Conformal blocks for the four-point function in conformal
quantum mechanics
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Jackiw, R., and S.-Y. Pi. “Conformal Blocks for the Four-point
Function in Conformal Quantum Mechanics.” Physical Review D
86.4 (2012). © 2012 American Physical Society
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http://dx.doi.org/10.1103/PhysRevD.86.045017
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PHYSICAL REVIEW D 86, 045017 (2012)
Conformal blocks for the four-point function in conformal quantum mechanics
R. Jackiw1 and S.-Y. Pi2
1
2
Department of Physics, MIT, Cambridge, Massachusetts 02139, USA
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
(Received 7 May 2012; published 13 August 2012)
Extending previous work on two- and three-point functions, we study the four-point function and its
conformal block structure in conformal quantum mechanics CFT1 , which realizes the SOð2; 1Þ symmetry
group. Conformal covariance is preserved even though the operators with which we work need not be
primary and the states are not conformally invariant. We find that only one conformal block contributes to
the four-point function. We describe some further properties of the states that we use and we construct
dynamical evolution generated by the compact generator of SOð2; 1Þ.
DOI: 10.1103/PhysRevD.86.045017
PACS numbers: 11.10.Kk
I. INTRODUCTION AND REVIEW
A recent paper [1] initiated research on the
AdSdþ1 =CFTd correspondence for the special case d¼1.
This dimension corresponds to the lowest ‘‘rung’’ on the
dimensional ‘‘ladder’’ of SOðd þ 1; 1Þ conformally invariant scalar field theories in d dimensions.
1
Ld ¼ @ @ gð2d=d2Þ :
2
(1.1)
At d ¼ 1 ½ðt; rÞ ! qðtÞÞ, L1 governs conformal quantum mechanics with a g=q2 potential [2], and supports an
SOð2; 1Þ symmetry, with generators H, D and K.
Their algebra
i½D; H ¼ H;
(1.2a)
i½D; K ¼ K;
(1.2b)
i½K; H ¼ 2D;
(1.2c)
when presented in Cartan basis,
1 K
R
þ aH ;
2 a
1 K
L aH iD;
2 a
However, in CFT1 normalized states are not invariant and
invariant states are not normalizable, rendering problematic calculation of expectation values. Furthermore, one
wonders which operators in conformal quantum mechanics
realize the primary operators ’ðtÞ, whose correlation functions arise from the AdS2 calculation.
These puzzles are resolved in the paper [1]. We focus on
the R operator, taken to be positive (g 0) and defined on
the half-line (q 0), with integer-spaced eigenvalues rn
and orthonormal eigenstates jni.
Rjni ¼ rn jni;
hnjn0 i ¼ nn0 ;
½L ; Lþ ¼ 2R:
(1.3a)
(1.6a)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L jni ¼ rn ðrn 1Þ r0 ðr0 1Þjn 1i:
OðtÞ ¼ NðtÞ expð!ðtÞLþ Þ;
!ðtÞ þ 1 2r0
NðtÞ ¼ ½ð2r0 Þð1=2Þ
;
2
a þ it
¼ ei where t ¼ a tan=2;
!ðtÞ ¼
a it
(1.3b)
(1.4a)
(1.4b)
(1.7)
and defined ‘‘t states’’ jti by the action of OðtÞ on the R
vacuum.
jti ¼ OðtÞjn ¼ 0i;
(1.8)
Rjn ¼ 0i ¼ r0 jn ¼ 0i:
(1.9)
From their definition (1.8) it follows that the jti states
satisfy 1
(1.5)
jti states that satisfy (by postulate) (1.10) were first presented
by dAFF Ref [2]. Subsequently in Ref. [1] they were constructed
by the action of the operator OðtÞ on the R vacuum, as in (1.7)
and (1.8).
1
where ’ðtÞ are primary operators in the boundary conformal theory, and the averaging state h. . .i is conformally
invariant, i.e. it is annihilated by the conformal generators.
1550-7998= 2012=86(4)=045017(5)
n ¼ 0; 1. .. ;
We need states that carry a representation of the SOð2; 1Þ
action. To this end we constructed the operator OðtÞ,
(a is a scaling parameter with dimension of time;
frequently we set it to 1.)
In spite of the natural position that d ¼ 1 enjoys, various
questions arise about the correspondence. AdS2 calculations allegedly produce boundary N-point correlation functions in CFT1 .
GN ðt1 ; . . . ; tN Þ h’1 ðt1 Þ . . . ’N ðtN Þi;
r0 > 0;
(1.6b)
reads
½R; L ¼ L ;
rn ¼ r0 þ n;
045017-1
Ó 2012 American Physical Society
R. JACKIW AND S.-Y. PI
d
Hjti ¼ i jti;
dt
d
Djti ¼ i t þ r0 jti;
dt
d
Kjti ¼ i t2 þ 2r0 t jti:
dt
PHYSICAL REVIEW D 86, 045017 (2012)
(1.10a)
d
’ðtÞ;
dt
d
i½D; ’ðtÞ ¼ t þ ’ðtÞ;
dt
d
i½K; ’ðtÞ ¼ t2 þ 2t ’ðtÞ:
dt
(1.15)
(1.10c)
The two ’ fields are taken to be identical, with scale
dimension . We demonstrate that conformal covariance
and block structure are maintained by our unconventional
realization of the conformal symmetry: once again defects
cancel.
In Sec. III, we study some further properties of the jti
states and of related energy eigenstates jEi of the
Hamiltonian H. Also we show how the R operator can
replace H as the evolution generator.
(1.11a)
(1.11c)
GN ðt1 ; t2 ; . . . ; tN1 ; tN Þ
¼ hn ¼ 0jOy ðt1 Þ’2 ðt2 Þ . . . ’N1 ðtN1 ÞOðtN Þjn ¼ 0i
(1.12)
In spite of the fact that the OðtÞ operators are not
primary, and the averaging state jn ¼ 0i is not conformally
invariant, the two ‘‘defects’’ cancel and the resultant
N-point functions satisfy conformal covariance conditions.
Consequently, in an operator-state correspondence we
may consider the operators OðtÞ, when acting on the states
jn ¼ 0i, as primary with dimension r0 .
In this way one establishes that2 [3]
G2 ðt1 ; t2 Þ ¼ ht1 jt2 i ¼ hn ¼ 0jOy ðt1 ÞOðt2 Þjn ¼ 0i
¼
ð2r0 Þa2r0
;
½2iðt1 t2 Þ2r0
(1.13)
G3 ðt1 ; t; t2 Þ ¼ ht1 j’ðtÞjt2 i ¼ hn ¼ 0jOy ðt1 Þ’ðtÞOðt2 Þjn ¼ 0i
2r þ
i 0
¼ hn ¼ 0j’ð0Þjn ¼ 0i
2
ð2r0 Þa2r0
:
(1.14)
ðt1 tÞ ðt t2 Þ ðt2 t1 Þ2r0 The expressions (1.13) and (1.14) also arise from calculations based on a scalar field in AdS2 , at the boundary of
the AdS2 bulk.
In Sec. II, we extend the investigation to the quantum
mechanical four-point function.
2
dAFF Collaboration Ref. [2].
II. CORRELATION FUNCTION AND
CONFORMAL BLOCK
(1.11b)
Thus an N-point function involves the jti states.
¼ ht1 j’2 ðt2 Þ . . . ’N1 ðtN1 ÞjtN i:
¼ hn ¼ 0jOy ðt1 Þ’ðt2 Þ’ðt3 ÞOðt4 Þjn ¼ 0i:
(1.10b)
N-point functions are constructed from the jti states. For
GN ðt1 ; . . . ; tN Þ, the averaging state h. . .i is the R vacuum
jn ¼ 0i. The first and last operators are taken to be Oy ðt1 Þ
and OðtN Þ, while the remaining N 2 operators are conventional but unspecified primary operators ’, with scale
dimension .
i½H; ’ðtÞ ¼
G4 ðt1 ; t2 ; t3 ; t4 Þ ¼ ht1 j’ðt2 Þ’ðt3 Þjt4 i
A. Four-point function in CFT1
To calculate G4 in (1.15), insert complete sets of jni states
between the operators. Also without loss of generality evaluate the sums at special values: t1 ¼ ia, t4 ¼ ia. [This may
always be achieved by a complex SOð2; 1Þ transformation.]
One is left with a single sum. It remains to reduce matrix
elements hnj’ðtÞjn0 i to hn ¼ 0j’ð0Þjn0 ¼ 0i. This was accomplished by dAFF [2] with the SOð2; 1Þ Wigner-Eckart
theorem. This procedure leads to [4]
G4 ðt1 ;t2 ;t3 ; t4 Þ
¼ jhn ¼ 0j’ð0Þjn ¼ 0ij2
1
X
2 ð1 Þ
2 ð2r0 Þ
22þ2r0 ðt13 t24 Þ2 ðt14 Þ2r0 2
1
xn
;
2
n¼0 ð2r0 þ nÞ ð1 nÞ n!
t t
tij ti tj ; x 12 34 :
t13 t24
(2.1)
(The scaling parameter a is set to unity.)
Remarkably, the sum may be evaluated in terms of the
hypergeometric function 2 F1 . The final expression for G4 is
G4 ðt1 ; t2 ; t3 ; t4 Þ
¼ jhn ¼ 0j’ð0Þjn ¼ 0ij2
1
2þ2r0
2
ð2r0 Þ
xr0 F ð; ; 2r0 ; xÞ:
ðt13 t24 Þr0 ðt12 t34 Þþr0 2 1
(2.2)
The polynomial in tij provides conformal covariance, while
the x dependence is conformally invariant. (In one dimension four points lead to a single invariant, as opposed to two
invariants in higher dimensions.)
The four-point function may be presented by a Mellin
transform since 2 F1 possesses a Mellin-Barnes representation.
045017-2
CONFORMAL BLOCKS FOR THE FOUR-POINT FUNCTION . . .
2 F1 ð; ; 2r0 ; xÞ
¼
ð2r0 Þ Z i1
2 ð þ sÞðsÞ
ðxÞs :
ds
2
ð2r0 þ sÞ
ðÞ i1
(2.3)
The sum in (2.1) arises from the poles of ðsÞ in (2.3).
A single Mellin integral suffices at d ¼ 1 because there is
only a single invariant.
B. Conformal block in CFT1
In general one expects that the four-point function G4
may be presented as a superposition of ‘‘conformal blocks.’’
These quantities are kinematically determined by the eigenfunctions of the SOð2; 1Þ Casimir. This is like a partial wave
expansion of a scattering amplitude—indeed ‘‘conformal
partial waves’’ is an alternative nomenclature.
Conformal blocks at arbitrary d for SOðd þ 1; 1Þ have
been extensively studied by Dolan and Osborn. Recently
they have constructed the d ¼ 1, SOð2; 1Þ quantities by
passing to the (somewhat singular) limit d ! 1 for a block
coming from a single operator and its descendants [5]. In
contrast, from the start we work directly with the SOð2; 1Þ
symmetry at d ¼ 1.
We present the general four-point function.
PHYSICAL REVIEW D 86, 045017 (2012)
H ¼ H1 þ H2 ; K ¼ K1 þ K2 ; D ¼ D1 þ D2
@
@
@
Hi ¼ i ; Di ¼ i ti þ i ; Ki ¼ i t2i
þ 2i ti :
@ti
@ti
@ti
(2.8)
c is the eigenvalue. Thus the derivative operator D corresponding to C,
@2
@
@
D t212
þ 2t12 2
1
@t1
@t2
@t1 @t2
þ ð1 þ 2 Þ2 ð1 þ 2 Þ;
acts on pB as
DðpBÞ ¼ pðx2 ð1 xÞB00
þ ð1 þ 12 34 Þx2 B0 þ 12 34 xBÞ
¼
1 þ2
ðt12 Þ
3 þ4
ðt34 Þ
1
FðxÞ
ðt13 Þ34 ðt14 Þ12 34 ðt24 Þ12
¼ pðt1 ; t2 ; t3 ; t4 ÞFðxÞ:
x2 ð1 xÞB00 þ ð1 þ 12 34 Þx2 B0 þ 12 34 xB ¼ cB;
(2.11)
and is solved by
B ¼ x 2 F1 ð 12 ; þ 34 ; 2; xÞ;
c ¼ ð 1Þ:
(2.12a)
(2.12b)
In order to match this block to the four-point function (2.2)
where 1 ¼ 4 ¼ r0 , 2 ¼ 3 ¼ we must set ¼ r0 ,
so that
B ¼ xr0 2 F1 ð; ; 2r0 ; xÞ:
(2.4)
The t-polynomial p carries the conformal transformation
property of G4 , while F is invariant. i is the dimension of
’i and ij i j . (This expression is more general
than the one we used in our previous discussion, which is
specialized to 1 ¼ 4 ¼ r0 , 2 ¼ 3 ¼ , ’1 ¼ Oy ,
’4 ¼ O, ’2;3 ¼ ’.)
The block decomposition states
X
(2.5)
FðxÞ ¼ bi Bi ðxÞ;
(2.10)
d
). The eigenvalue equation reads
(dash signifies dx
G4 ðt1 ; t2 ; t3 ; t4 Þ
¼ h’1 ðt1 Þ’2 ðt2 Þ’3 ðt3 Þ’4 ðt4 Þi
(2.9)
(2.13)
Evidently the single block (2.13) reproduces the four-point
function. It is a surprise that one block suffices.
The usual route to conformal blocks is through the shortdistance expansion for ’1 ðt1 Þ’2 ðt2 Þ. In our construction
’1 ðt1 Þ is replaced by Oy ðt1 Þ, which does not have an evident
short distance expansion with ’2 ðt2 Þ. Nevertheless, within
our approach we are able to derive a block representation for
the four-point function. This puts into evidence once again
that our method, with its cancellation of defects, preserves
conformal covariance.
i
where i labels the kinematical variety of blocks Bi . Each Bi
is constructed from a specific primary operator and its
descendants. The bi ’s contain dynamical data. The blocks
are eigenfunctions of the Casimir.
1
C ¼ ðHK þ KHÞ D2 ;
2
CðpBÞ ¼ cðpBÞ:
(2.6)
III. VARIOUS OBSERVATIONS
ON THE FORMALISM
The construction of the states jti in (1.7) and (1.8) has
found response in the literature [6]. Therefore, we elaborate some of their further properties, which follow from
(1.2) and (1.10).
A. Energy eigenstates
(2.7)
Since the action of H on jti is known from (1.10a), it is
readily seen that3
In (2.6) and (2.7), the individual generators are sums of the
corresponding derivative operators
045017-3
3
Energy eigenstates were defined by dAFF, Ref. [2].
R. JACKIW AND S.-Y. PI
jEi ¼ 2r0
PHYSICAL REVIEW D 86, 045017 (2012)
E1=2 Z 1 dt iEt
e
jti
ðaEÞr0 1 2
(3.1)
Then (3.7) and (3.8) show that
1 1=2
ð2r0 Þjn ¼ 0i;
22r0
1
jt ¼ 0i ¼ 2r0 1=2 ð2r0 ÞeHa jn ¼ 0i:
2
eHa jt ¼ 0i ¼
is an orthonormal energy eigenstate. The prefactor ensures
normalization.
hEjE0 i ¼ ðE E0 Þ:
(3.2)
The SOð2; 1Þ generators act as
HjEi ¼ EjEi;
d
1
þ jEi;
DjEi ¼ i E
dE 2
d2
d
ðr0 1=2Þ2
þ
jEi:
KjEi ¼ E 2 dE
E
dE
ðaEÞr0 iEt
e
:
E1=2
(3.4)
B. (In)completeness of the jti states
Combining (3.1) with (3.4) gives
E Z 1 dt
jtihtjEi;
jEi ¼ 22r0
ðaEÞ2r0 1 2
jti ¼ eiHt jt ¼ 0i ¼
(3.5a)
1=2 ð2r0 Þ ðaþitÞH
e
jn ¼ 0i:
22r0
(3.9b)
This is an interesting alternative to (1.7) and (1.8).
(3.3c)
The jEi states allow establishing further properties of the
jti states, whose overlap with jEi is determined from (1.13)
and (3.1).
htjEi ¼ 2r0
Since H generates t evolution, a further consequence is4
(3.3a)
(3.3b)
(3.9a)
D. Alternative evolution
In our treatment evolution takes place in t time and is
generated by H. This is seen in (1.10a) and (1.11a), where
the action of H is time derivation, i.e. infinitesimal time
translation.
However, our formalism is based on R, rather than H.
Thus recasting evolution so that it is generated by R
becomes an interesting alternative. This is accomplished
by redefining time t.
Observe from (1.10) that
1
K
1
d rt
Rjti ¼ aH þ jti ¼ i ½a þ t2 =a þ 0 jti:
2
a
2
dt
a
(3.10)
Upon defining a new ‘‘time’’ ,
or
22r0
Z 1 dt
ðaHÞ2r0
jtihtjEi:
jEi ¼
H
1 2
Since the energy eigenstates are complete, we arrive at an
(in)complete relation for the jti states.
1 aH 2r0 Z 1 dt
jtihtj:
(3.6)
¼
H 2
1 2
ðcos=2Þ2r0
d
ððcos=2Þ2r0 jt ¼ a tan=2iÞ:
d
Hence if we define new time states ji
(3.12)
it follows that R translates infinitesimally.
In the paper [1] it is shown that
Rji ¼ i
(3.7)
satisfies Rj c i¼r0 j c i; hence j c i is proportional to jn¼0i.
Naming the proportionality constant N , we have
~ expðei Lþ Þjn ¼ 0i;
ji ¼ NðÞ
~
NðÞ
¼ ðcos=2Þ2r0 Nðt ¼ a tan=2Þ;
(3.8a)
¼ ½ð2r0 Þ1=2 eir0 :
0
The matrix element (with a restored) is given by (3.4).
Therefore
Z1
1 aE 2r0 ð2r0 Þ
jN j2 ¼
dEe2Ea
¼ 2r0 :
(3.8b)
E 2
4
0
d
ji:
d
(3.13)
Explicitly the state ji is given by
j c i ¼ N jn ¼ 0i;
jN j2 ¼ h c j c i ¼ ht ¼ 0je2Ha jt ¼ 0i;
Z1
¼
dEe2Ea jht ¼ 0jEij2 :
(3.11)
[compare (1.7)] the expression in the last parenthesis of
(3.10) may be rewritten as
ji ¼ ðcos=2Þ2r0 jt ¼ a tan=2i;
C. State-operator correspondence
j c i eHa jt ¼ 0i
t ¼ a tan=2
(3.5b)
(3.14a)
(3.14b)
The spectrum of H is continuous and the conjugate time
variable is unrestricted. On the other hand, the spectrum of
R is discrete, equally spaced, and the conjugate variable
is periodic.
045017-4
4
Nakayama, Ref [6].
CONFORMAL BLOCKS FOR THE FOUR-POINT FUNCTION . . .
In terms of the new variable, the two-point function
becomes (see footnote 4)
G2 ð0 ; Þ ¼
ð2r0 Þ
:
0
2r0
½2ifsin½
2 g
(3.15)
One may also consider evolution generated by 12 ðaH KaÞ.
This development begins when the new time is defined as
t ¼ a tanh=2, which leads to similar replacement in (3.11),
(3.12), (3.13), (3.14), and (3.15) of trigonometric functions
by hyperbolic ones.
PHYSICAL REVIEW D 86, 045017 (2012)
are not invariant [R vacuum jn ¼ 0i]. Nevertheless results
obey the conformal constraints.
For the two- and three-point functions an AdS2 bulk
dual can be identified [1]. We have not accomplished that
for the four-point function. But the simplicity of the block
structure—just one block is needed to reproduce the fourpoint function—gives the hope that a dual model in the
AdS2 bulk can be found. It is interesting to observe that
the AdS2 bulk propagator is given by a hypergeometric
function, just as G4 and its conformal block are.
ACKNOWLEDGMENTS
IV. CONCLUSION
We have studied the four-point function and its conformal block for CFT1 conformal quantum mechanics. We
used operators that are not primary ½OðtÞ and states that
We acknowledge conversations with S. Behbahani,
C. Chamon, D. Harlow and L. Santos. This research is
supported by the DOE under Grant Nos. DE-FG0205ER41360 (R. J.) and DE-FG02- 91ER40676 (S. Y. P).
[1] C. Chamon, R. Jackiw, S.-Y. Pi, and L. Santos, Phys. Lett.
B 701, 503 (2011). Corrections: Eq. (2.9), first line, add
‘‘R.’’ Eq. (3.15), replace second equals sign by proportionality sign.
[2] Scale-free quantum mechanics was investigated in response to MIT/SLAC deep inelastic scattering results.
The g=q2 potential appeared in a pedagogical article on
scale symmetry: R. Jackiw, Phys. Today 25, No. 1, 23
(1972).The model was thoroughly investigated by V. de
Alfaro, S. Fubini, and G. Furlan (dAFF Collaboration),
Nuovo Cimento Soc. Ital. Fis. A 34, 569 (1976).
[3] A careful evaluation shows that the singularity in G2 ðt1 ; t2 Þ
at, t1 ¼ t2 is regulated as t1 t2 ! t1 t2 i".
[4] S. Behbahani and D. Harlow (unpublished). Eq. (2.1) was
communicated to us by S. Behbahani. Some small but
crucial sign errors needed correcting.
[5] F. Dolan and H. Osborn, arXiv:1108.6194.
[6] R. Nakayama, Prog. Theor. Phys. 127, 393 (2012);
B. Freivogel, J. McGreevy, and S. J. Suh, Phys. Rev. D
85, 105002 (2012); D. Anninos, S. A. Hartnoll, and
D. M. Hofman, Classical Quantum Gravity 29, 075002
(2012).
045017-5