-
der time evolution
The time derivative of the Hamiltonian constraint, the third equation of
(26), is given by
((detB)1/2(detΨ)1/2
(Λ + trΨ−1) +
(Λ + trΨ−1) (43)
which has split up into two terms. The first term is directly proportional
to the Hamiltonian constraint, therefore it is already consistent. We will
nevertheless expand it using (21) and (25)
(B−1)di Ḃ
d + (Ψ
−1)aeΨ̇ae
(detB)1/2(detΨ)1/2(Λ + trΨ−1)
(B−1)di
−δ~θB
d − iǫijkDjHdk
+(Ψ−1)ae
−δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
H. (44)
We will be content to compute the δ~θ terms of (44). These are
(B−1)di δ~θB
d = (B
−1)di fdbfB
0 = δdbfdbfA
0 = 0 (45)
on account of antisymmetry of the structure constants, and
(Ψ−1)eaδ~θΨae = (Ψ
−1)ea
fabfΨfe + febgΨag
= 0, (46)
also due to antisymmetry of the structure constants. We have shown that
the first term on the right hand side of (43) is consistent with respect to
time evolution. To verify consistency of the Hamiltonian constraint under
time evolution, it remains to show that the second term is weakly equal to
zero. It suffices to show this just for the second term, in brackets, of (43)
(Λ + trΨ−1) = −(Ψ−1Ψ−1)feΨ̇ef
= (Ψ−1Ψ−1)ef
δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
, (47)
where we have used (25). Equation (47) has split up into two terms, of
which the first term is
(Ψ−1Ψ−1)eaδ~θΨae = (Ψ
−1Ψ−1)ea
fAbfΨfe + febgΨag
fabf (Ψ
−1)fa + febg(Ψ
−1)eg
Ab0 = m(
~ψ) ∼ 0 (48)
which vanishes weakly since it is a nonlinear function of at least linear order
in ψd. The second term of (47) splits into two terms which we must evaluate.
The first contribution is proportional to
(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(H(1))
−g(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(B−1)dk(Ψ−1Ψ−1)df
−g(Ψ−1Ψ−1)ea(Ψ−1Ψ−1)df (detB)−1ǫedgBjgDjΨaf
−g(detB)−1ǫedg(Ψ−1Ψ−1)ea(Ψ−1Ψ−1)dfvg{Ψaf} ≡ v{~ψ} (49)
for some vector field v. We have used the fact that the term in (49) quartic
in Ψ−1 in antisymmetric in a and f due to the epsilon symbol. Hence Ψaf
as acted upon by vg can only appear in an antisymmetric combination, and
is therefore proportional to the diffeomorphism constraint ψd whose spatial
derivatives weakly vanish. Hence (49) presents a consistent contribution to
the time evolution of H, which leaves remaining the second contribution to
the second term of (47). This term is proportional to
(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(H(2))
= (Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )ǫmnkN
n − δinδjm
(B−1)eiB
−1Ψ−1)ea(DjΨaf )
N i(B−1)eiB
f − δefN
(Ψ−1Ψ−1)ea(DjΨaf )
= (−g)−1/2N iHai vf{Ψaf} − (Ψ−1Ψ−1)fa(N jDjΨaf )
= (−1)−1/2N iHai Ga −N jDj(Λ + trΨ−1). (50)
The first term on the final right hand side of (50) is proportional to the
Gauss’ law constraint, and the second term is proportional to the derivative
of the Hamiltonian constraint. To obtain this second term we have added
in Λ as a constant of differentiation with respect to ∂j . Substituting (48),
(49) and (50) into (47), then we have
Ḣ =∼ Ô(~ψ) + (−g)−1/2N iHai Ga + T̂ ((−g)−1/2H), (51)
where Ô and T̂ are operators consisting of spatial derivatives acting to the
right and c numbers. The time derivative of the Hamiltonian constraint
is a linear combination of the Gauss’ law and Hamiltonian constraints and
its spatial derivatives, plus terms of linear order and higher in the diffeo-
morphism constraint and its spatial derivatives. Hence the Hamiltonian
constraint is consistent under time evolution.
7 Recapitulation
The final equations governing the time evolution of the initial value con-
straints are given weakly by
ψ̇d =
i(detB)−1(Ψ−1Ψ−1)da + ǫdaeN
i(B−1)ei
Ab0ǫbdh − δdhN jDj
ψh + i(detB)
vd{Λ + trΨ−1};
Ġa = −fabcAb0Gc + δ(1)a (~ψ);
ǫijk(B−1)di (DjH
k ) + ǫ
ijk(B−1)ei (Ψ
−1)ae(DjΨaf )H
−N j∂j
(Λ + trΨ−1)
+(−g)−1/2N iHai Ga −
−g(detB)−1ǫedg(Ψ−2Ψ−1)ea(Ψ−1Ψ−1)dfvg{ǫafhψh}+m(~ψ).(52)
Equations (52) show that all constraints derivable from the the action (1)
are preserved under time evolution, since their time derivatives yield linear
combinations of the same set of constraints and their spatial derivatives.
There are no additional constraints generated which implies that the action
(1) is consistent in the Dirac sense. On the other hand, we have not defined
the canonical structure of (52) or any Poisson brackets.
Equations (52) can be written schematically in the following form
~̇H ∼ ~H + ~G+H; ~̇G ∼ ~G+Φ( ~H); Ḣ ∼ H + ~G+Φ( ~H), (53)
where Φ is some nonlinear function of the diffeomorphism constraint ~H,
which is of at least first order in ~H. In the Hamiltonian formulation of a
theory, one identifies time derivatives of a variable f with via ḟ = {f,H}
the Poisson brackets of the variable with the Hamiltonian H . So while we
have not specified Poisson brackets, equation (53) implies the existence of
Poisson brackets associated to some Hamiltonian HInst for the action (1),
{ ~H,HInst} ∼ ~H + ~G+H; {~G,HInst} ∼ ~G+Φ( ~H);
{H,HInst} ∼ H +Φ( ~H) + ~G. (54)
So the main result of this paper has been to demonstrate that the instanton
representation of Plebanski gravity forms a consistent system, in the sense
that the constraint surface is preserved under time evolution. As a direction
of future research we will compute the algebra of constraints for (1) directly
from its canonical structure. Nevertheless it will be useful for the present
paper to think of equations (52) in the Dirac context, mainly for compari-
son with other formulations of general relativity. This will bring us to the
Ashtekar variables.
8 Discussion: Relation of the instanton represen-
tation to the Ashtekar variables
We will now provide the rationale for not following the Dirac procedure for
constrained systems [2] with respect to (1), by comparison with the Ashekar
formulation of GR. The action for the instanton representation (1) can be
written in the following 3+1 decomposed form
IInst =
0we{Ψae} − ǫijkN iBjaBkeΨae
−iN(detB)1/2(detΨ)1/2
Λ+ trΨ−1
, (55)
which regards Ψae and A
i as phase space variables. But the phase space of
(55) is noncanonical since its symplectic two form
δθInst = δ
d3xΨaeB
d3xBieδΨae ∧ δAai +
d3xΨaeǫ
ijkDj(δA
k) ∧ δAai , (56)
is not closed owing to the presence of the second term on the right hand
side. The initial stages of the Dirac procedure applied to (55) state that the
momentum conjugate to Aai yields the primary constraint
Πia =
δIInst
δȦai
= ΨaeB
e. (57)
Then making the identification σ̃ia = Π
a and upon substitution into (57)
and into (55), one obtains the action
IAsh =
σ̃iaȦ
0Ga −N iHi −
, (58)
which is the action for the Ashtekar complex formalism of general relativity
[5], [6], with σ̃ia being the densitized triad. This is a totally constrained sys-
tem with (Aa0, N
i, N), respectively the SO(3, C) rotation angle Aa0, the shift
vector N i and the densitized lapse function N = N(detσ̃)−1/2 as auxilliary
fields. The constraints in (58) smearing the auxilliary fields are the Gauss’
law, vector and Hamiltonian constraints
Ga = Diσ̃
a; Hi = ǫijkσ̃
a ; H = ǫijkǫ
abcσ̃iaσ̃
σ̃kc +B
. (59)
From (58) one reads off the symplectic two form ΩAsh given by
ΩAsh =
d3xδσ̃ia ∧ δAai = δ
d3xσ̃iaδA
= δθAsh, (60)
which is the exact functional variation of the canonical one form θAsh.
The actions (55) and (58) are transformable into each other only under
the condition (detB) 6= 0 and (detΨ) 6= 0. In (58) it is clear that σ̃ia and Aai
form a canonically conjugate pair, which suggests that (55) is a noncanonical
version of (58). The constraints algebra for (59) is
{ ~H[ ~N ], ~H [ ~M ]} = Hk
N i∂kMi −M i∂kNi
{ ~H[N ], Ga[θa]} = Ga[N i∂iθa];
{Ga[θa], Gb[λb]} = Ga
fabcθ
{H(N), ~H [ ~N ]} = H[N i∂iN
{H(N ), Ga(θa)} = 0;[
H(N),H(M )
= Hi[
N∂jM −M∂jN
H ij], (61)
which is first class due to closure of the algebra, and is therefore consistent
in the Dirac sense. Let us consider (61) for each constraint with the total
Hamiltonian HAsh and compare with (54). This is given schematically by
{ ~H,HAsh} ∼ ~H + ~G+H; {~G,HAsh} ∼ ~G+ ~H;
{H,HAsh} ∼ H + ~H. (62)
Comparison of (62) with (54) shows an essentially similar structure for the
top two lines involving ~H and ~G.7 But there is a marked dissimilarity with
respect to the Hamiltonian constraint H. Note that there is a Gauss’ law
constraint appearing in the right hand side of the last line of (54) whereas
there is no such constraint on the corresponding right hand side of (62).
This means that while the Hamiltonian constraint is gauge-invariant under
SO(3, C) gauge-transformations as implied by (61) and (62), this is not
the case in (54). This means that the action (1), which as shown in [1]
describes general relativity for Petrov Types I, D and O, has a different
role for the Gauss’ law and Hamiltonian constraints than the action (58),
which also describes general relativity. Therefore IInst and IAsh at some
level correspond to genuinely different descriptions of GR, a feature which
would have been missed had we applied the step-by-step Dirac procedure.
9 Appendix: Commutation relations for IInst
We will now infer the Poisson brackets for (55) by inference from the corre-
sponding canonical Ashtekar Poisson brackets
{Aai (x), σ̃
b (y)} = δ
(3)(x,y) (63)
along with the vanishing brackets
{Aai (x), Abj(y)} = {σ̃ia(x), σ̃
(x)} = 0. (64)
To find the analogue of (63) and (64) for (55), we will use the tranformation
equation
σ̃ia = ΨaeB
e, (65)
which corresponds to a noncanonical transformation. Substitution of (65)
into (63) yields
{Aai (x),Ψbf (y)Bif (y)} = δ
(3)(x,y)
{Aai (x),Ψbf (y)}B
(y) + Ψbf (x){Aai (x), B
(y)}. (66)
The second term on the right hand side of (66) vanishes on account of the
first relations of (64), and upon multiplying (66) by the inverse magnetic
field (B−1)ei , assumed to be nondegenerate, we obtain
{Aai (x),Ψbf (y)} = δab (B−1(y))
(3)(x,y). (67)
7The linearly versus nonlinearly of the diffeomorphism constraints on the right hand
side is just a minor difference.
This gives us the Poisson brackets {A,A} ∼ 0 and {A,Ψ} ∼ B−1, which
leaves remaining the brackets {Ψ,Ψ}. To obtain these, we substitute (65)
into the second equation of (64), yielding
{σ̃ia(x), σ̃bj(y)} = {Ψae(x)Bie(x),Ψbf (y)B
f (y)}
= Ψae(x){Bie(x),Ψbf (y)}B
(y) + {Ψae(x),Ψbf (y)}Bie(x)B
+Ψbf (x)Ψae(x){Bie(x), B
f (y)}+Ψbf (y){Ψae(x), B
f (y)}B
e(x) = 0. (68)
Noting that the third term vanishes on account of the first equation of (64),
equation (68) reduces to
{Ψae(x),Ψbf (y)}Bie(x)B
f (y)
+Ψae(x){Bie(x),Ψbf (y)}B
(y)−Ψbf (y){Bjf (y),Ψae(x)}B
e(x) = 0. (69)
The bottom two terms of (69) can be computed using (67)
{Bie(x),Ψbf (y)} = ǫimnDxm{Aen(x),Ψbf (y)} = ǫimnDxm(δeb (B−1(y))fnδ(3)(x,y)).(70)
Substituting (70) into (69) and cancelling a pair of magnetic fields, then we
have that
{Ψae(x),Ψbf (y)}Bie(x)B
f (y) = ǫ
Ψae(x)D
m +Ψba(y)D
δ(3)(x,y). (71)
Left and right multiplying (71) by the inverse of the magnetic fields, we have
{Ψae(x),Ψbf (y)} = ǫijm
(B−1(y))
mΨab(x)(B
−1(x))ei
+(B−1(x))eiD
mΨba(y)(B
−1(y))
δ(3)(x,y). (72)
One sees that the internal components of Ψae have nontrivial commutation
relations with themselves.
References
[1] Eyo Ita ‘Instanton representation of Plebanski gravity. Gravitational
instantons from the classical formulation.’ arXiv: gr-qc/0703057
[2] Paul Dirac ‘Lectures on quantum mechanics’ Yeshiva University Press,
New York, 1964
[3] Hans Stephani, Dietrich Kramer, Maclcolm MacCallum, Cornelius
Hoenselaers, and Eduard Herlt ‘Exact Solutions of Einstein’s Field
Equations’ Cambridge University Press
[4] R. Penrose and W. Rindler ‘Spinors and space-time’ Cambridge Mono-
graphs in Mathematical Physics
[5] Ahbay Ashtekar ‘New Hamiltonian formulation of general relativity’
Phys. Rev. D36(1987)1587
[6] Ahbay Ashtekar ‘New variables for classical and quantum gravity’ Phys.
Rev. Lett. Volume 57, number 18 (1986)

  The instanton representation of Plebanski gravity provides as equations of
motion a Hodge self-duality condition and a set of `generalized' Maxwell's
equations, subject to gravitational degrees of freedom encoded in the initial
value constraints of general relativity. The main result of the present paper
will be to prove that this constraint surface is preserved under time
evolution. We carry this out not using the usual Dirac procedure, but rather
the Lagrangian equations of motion themsleves. Finally, we provide a comparison
with the Ashtekar formulation to place these results into overall context.

Introduction
In [1] a new formulation of general relativity was presented, named the
instanton representation of Plebanski gravity. The basic dynamical variables
are an SO(3, C) gauge connection Aaµ and a matrix Ψae taking its values in
two copies of SO(3, C).1 The consequences of the associated action IInst
were determined via its equations of motion, which hinge crucially on weak
equalities implied by the the initial value constraints. For these consequences
to be self-consistent, the constraint surface must be preserved for all time
by the evolution equations. The present paper will demonstrate that this
is indeed the case. We will not use the usual Hamiltonian formulation for
totally constrained systems [2], since we will not make use of any canonical
structure implied by IInst. Rather, we will deduce the time evolution of the
dynamical variables directly from the equations of motion of IInst.
Sections 2 and 3 of this paper present the instanton representation ac-
tion and derive the time evolution of the basic variables. Sections 4, 5 and 6
demonstrate that the nondynamical equations, referred to as the diffeomor-
phism, Gauss’ law and Hamiltonian constraints, evolve into combinations of
the same constraint set. The result is that the time derivatives of these con-
straints are weakly equal to zero with no additional constraints generated on
the system. While we do not use the usual Dirac method in this paper, the
result is still that the instanton representation is in a sense Dirac consistent.
We will make this inference clearer by comparison with the Ashtekar vari-
ables in the discussion section. On a final note, the terms ‘diffeomorphism’
and ‘Gauss’ law’ constraints are used loosely in this paper, in that we have
not specified what transformations of the basic variables these constraints
generate. The use of these terms is mainly for notational purposes, due to
their counterparts which appear in the Ashtekar variables.
2 Instanton representation of Plebanski gravity
The starting action for the instanton representation of Plebanski gravity is
given by [1]
IInst =
d3xΨaeB
F a0i + ǫkjmB
−iN(detB)1/2
Λ+ trΨ−1
, (1)
1Index labelling conventions for this paper are that symbols a, b, . . . from the begini-
ing of the Latin alphabet denote internal SO(3, C) indices while those from the middle
i, j, k, . . . denote spatial indices. Both of these sets of indices take takes 1, 2 and 3. The
Greek symbols µ, ν, . . . refer to spacetime indices which take values 0, 1, 2, 3.
where Nµ = (N,N i) are the lapse function and shift vector from metric
general relativity, and Λ is the cosmological constant. The basic fields are
Ψae and A
i , and we action (1) is defined only on configurations restricted
to (detB) 6= 0 and (detΨ) 6= 0.2 In the Dirac procedure one refers to Nµ
as nondynamical fields, since their velocities do not appear in the action.
While the velocity Ψ̇ae also does not appear, we will distinguish this field
from Nµ since the action (1), unlike for the latter, is nonlinear in Ψae.
The equation of motion for the shift vector N i, the analogue of the
Hamilton equation for its conjugate momentum Π ~N , is given by
δIInst
= ǫmjkB
eΨae = (detB)(B
−1)diψd ∼ 0, (2)
where ψd = ǫdaeΨae is the antisymmetric part of Ψae. This is equivalent to
the diffeomorphism constraint Hi owing to the nondegeneracy of B
a, and
we will often use Hi and ψd interchangeably in this paper. The equation of
motion for the lapse function N , the analogue of the Hamilton equation for
its conjugte momentum ΠN , is given by
δIInst
= (detB)1/2
Λ+ trΨ−1
= 0. (3)
Nondegeneracy of Ψae and the magnetic field B
e implies that on-shell, the
following relation must be satisfied
Λ + trΨ−1 = 0, (4)
which we will similarly take as synonymous with the Hamiltonian constraint.
The equation of motion for Ψae is
δIInst
= BkeF
0k + ǫkjmB
m + iN
detΨ(Ψ−1Ψ−1)ea ∼ 0, (5)
up to a term proportional to (4) which we have set weakly equal to zero.
One could attempt to define a momentum conjugate to Ψae, for which (5)
would be the associated Hamilton’s equation of motion. But since Ψae forms
part of the canonical structure of (1), then our interpretation is that this is
not technically correct.3
The equation of motion for the connection Aaµ is given by
δIInst
∼ ǫµσνρDσ(ΨaeF eνρ)−
4ǫmjkN
mBkeΨ[de]
+N(B−1)dj
Λ+ trΨ−1
, (6)
2The latter case limits the application of our results to spacetimes of Petroc Types I,
D and O (See e.g. [3] and [4].
3This is because (5) contains a velocity Ȧa
within F a
and will therefore be regarded
as an evolution equation rather than a constraint. This is in stark contrast with (2) and
(3), which are genuine constraint equations due to the absence of any velocities.
where we have defined
ea(x, y) ≡
δAai (x)
Bje(y) = ǫ
−δae∂k + fedaAdk
δ(3)(x, y); D
ea ≡ 0. (7)
The terms in large round brackets in (6) vanish weakly, since they are pro-
portional to the constraints (2) and (4) and their spatial derivatives. For
the purposes of this paper we will regard (6) as synonymous with
ǫµσνρDσ(ΨaeF
νρ) ∼ 0. (8)
In an abuse of notation, we will treat (5) and (8) as strong equalities in
this paper. This will be justified once we have completed the demonstration
that the constraint surface defined collectively by (2), (3) and the Gauss’
constraint from (8) is indeed preserved under time evolution. As a note
prior to proceeding we will often make the identification
N(detB)1/2
detΨ ≡
−g (9)
as a shorthand notation, to avoid cluttering many of the derivations which
follow in this paper.
2.1 Internal consistency of the equations of motion
Prior to embarking upon the issue of consistency of time evolution of the
initial value constraints, we will check for internal consistency of IInst, which
entails probing of the physical content implied by (8) and (5). First, equation
(8) can be decomposed into its spatial and temporal parts as
Di(ΨbfB
f ) = 0; D0(ΨbfB
f ) = ǫ
ijkDj(ΨbfF
0k). (10)
The first equation of (10) is the Gauss’ law constraint of a SO(3) Yang–
Mills theory, when one makes the identification of ΨbfB
f ∼ Eib with the
Yang–Mills electric field. The Maxwell equations for U(1) gauge theory
with sources (ρ, ~J), in units where c = 1, are given by
~∇ · ~B = 0; Ḃ = −~∇× ~E = 0; ~∇ · ~E = ρ; ~̇E = − ~J + ~∇× ~B. (11)
Equations (10) can be seen as a generalization of the first two equations of
(11) to SO(3) nonabelian gauge theory in flat space when one: (i) identifies
with the SO(3) generalization of the electric field ~E, and (ii) one
chooses Ψae = kδae for some numerical constant k.
When ρ = 0 and ~J = 0, then one has the vacuum theory and equations
(11) are invariant under the transformation
( ~E, ~B) −→ (− ~B, ~E). (12)
Then the second pair of equations of (11) become implied by the first pair.
This is the condition that the Abelian curvature Fµν , where F0i = Ei and
ǫijkFjk = Bi, is Hodge self-dual with respect to the metric of a conformally
flat spacetime. But equations (10) for more general Ψae encode gravitational
degrees of freedom, which as shown in [1] generalizes the concept of self-
duality to more general spacetimes solving the Einstein equations. Let us
first attempt to derive the analogue for (10) of the second pair of (11) in the
vacuum case. Acting on the first equation of (10) with D0 yields
D0Di(ΨbfB
f ) = DiD0(ΨbfB
f ) + [D0,Di](ΨbfB
f ) = 0. (13)
Substituting the second equation of (10) into the first term on the right hand
side of (13) and using the definition of temporal curvature as the commutator
of covariant derivatives on the second term we have
ijkDj(ΨbfF
0k)) + fbcdF
0iΨdfB
f = fbcd
0k +B
Ψdf = 0 (14)
where we have also used the spatial part of the commutator ǫijkDiDjva =
fabcB
b vc. Note that the term in brackets in (14) is symmetric in f and c,
and also forms the symmetric part of the left hand side of (5)
0i + i
−g(Ψ−1Ψ−1)fb + ǫijkBifB
k = 0, (15)
re-written here for completeness. To make progress from (14), we will sub-
stitute (15) into (14). This causes the last term of (15) to drop out due to
antisymmetry, which leaves us with
−gfbcd
Ψdf (Ψ
−1Ψ−1)fc +Ψdf (Ψ
−1Ψ−1)fc
= −2i
−gfbcdΨ−1dc . (16)
The equations are consistent only if (16) vanishes, which is the requirement
that Ψae = Ψea be symmetric. This of course is the requirement that the
diffeomorphism constraint (2) be satisfied. So the analogue of the second
pair of (11) in the vacuum case must be encoded in the requirement that
Ψae = Ψea be symmetric.
3 The time evolution equations
We must now verify that the initial value constraints are preserved under
time evolution defined by the equations of motion (5) and (6). These equa-
tions are respectively the Hodge duality condition
0k + i
−g(Ψ−1Ψ−1)fb + ǫijkN iBjbB
f = 0, (17)
and one of the Bianchi identity-like equations
ǫijkDj(ΨaeF
ok) = D0(ΨaeB
e). (18)
Since the initial value constraints were used to obtain the second line of (17)
from (1), then we must verify that these constraints are preserved under
time evolution as a requirement of consistency. Using F b0i = Ȧ
i −DiAb0 and
defining
−g(B−1)fi (Ψ
−1Ψ−1)fb + ǫmnkN
mBnb ≡ iHbk, (19)
Then equation (17) can be written as a time evolution equation for the
connection, which is not the same as a constraint equation as noted earlier
F b0i = −iHbi −→ Ȧbi = DiAb0 − iHbi . (20)
From equation (20) we can obtain the following equation governing time
evolution equation for the magnetic field
Ḃie = ǫ
ijkDjȦ
k = ǫ
ijkDj
0 − iHek
= febcB
0 − iǫijkDjHek = −δ~θB
e − iǫijkDjHek, (21)
which will be useful. On the first term on the right hand side of (21) we
have used the definition of the curvature as the commutator of covariant
derivatives. The notation δ~θ in (21) suggests that that B
e transforms as
a SO(3, C) vector under gauge transformations parametrized by θb ≡ Ab0.4
Since we have not specified anything about the canonical structure of IInst,
then δ~θ as used in (21) and in (24) should at this stage simply be regarded
as a definition useful for shorthand notation.
We will now apply the Liebnitz rule in conjunction with the definition
of the temporal covariant derivatives to (18) to determine the equation gov-
erning the time evolution of Ψae. This is given by
D0(ΨaeB
e) = B
eΨ̇ae +ΨaeḂ
e + fabcA
0(ΨceB
e) = ǫ
ijkDj(ΨaeF
0k). (22)
Substituting (21) and (20) into the left and right hand sides of (22), we have
BieΨ̇ae +Ψae
febcB
0 − iǫijkDjHek
+ fabcA
0(ΨceB
e) = −iǫijkDj(ΨaeHek).(23)
In what follows, it will be convenient to use the following transformation
properties for Ψae as A
i under SO(3, C) gauge transformations
δ~θΨae =
fabcΨce + febcΨac
Ab0; δ~θA
i = −DiAa0; δ~θB
e = −febcBibAc0. (24)
4We will make the identification with SO(3, C) gauge transformations later in this
paper when we bring in the relation of IInst with the Ashtekar variables.
Then using (24), the time evolution equations for the phase space variables
ΩInst can be written in the following compact form
Ȧbi = −δ~θA
i − iHbi ; Ψ̇ae = −δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
k . (25)
We have found evolution equations for Ψae and A
i from the covariant equa-
tions of Aaµ and the Hodge-duality condition We have obtained these without
using Poisson brackets, and by assuming that the Hamiltonian and diffeo-
morphism constraints are satisfied. Therefore the first order of business is
then to check for the preservation of the initial value constraints under the
time evolution generated by (25). This means that we must check that
the time evolution of the diffeomorphism, Gauss’ law and Hamiltonian con-
straints are combinations of terms proportional to the same constraints and
their spatial derivatives, and terms which vanish when the constraints hold.5
These constraints are given by
we{Ψae} = 0; (detB)(B−1)diψd = 0; (detB)1/2
Λ+ trΨ−1
= 0(26)
where (detB) 6= 0 and (detΨ) 6= 0. We will occasionally make the identifi-
cation
N(detB)1/2(detΨ)1/2 ≡
−g (27)
for a shorthand notation. Additionally, the following definitions are provided
for the vector fields appearing in the Gauss’ constraint
we = B
eDi; ve = B
e∂i (28)
where Di is the SO(3, C) covariant derivative with respect to the connection
Aai . Equations (26) are the equations of motion for the auxilliary fields A
N i and N .
4 Consistency of the diffeomorphism constraint un-
der time evolution
The diffeomorphism constraint is directly proportional to ψd = ǫdaeΨae, the
antisymmetric part of Ψae. So to establish the consistency condition for
this constraint, it suffices to show that the antisymmetric part of the second
equation of (25) weakly vanishes. This is given by
ǫdaeΨ̇ae = −δ~θ(ǫdaeΨae)− iǫdaeǫ
ijk(B−1)ei (DjΨaf )H
k , (29)
5This includes any nonlinear function of linear order or higher in the constraints, a
situation which involves the diffeomorphism constraint.
which splits into two terms. Using (24), one finds that the first term of (29)
is given by
−ǫdaeδ~θΨae = −ǫdae
fabcΨce +Ψacfebc
δebδdc − δecδbd
Ψce +
δdbδac − δdcδab
Ψdb − δbdtrΨ + δdbtrΨ−Ψbd
Ab0 = 2Ψ[bd]A
0 = −ǫdbhAb0ψh, (30)
which is proportional to the diffeomorphism constraint. The second term of
(30) has two contributions due to H
k as defined in (19). The first contribu-
tion reduces to
−iǫdaeǫijk(B−1)ei (DjΨaf )(H(1))
= −iǫdaeǫijk(B−1)ei (DjΨaf )
−g(B−1)gk(Ψ
−1Ψ−1)gf
= iǫdae(detB)
−1ǫegh(Ψ−1Ψ−1)gfB
hDjΨaf
= i(detB)−1(Ψ−1Ψ−1)gf
a − δgaδhd
vv{Ψaf}
= i(detB)−1(Ψ−1Ψ−1)gf
dva{Ψaf} − vd{Ψgf}
= i(detB)−1
(Ψ−1Ψ−1)dfGf + vd{Λ+ trΨ−1}
. (31)
The first term on the final right hand side of (31) is the Gauss’ constraint
and the second term is the derivative of a term direction proportional to
the Hamiltonian constraint.6 The second contribution to the second term
of (29) is given by
ǫdaeǫ
ijk(B−1)ei (DjΨaf )(H(2))
k = ǫdaeǫ
ijk(B−1)ei (DjΨaf )ǫmnkN
= ǫdac
n − δinδjm
(B−1)ei (DjΨaf )N
= ǫdaeN
i(B−1)eivf{Ψaf} −N jDj(ǫdaeΨae) = ǫdaeN i(B−1)efGa −N jDjψd.(32)
The result is that the time evolution of the diffeomorphism constraint is
directly proportional to
ψ̇d =
i(detB)−1(Ψ−1Ψ−1)da + ǫdaeN
i(B−1)ei
Ab0ǫbdh − δdhN jDj
ψh + i(detB)
vd{(−g)−1/2H}, (33)
which is a linear combination of terms proportional to the constraints (26)
and their spatial derivatives. The result is that the diffeomorphism con-
straint Hi = 0 is consistent with respect to the Hamiltonian evolution gen-
erated by the equations (25). So it remains to verify consistency of Gauss’
law and the Hamiltonian constraints Ga and H.
6We have added in a term Λ, which can be regarded as a constant of integration with
respect to the spatial derivatives from vd.
5 Consistency of the Gauss’ constraint under time
evolution
Having verified the consistency of the diffeomorphism constraint under time
evolution, we now move on to the Gauss’ constraint. Application of the
Liebnitz rule to the first equation of (26) yields
Ġa = Ḃ
eDiΨae +B
eDiΨ̇ae +B
fabfΨfe + febgΨag
Ȧai . (34)
Upon substituion of (21) and (25) into (34), we have
Ġa =
−δ~θB
e − iǫijkDjHek
DiΨae +B
−δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
fabfΨfe + febgΨag
−δ~θA
i − iHbi
.(35)
Using the Liebniz rule to combine the δ~θ terms of (35), we have
Ġa = −δ~θGa − iǫ
k)DiΨae +B
e Dm((B
−1)ei (DjΨaf )H
fabfΨfe + febgΨag
i . (36)
The requirement of consistency is that we must show that the right hand
side of (36) vanishes weakly. First, we will show that the third term on the
right hand side of (36) vanishes up to terms of linear order and higher in
the diffeomorphism constraint. This term, up to an insignificant numerical
factor, has two contributions. The first contribution is
fabfΨfe + febgΨag
Bie(H(1))
fabfΨfe + febgΨag
(Ψ−1Ψ−1)eb
fabf (Ψ
−1)fb + febg(Ψ
−1Ψ−1)ebΨag
∼ δ(1)a (~ψ) ∼ 0, (37)
which is directly proportional to a nonlinear function of first order in ψd
which is proportional to the diffeomorphism constraint. The second contri-
bution to the third term on the right hand side of (36) is
fabfΨfe + febgΨag
Bie(H(2))
fabfΨfe + febgΨag
ǫkmnN
kBme B
fabfΨfe + febgΨag
(detB)Nk(B−1)dkǫdeb
= (detB)Nk(B−1)dk
δfdδae − δfeδad
Ψfe + 2δdgΨag
= (detB)Nk(B−1)dk
Ψda − δadtrΨ + 2Ψad
≡ δ(2)a ( ~N) (38)
which does not vanish, and neither is it expressible as a constraint. For
the Gauss’ law constraint to be consistent under time evolution, a necessary
condition is that this δ
a ( ~N) term must be exactly cancelled by another
term arising from the variation.
Let us expand the terms in square brackets in (36). This is given, using
the Liebniz rule on the second term, by
ǫijk(DjH
k)(DiΨae) + ǫ
ijkBme Dm((B
−1)ei (DjΨae)H
= ǫijk(DjH
k)(DiΨae)− ǫijkBme (B−1)en(DmBng )(B−1)
i (DjΨaf )H
+ǫmjk(DmDjΨaf )H
+ ǫmjk(DjΨaf )(DmH
). (39)
The first and last terms on the right hand side of (39) cancel, which can be
seen by relabelling of indices. Upon application of the definition of curvature
as the commutator of covariant derivatives to the third term, then (39)
reduces to
−ǫijk(DnBng )(B−1)
i (DjΨaf )H
fabcΨcf + ffbcΨac
. (40)
The first term of (40) vanishes on account of the Bianchi identity and the
second term contains two contributions which we must evaluate. The first
contribution is given by
(H(2))
fabcΨcf + ffbcΨac
= (detB)Nk(B−1)dkǫdbf
fabcΨcf + ffbcΨac
= (detB)Nk(B−1)dk
δdaδfc − δdcδfa
Ψcf − 2δdcΨac
= (detB)Nk(B−1)dk
δdatrΨ−Ψda − 2Ψad
= −δ(2)a ( ~N ),(41)
with δ
a ( ~N) as given in (37). So putting the results of (39), (40) and (41)
into (36), we have
Ġa = −δ~θGa + δ
~N) + δ(1)a (
~ψ) + δ(1)a (
~ψ)− δ(2)a ( ~N) = −δ~θGa + 2δ
(1)(~ψ).(42)
The velocity of the Gauss’ law constraint is a linear combination of the
Gauss’ constraint with terms of the diffeomorphism constraint of linear or-
der and higher. Hence the time evolution of the Gauss’ law constraint is con-
sistent in the sense that we have defined, since δ(1)(~ψ) vanishes for ψd = 0.
6 Consistency of the Hamiltonian constraint un-
der time evolution
The time derivative of the Hamiltonian constraint, the third equation of
(26), is given by
((detB)1/2(detΨ)1/2
(Λ + trΨ−1) +
(Λ + trΨ−1) (43)
which has split up into two terms. The first term is directly proportional
to the Hamiltonian constraint, therefore it is already consistent. We will
nevertheless expand it using (21) and (25)
(B−1)di Ḃ
d + (Ψ
−1)aeΨ̇ae
(detB)1/2(detΨ)1/2(Λ + trΨ−1)
(B−1)di
−δ~θB
d − iǫijkDjHdk
+(Ψ−1)ae
−δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
H. (44)
We will be content to compute the δ~θ terms of (44). These are
(B−1)di δ~θB
d = (B
−1)di fdbfB
0 = δdbfdbfA
0 = 0 (45)
on account of antisymmetry of the structure constants, and
(Ψ−1)eaδ~θΨae = (Ψ
−1)ea
fabfΨfe + febgΨag
= 0, (46)
also due to antisymmetry of the structure constants. We have shown that
the first term on the right hand side of (43) is consistent with respect to
time evolution. To verify consistency of the Hamiltonian constraint under
time evolution, it remains to show that the second term is weakly equal to
zero. It suffices to show this just for the second term, in brackets, of (43)
(Λ + trΨ−1) = −(Ψ−1Ψ−1)feΨ̇ef
= (Ψ−1Ψ−1)ef
δ~θΨae − iǫ
ijk(B−1)ei (DjΨaf )H
, (47)
where we have used (25). Equation (47) has split up into two terms, of
which the first term is
(Ψ−1Ψ−1)eaδ~θΨae = (Ψ
−1Ψ−1)ea
fAbfΨfe + febgΨag
fabf (Ψ
−1)fa + febg(Ψ
−1)eg
Ab0 = m(
~ψ) ∼ 0 (48)
which vanishes weakly since it is a nonlinear function of at least linear order
in ψd. The second term of (47) splits into two terms which we must evaluate.
The first contribution is proportional to
(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(H(1))
−g(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(B−1)dk(Ψ−1Ψ−1)df
−g(Ψ−1Ψ−1)ea(Ψ−1Ψ−1)df (detB)−1ǫedgBjgDjΨaf
−g(detB)−1ǫedg(Ψ−1Ψ−1)ea(Ψ−1Ψ−1)dfvg{Ψaf} ≡ v{~ψ} (49)
for some vector field v. We have used the fact that the term in (49) quartic
in Ψ−1 in antisymmetric in a and f due to the epsilon symbol. Hence Ψaf
as acted upon by vg can only appear in an antisymmetric combination, and
is therefore proportional to the diffeomorphism constraint ψd whose spatial
derivatives weakly vanish. Hence (49) presents a consistent contribution to
the time evolution of H, which leaves remaining the second contribution to
the second term of (47). This term is proportional to
(Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )(H(2))
= (Ψ−1Ψ−1)eaǫijk(B−1)ei (DjΨaf )ǫmnkN
n − δinδjm
(B−1)eiB
−1Ψ−1)ea(DjΨaf )
N i(B−1)eiB
f − δefN
(Ψ−1Ψ−1)ea(DjΨaf )
= (−g)−1/2N iHai vf{Ψaf} − (Ψ−1Ψ−1)fa(N jDjΨaf )
= (−1)−1/2N iHai Ga −N jDj(Λ + trΨ−1). (50)
The first term on the final right hand side of (50) is proportional to the
Gauss’ law constraint, and the second term is proportional to the derivative
of the Hamiltonian constraint. To obtain this second term we have added
in Λ as a constant of differentiation with respect to ∂j . Substituting (48),
(49) and (50) into (47), then we have
Ḣ =∼ Ô(~ψ) + (−g)−1/2N iHai Ga + T̂ ((−g)−1/2H), (51)
where Ô and T̂ are operators consisting of spatial derivatives acting to the
right and c numbers. The time derivative of the Hamiltonian constraint
is a linear combination of the Gauss’ law and Hamiltonian constraints and
its spatial derivatives, plus terms of linear order and higher in the diffeo-
morphism constraint and its spatial derivatives. Hence the Hamiltonian
constraint is consistent under time evolution.
7 Recapitulation
The final equations governing the time evolution of the initial value con-
straints are given weakly by
ψ̇d =
i(detB)−1(Ψ−1Ψ−1)da + ǫdaeN
i(B−1)ei
Ab0ǫbdh − δdhN jDj
ψh + i(detB)
vd{Λ + trΨ−1};
Ġa = −fabcAb0Gc + δ(1)a (~ψ);
ǫijk(B−1)di (DjH
k ) + ǫ
ijk(B−1)ei (Ψ
−1)ae(DjΨaf )H
−N j∂j
(Λ + trΨ−1)
+(−g)−1/2N iHai Ga −
−g(detB)−1ǫedg(Ψ−2Ψ−1)ea(Ψ−1Ψ−1)dfvg{ǫafhψh}+m(~ψ).(52)
Equations (52) show that all constraints derivable from the the action (1)
are preserved under time evolution, since their time derivatives yield linear
combinations of the same set of constraints and their spatial derivatives.
There are no additional constraints generated which implies that the action
(1) is consistent in the Dirac sense. On the other hand, we have not defined
the canonical structure of (52) or any Poisson brackets.
Equations (52) can be written schematically in the following form
~̇H ∼ ~H + ~G+H; ~̇G ∼ ~G+Φ( ~H); Ḣ ∼ H + ~G+Φ( ~H), (53)
where Φ is some nonlinear function of the diffeomorphism constraint ~H,
which is of at least first order in ~H. In the Hamiltonian formulation of a
theory, one identifies time derivatives of a variable f with via ḟ = {f,H}
the Poisson brackets of the variable with the Hamiltonian H . So while we
have not specified Poisson brackets, equation (53) implies the existence of
Poisson brackets associated to some Hamiltonian HInst for the action (1),
{ ~H,HInst} ∼ ~H + ~G+H; {~G,HInst} ∼ ~G+Φ( ~H);
{H,HInst} ∼ H +Φ( ~H) + ~G. (54)
So the main result of this paper has been to demonstrate that the instanton
representation of Plebanski gravity forms a consistent system, in the sense
that the constraint surface is preserved under time evolution. As a direction
of future research we will compute the algebra of constraints for (1) directly
from its canonical structure. Nevertheless it will be useful for the present
paper to think of equations (52) in the Dirac context, mainly for compari-
son with other formulations of general relativity. This will bring us to the
Ashtekar variables.
8 Discussion: Relation of the instanton represen-
tation to the Ashtekar variables
We will now provide the rationale for not following the Dirac procedure for
constrained systems [2] with respect to (1), by comparison with the Ashekar
formulation of GR. The action for the instanton representation (1) can be
written in the following 3+1 decomposed form
IInst =
0we{Ψae} − ǫijkN iBjaBkeΨae
−iN(detB)1/2(detΨ)1/2
Λ+ trΨ−1
, (55)
which regards Ψae and A
i as phase space variables. But the phase space of
(55) is noncanonical since its symplectic two form
δθInst = δ
d3xΨaeB
d3xBieδΨae ∧ δAai +
d3xΨaeǫ
ijkDj(δA
k) ∧ δAai , (56)
is not closed owing to the presence of the second term on the right hand
side. The initial stages of the Dirac procedure applied to (55) state that the
momentum conjugate to Aai yields the primary constraint
Πia =
δIInst
δȦai
= ΨaeB
e. (57)
Then making the identification σ̃ia = Π
a and upon substitution into (57)
and into (55), one obtains the action
IAsh =
σ̃iaȦ
0Ga −N iHi −
, (58)
which is the action for the Ashtekar complex formalism of general relativity
[5], [6], with σ̃ia being the densitized triad. This is a totally constrained sys-
tem with (Aa0, N
i, N), respectively the SO(3, C) rotation angle Aa0, the shift
vector N i and the densitized lapse function N = N(detσ̃)−1/2 as auxilliary
fields. The constraints in (58) smearing the auxilliary fields are the Gauss’
law, vector and Hamiltonian constraints
Ga = Diσ̃
a; Hi = ǫijkσ̃
a ; H = ǫijkǫ
abcσ̃iaσ̃
σ̃kc +B
. (59)
From (58) one reads off the symplectic two form ΩAsh given by
ΩAsh =
d3xδσ̃ia ∧ δAai = δ
d3xσ̃iaδA
= δθAsh, (60)
which is the exact functional variation of the canonical one form θAsh.
The actions (55) and (58) are transformable into each other only under
the condition (detB) 6= 0 and (detΨ) 6= 0. In (58) it is clear that σ̃ia and Aai
form a canonically conjugate pair, which suggests that (55) is a noncanonical
version of (58). The constraints algebra for (59) is
{ ~H[ ~N ], ~H [ ~M ]} = Hk
N i∂kMi −M i∂kNi
{ ~H[N ], Ga[θa]} = Ga[N i∂iθa];
{Ga[θa], Gb[λb]} = Ga
fabcθ
{H(N), ~H [ ~N ]} = H[N i∂iN
{H(N ), Ga(θa)} = 0;[
H(N),H(M )
= Hi[
N∂jM −M∂jN
H ij], (61)
which is first class due to closure of the algebra, and is therefore consistent
in the Dirac sense. Let us consider (61) for each constraint with the total
Hamiltonian HAsh and compare with (54). This is given schematically by
{ ~H,HAsh} ∼ ~H + ~G+H; {~G,HAsh} ∼ ~G+ ~H;
{H,HAsh} ∼ H + ~H. (62)
Comparison of (62) with (54) shows an essentially similar structure for the
top two lines involving ~H and ~G.7 But there is a marked dissimilarity with
respect to the Hamiltonian constraint H. Note that there is a Gauss’ law
constraint appearing in the right hand side of the last line of (54) whereas
there is no such constraint on the corresponding right hand side of (62).
This means that while the Hamiltonian constraint is gauge-invariant under
SO(3, C) gauge-transformations as implied by (61) and (62), this is not
the case in (54). This means that the action (1), which as shown in [1]
describes general relativity for Petrov Types I, D and O, has a different
role for the Gauss’ law and Hamiltonian constraints than the action (58),
which also describes general relativity. Therefore IInst and IAsh at some
level correspond to genuinely different descriptions of GR, a feature which
would have been missed had we applied the step-by-step Dirac procedure.
9 Appendix: Commutation relations for IInst
We will now infer the Poisson brackets for (55) by inference from the corre-
sponding canonical Ashtekar Poisson brackets
{Aai (x), σ̃
b (y)} = δ
(3)(x,y) (63)
along with the vanishing brackets
{Aai (x), Abj(y)} = {σ̃ia(x), σ̃
(x)} = 0. (64)
To find the analogue of (63) and (64) for (55), we will use the tranformation
equation
σ̃ia = ΨaeB
e, (65)
which corresponds to a noncanonical transformation. Substitution of (65)
into (63) yields
{Aai (x),Ψbf (y)Bif (y)} = δ
(3)(x,y)
{Aai (x),Ψbf (y)}B
(y) + Ψbf (x){Aai (x), B
(y)}. (66)
The second term on the right hand side of (66) vanishes on account of the
first relations of (64), and upon multiplying (66) by the inverse magnetic
field (B−1)ei , assumed to be nondegenerate, we obtain
{Aai (x),Ψbf (y)} = δab (B−1(y))
(3)(x,y). (67)
7The linearly versus nonlinearly of the diffeomorphism constraints on the right hand
side is just a minor difference.
This gives us the Poisson brackets {A,A} ∼ 0 and {A,Ψ} ∼ B−1, which
leaves remaining the brackets {Ψ,Ψ}. To obtain these, we substitute (65)
into the second equation of (64), yielding
{σ̃ia(x), σ̃bj(y)} = {Ψae(x)Bie(x),Ψbf (y)B
f (y)}
= Ψae(x){Bie(x),Ψbf (y)}B
(y) + {Ψae(x),Ψbf (y)}Bie(x)B
+Ψbf (x)Ψae(x){Bie(x), B
f (y)}+Ψbf (y){Ψae(x), B
f (y)}B
e(x) = 0. (68)
Noting that the third term vanishes on account of the first equation of (64),
equation (68) reduces to
{Ψae(x),Ψbf (y)}Bie(x)B
f (y)
+Ψae(x){Bie(x),Ψbf (y)}B
(y)−Ψbf (y){Bjf (y),Ψae(x)}B
e(x) = 0. (69)
The bottom two terms of (69) can be computed using (67)
{Bie(x),Ψbf (y)} = ǫimnDxm{Aen(x),Ψbf (y)} = ǫimnDxm(δeb (B−1(y))fnδ(3)(x,y)).(70)
Substituting (70) into (69) and cancelling a pair of magnetic fields, then we
have that
{Ψae(x),Ψbf (y)}Bie(x)B
f (y) = ǫ
Ψae(x)D
m +Ψba(y)D
δ(3)(x,y). (71)
Left and right multiplying (71) by the inverse of the magnetic fields, we have
{Ψae(x),Ψbf (y)} = ǫijm
(B−1(y))
mΨab(x)(B
−1(x))ei
+(B−1(x))eiD
mΨba(y)(B
−1(y))
δ(3)(x,y). (72)
One sees that the internal components of Ψae have nontrivial commutation
relations with themselves.
References
[1] Eyo Ita ‘Instanton representation of Plebanski gravity. Gravitational
instantons from the classical formulation.’ arXiv: gr-qc/0703057
[2] Paul Dirac ‘Lectures on quantum mechanics’ Yeshiva University Press,
New York, 1964
[3] Hans Stephani, Dietrich Kramer, Maclcolm MacCallum, Cornelius
Hoenselaers, and Eduard Herlt ‘Exact Solutions of Einstein’s Field
Equations’ Cambridge University Press
[4] R. Penrose and W. Rindler ‘Spinors and space-time’ Cambridge Mono-
graphs in Mathematical Physics
[5] Ahbay Ashtekar ‘New Hamiltonian formulation of general relativity’
Phys. Rev. D36(1987)1587
[6] Ahbay Ashtekar ‘New variables for classical and quantum gravity’ Phys.
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