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Moretii V,,1 Pinamonti N,2 ., ,
HOLOGRAPHY AND CONFORMAL SYMMETRY Nf-i i- 4CK HOLE HORIZONS
Department of Mathematics, Faculty of Science, University of Trento, Istituto Nazionale di Aita Matematica “F.Severi", unita locale di Trento & Istituto Nazionale di Fisica Nucleate, Gruppo Collegato di Trento, via Sommarive 14,1-38050
Povo (TN), Italy
Maldacena [4] conjectured that the quantum field theory in a, asymptotically AdS, d + 1 dimensional spacetime (the “bulk”) is in correspondence with a conformal theory in a d dimensional manifold (the (conformal) “boundary” at spacelike infinity). Notice that the d dimensional conformal group on the boundary acts as the asymptotic isometry group on the bulk. Afterwards, Witten [5] showed that that correspondence can be reset in terms of observables of the two theories. More recently Rehren [6.7] proved rigorously some holographic theorems concerning boundary and bulk observables in AdS background,
1. Introduction
In the last fifty years much work was done in order to understand the statistical origin of black-hole entropy. The Holographic principle, proposed for the first time by't Hooft and Susskind [1,2,3] is one of the most promising idea to deal with that problem. In few words the quantum theory responsible for the statistical black hole entropy should be suited on the event horizon, moreover it has to describe the events that take place in the spacetime. In some sense as a photograph describe a landscape. Starting from these ideas and using the machinery of string theory,
1 E-mail: [email protected]
2 E-mail: [email protected]
without using string theory. In the last, year we have shown that there is also a bulk-boundary
correspondence for QFT on Schwarzschild-like black holes [8,9]. Similar ideas, concerning algebraic QFT on spaeetimes with bifurcate Killing horizons and conformal symmetry, were presented by Guido, Longo, Roberts and Verch [10]. Also Schroer and Wiesbrock [11] have studied 'the relationship between horizons arid ambient QFT. Moreover, they use the term “hidden symmetry” in a sense similar as we do here and we done in [12]. Schroer [13] and Schroer and Fassarella [14] by means of the Ligfatfront formalism they presented holography for Minkowski spacetime.
In this letter, discarding the technical details and stressing some of their physical, implications, we want to summarize some of our results. The near horizon structure of every Schwarzschild-like spacetime is similar to the Cartesian product of a two dimensional Kindler spacetime and a sphere. The holographic properties of this spacetime are already exhibited discarding the sphere. That’s because in the first part of that letter we deal with two dimensional spaeetimes. We have shown recently [12] that the quantum field theory in a. two-dimensional Rindler spacetime presents a “'hidden” SL(2Ji) symmetry. S.£{2,1I) symmetry is the one dimensional conformal group. This, suggests that, as in ihe AdS Cd.se, Rindler quantum fields are in holographic relation with one dimensional conformal fields, But here rhe situation is a idde bit different, in fact, even if the quantum theory is Invariant under S !„(?■. R), the symmetry does not descend from the ssometries of the spaeedme, (hat is because we say dial the symmetry is “hidden".
We argue that these symmetry acquires a geometrical meaning in the, holographic-dual conformal theory. We search for the dual conformal theory on the horizon. In fed on the horizon the metric is degenerate and SL(2,3?) can be seen as a subgroup of the horizon diffeomorphism. Moreover, compactifiing the horizon, it is possible to extend the symmetry generated by the Lie algebra sl(2,W) to the
whole Virasoro algebra, with a central charge equal to one.
In the last section we consider the four dimensional case, we show that the holographic relation holds also in this case, even if, at least in general, the extension of the symmetry to the Virasoro algebra does not take place.
2 &l(2,K) symmetry from energy spectrum: a Hidden and a Manifest case
We start our discussion with an abstract problem: Consider a quantum Hamiltonian H whose spectrum goes from zero to infinity, (for simplicity without degeneration). The corresponding Hilbert space is
clearly H:=I?(M.+,dE). In [12] we have shown that ft is irreducible under unitary representation of SL(2,R). The corresponding generators, enjoying the sl(2M) commutation relation, are the selfadjoint extensions of i%, iD and iC, defined below.
H:=E, D:=~i\— + E~
12 dE
C
, , k — —
._A/rA+_____2
dE dE E
(1)
k can be fixed arbitrarily in {1/2,1,3/2,...}. In the Heisenberg representation of (1), the expectation value of H,D,C are constant of motion. See [8] for details. Notice that, up to now, only H has a physical meaning
as the Hamiltonian, (the generator of time translation) of the system, whereas the physical (geometrical) meaning of D and C has to be discussed in every particular case. We say that the found SL(2,K) symmetry is manifest when D and C have geometrical meaning on the contrary we say that the symmetry is hidden. Notice that, since the action of SL(2,K) is closed in the one particle Hilbert space H, the §1,(2,M) symmetry is inherited by the Pock space ,J(H). In the following we shall analyze some particular case.
a) free particles in Rindler space time
We remind here that every Schwaresehdddike metric ds$ = -A(r)dl1 + A"1(r)dr'' + rdQr, where Q are the angular coordinates, reduces to the metric of a two-dimensional Rindler R space time near the bifurcate horizon at r = rh. ds^ =-Kly2dt1 +dy2 with with A'(r/I) = 2k, and Ky2 = 2{r-rh), the angular coordinates are dropped in first, approximation. Consider the free Klein-Gordon equation,
■ K2(y3>,y3), - y2mz)§ = 0 for a free scalar field (js The one particle Hilbert space of the free quantum particle arises by decomposing any real solution \|i of the Klein-Gordon equation in t -stationary modes as follows.
VO.y) = f <*, (E)dE + c.c.
(2)
E g [0,-H>°) = Ii+ is an element of the spectrum of the Rindler Hamiltonian H associated with 3, evolution. If m = 0 there are two values of a, corresponding to
ingoing and outgoing modes, <t>'
(in)t(out)
whose
expression are e
~iE(t±ln(Ky)l k)
/■74ftE . On the other hand if m < 0 there is a unique mode #“ = <t>B whose
— 11.0 —
expression is sinh(TtE / k) / V2T?KEe~m> KiE/K (my).
Notice that in the massive case there is no energy degeneration and the ne-particle Hilbert space H is isomorphic to L2(R+,dE). In the other case (m = 0), twofold degeneracy implies that
H = L2 (M+, dE) ® L2 (R+, dE). Quantum field operators, acting in the symmetrized Fock space &(H) and referred to the Rindler vacuum j 0) — that is 10)(„010)OTr if m = 0 — read
W. >') = f Z (t, y)aEa + (t, y)aldE, (69)
a
As usual, the causal propagator A satisfies [<j>(x),<j>(x')] = —iA(x,x).
In this cases, in the sense discussed above, there is a hidden SL(2, R) symmetry. Indeed, at least for the massive case, D and C have no local action on the Rindler wedge R.
b) Free particles in AdS, space time.
We analyze here a free massive particle moving in the portion of AdS2, delimited by the (non-bifurcate)
Killing horizon. This particular chart of AdS1 describe a near horizon approximation of an extremal Reisner-Nordstrum black hole. We write the AdS2 metric in
the form dsl = -x2 //dt" + //x1dxl, where / is related with the cosmological constant. As in the Rindler case, consider the free Klein-Gordon field <j> satisfying the motion equation
—/23,2<t> + (x2 / /2dxx2d x - x'm2 )<j> = 0. The decomposition in stationary modes of the real solution of the Klein-Gordon field y reads
¥(f, >’) = (t, x)\pf+ (E)dE + c.c. (70)
generated by the following vector fields basis
. a , a 3 h\=—, d:=t--—x—,
31 dt dx
c:-
2 1‘ t +—
3 „ 3 -— 2 tx—.
dt dx
(72)
They satisfy the commutation relation of the sl(2,R) Lie algebra. There is an isomorphism between the generators of isometrics and the generators of a particular unitary representation of SL(2,W) (1). This isomorphism singles out a particular 51(2,1) representation and a particular value of k > 0 in C i (k -1/2)1 =1/4 + m2/1, Notice that in the classless case k-l. The SI,(2,R) is manifest.
3, Quantum fields on the horizon
a) The future horizon
The hidden 51(2, R) symmetry on Rindler particle is unsatisfactory, we have to deeper analyze the nature, of that symmetry. To do that we want to show what happens exactly on fh“ symmetry takes a gee Consider the Rindler i in a Minkowski spaceiime. i r. coordinates u = t - log( icy) / K,
(where M.velS ) that cover the Rind! are
respectively well defined on the past i tore
horizon F, (see figure). First of all, we consider the theory restricted on the future horizon F and we shall show that it is a well defined quantum theory. Take the wavefunction in (2) and consider the limit on the future horizon u +oo. That is equivalent to restrict the wavefunction on the horizon when it is considered as a wavefunction in Minkowski spacetime, obtaining
where <3>£ (?, x) := Jv (~/2E / x)^/2 l(2x) , and
v = yl/4 + m2 /i1 . As in the Rindler case, there is a single mode for every value of E in M+ . Moreover the modes are complete, then the one-particle Hilbert
space HA is isomorphic to L2 (R+, dE) too. In the symmetrized Fock space if(H), equipped with the vacuum 10)A, the quantum field operators, read
§(t,x) = jT <&E(t,x)bE + <$?E{t,x)b\dE. (71)
Notice that Hk is a unitary representation of SL(2, R) as above, but in this case, studying the isometrics of AdS2 is possible to give geometrical meaning at the generators D and C too. The isometries of AdS2 are
Y(i>)= j-
-i£v
■eip"AE)f+(E)dE + c.c.
(73)
is a pure phase (see [8] for details). In coordinate lie S., the restriction of \|/ to P is similar with the v replaced for u and pm_K(E) replaced by ~p..„(£) • If m = 0 the restrictions to F and P read respectively
-iEv
w(v) ~ j*—(E) + c.c.,
V4ilE
~i£v
W) = H=fr)№) + c^
(74)
Discarding the phase it is possible to consider the following real “field on the future Horizon”:
9(v)=l
‘i
-iEv
V4Tt£
<p+(E)dE +
(75)
^4%E
f ,(E)dE
as the basic object in defining a quantum field theory on the future event horizon. The same can be done for the past event horizon. The one-particle Hilbert space ?{f is defined as the space generated by positive frequency parts tj/+ (E) and turns out to be isomorphic to lJ(Bl,dE) once again. The field operator reads, on the symmetrized Fock space t¥{%) with vacuum j 0)F,
a/4k£
Or + -
dE.
(76)
%E
The causal propagator AF is defined by imposing
l#(v),{Kv')] = ~£AF(y»v') and it takes the form (1./ 4)sign(v - v ). In spite of the absence of any motion equation the essential features of free quantum field theory are preserved by that definition as proven ia [8].
+
l/Evn(v)
(77)
is well defined and diffeoraorphism invariant. Ia a suitable domain the map
tl(v) h> AK (ri) = i £ sign(v - v')ti(v') = \|/„ (v) defines
a one-to-one correspondence between exact one-forms and horizon wavefunctions of the form (2) and il = 2d\yn. Finally, similarly to usual quantum field theory [16], it holds [$F Cn). If Ol')] = ~dAf (11, rf) =
’The last term define a diffeomorphism-invariant
symplectic form on horizon wavefunctions.
Concerning locality, notice that |(T|) commute with
<J>(if) if supports of T| and tj' are contained in disjoint
segments.
b) The compactified horizon.
As pointed out above on F there is no preferred measure, then we can consider the compactified case S1 =Fu{«}. Everything described above, can be translated in this case. Parametrize S1 by 0€ [—tc, tc] . The circle wavefunction are
«, -me
p(e) = ^]^==p(n) + c.c. (78)
The complex combination of the positive frequency pail of p(0) form a Hilbert space isomoiphic to
/2 (€). Moreover in [9] we have shown that Ti^ turns to be isomorphic to Ti? . The isomorphism acts in this way on the wavefunctions: p(0) = ty(v(0)), where v(0) = ptan8/2, with p a positive constant. Moreover exists a basis {ZJE)}1 on E(M,dE) that realizes the isomorphism: P+(«) = (Zn (£),vj/+ (£)) and
yt(£;-T'J, tZH(E"}p+(n). Notice that positive frequencies on the horizon F correspond to positive frequencies on the circle S1. In the Fock space of the circle 5?(W„;), with respect to the vacuum 10),,,, the quantum field operator reads
CO
' ' > -i . ( / V;
'„=i \/4n>i \i4nn The causal propagator is computed trough the relation !#„• (©).<!>,..» (ff‘)]'--fA„:(0,C)'*) and it take the simple form: (l/4)sign(Q~Q')-(Q-Q')/n. As before, to get local quantities, and doe to ill-definiteness of the metric, the quantum fields need to lie smeared by 1-forms.
a..
■J4m
+
where r| = df .
a„
’/Sitn
4. The SL(2,R) symmetry becomes manifest on the
compactified horizon
Consider a quantum field theory on F (or equivalently on P) mapped as explained above on S1. On Hilbert space 7i^, = E(R* ,dE) acts the tree operators H,D,C as defined on the right side in (1). As discussed above, the operators iH ,iC, ID generates a unitary representation {Ug}^SU2M) of SL(2,31). We
1 The explicit form of Zn{E) involves the Laguerre polynomials, see
[12] for details.
want to discuss here the “geometrical nature” of that representation. Consider the new basis of SL(2,R):
K:=~Uh+^
A p.
S:=
p H-
D.
(80)
As shown in [9], in the case when k = l in (1) the representation Ug has the following geometrical meaning: consider a wavefunction p(0), the state pf(n):= Ugp+(n), with g e SL(2,M), corresponds to
the wavefunction p* (0). Moreover p* (0), satisfy a geometric transformation: p*(0) = p (8i)
Where dg is a diffeomorphism of the circle S1, In
particular, the following relation, between the Lie algebra of the unitary representation of SL(2,1!) and the diffeomorphisms group, holds. iK 3e, iS o sin(0)3e, iD cos(0)39. (82)
The theory on the horizon F,
For completeness we analyze the geometrical nature of the geSL( 2,R) symmetry on F. On the
horizon Hilbert space li,, s l.}(WJ,dE) acts unitarily a
representation {£/?} of g e SL(2,R) generated by (1).
In the case of k = 1 the wavefunction \|/ (v)
associated with U reads:
CIV +
f (v) = \)/
b
V cv + d j ^{d
(83)
Consider its Lie algebra: Viec^S1) of its vector field. A algebraic basis the vector fields is made of the real part of the following smooth fields:
4 :=iV”e3r (85)
The element of this basis, equipped with the usual bracket satisfy the Virasoro commutation rules with vanishing central charge:
{£n,£a} = {n~m)£M. (86)
Moreover a particular Hermiticity condition is fulfilled, by means of the involution ok X (-4 -X : otCJ = £m.
The tree generator satisfy the sl(2,№)
commutation relation, moreover they are a linear combination of the generators of the SL(2,W) on the circle defined above (16)
— 39,
= sin03a
(87)
£] +£4 n-s
—I------L = COS Qoe,
2 i
then } are in relation with the tree particular
operators [K,S,D] generating a unitary representation of SL(2,11) on the Hilbert space 7f, . Now a question arises. Is it possible to extend the Si A 2, S.)
representation to the whole Diff'Alf)
Instead
a b"' vc d j
Notice that in the action of SL(2,R) is not exactly geometric due to the term -y(b/d). The added term disappears considering local 1-form dy instead of wavefunction \jr. In this case: iH <-» 3V, iD <-> vdv. (84)
The action of iC is not exactly geometric due to the term ~\|i(bld). Apart this term it corresponds to
corresponds to v23,.
5. Virasoro algebra
Above we have seen that the unitary SL(2,W) symmetry is manifest on the circle S‘ = Fu{<»} without a measure. But due to the degeneracy of the metric on F and then on §', we expect more symmetry. In fact SL(2,R) is only a little part of a greater group of symmetry: the group of
diffeomorphism preserving orientation S1).
of answering that question, we want tc ere that,
at least in sortie particular situations, the quantum sl(2,R) algebra can be extended to the whole Virasoro algebra. Moreover, a central charge arises through this extension. In the case when k = \ in (1), on the Fock
space ) > consider an=i4na„, a_n = -i\lnarn if
n > 0 and a0 = 0 written in terms of the creation and
annihilation operators. {an} satisfy the commutation
relation: [aR,am] = n?>n „J and a\ = a_„. {an} with
that commutation relation form the so called oscillator algebra [17]. The quantum field (13) take the simplest form
M0)=-
1
(88)
i-s/4k ~£ n
Define the operator as in [17]
Lm-=^Taln+ Z meZ> (89)
** n>mi2
where €IB is equal to one if m is even and 0 otherwise. Notice that on the Fock space 3'('fts,), since the
number of particles is finite, the sum in (23) are finite. Moreover it is possible to show that they are well defined quantum operators on 301$ 1) - In the
following we analyze some properties of [LJ . First of all we notice that they satisfy the Virasoro
commutation relation with central charge c equal to
one,
lln,LJ = (n~m)Ln+m+5FA.m^L
Then the Hermiticity condition LJ, = L holds. The spectrum of Lg is discrete and positive definite
a(L0) = {a^/2 + N}, Ne N.
Notice chat, if ao=0, only the action of is
closed in the one particle Hilbert space 7i's, . By means of a difficult proof, it is possible to show that the action of {Ln) on the fields #si (9) correspond to the action of {£„} on the point 0 of the circle S'.
6, Holography
Above we have shown that there are some particular unitary relations between some different quantum theories. In particular these unitary relations were used to show the existence of SL(2,X) “symmetry” also in some cases were it was not expected as for example on the Rindler free fields. We ■ i to show here that " r, r :• "alation ;n the Rindler R . ■ «■ ■ fields
d. on the line thou,.:., „11.... dtndler
■n F
A (f
A /. / ■' ■:■■■ ■■ ' ... -- ^oastim \
df — mtoly
For simplicity we shall discuss here only the case of Rindler free massive fields. As mentioned above there is a unitary map UF : $(?4) -» g(Hf) between the Rindler Fock lS(Tifl) space and the horizon Fock space S(Tif). UT maps the Rindler vacuum horizon vacuum
10) H 0)f ar>d Uf%fUr =|(/) for any smooth compactly supported function / used to smear the bulk field, r| = 2d(A(f) }F). (See figure.)
We want to show here the main idea of the above proposition. Details on the construction of are presented in [8]. The unitary action Uf has the
following geometrical meaning: Consider a local function f used to smear the Rindler field cj>R, then yf = A(/) is the associated wavefunction. Taking its
restriction on the horizon we obtain an horizon wavefunction as in (7) whose positive frequency part
reads epm,*(E>\\(f+(E). Then define a horizon
wavefunction q>f as in (9) with <p+ replaced by \jr/+.
The map \yf t-> q>f corresponds to the unitary operator
Um from H to Hg . Imposing U¥ j 0) =] 0>r, by
taking tensor products of UWH, this map extends to a
unitary map U\,: $(H) -» • Finally, by direct
inspection one finds that, if t| = 2dq>f, one also has
u;%myur=fcf).
As a consequence, one has the holographic relation: the invariance of vacuum expectation values,
f<o) i di,) ■■ • ■■ (ii,) I o>F = (o i #(/,) • • • |(/„) 1 o). (%)
There is an analogous relation between the Rindler free fV\Wc- and quantum operators ti-»
i1 l P . The massless ease is ,1 •
i • : one has to decompose I -
ingoing and outgoing modes. Tfr, . can be mapped, with a similar pr horizon F whereas the outgo»" relation with a quantum field thee horizon P. Before ending this section we want to remind here that the unitary holographic relation presented above is a particular case of the holographic relation existing between the abstract bulk observable algebra Ar and the horizon observable algebra Af Such that 4>r (/) is mapped onto c|)F (r|), where n = 2d(A(f)ff). The key point is that the algebraic holography preserve the causal propagator:
-iA(/,g) = -!AF(ii/,^).
Details on that can be found in [8,8]. This analysis suggests that similar holographic relations seems to hold also for theories on more complex spacetime as for example Schwarzschild spacetime.
7. Four dimensional case
We want to extend the result suited above, concerning Rindler holography, to the four dimensional case. For this purpose we do not discard the angular coordinates 0,<j> in the near horizon approximation of a Schwarzschild-like spacetime as discussed in section 2.
The metric reads:
ds2 = -K2y2dt2 +dy2 + r2dQ2.
Every field takes an angular part described by the usual spherical harmonics ^‘(0,<j>). QFT in the bulk involves the one-particle Hilbert space ©“„(^ ®C2M) with H,=L2(M\dE) if I >0, C2M being the space at fixed total angular momentum I and 1% =L2(§L\dE) in the massive case but = L2 (®+, dE) © L2 (M+, dE) in the massless case. For wavefunctions with components in a fixed space C2l+1 ®Lz(R+,dE) Klein-Gordon equation reduces to the two-dimensional one with a positive contribution to the mass depending on I. Quantum field theory can also be constructed on the compactified future horizon F = (lu{oo})xS2. The quantum fields on F take the following form:
-inQ
n>l,~l£m<l ‘sJA-Tttl
a/4 nn
The appropriate causal propagator reads
Af(x,x) = | sign(d-0')j8(0-0')8(<j>-§')JgS2(0,#).
Also case the holographic relation between the
bull .• > . horizon fields holds. We notice eventually
that a difference concerning the extension of
the S£(2,R) symmetry to the Virasoro algebra. The Virasoro generators can be defines as follows:
Ln ’ anfllm Z a(-k)lma(k+n)lm, Zi, (92)
^ k>n/2
where anlm := i4nanlm if n > 0 and
aalm := if n<0 and a0lm = 0. Notice that
fixing (l,m) one gets a Virasoro algebra acting on the subspace ft,. The Virasoro algebra acting on is
if := E 1?, (93)
notice that {lln} forms a reducible Virasoro algebra on ® C2i+1) whose central charge is c, := 21 +1. To define the Virasoro generators on the whole Fock space we have to perform an infinite sum
4:=Z4f (94)
!
the corresponding central charge c = ^T;c, becomes
infinite and then the Virasoro algebra is not well defined.
Acknowledgments
Part of this work due to N.P., it has '
Provincia Autonoma di Trento by the Rif. 2003-S! 16-00047 Reg. delib. n. :
B.
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