Construction of Bulk Fields with Gauge Redundancy

0.32 MB·English

by  Idse Heemskerk

Construction of Bulk Fields with Gauge Redundancy
Idse Heemskerk1
Department of Physics
University of California
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Santa Barbara, CA 93106
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Abstract
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We extend the construction of field operators in AdS as smeared single trace oper-
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h ators in the boundary CFT to gauge fields and gravity. Bulk field operators in a fixed
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- gauge can be thought of as non-local gauge invariant observables. Non-local commu-
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e tators result from the Gauss’ law constraint, which for gravity implies a perturbative
h notion of holography. We work out these commutators in a generalized Coulomb gauge
[
and obtain leading order smearing functions in radial gauge.

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[email protected]
1 Introduction
In the supergravity limit of AdS/CFT, a large N, strong coupling gauge theory is dual to a
theory of gravity which is local in the sense that it is described by a local Lagrange density.
The latter implies the canonical notion of locality that observables with spacelike separated
support commute, so that no measurement can influence another measurement outside its
future lightcone. Understanding bulk locality from the perspective of the lower dimensional
boundary gauge theory is a challenge [5, 6]. A canonical approach is the 1/N perturbative
construction of bulk field operators from smeared single trace boundary operators [1, 2, 3, 4].
The only case that has been worked out so far is the scalar field, for which one can find
smearing functions K(n) such that the following operator identity holds
(cid:90)
φ(y) = ddx K(0)(y|x )O(x ) (1.1)
1 1 1
(cid:90)
g (cid:112)
+ ddx ddx −g(cid:48)K(1)(y|x ,x )O(x )O(x )+O(1/N2).
1 2 1 2 1 2
N
In this paper we work out the cases of the gauge field and gravitation field. Technically,
the construction is now complicated by the need to fix a gauge and solve the constraints
before the field operators can be expressed in terms of gauge invariant boundary currents.
Conceptually, one may wonder if the constructed field operators have physical meaning and
if in the gravitational theory one can talk about locality in the sense of local commuting
observables, given the mantra that there are no local diffeomorphism invariant observables.
We begin by reviewing that non-local gauge invariant operators can be traded for local
operators in a fixed gauge, but that these operators can have local commutators only up to
the non-locality required by the constraint. We then explicitly carry out the canonical quan-
tization of the Maxwell field in AdS, in an appropriate generalization of Coulomb gauge. We
work out the non-local commutators due to Gauss’ law in detail and from mode expansions
we obtain leading order smearing functions in Coulomb and radial gauge. For the gravita-
tional field we work out the Gauss’ law commutators in transverse gauge, which in this case
implies a perturbative notion of holography. Finally we obtain the leading order smearing
function in Pfefferman-Graham gauge. In the discussion we comment on future directions.
2 Local observables and gauge invariance
A charged scalar field field φ is not gauge invariant. However, if there is a place where
the potential is fixed we can contruct a gauge invariant operator by connecting the field to
this place with a Wilson line. For example, consider a half space that has a flat boundary
on which the gauge field A vanishes. Let xµ be the coordinates on the boundary and z a
µ
1
Figure 1: A Wilson line along C connects a charged scalar to the boundary. The gauge
1
invariant integral of A along C reduces to the scalar potential A in radial gauge.
2 0
“radial coordinate” away from the boundary (see fig. 1). Then one gauge invariant non-local
operator is
(cid:18) (cid:90) (cid:19) (cid:18) (cid:90) z (cid:19)
e(z,x) ≡ φ(z,x)exp i A = φ(z,x)exp i A (z(cid:48),x)dz(cid:48) . (2.2)
z
C1 0
Another gauge invariant operator that we could construct is the Wilson loop in figure 1
1 (cid:73) (cid:90) z
V(z,x) ≡ lim A = E (z,x)dz, (2.3)
z
δ→0 δ
C2 0
which measures the potential difference between us and the boundary. It is clear that
[e(z,t,xi),V(z(cid:48),t,xi)] (cid:54)= 0,
even though (z,t,xi) and (z(cid:48),t,xi) are spacelike to each other because these non-local oper-
ators will overlap. However, these operators are precisely φ(z,x) and A (z,x) if we choose
0
radial gauge A = 0. From the gauge fixed perspective the non-locality in the commutator
z
comes from having to impose the Gauss’ law constraint. As an illustration, consider flat
space quantum electrodynamics in Coulomb gauge. The gauge field is fixed to go to zero at
spacelike infinity and we can then define non-local gauge invariant operators [8] that have
“Wilson lines” connecting the charged field to infinity or equivalently work in the gauge
fixed theory [9]. Gauge fixing ∇·A = 0 yields the constraint −∇2A = eψ¯γµψ, which after
0
canonical quantization of ψ leads to the non-local equal time commutator
e
[A (x,t),ψ(x(cid:48),t)] = − ψ(x(cid:48),t), (2.4)
0 4π|x−x(cid:48)|
and consequently a non-local commutator between the electric field and the electron.
For gravity, the situation is similar. It is clear that it is useful to talk about localized
events even in gravitational theories, but that position should be defined in a coordinate
independent, relative way. If space had a boundary, that boundary would be a convenient
2
reference point. So we consider again half space with a flat boundary, where now the metric
is held fixed (we have in mind asymptotically Poincare AdS with a cut-off2). Then we can
define a scalar field φ(z,x) in that space with
(cid:114)
(cid:90)
dyM dyN
lnz = −g (yL(s)) ds, yM = (z,xµ) = (z,t,xi) = (t,ua), (2.5)
MN
ds ds
where z measures the log of the geodesic distance along a geodesic that starts at x normal
to the boundary. This diffeomorphism invariant field operator is a non-local functional of
g . However, this is exactly the same thing as imposing the gauge fixing condition that
MN
the metric take the form
dz2
ds2 = g dyMdyN = +γ (z,x)dxµdxν, limz2γ (z,x) = η , (2.6)
MN z2 µν z→0 µν µν
analogous to Gaussian normal coordinates [7] and then considering the diffeomorphism vari-
ant field φ(z,x) in these coordinates. One can also define gauge invariant operators that
reduce to particular metric components in some gauge, namely the length of curves with two
ends at the boundary. Just like before those will have non-local commutators due to the
constraint, which we will work out in section 4. An important difference between gravity
and gauge theory is that Wilson loops in the bulk, which in the gauge theory are observ-
ables of compact support that commute with the electric field at infinity, have no gravitional
analogue.
To summarize: for all intents and purposes we can talk about local fields after gauge
fixing as physical observables, but these will have commutators that are local only to the
extent allowed by the constraint. We will see this explicitly for the U(1) gauge field and
gravity in AdS.
3 Maxwell field in AdS
The Maxwell field coupled to a charged scalar in AdS has action
d+1
(cid:90) √ (cid:20)1 (cid:21)
S = − dd+1y −g F FMN +(D φ)∗DMφ+m2φ∗φ , (3.7)
MN M
4
and the equations of motion for the gauge field are
1 (cid:16) √ (cid:17)
√ ∂ gMM(cid:48)gNN(cid:48) −g(∂ A −∂ A ) =JN, (3.8)
M M(cid:48) N(cid:48) N(cid:48) M(cid:48)
−g
JN ≡ iq(φ∗DNφ−DNφ∗φ), (3.9)
2There is no essential difference if we take the cut-off to zero so that boundary becomes asymptotic
boundary if we make the appropriate subtractions.
3
where D denotes the AdS gauge covariant derivative D φ = (∂ −iqA )φ and ∇ the
M M M M M
covariant derivative with respect to the metric on the Poincar´e patch, given by (2.6) with
h = z−2η . For later use we define the electric field, whose flux is conserved by Gauss’
µν µν
law (see (2.5) for index conventions)
1 √
Ea ≡ gabg00F , √ ∂ ( −gEa) = J0. (3.10)
0b a
−g
The goal is to fix a gauge, solve the constraint, and canonically quantize the fields. We
can then calculate the commutator analogous to (2.4) and obtain the leading order smearing
function. Radial gauge A = 0 and Lorentz gauge ∇ AM = 0 are usually convenient gauges
z M
because they respectively preserve the full AdS and Poincar´e symmetry of the metric, and
we will obtain the smearing function in radial gauge at the end of this section. However,
these gauge conditions make it harder to see that Gauss’ law leads to non-local commutators.
We therefore generalize the Coulomb gauge condition, which in flat space is ∂iA = 0, and
i
eliminatestheterminvolvingthespatialcomponentsfromtheconstraint. Thegeneralization
of this condition to AdS is not ∇aA = 0. Instead we see from (3.8) that we can eliminate
a
the spatial components from the constraint by imposing
√
(cid:0) (cid:1)
−∂ gabg00 −gA ) = ∂ (z3−dA ) = 0. (3.11)
a b a a
This reduces the constraint to
zd+1∂ (z−d−1Ea) = −zd+1∂ (z3−d∂ A )−z4∂2A = J0, (3.12)
a z z 0 i 0
and the remaining equations to
(cid:0) (cid:1)
∂ z3−d(∂ A −∂ A ) −z3−d∂2A =P (z−d−1Jc), (3.13)
a a d d a 0 d dc
where ∂2 = δij∂ ∂ and P is a generalized transverse projector
i i j
P ≡ δ −z3−d∂ (∂ z3−d∂ )−1∂ . (3.14)
ad ad a c c d
The spatial equations can be decoupled using the gauge condition (3.11), yielding
∂2A +∂ (zd−3∂ (z3−dA )) = zd−3P (z−d−1Jc), (3.15)
z z z z zc
∂2A +zd−3∂ (z3−d∂ A ) = zd−3P (z−d−1Jc), (3.16)
j z z i ic
where ∂2 = ηµν∂ ∂ . The gauge condition and resulting equations of motion are not AdS
µ ν d+1
covariant, nor are they covariant under the Euclidean AdS symmetries of a constant times-
d
lice. Green’s functions for these equations will therefore not be functions of a single invariant
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distance. With boundary conditions that we will discuss in the shortly, the constraint de-
termines A uniquely in terms of the charge density. We can now canonically quantize the
0
remaining physical degrees of freedom subject to the gauge fixing condition,
[Aˆ (u),Πˆ (u(cid:48))] = iδtr(u−u(cid:48)), Π =z3−d∂ A , (3.17)
a b ab a 0 a
√
[φˆ(u(cid:48),t),πˆ(u,t)] = iδd(u−u(cid:48)), π =− −g(D0φ)∗, (3.18)
where the transverse delta generalizes the one in flatspace [9],
δtr(u−u(cid:48)) = zd−3P z3−dδd(u−u(cid:48)). (3.19)
ab ab
The commutator between the current operator and the scalar field is then
q
[Jˆ0(u,t),φˆ(u(cid:48),t)] = √ φˆ(u)δd(u−u(cid:48)). (3.20)
−g
Gauge/gravity duality maps these operators in the bulk of AdS to operators in a boundary
theory through the extrapolate dictionary [10]
√
lim −gFˆzµ(z,x) = ˆjµ(x), limz∆+φˆ(z,x) = O(x), (3.21)
z→0 z→0
The explicit solution for A in terms of J is slightly different for the cases d = 1 and d > 1.
0 0
In d = 1 the Maxwell equations are pure constraint and the solutions are simple. We will
therefore solve it first.
3.1 AdS
2
In AdS , the Coulomb gauge condition is ∂ (z2A ) = 0. We can consistently impose the
2 z z
slightly stronger condition A = 0 (radial gauge), after which the equations take the form
z
z2∂ (z2∂ A ) = −J0, z4∂ ∂ A = Jz, (3.22)
z z 0 0 z 0
with the homogeneous solution
a
A = +b(t). (3.23)
0
z
The boundary conditions in d = 1 are less intuitive than in higher dimensions. Spatial
infinity consists of two disconnected points so flux can come in from z = ∞. We demand
that there is not so z2∂ A | = a = 0. On the other side, at z = 0, we fix b = 0 so that
z 0 z=∞
the unique homogeneous solution A = 0. Solving the Gauss’ law constraint (3.22) for a
0
point charge a point charge
J0(z(cid:48)) = δ(z(cid:48)(cid:48) −z(cid:48))z(cid:48)2
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with these boundary conditions then yields the Coulomb potential
(cid:18) (cid:19)
1 1
Φ(z,z(cid:48)) = Θ(z −z(cid:48)) − . (3.24)
z z(cid:48)
The solution for an arbitrary localized charge distribution is
(cid:90) dz(cid:48)
A (z) = Φ(z,z(cid:48))J0(z(cid:48),t). (3.25)
0 z(cid:48)2
The second equation (3.22) is implied by current conservation. From (3.10), (3.20) we then
find the non-local commutator
[Eˆ(z),φˆ(z(cid:48))] = qz2Θ(z −z(cid:48))φˆ(z(cid:48)), (3.26)
between the electric field and the charged scalar. As explained in the previous section one
way to understand this is that φ(z(cid:48)) is a secretly non-local gauge invariant operator that
creates both the charged scalar and its (longitudinal) electric field. Alternatively we simply
notice that (3.26) is required by charge conservation. Gauss’ law relates the integral of
electric flux through the boundary to the total charge in the bulk so that in d = 1 the
electric field is proportional to the total charge operator
Qˆ = limz−2Ez. (3.27)
bulk
z→0
ˆ ˆ
Theextrapolatedictionary(3.21)nowjustsaysQ = Q , aswemighthaveexpected.
bulk boundary
3.2 AdS
d+1>2
The general case of AdS , while conceptually the same, is technically useful as a warmup
d+1
for gravity. Fourier transforming along the spatial boundary directions, the homogeneous
solution to the constraint (3.12) is
A (k ,z) = a (k )I(k,z)+b (k )K(k,z), k ≡ (δ kikj)1/2, (3.28)
0 i 0 i 0 i ij
where the two modes
Γ(d/2) (cid:18)2z(cid:19)d/2−1 2(kz/2)d/2−1
I(k,z) = I (kz), K(k,z) = K (kz), (3.29)
d−2 k d/2−1 Γ(d −1) d/2−1
2
are normalized such that for small z we have I(kz) ∼ zd−2/(d − 2) and K(kz) ∼ 1. To
obtain the Green’s function we solve
zd+1∂ (z3−d∂ Φ(k;z,z(cid:48)))−z4k2Φ(k;z,z(cid:48)) = −zd+1δ(z −z(cid:48)), (3.30)
z z
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with the boundary condition that Φ = 0 at z = 0 and z = ∞. The unique solution is
Φ(k;z,z(cid:48)) = K(k,z(cid:48))I(k,z)Θ(z(cid:48) −z)+K(k,z)I(k,z(cid:48))Θ(z −z(cid:48)), (3.31)
Fourier transforming back and integrating with the current operator to get the general so-
lution as before, we find
[Aˆ (t,u),φˆ(t,u(cid:48))] = qΦ(u,u(cid:48))φˆ(t,u(cid:48)). (3.32)
0
Taking the electric field to the boundary and applying the dictionary (3.21), we get
(cid:90) dd−1k
[ˆj0(x),φˆ(x(cid:48),z(cid:48))] = qφˆ(t,u(cid:48)) eikjxjK(k,z(cid:48)). (3.33)
(2π)d−1
We have a single trace local operator in the CFT that does not commute with the bulk scalar
field. Again this can be understood as a consequence of charge conservation and (3.33) can
be integrated over a spatial slice through the boundary to yield the commutator of the total
charge with the field.
Withthebulkdimensiongreaterthantwotherearephotonsandfromthemodeexpansion
we can get an expression for the field operator in terms of the boundary current, analogous
to (1.1). For timelike boundary momenta k2 = ηµνk k < 0 the mode solutions to (3.15) are
µ ν
A = a (k )F(k,z)+b (k )Y(k,z), A = a (k )F (k,z)+b (k )Y (k,z), (3.34)
i i µ i µ z z µ z z µ z
where we have defined
Γ(d) (cid:18)2z(cid:19)d2−1 d (cid:18)2z(cid:19)d2−1
F(k,z) = 2 J (|k|z), F (k,z) = Γ( −1) J (|k|z). (3.35)
d−2 |k| d−1 z 2 |k| d−2
2 2
and similarly Y and Y with the substitution J → Y . The normalization is such that near
z ν ν
the bounary we have
∂ F ∼ F ∼ zd−3.
z z
For the solutions Y normalization does not matter as we now impose the standard boundary
a
condition b = 0. From the dictionary (3.21) for timelike boundary momentum we then find
a
j (k ) = a (k )−k a (k ). (3.36)
i µ i µ i z µ
Substituting the solution into the gauge condition we get k a (k ) = 0: the spatial compo-
j j µ
nents along the boundary are transverse with respect to the spatial momentum along the
boundary. Therefore we identify k a with the longitudinal part of the boundary current and
i z
a with the transverse part
i
(cid:18) (cid:19)
k k k j (k )
i j i i µ
a (k ) = δ − j (k ), a = . (3.37)
i µ ij k2 j µ z k2
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When we plug this into the mode expansion, we obtain after inverse Fourier transforming
j (k ) and swapping the integrals
i µ
(cid:90)
Aˆ (z,x) = ddx(cid:48)K(0)(z;x−x(cid:48))ˆj (x(cid:48)), (3.38)
a aj j
where the leading order smearing functions are
(cid:90) ddk (cid:18) k k (cid:19)
K(0)(z,x−x(cid:48)) = Re δ − i j F(k,z)eikµ(xµ−x(cid:48)µ) (3.39)
ij (2π)d ij k2
k0>k
(cid:90) ddk k
K(0)(z,x−x(cid:48)) = Re j F (k,z)eikµ(xµ−x(cid:48)µ) (3.40)
zj (2π)dk2 z
k0>k
The restriction of the integration range to timelike momenta is because from Lorentz invari-
ance ˆjµ(k ) = 0 for spacelike momenta.
µ
We can also easily obtain the smearing function in radial gauge A = 0, where the
z
Maxwell equations take the form
zd+1∂ (z3−d∂ Aµ)+z4∂ρ(∂ Aµ −∂µA ) = Jµ, −z4∂ ∂ Aµ = Jz (3.41)
z z ρ ρ z µ
If we impose the supplementary gauge condition that ∂ Aµ = 0 on the boundary3, and solve
µ
the sourceless equations of motion, we find the general solution
Aµ(z,k) = a (k )F(k,z)+b (k )Y(k,z), kµa = kµb = 0. (3.42)
µ µ µ µ µ µ
Setting b = 0 we find from the dictionary that a = j so that
µ µ µ
(cid:90)
Aˆ (z,x) = ddx(cid:48)K(0)(z,x−x(cid:48))ˆj (x(cid:48)), (3.43)
µ rad µ
with the simple smearing function
(cid:90) ddk
K(0)(z,x−x(cid:48)) = Re F(k,z)eikµ(xµ−x(cid:48)µ). (3.44)
rad (2π)d
k0>k
We could have obtained the same smearing functions by solving for the Green’s functions
forthespatialequationsandapplyingGreen’stheorem. Forfindingtheresultsinthissection
that method is less direct but in general it makes it possible to find nicer smearing functions.
3Alternatively, andfromsomeperspectivemorenaturally, wecanfixthenon-normalizablemodewithout
imposing transversality on the boundary and transversality for the normalizable mode will be implied by
the equation of motion.
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4 Linearized Einstein gravity
Although there is significantly more gauge redundancy in the case of gravity, perturbatively
the structure is similar to electromagnetism. We will consider gravity coupled to a free
massless scalar
(cid:90) (cid:90)
1 1
S = dd+1y(R−2Λ)− dd+1ygMN∂ φ∂ φ (4.45)
M N
2κ 2
with Λ = −d(d−1)/2 and work to linearized order in metric perturbations around the AdS
background
g = g¯ +h .
MN MN MN
Since gravity is trivial in d = 1 we will immediately work in arbitrary dimension d > 1. The
expansion of the Einstein-Hilbert action with cosmological constant to second order in h
MN
is given in [11], and from it we obtain the linearized Einstein equations on AdS, which can
be written as
2κT = ∇¯ ∇¯ h˜K +∇ ∇Kh˜ −∇¯2h˜ −∇¯ ∇¯Lh˜K g¯ (4.46)
MN K (M N) (M N)K MN K L MN
˜ ˜
+hg +(d−1)h .
MN MN
Here h˜ = h − 1hg¯ , known as the trace reversed metric in four dimensions. The
MN MN 2 MN
stress tensor for the scalar field is given by
1
T = ∇ φ∇ φ− (∇φ)2g . (4.47)
MN M N MN
2
and using the canonical commutators we find
[T (t,u),φ(t,u(cid:48))] = −izd−1δd(u−u(cid:48))∂(cid:48) φ(u(cid:48)). (4.48)
0M M
4.1 Gauss’ law and holography
To solve the gravitational Gauss’ law and see that it leads to a non-local commutator it is
again easiest to work in transverse gauge. While this simplifies Gauss’ law, which in this
context we will refer to as the Hamiltonian constraint, the other equations are hard to solve
and the AdS/CFT dictionary is messy. Therefore, we will not work out the full canonical
quantization and smearing functions in this gauge but only discuss the gravitational Gauss’
law. To generalize the flat space transverse gauge we first decompose the metric as follows
h = 2Φg¯ , h = w n¯ , h = 2s −2Ψg¯ , s g¯ab = 0, (4.49)
00 00 0a a 0 ab ab ab ab
with n¯M the unit timelike normal in the background. The Hamiltonian constraint (the 00
component of (4.46)) now takes the form
κT g¯00 = −(d−1)∇¯a∇¯ Ψ−∇¯ ∇¯ sab +(d−1)dΨ, (4.50)
00 a a b
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