arXiv:0704.1289v1  [gr-qc]  10 Apr 2007
Preprint typeset in JHEP style - HYPER VERSION
Brane-world Quantum Gravity
M.D. Maia∗, Nildsen Silva †and M.C.B. Fernandes‡
Instituto de F́ısica, Universidade de Braśılia,
70919-970, Braśılia, D. F., Brazil
Abstract: The Arnowitt-Deser-Misner canonical formulation of general relativity is ex-
tended to the covariant brane-world theory in arbitrary dimensions. The exclusive probing
of the extra dimensions makes a substantial difference, allowing for the construction of a
non-constrained canonical theory. The quantum states of the brane-world geometry are
defined by the Tomonaga-Schwinger equation, whose integrability conditions are deter-
mined by the classical perturbations of submanifolds contained in the Nash’s differentiable
embedding theorem. In principle, quantum brane-world theory can be tested by current
experiments in astrophysics and by near future laboratory experiments at Tev energy. The
implications to the black-hole information loss problem, to the accelerating cosmology, and
to a quantum mathematical theory of four-sub manifolds are briefly commented.
Keywords: Quantum Gravity Brane-world Nash Theorem Tomonaga-Schwinger.
∗maia@unb.br
†nildsen@hotmail.com
‡mcezar@fis.unb.br
http://arxiv.org/abs/0704.1289v1
http://jhep.sissa.it/stdsearch
Contents
1. Quantizing the Brane-World 1
2. Covariant Brane-world Gravity 2
3. Canonical Equations 6
4. Tomonaga-Schwinger Quantum States 8
5. Overview and Perspectives 10
1. Quantizing the Brane-World
If gravity is to occupy a significant place in modern physics,
it can do so only by being qualitatively different from other
fields. As soon as we assume that gravity behaves qualitatively
like other fields, we find that it is quantitatively insignificant
C.W. Misner (1957)
After analyzing perturbative quantum gravity, Misner reached the interesting conclu-
sion that an effective quantum gravity must have qualities which makes it different from
gauge theories [1]. Translating quantitative significance in terms of energy level, Misner’s
conclusion suggests that the problem of quantization of the gravitational field should be
solved concomitantly with the hierarchy problem of the fundamental interactions. In what
follows, we apply this criterium to brane-world gravity.
Brane-world gravity is based on a higher dimensional solution of the hierarchy prob-
lem. In a seminal paper N. Arkani-Hamed, G. Dvali and S. Dimopolous questioned the
currently accepted hypothesis that gravitons are quantitatively relevant only at the Planck
scale of energies, essentially because this is an assumption devoid of experimental sup-
port. They proposed that the known gauge fields (and hence all ordinary matter) are
to be confined within the four-dimensional brane-world, but gravitons can propagate in a
higher-dimensional space, the bulk, at the same Tev scale of energies of the gauge fields [2]
(For historical papers on the development of the theory see also [3, 4, 5, 6, 7, 8]). Accord-
ing to this view, brane-world gravity is qualitatively distinct from, but it is quantitatively
equivalent to the gauge fields of the standard model.
Brane-world gravity predicts the existence of short lived Tev mini black holes, which
in principle can be produced at the laboratory by a high energy proton-proton collision,
with implications to the black hole information loss problem at the quantum level. The
– 1 –
proposed experiment is set in Minkowski space-time, but it ends in a Schwarzschild (or
Reissner-Nordstrom) space-time [9]. Therefore, the theory supporting this experiment
must be compatible with cross sections of the order of the Schwarzschild radius, and also
with an explanation on how the original Minkowski space-time deforms into black hole,
and back in a short period of time.
Brane-world gravity may also explain the acceleration of the universe (see eg [10] and
references therein). In short, due to the presence of the extrinsic curvature, the vacuum in
brane-world gravity is richer than the vacuum in general relativity. Besides the cosmological
constant, it also contain a conserved geometric tensor built from the extrinsic curvature.
Consequently, when studying the quantum fluctuations of such vacuum we may obtain
a different estimate for the vacuum energy density as compared with the case of general
relativity.
Since most of the current research on brane-world theory is based on models defined in
a five-dimensional bulk, using specific coordinates and particular symmetries (see eg [11]),
we find it necessary to review in the next section the covariant equations of motion of a
brane-world defined in an arbitrary bulk, with an arbitrary number of dimensions. Those
equations can be found elsewhere [12, 10], but here we have included some details which
are required for the quantum description. Readers who are familiar with this may jump
to section 3, where the canonical equations of the brane-world with respect to the extra
dimensions are discussed. In section 4 we introduce the Tomonaga-Schwinger equation for
the brane-world with respect to the extra dimensions and comment on its integrability.
2. Covariant Brane-world Gravity
There are essentially three basic postulates in brane-world theory: (1) The bulk geometry
is defined by Einstein’s equations; (2) The brane-world is a sub manifold embedded in
that bulk ; (3) The gauge fields and ordinary matter are confined to four dimensions, but
gravitons propagates along the extra dimensions at Tev energy [2].
The embedding of the brane-world in the bulk plays an essential role on the covariant
(that is, model independent) formulation of the brane-world gravity, because it tells how
the Einstein-Hilbert dynamics of the bulk is transferred to the brane-world. However,
there a are many different ways to embed a manifold into another, classified as local,
global, isometric, conformal (or more generally defined by a collineation), rigid, deformable,
analytic or differentiable. The choice of one or another depend on what the embedded
manifold is supposed to do.
In string theory the action principle is defined on the world-sheets, with additional
boundary conditions, so that the embedding is necessarily global. Since the world sheets
are 2-dimensional they are all conformally flat and their global embedding is not difficult to
achieve. However, if higher-dimensional objects such as p-branes are to be considered, then
the global embedding may turn out to be difficult to realize in 10 or even in 11 dimensions
[13].
Differently from string theory, the Einstein-Hilbert action in brane-world theory is set
on the bulk, which is therefore the primary dynamical object. Furthermore, the embedding
– 2 –
is locally defined, meaning that the bulk is a local fiber bundle whose fibers are the direct
sum of the tangent and normal spaces at each point of the brane-world taken as the base
space. If we want to draw a picture, the bulk can be seen as as a locally constructed space
around each point of the brane-world.
A local differentiable embedding requires only that the embedding functions are dif-
ferentiable and regular. This follows from Nash’s embedding theorem, an important im-
provement over the traditional analytic embedding theorems of Janet and Cartan [14, 15],
which demand that the embedding functions are represented by convergent positive power
series. Furthermore, Nash’s theorem shows that any sub manifold can be generated by
a continuous sequence of small perturbations of an arbitrarily given sub manifold 1. Al-
though the theorem was originally demonstrated for the case of an Euclidean bulk, it was
later generalized to pseudo Riemannian manifolds [18, 19].
Given a particular Riemannian sub-manifold σ̄4, its local isometric embedding in a
certain bulk MD, is given by D = 4 + N differentiable and regular embedding maps
X̄A : σ̄4 → MD, such that 2
X̄A,µX̄B,ν GAB = ḡµν , X̄A,µ η̄Bb GAB = 0, and η̄Aa η̄Bb GAB = ḡab (2.1)
where η̄Aa are the components of the N linearly independent vector fields in the same
coordinates of the bulk where the components GAB of the bulk metric are defined. The
vectors {X̄A,µ , η̄Ba } define a Gaussian reference frame called here the embedding frame. The
derivatives of the vectors η̄a is expressed in terms of the second and third fundamental
forms k̄µνa, Āµab respectively by the Gauss-Weingerten equations [20]
η̄Aa,α = ḡ
µν k̄αµaX̄A,ν + ḡmnĀαamη̄An (2.2)
Without loss of generality we may chose the normal vectors η̄a to be orthogonal to each
other, so that ḡab = ǫaδab, where ǫa = ±1 depending on the signature of the bulk [19].
Nash’s perturbative approach to embedding consists in subjecting the fundamental
forms of σ̄4 to small parametric deviations along each normal vector. It can be also
described by introducing a small perturbation with parameter δya, of the base vectors
{X̄A,µ , ηAa } along each normal η̄Aa evaluated on σ̄4, obtaining another set of vectors (no sum
on a)
ZA,µ = X̄A,µ + (δya£η̄aX̄A),µ = X̄A,µ − δya[X̄ , η̄a]A,µ = X̄A,µ + δyaη̄Aa,µ, (2.3)
ηAa = η̄
a + (δy
£η̄a η̄a)
A = η̄Aa + δy
a[η̄a, η̄a]
A = η̄Aa (2.4)
1The perturbative approach to the embedding was originally proposed by J. E. Campbell in 1926.
However, his result differs from Nash’s theorem because analytic conditions where implicitly used [16,
17]. Since the perturbation procedure is based on regular and differentiable functions, the differentiable
embedding is less restrictive to the geometry than the analytic embeddings.
2Capital Latin indices refer to the bulk, which is a Riemannian geometry with metric GAB in arbitrary
coordinates. Small case Latin indices refer to the extra dimensions going from 5 to D, and all Greek indices
refer to the brane, from 1 to 4. A curly R always denotes bulk curvatures, like in RABCD . Ordinary capital
R like in Rµν denotes brane-world curvatures. Covariant derivatives need to be specified, for the bulk or
the brane-world metrics. For a vector V A in the bulk its covariant derivative with respect to GAB is denoted
as V A;B. On the other hand, from the point of view of the brane-world metric, the components V
A behave
as a set of N scalar functions as in [20]. For generality we denote G = |det(GAB)|.
– 3 –
which define a perturbed embedding frame {ZA,µ, ηAa } in the bulk. Admitting that these
new functions remain differentiable and regular and that they satisfy the equations similar
to (2.1),
ZA,µZB,νGAB = gµν , ZA,µηBa GAB = gµa, ηAa ηBb GAB = gab = ḡab (2.5)
we obtain a N -parameter local family of submanifolds σ4 generated by local perturbations
of σ̄4, by a continuous variations of the parameters δy
The next problem is to find a solution of these equations. However, instead of finding
the coordinates ZA, it is more convenient to write the perturbed solution in terms of the
fundamental forms, expressed in terms of the initial geometry of σ̄4. By direct substitution
of Z,µ and ηAa derived from (2.3) in equations (2.5) we obtain
gµν(x, y) = ZA,µZB,νGAB = ḡµν− 2δyak̄µνa
+ δyaδyb[ḡαβ k̄µαak̄νβb + g
cdĀµcaĀνdb], (2.6)
gµa(x, y) = ZA,µηBa GAB =δybAµab, (2.7)
gab(x, y) = η
b GAB = ḡab (2.8)
kµνa(x, y) = −ηAa,µZB,νGAB
= k̄µνa− δybḡαβ k̄µαak̄νβb −gcdδybĀµcaĀνdb, (2.9)
Aµab(x, y) = η
b GAB=Āµab(x) (2.10)
The contravariant components of the perturbed geometry must be consistent with GACGCB =
δAB , which can be realized by setting gµρg
ρν = δνµ, gacg
cb = δba. Since the indices µ and b
can never be equal, we must nave gµρg
ρb+ gµcg
cb = δbµ = 0. After some algebra we see that
this corresponds to an identity ymyngabAµamAνbn = −ybyngmaAµ[am]Aν[bn] ≡ 0.
Comparing (2.6) and (2.9) we obtain
kµνa = −
(2.11)
Consequently, the local bulk defined in a neighborhood around σ̄4, is foliated by this
perturbed geometry, so that the Riemann curvature of the bulk may be expressed in the
perturbed embedding frame. For any fields in the bulk ξ and ζ, the covariant derivative
Dξζ is defined by the metric affine connection ΓABC , with the Riemann tensor given by
R(ξ, ζ) = [Dξ,Dζ ]. Writing the components of this tensor in the embedding frame we
obtain the Gauss, Codazzi and Ricci equations, respectively:
RABCDZA,αZB,βZC,γZD,δ = Rαβγδ − 2gmnkα[γmkδ]βn (2.12)
RABCDZA,αηBb ZC,γZD,δ = kα[γb;δ] − gmnA[γmbkαδ]n (2.13)
RABCDηAa ηBb ZC,γZD,δ = −2gmnA[γmaAδ]nb − 2A[γab;δ] − gµνk[γµakδ]νb (2.14)
which are the integrability conditions for the embedding. The differentiable embedding
occurs when for a given Riemann tensor for the bulk these equations can be solved without
appeal to analyticity. A substantial part of Nash’s theorem consists in showing that the
solution requires that the functions appearing in the right hand side must be regular.
– 4 –
The expression (2.11) shows that besides the brane-world gravitational field the extrin-
sic curvature kµνa also propagate in the bulk. The implication of this is that the imposition
of any restrictive conditions on kµνa also implies on restrictions on the propagation of the
gravitational field of the brane-world. On the other hand, from (2.10) it follows that the
third fundamental form Aµab does not propagate at all in the bulk, behaving as if it is a
confined field.
The equations of motion of the brane-world follow directly from the Einstein-Hilbert
principle on the bulk and from the integrability conditions (2.12)-(2.14). To see how this
works take the trace of the first equation (2.5): gµνZA,µZ
,νGAB = D−N = GABGAB−gabgab,
and replace gab from (2.5), obtaining
gµνZA,µZ
,ν = GAB − gabηAa ηBb (2.15)
The contractions of (2.12) with gµν , and using (2.15) gives the the Ricci tensor and Ricci
scalar of the brane-world respectively expressed as
Rµν = g
cd(gαβkµαckνβd − hckµνd) +RABZA,µZB,ν − gabRABCDηAa ZB,µZC,νηDb (2.16)
After another contraction with gµν , using again (2.15), and noting that
gadgbcRABCDηAa ηBb ηCc ηDd = 0, we obtain the Ricci scalar
R = (K2 − h2) +R− 2gabRABηAa ηBb (2.17)
where K2 = gabkµνakµνb. ha = g
µνkµνa and h
2 = gabhahb. Therefore the Einstein-Hilbert
action for the bulk geometry in D-dimensions can be written as
GdDv =
R− (K2 − h2) + 2gabRABηAa ηBb
GdDv (2.18)
where α∗ denotes the fundamental energy scale in the bulk and L∗ is the source Lagrangian.
The Euler-Lagrange equations of (2.18) with respect to GAB are Einstein’s equations in D
dimensions:
RAB −
RGAB = α∗T ∗AB (2.19)
Here T ∗AB denote the components of the energy-momentum tensor of the sources.
The equations of motion of the embedded brane-world can be derived directly from
the components of (2.19) written in the embedding frame. The tangent components follow
from the contractions of (2.19) with ZA,µZB,ν . After using (2.16) and (2.17) we obtain
Rµν −
Rgµν −Qµν −Wµν − gabRABηAa ηBb = α∗T ∗µν (2.20)
where we have denoted
Qµν = g
abkρµakρνb − gabhakµνb −
(K2 − h2)gµν (2.21)
Wµν = g
adRABCDηAa ZB,µZC,νηDd
– 5 –
By a direct calculation we can see that the extrinsic tensor Qµν is an independently con-
served quantity with respect to the brane-world metric.
The contraction of (2.19) with ZAµ ηBb gives a vectorial equation. Using (2.17) and
(2.13) we obtain
kρµa;ρ−ha,µ+Aρcakρ cµ −Aµcahc+ 2Wµa = −2α∗(T ∗µa −
N + 2
T ∗gµa) (2.22)
where we have denoted
Wµa = g
bdRABCDηAa ηBb ZC,µηDd (2.23)
Finally, contracting (2.19) with ηAa η
b we obtain N(N +1)/2 scalar equations involving the
so called Hawking-Gibbons term Sab = RABηAa ηBb and its trace S = gabSab
Sab − Sgab −
[R−K2 + h2]gab = α∗T ∗ab (2.24)
In its most general form, without assuming extra dimensional matter, the confinement hy-
pothesis states that the only non-vanishing components of TAB are the tangent components
Tµν representing the confined sources [2]. Therefore we set
ABZA,µZB,ν = α∗T ∗µν = −8πGTµν (2.25)
ABZA,µηBa = α∗T ∗µa = 0 (2.26)
b = α∗T
ab = 0 (2.27)
Equations (2.20), (2.22) and (2.24) with confinement conditions are sometimes called the
gravi-tensor, gravi-vector and gravi-scalars (Usually a single gravi-scalar equation in the
5-dimensional models [21]) equations respectively. These represent generalizations of Ein-
stein’s equations of general relativity, in the sense that they describe the evolution of all
geometrical components gµν , Aµab and kµνa of the brane-world. Clearly, the usual Ein-
stein’s equations are recovered when all elements of the extrinsic geometry are removed
from those equations.
3. Canonical Equations
The standard ADM canonical quantization of the gravitational field in general relativity
was originally intended to describe the quantum fluctuations of 3-dimensional hypersurfaces
in a space-time [22]. The space-time metric is decomposed in 3-surface components, plus
a shift vector and a lapse function defined in a Gaussian reference frame defined on the 3-
dimensional hypersurface. After writing the Einstein-Hilbert Lagrangian in this Gaussian
frame, the Euler-Lagrange equations with respect to the shift leads to the vanishing of the
Hamiltonian. This is not a real problem because in principle the system could be solved by
use of Dirac’s standard procedure for constrained systems. However, as it is well known,
the Poisson bracket structure does not propagate covariantly as it would be expected. In
spite of all efforts made up to the present, this problem remains unsolved [23, 24, 25, 26].
It is possible to describe a non-constrained canonical system in a special frame defined by a
– 6 –
3-dimensional hypersurface orthogonal Gaussian coordinate system. In such special frame
the shift vector vanishes and the Hamiltonian constraint does not apply [27]. Nonetheless,
this has been regarded as of little value for general relativity itself, essentially because the
diffeomorphism group of the theory is one of the fundamental postulates of the theory [28].
The extension of the ADM canonical formulation to the brane-world is straightfor-
ward but it requires a few adaptations: First, the bulk is locally foliated by a continuous
sequence of brane-worlds propagating along the extra dimensions rather than by a 3-surface
propagating along a single time direction. Secondly, the confinement hypothesis implies
that the diffeomorphism invariance do not extend to the extra dimensions, otherwise a
coordinate transformation in the bulk would have the effect of introducing a component
of the energy-momentum tensor of the confined fields and ordinary matter in the bulk.
Therefore, in order to maintain the intended solution of the hierarchy problem, the diffeo-
morphism of the brane-world must be restricted as a confined symmetry. Actually this can
be regarded as one of the merits of brane-world theory, which differentiates it from being
just a higher dimensional version of general relativity. However, to deserve such merit
the extra dimensions need to be taken seriously as true physical degrees of freedom in the
canonical formulation of the theory. The momentum conjugated to the metric field GAB ,
with respect to the displacement along ηa is defined as usual
pAB(a) =
where L is the Einstein-Hilbert Lagrangian of the bulk in (2.18). Noting that in the em-
bedding frame we can write 2gabRABηAa ηBb = K2−gabha,b, after eliminating the divergence
term gabha,b, the Lagrangian can be simplified to
L = [R+ (K2 + h2)]
G (3.1)
Therefore, using (2.11) we obtain the canonically conjugated momenta
pµν (a) =
) = −
∂kµνa
= −(kµνa + hagµν)
G (3.2)
pµa(b) =
∂Aµab
= 0, pab(c) =
) = 0 (3.3)
The last two components are equal to zero because the Lagrangian (3.1) does not depend
explicitly on Aµab and on gab,c.
Using the above expressions, the Hamiltonian Ha corresponding to the displacement
along each orthogonal direction ηa separately, follows from a partial Legendre transforma-
tion (no sum on (a)):
Ha(g, p(a)) = pAB(a)gAB,a − L = −R
G − [(K2 + h2 + 2(K2a + h2a)]
– 7 –
where we have denoted K2a = k
a kµνa, K
2 = gabKaKb, and p(a) = g
µνpµν(a). After
replacing ha =
−p(a)
G it follows that
Ha = −R
pµν(a)
(3.4)
For a given functional F(gµν , pµν) defined in the phase space of the brane-world, the prop-
agation of F along ya is given by the Poisson brackets with each Hamiltonian separately
= [F ,Ha] =
δ̃gµν
δ̃pµν(a)
δ̃pµν(a)
δ̃gµν
(3.5)
Here δ̃ denotes the standard functional derivative in phase space.
Hamilton’s equations for the brane-world with respect to each extra coordinate ya may
be written as
δ̃pµν(a)
= [gµν ,Ha] = −2kµνa, (3.6)
= − δ̃Ha
δ̃gµν
,Ha] (3.7)
As it can be seen, the differences between the brane-world canonical formulation and the
ADM formulation of general relativity follow from the non-vanishing of the Hamiltonians
Ha, as a consequence of the brane-world scheme for solving the hierarchy problem. With
this result, the ADM quantization program can be retaken, with the difference that the
quantum equation should describe the ”states” of four-dimensional submanifolds in the
bulk, with respect to the extra dimensions.
4. Tomonaga-Schwinger Quantum States
The Tomonaga-Schwinger equation originated from Dirac’s many fingered time formalism
for relativistic quantum theory, in which a set of N electrons was associated to N proper
times satisfying N Schrodinger’s-like equations [29]. The continuous limit of this equa-
tion was formulated by Tomonaga for a relativistic field defined in a region of space-time
characterized by an evolving space-like 3-hypersurface σ with a time direction attached to
each of its point. This geometric extension of Dirac’s many fingered time, which was soon
realized to be equivalent to the interaction representation of quantum mechanics developed
by Schwinger [30, 31, 32]. Here, it is more convenient to look at the Tomonaga-Schwinger
equation from the geometrical point of view written as
= ĤσΨ (4.1)
which represents a generalization of Schrodinger’s equation, describing the quantum state
functional Ψ of a space-like 3-hypersurface σ embedded in Minkowski space-time. In the
– 8 –
right hand side, the Hamiltonian operator describes the translational operator along a time-
like direction orthogonal to σ. The functional derivative in the left hand side is defined by
the limit
= lim
Ψ(σ′)−Ψ(σ)
(4.2)
where ∆V denotes the local volume element between two neighboring hypersurfaces σ and
The main difficulty of (4.1) is that it is not easily integrable. In the particular case
where [Ĥσ, Ĥσ′ ] = 0, the equation can be integrated, but the hypersurfaces σ and σ′ are
necessarily flat. In the general case where the hypersurfaces are not flat, the solutions
of (4.1) can be determined as an approximate solution after the application of the Yang-
Feldmann formalism and Dyson’s expression for the S matrix [32, 33]. The difficulty in
solving (4.1) can be traced back to the fact that the limit operation in (4.2) was not
defined. In fact, the conditions to decide how close σ and σ′ are were not given previously,
and it can be decided only after solving the quantum equation itself using some quantum
approximation method.
In the application of (4.1) to the brane-world, the limit operation between two four-
dimensional brane-worlds σ4 and σ
4 is improved because Nash’s theorem shows at the
classical level how to tell the separation between the two sub manifolds. In other words,
since each brane-world was generated by classical perturbations of an initial sub manifold
σ4, the volume element in (4.2) has been already specified by the parameter δy
a of the
perturbed geometry. In practice, we may split the bulk volume ∆V between σ4 and σ
into a product of the volume ∆v of a a small compact region in σ4, times the variation ∆y
of the extra dimensional coordinates ya. Therefore, it sufficient to specify only the limit
operation ∆ya → 0 and the functional derivative (4.2) for density functions with compact
support on the brane-world, with respect to each extra dimension can be simplified to
= lim
∆ya→0
Ψ(σ′)−Ψ(σ)
Repeating for all extra dimensions, we find that the Tomonaga-Schwinger equation (4.1)
can be extended the brane-world, as a system of N partial equations, one for each extra
dimension
= ĤaΨa, a = 5..D (4.3)
which gives to Ĥa the interpretation of the extra dimensional translational operator.
The final quantum state is given by the superposition of the N separates states Ψa as
BaΨa. The state functional density Ψ represents the quantum fluctuation of the
brane-world sub manifold in the bulk at the (Tev) energy scale, subjected to quantum
uncertainties and a state probability given by
||Ψ||2 =
Ψ†Ψδyδv
– 9 –
An observer confined to the brane-world may evaluate the quantum expectation values of
the brane-world metric and the extrinsic curvature are given by
< Ψ|ĝµν |Ψ >=
Ψ†ĝµνΨδyδv, < Ψ|k̂µν |Ψ >=
Ψ†k̂µνΨδyδv
Since the classical kµνa is the derivative of the metric gµν with respect to ya, we may set
boundary conditions on these quantities at the initial brane-world ya = 0 to determine the
final solution.
5. Overview and Perspectives
We have shown that the Einstein-Hilbert principle applied to the bulk geometry plus the
differentiable embedding conditions are sufficient to determine the classical and quantum
structures of the brane-world in D-dimensions. In particular, it was shown that Nash’s
theorem makes it possible to generate any embedded sub manifold by a continuous se-
quence of infinitesimal perturbations of an arbitrarily given embedded geometry along the
extra dimensions. Using the classical perturbative embeddings, and the basic principles of
the brane-world theory we have obtained a canonical structure very much like the ADM
canonical formulation of general relativity, with the exception that the Hamiltonians do
not vanish.
The definition of the functional derivatives was improved with respect to four-dimensional
field theory, by using the previously defined perturbative embedding structure of the brane-
world. The quantization of the brane-world was described the Tomonaga-Schwinger equa-
tion defined for brane-world sub manifolds, calculated for each extra dimension. Actually,
as a result of the the the classical perturbation theory, the Tomonaga-Schwinger equation
becomes exactly integrable.
In view of current astrophysical observations and the near future high energy exper-
iments, there are some applications of the quantum brane-world theory to be detailed in
subsequent papers:
(a) Brane-world Cosmology
Brane-world cosmology offers a possible explanation to the accelerated expansion of the
universe, resulting from the modification of the Friedman’s equation by the presence of the
extrinsic curvature included in the tensor Qµν given by (2.21) [10]. The presence of this
tensor has the meaning that the brane-world vacuum is more complex than the vacuum in
general relativity. In fact, for a constant curvature bulk with curvature Λ∗, after eliminating
redundant terms, the gravi-tensor vacuum equation becomes Rµν−1/2Rgµν−Qµν+Λ∗gµν =
Therefore, the vacuum energy density < ρv > resulting from gravitationally coupled
fields must be revaluated, including the extrinsic curvature component. This suggests that
in some epochs, say at the early inflationary period, the extrinsic curvature may contribute
to the vacuum energy, differently from other periods.
The particular case where we have only one extra dimensions (D = 5), has some limita-
tions with respect to the differentiable embedding. However, some cosmological models like
– 10 –
the FRW, deSitter and anti deSitter solutions of Einstein’s equations in four dimensions
can be embedded in five dimensional bulks without restrictions, in accordance with the
perturbative embedding equations previously shown. Consequently, in such brane-world
cosmological models the conditions required for a proper definition of the functional deriva-
tives in the Tomonaga-Schwinger equation are well established, and the equation can can
be integrated without difficulty.
(b) Laboratory production of mini black holes:
Brane-world gravity predict the generation of short lived mini black holes produced at the
Tev energy in the laboratory, resulting from proton-proton collisions [34]. However, using
semi-classical quantum gravity in four dimensions, we have learned that quantum unitarity
does not necessarily hold true during the black hole evaporation. On the other hand, using
Euclidean path integral, it was shown that the unitarity can be restored with the aid of
the ADS/CFT correspondence in the framework of AdS5 × S5 string theory [35].
Since the generation of mini black holes are possible only in the brane-world context,
the whole process includes the original Minkowski’s space-time where the experiment is
devised. Soon after the collision, the space-time must be deformable into a Schwarzschild
or a Reissner-Nordstrom black hole. Finally, after a short period of evaporation the space-
time may be back to Minkowski’s configuration or else leaves a curved remnant. The
description of such process can start with the classical perturbations in accordance with
Nash’s embedded geometries, but the unitarity is has to be decided at the quantum level.
In this respect we notice that both Schwarzschild and Reissner-Nordstrom black holes are
well defined submanifolds embedded in a six-dimensional flat bulk with signature (4, 2).
In this case, the bulk isometry group SO(4, 2) is isomorphic to the conformal group in
Minkowski’s space-time, compatible with the ADS/CFT correspondence adapted to the
brane-world [36]. Therefore, the quantum unitarity implicitly assumed in the Tomonaga-
Schwinger equation, must be consistent with the black hole evaporation theorems in six
dimensions.
(c) Quantum Four-manifold Theory:
The above description of quantum theory of the brane-world is based almost entirely on
the general theory of differentiable sub manifolds. This suggests a quantum theory of
four-dimensional sub manifolds. It starts with the classical perturbations of embedded
geometries, but ends with a quantum version of the embedding theorem, including the
fluctuations of the embedding as described by the Tomonaga-Schwinger equation. This
quantum theory of submanifolds would be particularly interesting when the bulk has di-
mensions greater than five, where the third fundamental form behaves similarly to a gauge
field with respect to the extra dimensional group of isometries. The identification of the
third fundamental form as a gauge field with the symmetry of the extra dimensions plying
the role of the gauge group is old, but it was never taken seriously [37].
One frequent criticism to string theory is that it depends on a pre-existing background
space-time with 10 (or 11) dimensions, acting as the host space for all possible dynamics
[38, 39]. On the other hand, loop quantum gravity does not require such background, but
it depends on a previously existing spin network structure [40]. Quantum brane-world
– 11 –
gravity does not have a background space in the same sense of string theory because the
bulk is the primary dynamical object. The Einstein-Hilbert principle applied to the bulk
geometry provides all dynamics of the brane-world, without requiring any new algebraic
structure besides the theory of differentiable manifolds, where our basic notions of space,
topology and analysis begin and make sense.
References
[1] C. Misner, Rev. Mod. Phys. 29, 497 (1957).
[2] N. Arkani-Hamed et Al, Phys. Lett. B429,263 (1998), hep-ph/9807344 Phys. Rev. Lett. 84,
586, (2000)
[3] I. Antoniadis et al, Phys. Lett. ibid B246, 377, (1990); B436, 257 (1998).
[4] K. Akama, Pregeometry, in Gauge Theory and Gravitation, Lecture notes in Physics 176
(Springer Verlag (1983)), hep-th/0001113;
[5] V. A. Rubakov & M. E. Shaposhnikov, Phys. Lett. 125B, 136, (1983);
[6] T. Regge & C. Teitelboim, General Relativity a la String, proc. II Marcel Grossmann
meeting, ICTP, Trieste (1972).
[7] M. Visser, Phys. Lett. 159B, 22 (1985), hep-th/9910093.
[8] P. Horava & E. Witten, Nuc. Phys. B475, 94 (1996)
[9] G. Landsberg, J.Phys. G32, R337, (2006), hep-ph/0607297
[10] M. D. Maia et al, Class.Quant.Grav. 22, 1623 (2005), astro-ph/0403072
[11] L. Randall & R. Sundrum, Phys. Rev. Lett. 83, 4690 (1998); ibid 83, 3370 (1999)
[12] K. Maeda, Phys. Rev. D64, 123525 (2001). astro-ph/0012313,
[13] A. Friedman, Rev. Mod. Phys. 37, 201, 1965
[14] M. Janet, Ann. Soc. Pol. Mat 5, (1928)
[15] E. Cartan, Ann. Soc. Pol. Mat 6, 1, (1927)
[16] J. E. Campbell, A course of Differential Geometry, Claredon Press, Oxford (1926)
[17] F. Dahia, C. Romero, Braz.J.Phys. 35, 1140, (2005)
[18] J. Nash, Ann. Maths. 63, 20 (1956)
[19] R. Greene, Memoirs Amer. Math. Soc. 97, (1970)
[20] L.P. Eisenhart, Princeton U. P. Sixth print, p.159 ff, (1966).
[21] R. Durrer & P. Kocian, Class. Quant. Grav. 21, 2127, (2004), hep-th/030518
[22] R. Arnowitt, S. Deser and C. Misner, The Dynamics of General Relativity. In Gravitation:
An introduction to Current Research, L. Witten, Ed. John Willey & Sons, p. 227, (1962)
[23] K. V. Kuchar, Canonical Quantum Gravity, 13th Intl. Conf. on General Relativity &
Gravitation, Cordoba, Argentina (1992). Ibid. Time and Interpretations of Quantum Gravity
4th Can. Conf. on general Relativity and relati. Astrophys. World. Scientific (1991).
– 12 –
[24] C. J. Isham. Canonical Quantization and the Problem of Time NATO Advanced Studies
Institute, Salamanca (1992),
[25] S. Hojman et al, Ann. of Phys. 96, 88, (1976).
[26] E. Alvarez, Lect.Notes Phys. 669, 31-58 (2005), gr-qc/0405107
[27] P. A. M. Dirac Phys. Rev. 114, 924, (1959)
[28] P A. M. Dirac Proc.Roy.Soc.Lond. A246, 333, (1958).
[29] P. A. M. Dirac, Proc. Roy. Soc. London 136, 453 (1932)
[30] S. Tomonaga, Prog. Theor. Phys. 1, 27 (1946)
[31] J. Schwinger, Phys. Rev. 74, 1439 (1948)
[32] K. Nishijima, Fields and Particles, W. A. Benjamin, (1969).
[33] W. Pauli, Selected Topics in Field Quantization, MIT Press, (1973); Dover edition (1973), p.
[34] S. Dimopolous & G. Landsberg, Phys. Rev. Lett. 87, 161602 (2001), hep-ph/0106295
[35] S. Hawking, Phys. Rev. D72, 084013 (2005), hep-th/0507171
[36] M. D. Maia, Int.J.Mod.Phys. D14, 2251, (2005), gr-qc/0505119
[37] Y. Ne’emann, Rev. Mod. Phys., 37, 227 (1965)
[38] A. Ashtekar & Jerzy Lewandowski, Class.Quant.Grav.21, R53(2004), gr-qc/0404018
[39] T. Thiemann, AEI-2006-066, hep-th/0608210
[40] H. Nicolai, K. Peeters & M. Zamaklar, hep-th/0501114
– 13 –

  The Arnowitt-Deser-Misner canonical formulation of general relativity is
extended to the covariant brane-world theory in arbitrary dimensions. The
exclusive probing of the extra dimensions makes a substantial difference,
allowing for the construction of a non-constrained canonical theory. The
quantum states of the brane-world geometry are defined by the
Tomonaga-Schwinger equation, whose integrability conditions are determined by
the classical perturbations of submanifolds contained in the Nash's
differentiable embedding theorem. In principle, quantum brane-world theory can
be tested by current experiments in astrophysics and by near future laboratory
experiments at Tev energy. The implications to the black-hole information loss
problem, to the accelerating cosmology, and to a quantum mathematical theory of
four-sub manifolds are briefly commented.

arXiv:0704.1289v1  [gr-qc]  10 Apr 2007
Preprint typeset in JHEP style - HYPER VERSION
Brane-world Quantum Gravity
M.D. Maia∗, Nildsen Silva †and M.C.B. Fernandes‡
Instituto de F́ısica, Universidade de Braśılia,
70919-970, Braśılia, D. F., Brazil
Abstract: The Arnowitt-Deser-Misner canonical formulation of general relativity is ex-
tended to the covariant brane-world theory in arbitrary dimensions. The exclusive probing
of the extra dimensions makes a substantial difference, allowing for the construction of a
non-constrained canonical theory. The quantum states of the brane-world geometry are
defined by the Tomonaga-Schwinger equation, whose integrability conditions are deter-
mined by the classical perturbations of submanifolds contained in the Nash’s differentiable
embedding theorem. In principle, quantum brane-world theory can be tested by current
experiments in astrophysics and by near future laboratory experiments at Tev energy. The
implications to the black-hole information loss problem, to the accelerating cosmology, and
to a quantum mathematical theory of four-sub manifolds are briefly commented.
Keywords: Quantum Gravity Brane-world Nash Theorem Tomonaga-Schwinger.
∗maia@unb.br
†nildsen@hotmail.com
‡mcezar@fis.unb.br
http://arxiv.org/abs/0704.1289v1
http://jhep.sissa.it/stdsearch
Contents
1. Quantizing the Brane-World 1
2. Covariant Brane-world Gravity 2
3. Canonical Equations 6
4. Tomonaga-Schwinger Quantum States 8
5. Overview and Perspectives 10
1. Quantizing the Brane-World
If gravity is to occupy a significant place in modern physics,
it can do so only by being qualitatively different from other
fields. As soon as we assume that gravity behaves qualitatively
like other fields, we find that it is quantitatively insignificant
C.W. Misner (1957)
After analyzing perturbative quantum gravity, Misner reached the interesting conclu-
sion that an effective quantum gravity must have qualities which makes it different from
gauge theories [1]. Translating quantitative significance in terms of energy level, Misner’s
conclusion suggests that the problem of quantization of the gravitational field should be
solved concomitantly with the hierarchy problem of the fundamental interactions. In what
follows, we apply this criterium to brane-world gravity.
Brane-world gravity is based on a higher dimensional solution of the hierarchy prob-
lem. In a seminal paper N. Arkani-Hamed, G. Dvali and S. Dimopolous questioned the
currently accepted hypothesis that gravitons are quantitatively relevant only at the Planck
scale of energies, essentially because this is an assumption devoid of experimental sup-
port. They proposed that the known gauge fields (and hence all ordinary matter) are
to be confined within the four-dimensional brane-world, but gravitons can propagate in a
higher-dimensional space, the bulk, at the same Tev scale of energies of the gauge fields [2]
(For historical papers on the development of the theory see also [3, 4, 5, 6, 7, 8]). Accord-
ing to this view, brane-world gravity is qualitatively distinct from, but it is quantitatively
equivalent to the gauge fields of the standard model.
Brane-world gravity predicts the existence of short lived Tev mini black holes, which
in principle can be produced at the laboratory by a high energy proton-proton collision,
with implications to the black hole information loss problem at the quantum level. The
– 1 –
proposed experiment is set in Minkowski space-time, but it ends in a Schwarzschild (or
Reissner-Nordstrom) space-time [9]. Therefore, the theory supporting this experiment
must be compatible with cross sections of the order of the Schwarzschild radius, and also
with an explanation on how the original Minkowski space-time deforms into black hole,
and back in a short period of time.
Brane-world gravity may also explain the acceleration of the universe (see eg [10] and
references therein). In short, due to the presence of the extrinsic curvature, the vacuum in
brane-world gravity is richer than the vacuum in general relativity. Besides the cosmological
constant, it also contain a conserved geometric tensor built from the extrinsic curvature.
Consequently, when studying the quantum fluctuations of such vacuum we may obtain
a different estimate for the vacuum energy density as compared with the case of general
relativity.
Since most of the current research on brane-world theory is based on models defined in
a five-dimensional bulk, using specific coordinates and particular symmetries (see eg [11]),
we find it necessary to review in the next section the covariant equations of motion of a
brane-world defined in an arbitrary bulk, with an arbitrary number of dimensions. Those
equations can be found elsewhere [12, 10], but here we have included some details which
are required for the quantum description. Readers who are familiar with this may jump
to section 3, where the canonical equations of the brane-world with respect to the extra
dimensions are discussed. In section 4 we introduce the Tomonaga-Schwinger equation for
the brane-world with respect to the extra dimensions and comment on its integrability.
2. Covariant Brane-world Gravity
There are essentially three basic postulates in brane-world theory: (1) The bulk geometry
is defined by Einstein’s equations; (2) The brane-world is a sub manifold embedded in
that bulk ; (3) The gauge fields and ordinary matter are confined to four dimensions, but
gravitons propagates along the extra dimensions at Tev energy [2].
The embedding of the brane-world in the bulk plays an essential role on the covariant
(that is, model independent) formulation of the brane-world gravity, because it tells how
the Einstein-Hilbert dynamics of the bulk is transferred to the brane-world. However,
there a are many different ways to embed a manifold into another, classified as local,
global, isometric, conformal (or more generally defined by a collineation), rigid, deformable,
analytic or differentiable. The choice of one or another depend on what the embedded
manifold is supposed to do.
In string theory the action principle is defined on the world-sheets, with additional
boundary conditions, so that the embedding is necessarily global. Since the world sheets
are 2-dimensional they are all conformally flat and their global embedding is not difficult to
achieve. However, if higher-dimensional objects such as p-branes are to be considered, then
the global embedding may turn out to be difficult to realize in 10 or even in 11 dimensions
[13].
Differently from string theory, the Einstein-Hilbert action in brane-world theory is set
on the bulk, which is therefore the primary dynamical object. Furthermore, the embedding
– 2 –
is locally defined, meaning that the bulk is a local fiber bundle whose fibers are the direct
sum of the tangent and normal spaces at each point of the brane-world taken as the base
space. If we want to draw a picture, the bulk can be seen as as a locally constructed space
around each point of the brane-world.
A local differentiable embedding requires only that the embedding functions are dif-
ferentiable and regular. This follows from Nash’s embedding theorem, an important im-
provement over the traditional analytic embedding theorems of Janet and Cartan [14, 15],
which demand that the embedding functions are represented by convergent positive power
series. Furthermore, Nash’s theorem shows that any sub manifold can be generated by
a continuous sequence of small perturbations of an arbitrarily given sub manifold 1. Al-
though the theorem was originally demonstrated for the case of an Euclidean bulk, it was
later generalized to pseudo Riemannian manifolds [18, 19].
Given a particular Riemannian sub-manifold σ̄4, its local isometric embedding in a
certain bulk MD, is given by D = 4 + N differentiable and regular embedding maps
X̄A : σ̄4 → MD, such that 2
X̄A,µX̄B,ν GAB = ḡµν , X̄A,µ η̄Bb GAB = 0, and η̄Aa η̄Bb GAB = ḡab (2.1)
where η̄Aa are the components of the N linearly independent vector fields in the same
coordinates of the bulk where the components GAB of the bulk metric are defined. The
vectors {X̄A,µ , η̄Ba } define a Gaussian reference frame called here the embedding frame. The
derivatives of the vectors η̄a is expressed in terms of the second and third fundamental
forms k̄µνa, Āµab respectively by the Gauss-Weingerten equations [20]
η̄Aa,α = ḡ
µν k̄αµaX̄A,ν + ḡmnĀαamη̄An (2.2)
Without loss of generality we may chose the normal vectors η̄a to be orthogonal to each
other, so that ḡab = ǫaδab, where ǫa = ±1 depending on the signature of the bulk [19].
Nash’s perturbative approach to embedding consists in subjecting the fundamental
forms of σ̄4 to small parametric deviations along each normal vector. It can be also
described by introducing a small perturbation with parameter δya, of the base vectors
{X̄A,µ , ηAa } along each normal η̄Aa evaluated on σ̄4, obtaining another set of vectors (no sum
on a)
ZA,µ = X̄A,µ + (δya£η̄aX̄A),µ = X̄A,µ − δya[X̄ , η̄a]A,µ = X̄A,µ + δyaη̄Aa,µ, (2.3)
ηAa = η̄
a + (δy
£η̄a η̄a)
A = η̄Aa + δy
a[η̄a, η̄a]
A = η̄Aa (2.4)
1The perturbative approach to the embedding was originally proposed by J. E. Campbell in 1926.
However, his result differs from Nash’s theorem because analytic conditions where implicitly used [16,
17]. Since the perturbation procedure is based on regular and differentiable functions, the differentiable
embedding is less restrictive to the geometry than the analytic embeddings.
2Capital Latin indices refer to the bulk, which is a Riemannian geometry with metric GAB in arbitrary
coordinates. Small case Latin indices refer to the extra dimensions going from 5 to D, and all Greek indices
refer to the brane, from 1 to 4. A curly R always denotes bulk curvatures, like in RABCD . Ordinary capital
R like in Rµν denotes brane-world curvatures. Covariant derivatives need to be specified, for the bulk or
the brane-world metrics. For a vector V A in the bulk its covariant derivative with respect to GAB is denoted
as V A;B. On the other hand, from the point of view of the brane-world metric, the components V
A behave
as a set of N scalar functions as in [20]. For generality we denote G = |det(GAB)|.
– 3 –
which define a perturbed embedding frame {ZA,µ, ηAa } in the bulk. Admitting that these
new functions remain differentiable and regular and that they satisfy the equations similar
to (2.1),
ZA,µZB,νGAB = gµν , ZA,µηBa GAB = gµa, ηAa ηBb GAB = gab = ḡab (2.5)
we obtain a N -parameter local family of submanifolds σ4 generated by local perturbations
of σ̄4, by a continuous variations of the parameters δy
The next problem is to find a solution of these equations. However, instead of finding
the coordinates ZA, it is more convenient to write the perturbed solution in terms of the
fundamental forms, expressed in terms of the initial geometry of σ̄4. By direct substitution
of Z,µ and ηAa derived from (2.3) in equations (2.5) we obtain
gµν(x, y) = ZA,µZB,νGAB = ḡµν− 2δyak̄µνa
+ δyaδyb[ḡαβ k̄µαak̄νβb + g
cdĀµcaĀνdb], (2.6)
gµa(x, y) = ZA,µηBa GAB =δybAµab, (2.7)
gab(x, y) = η
b GAB = ḡab (2.8)
kµνa(x, y) = −ηAa,µZB,νGAB
= k̄µνa− δybḡαβ k̄µαak̄νβb −gcdδybĀµcaĀνdb, (2.9)
Aµab(x, y) = η
b GAB=Āµab(x) (2.10)
The contravariant components of the perturbed geometry must be consistent with GACGCB =
δAB , which can be realized by setting gµρg
ρν = δνµ, gacg
cb = δba. Since the indices µ and b
can never be equal, we must nave gµρg
ρb+ gµcg
cb = δbµ = 0. After some algebra we see that
this corresponds to an identity ymyngabAµamAνbn = −ybyngmaAµ[am]Aν[bn] ≡ 0.
Comparing (2.6) and (2.9) we obtain
kµνa = −
(2.11)
Consequently, the local bulk defined in a neighborhood around σ̄4, is foliated by this
perturbed geometry, so that the Riemann curvature of the bulk may be expressed in the
perturbed embedding frame. For any fields in the bulk ξ and ζ, the covariant derivative
Dξζ is defined by the metric affine connection ΓABC , with the Riemann tensor given by
R(ξ, ζ) = [Dξ,Dζ ]. Writing the components of this tensor in the embedding frame we
obtain the Gauss, Codazzi and Ricci equations, respectively:
RABCDZA,αZB,βZC,γZD,δ = Rαβγδ − 2gmnkα[γmkδ]βn (2.12)
RABCDZA,αηBb ZC,γZD,δ = kα[γb;δ] − gmnA[γmbkαδ]n (2.13)
RABCDηAa ηBb ZC,γZD,δ = −2gmnA[γmaAδ]nb − 2A[γab;δ] − gµνk[γµakδ]νb (2.14)
which are the integrability conditions for the embedding. The differentiable embedding
occurs when for a given Riemann tensor for the bulk these equations can be solved without
appeal to analyticity. A substantial part of Nash’s theorem consists in showing that the
solution requires that the functions appearing in the right hand side must be regular.
– 4 –
The expression (2.11) shows that besides the brane-world gravitational field the extrin-
sic curvature kµνa also propagate in the bulk. The implication of this is that the imposition
of any restrictive conditions on kµνa also implies on restrictions on the propagation of the
gravitational field of the brane-world. On the other hand, from (2.10) it follows that the
third fundamental form Aµab does not propagate at all in the bulk, behaving as if it is a
confined field.
The equations of motion of the brane-world follow directly from the Einstein-Hilbert
principle on the bulk and from the integrability conditions (2.12)-(2.14). To see how this
works take the trace of the first equation (2.5): gµνZA,µZ
,νGAB = D−N = GABGAB−gabgab,
and replace gab from (2.5), obtaining
gµνZA,µZ
,ν = GAB − gabηAa ηBb (2.15)
The contractions of (2.12) with gµν , and using (2.15) gives the the Ricci tensor and Ricci
scalar of the brane-world respectively expressed as
Rµν = g
cd(gαβkµαckνβd − hckµνd) +RABZA,µZB,ν − gabRABCDηAa ZB,µZC,νηDb (2.16)
After another contraction with gµν , using again (2.15), and noting that
gadgbcRABCDηAa ηBb ηCc ηDd = 0, we obtain the Ricci scalar
R = (K2 − h2) +R− 2gabRABηAa ηBb (2.17)
where K2 = gabkµνakµνb. ha = g
µνkµνa and h
2 = gabhahb. Therefore the Einstein-Hilbert
action for the bulk geometry in D-dimensions can be written as
GdDv =
R− (K2 − h2) + 2gabRABηAa ηBb
GdDv (2.18)
where α∗ denotes the fundamental energy scale in the bulk and L∗ is the source Lagrangian.
The Euler-Lagrange equations of (2.18) with respect to GAB are Einstein’s equations in D
dimensions:
RAB −
RGAB = α∗T ∗AB (2.19)
Here T ∗AB denote the components of the energy-momentum tensor of the sources.
The equations of motion of the embedded brane-world can be derived directly from
the components of (2.19) written in the embedding frame. The tangent components follow
from the contractions of (2.19) with ZA,µZB,ν . After using (2.16) and (2.17) we obtain
Rµν −
Rgµν −Qµν −Wµν − gabRABηAa ηBb = α∗T ∗µν (2.20)
where we have denoted
Qµν = g
abkρµakρνb − gabhakµνb −
(K2 − h2)gµν (2.21)
Wµν = g
adRABCDηAa ZB,µZC,νηDd
– 5 –
By a direct calculation we can see that the extrinsic tensor Qµν is an independently con-
served quantity with respect to the brane-world metric.
The contraction of (2.19) with ZAµ ηBb gives a vectorial equation. Using (2.17) and
(2.13) we obtain
kρµa;ρ−ha,µ+Aρcakρ cµ −Aµcahc+ 2Wµa = −2α∗(T ∗µa −
N + 2
T ∗gµa) (2.22)
where we have denoted
Wµa = g
bdRABCDηAa ηBb ZC,µηDd (2.23)
Finally, contracting (2.19) with ηAa η
b we obtain N(N +1)/2 scalar equations involving the
so called Hawking-Gibbons term Sab = RABηAa ηBb and its trace S = gabSab
Sab − Sgab −
[R−K2 + h2]gab = α∗T ∗ab (2.24)
In its most general form, without assuming extra dimensional matter, the confinement hy-
pothesis states that the only non-vanishing components of TAB are the tangent components
Tµν representing the confined sources [2]. Therefore we set
ABZA,µZB,ν = α∗T ∗µν = −8πGTµν (2.25)
ABZA,µηBa = α∗T ∗µa = 0 (2.26)
b = α∗T
ab = 0 (2.27)
Equations (2.20), (2.22) and (2.24) with confinement conditions are sometimes called the
gravi-tensor, gravi-vector and gravi-scalars (Usually a single gravi-scalar equation in the
5-dimensional models [21]) equations respectively. These represent generalizations of Ein-
stein’s equations of general relativity, in the sense that they describe the evolution of all
geometrical components gµν , Aµab and kµνa of the brane-world. Clearly, the usual Ein-
stein’s equations are recovered when all elements of the extrinsic geometry are removed
from those equations.
3. Canonical Equations
The standard ADM canonical quantization of the gravitational field in general relativity
was originally intended to describe the quantum fluctuations of 3-dimensional hypersurfaces
in a space-time [22]. The space-time metric is decomposed in 3-surface components, plus
a shift vector and a lapse function defined in a Gaussian reference frame defined on the 3-
dimensional hypersurface. After writing the Einstein-Hilbert Lagrangian in this Gaussian
frame, the Euler-Lagrange equations with respect to the shift leads to the vanishing of the
Hamiltonian. This is not a real problem because in principle the system could be solved by
use of Dirac’s standard procedure for constrained systems. However, as it is well known,
the Poisson bracket structure does not propagate covariantly as it would be expected. In
spite of all efforts made up to the present, this problem remains unsolved [23, 24, 25, 26].
It is possible to describe a non-constrained canonical system in a special frame defined by a
– 6 –
3-dimensional hypersurface orthogonal Gaussian coordinate system. In such special frame
the shift vector vanishes and the Hamiltonian constraint does not apply [27]. Nonetheless,
this has been regarded as of little value for general relativity itself, essentially because the
diffeomorphism group of the theory is one of the fundamental postulates of the theory [28].
The extension of the ADM canonical formulation to the brane-world is straightfor-
ward but it requires a few adaptations: First, the bulk is locally foliated by a continuous
sequence of brane-worlds propagating along the extra dimensions rather than by a 3-surface
propagating along a single time direction. Secondly, the confinement hypothesis implies
that the diffeomorphism invariance do not extend to the extra dimensions, otherwise a
coordinate transformation in the bulk would have the effect of introducing a component
of the energy-momentum tensor of the confined fields and ordinary matter in the bulk.
Therefore, in order to maintain the intended solution of the hierarchy problem, the diffeo-
morphism of the brane-world must be restricted as a confined symmetry. Actually this can
be regarded as one of the merits of brane-world theory, which differentiates it from being
just a higher dimensional version of general relativity. However, to deserve such merit
the extra dimensions need to be taken seriously as true physical degrees of freedom in the
canonical formulation of the theory. The momentum conjugated to the metric field GAB ,
with respect to the displacement along ηa is defined as usual
pAB(a) =
where L is the Einstein-Hilbert Lagrangian of the bulk in (2.18). Noting that in the em-
bedding frame we can write 2gabRABηAa ηBb = K2−gabha,b, after eliminating the divergence
term gabha,b, the Lagrangian can be simplified to
L = [R+ (K2 + h2)]
G (3.1)
Therefore, using (2.11) we obtain the canonically conjugated momenta
pµν (a) =
) = −
∂kµνa
= −(kµνa + hagµν)
G (3.2)
pµa(b) =
∂Aµab
= 0, pab(c) =
) = 0 (3.3)
The last two components are equal to zero because the Lagrangian (3.1) does not depend
explicitly on Aµab and on gab,c.
Using the above expressions, the Hamiltonian Ha corresponding to the displacement
along each orthogonal direction ηa separately, follows from a partial Legendre transforma-
tion (no sum on (a)):
Ha(g, p(a)) = pAB(a)gAB,a − L = −R
G − [(K2 + h2 + 2(K2a + h2a)]
– 7 –
where we have denoted K2a = k
a kµνa, K
2 = gabKaKb, and p(a) = g
µνpµν(a). After
replacing ha =
−p(a)
G it follows that
Ha = −R
pµν(a)
(3.4)
For a given functional F(gµν , pµν) defined in the phase space of the brane-world, the prop-
agation of F along ya is given by the Poisson brackets with each Hamiltonian separately
= [F ,Ha] =
δ̃gµν
δ̃pµν(a)
δ̃pµν(a)
δ̃gµν
(3.5)
Here δ̃ denotes the standard functional derivative in phase space.
Hamilton’s equations for the brane-world with respect to each extra coordinate ya may
be written as
δ̃pµν(a)
= [gµν ,Ha] = −2kµνa, (3.6)
= − δ̃Ha
δ̃gµν
,Ha] (3.7)
As it can be seen, the differences between the brane-world canonical formulation and the
ADM formulation of general relativity follow from the non-vanishing of the Hamiltonians
Ha, as a consequence of the brane-world scheme for solving the hierarchy problem. With
this result, the ADM quantization program can be retaken, with the difference that the
quantum equation should describe the ”states” of four-dimensional submanifolds in the
bulk, with respect to the extra dimensions.
4. Tomonaga-Schwinger Quantum States
The Tomonaga-Schwinger equation originated from Dirac’s many fingered time formalism
for relativistic quantum theory, in which a set of N electrons was associated to N proper
times satisfying N Schrodinger’s-like equations [29]. The continuous limit of this equa-
tion was formulated by Tomonaga for a relativistic field defined in a region of space-time
characterized by an evolving space-like 3-hypersurface σ with a time direction attached to
each of its point. This geometric extension of Dirac’s many fingered time, which was soon
realized to be equivalent to the interaction representation of quantum mechanics developed
by Schwinger [30, 31, 32]. Here, it is more convenient to look at the Tomonaga-Schwinger
equation from the geometrical point of view written as
= ĤσΨ (4.1)
which represents a generalization of Schrodinger’s equation, describing the quantum state
functional Ψ of a space-like 3-hypersurface σ embedded in Minkowski space-time. In the
– 8 –
right hand side, the Hamiltonian operator describes the translational operator along a time-
like direction orthogonal to σ. The functional derivative in the left hand side is defined by
the limit
= lim
Ψ(σ′)−Ψ(σ)
(4.2)
where ∆V denotes the local volume element between two neighboring hypersurfaces σ and
The main difficulty of (4.1) is that it is not easily integrable. In the particular case
where [Ĥσ, Ĥσ′ ] = 0, the equation can be integrated, but the hypersurfaces σ and σ′ are
necessarily flat. In the general case where the hypersurfaces are not flat, the solutions
of (4.1) can be determined as an approximate solution after the application of the Yang-
Feldmann formalism and Dyson’s expression for the S matrix [32, 33]. The difficulty in
solving (4.1) can be traced back to the fact that the limit operation in (4.2) was not
defined. In fact, the conditions to decide how close σ and σ′ are were not given previously,
and it can be decided only after solving the quantum equation itself using some quantum
approximation method.
In the application of (4.1) to the brane-world, the limit operation between two four-
dimensional brane-worlds σ4 and σ
4 is improved because Nash’s theorem shows at the
classical level how to tell the separation between the two sub manifolds. In other words,
since each brane-world was generated by classical perturbations of an initial sub manifold
σ4, the volume element in (4.2) has been already specified by the parameter δy
a of the
perturbed geometry. In practice, we may split the bulk volume ∆V between σ4 and σ
into a product of the volume ∆v of a a small compact region in σ4, times the variation ∆y
of the extra dimensional coordinates ya. Therefore, it sufficient to specify only the limit
operation ∆ya → 0 and the functional derivative (4.2) for density functions with compact
support on the brane-world, with respect to each extra dimension can be simplified to
= lim
∆ya→0
Ψ(σ′)−Ψ(σ)
Repeating for all extra dimensions, we find that the Tomonaga-Schwinger equation (4.1)
can be extended the brane-world, as a system of N partial equations, one for each extra
dimension
= ĤaΨa, a = 5..D (4.3)
which gives to Ĥa the interpretation of the extra dimensional translational operator.
The final quantum state is given by the superposition of the N separates states Ψa as
BaΨa. The state functional density Ψ represents the quantum fluctuation of the
brane-world sub manifold in the bulk at the (Tev) energy scale, subjected to quantum
uncertainties and a state probability given by
||Ψ||2 =
Ψ†Ψδyδv
– 9 –
An observer confined to the brane-world may evaluate the quantum expectation values of
the brane-world metric and the extrinsic curvature are given by
< Ψ|ĝµν |Ψ >=
Ψ†ĝµνΨδyδv, < Ψ|k̂µν |Ψ >=
Ψ†k̂µνΨδyδv
Since the classical kµνa is the derivative of the metric gµν with respect to ya, we may set
boundary conditions on these quantities at the initial brane-world ya = 0 to determine the
final solution.
5. Overview and Perspectives
We have shown that the Einstein-Hilbert principle applied to the bulk geometry plus the
differentiable embedding conditions are sufficient to determine the classical and quantum
structures of the brane-world in D-dimensions. In particular, it was shown that Nash’s
theorem makes it possible to generate any embedded sub manifold by a continuous se-
quence of infinitesimal perturbations of an arbitrarily given embedded geometry along the
extra dimensions. Using the classical perturbative embeddings, and the basic principles of
the brane-world theory we have obtained a canonical structure very much like the ADM
canonical formulation of general relativity, with the exception that the Hamiltonians do
not vanish.
The definition of the functional derivatives was improved with respect to four-dimensional
field theory, by using the previously defined perturbative embedding structure of the brane-
world. The quantization of the brane-world was described the Tomonaga-Schwinger equa-
tion defined for brane-world sub manifolds, calculated for each extra dimension. Actually,
as a result of the the the classical perturbation theory, the Tomonaga-Schwinger equation
becomes exactly integrable.
In view of current astrophysical observations and the near future high energy exper-
iments, there are some applications of the quantum brane-world theory to be detailed in
subsequent papers:
(a) Brane-world Cosmology
Brane-world cosmology offers a possible explanation to the accelerated expansion of the
universe, resulting from the modification of the Friedman’s equation by the presence of the
extrinsic curvature included in the tensor Qµν given by (2.21) [10]. The presence of this
tensor has the meaning that the brane-world vacuum is more complex than the vacuum in
general relativity. In fact, for a constant curvature bulk with curvature Λ∗, after eliminating
redundant terms, the gravi-tensor vacuum equation becomes Rµν−1/2Rgµν−Qµν+Λ∗gµν =
Therefore, the vacuum energy density < ρv > resulting from gravitationally coupled
fields must be revaluated, including the extrinsic curvature component. This suggests that
in some epochs, say at the early inflationary period, the extrinsic curvature may contribute
to the vacuum energy, differently from other periods.
The particular case where we have only one extra dimensions (D = 5), has some limita-
tions with respect to the differentiable embedding. However, some cosmological models like
– 10 –
the FRW, deSitter and anti deSitter solutions of Einstein’s equations in four dimensions
can be embedded in five dimensional bulks without restrictions, in accordance with the
perturbative embedding equations previously shown. Consequently, in such brane-world
cosmological models the conditions required for a proper definition of the functional deriva-
tives in the Tomonaga-Schwinger equation are well established, and the equation can can
be integrated without difficulty.
(b) Laboratory production of mini black holes:
Brane-world gravity predict the generation of short lived mini black holes produced at the
Tev energy in the laboratory, resulting from proton-proton collisions [34]. However, using
semi-classical quantum gravity in four dimensions, we have learned that quantum unitarity
does not necessarily hold true during the black hole evaporation. On the other hand, using
Euclidean path integral, it was shown that the unitarity can be restored with the aid of
the ADS/CFT correspondence in the framework of AdS5 × S5 string theory [35].
Since the generation of mini black holes are possible only in the brane-world context,
the whole process includes the original Minkowski’s space-time where the experiment is
devised. Soon after the collision, the space-time must be deformable into a Schwarzschild
or a Reissner-Nordstrom black hole. Finally, after a short period of evaporation the space-
time may be back to Minkowski’s configuration or else leaves a curved remnant. The
description of such process can start with the classical perturbations in accordance with
Nash’s embedded geometries, but the unitarity is has to be decided at the quantum level.
In this respect we notice that both Schwarzschild and Reissner-Nordstrom black holes are
well defined submanifolds embedded in a six-dimensional flat bulk with signature (4, 2).
In this case, the bulk isometry group SO(4, 2) is isomorphic to the conformal group in
Minkowski’s space-time, compatible with the ADS/CFT correspondence adapted to the
brane-world [36]. Therefore, the quantum unitarity implicitly assumed in the Tomonaga-
Schwinger equation, must be consistent with the black hole evaporation theorems in six
dimensions.
(c) Quantum Four-manifold Theory:
The above description of quantum theory of the brane-world is based almost entirely on
the general theory of differentiable sub manifolds. This suggests a quantum theory of
four-dimensional sub manifolds. It starts with the classical perturbations of embedded
geometries, but ends with a quantum version of the embedding theorem, including the
fluctuations of the embedding as described by the Tomonaga-Schwinger equation. This
quantum theory of submanifolds would be particularly interesting when the bulk has di-
mensions greater than five, where the third fundamental form behaves similarly to a gauge
field with respect to the extra dimensional group of isometries. The identification of the
third fundamental form as a gauge field with the symmetry of the extra dimensions plying
the role of the gauge group is old, but it was never taken seriously [37].
One frequent criticism to string theory is that it depends on a pre-existing background
space-time with 10 (or 11) dimensions, acting as the host space for all possible dynamics
[38, 39]. On the other hand, loop quantum gravity does not require such background, but
it depends on a previously existing spin network structure [40]. Quantum brane-world
– 11 –
gravity does not have a background space in the same sense of string theory because the
bulk is the primary dynamical object. The Einstein-Hilbert principle applied to the bulk
geometry provides all dynamics of the brane-world, without requiring any new algebraic
structure besides the theory of differentiable manifolds, where our basic notions of space,
topology and analysis begin and make sense.
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