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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Conformal blocks for the four-point function in conformal quantum mechanics
R. Jackiw1and S.-Y. Pi21Department of Physics, MIT, Cambridge, Massachusetts 02139, USA 2Department 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 invari-ant scalar field theories in d dimensions.
Ld¼ 1 2@@
gð2d=d2Þ: (1.1)
At d ¼ 1 ½ðt; rÞ ! qðtÞÞ, L1 governs conformal quan-tum 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, R 1 2 K aþ aH ; (1.3a) L 1 2 K a aH iD; (1.3b) reads ½R; L ¼ L; (1.4a) ½L; Lþ ¼ 2R: (1.4b) (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 calcula-tions allegedly produce boundary N-point correlation func-tions in CFT1.
GNðt1; . . . ; tNÞ h’1ðt1Þ . . . ’NðtNÞi; (1.5) where ’ðtÞ are primary operators in the boundary confor-mal theory, and the averaging state h. . .i is conforconfor-mally invariant, i.e. it is annihilated by the conformal generators.
However, in CFT1 normalized states are not invariant and invariant states are not normalizable, rendering problem-atic calculation of expectation values. Furthermore, one wonders which operators in conformal quantum mechanics realize the primary operators ’ðtÞ, whose correlation func-tions arise from the AdS2calculation.
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 rnjni; rn¼ r0þ n; r0> 0; n ¼ 0;1...; (1.6a) hnjn0i ¼ nn0; Ljni ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rnðrn 1Þ r0ðr0 1Þ q jn 1i: (1.6b)
We need states that carry a representation of the SOð2; 1Þ action. To this end we constructed the operatorOðtÞ,
OðtÞ ¼ NðtÞ expð!ðtÞLþÞ; NðtÞ ¼ ½ð2r0Þð1=2Þ !ðtÞ þ 1 2 2r 0 ; !ðtÞ ¼a þ it a it¼ e i where t ¼ a tan=2; (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 r0jn ¼ 0i: (1.9) From their definition (1.8) it follows that the jti states satisfy1
1jti 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 operatorOðtÞ on the R vacuum, as in (1.7) and (1.8).
Hjti ¼ i d dtjti; (1.10a) Djti ¼ i td dtþ r0 jti; (1.10b) Kjti ¼ i t2 d dtþ 2r0t jti: (1.10c)
N-point functions are constructed from the jti states. For GNðt1; . . . ; tNÞ, the averaging state h. . .i is the R vacuum jn ¼ 0i. The first and last operators are taken to be Oyðt
1Þ andOðtNÞ, while the remaining N 2 operators are con-ventional but unspecified primary operators ’, with scale dimension . i½H; ’ðtÞ ¼ d dt’ðtÞ; (1.11a) i½D; ’ðtÞ ¼ t d dtþ ’ðtÞ; (1.11b) i½K; ’ðtÞ ¼ t2 d dtþ 2t ’ðtÞ: (1.11c) Thus an N-point function involves the jti states.
GNðt1; t2; . . . ; tN1; tNÞ ¼ hn ¼ 0jOyðt
1Þ’2ðt2Þ . . . ’N1ðtN1ÞOðtNÞjn ¼ 0i ¼ ht1j’2ðt2Þ . . . ’N1ðtN1ÞjtNi: (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 operatorsOðtÞ, when acting on the states jn ¼ 0i, as primary with dimension r0.
In this way one establishes that2[3]
G2ðt1; t2Þ ¼ ht1jt2i ¼ hn ¼ 0jOyðt1ÞOðt2Þjn ¼ 0i ¼ ð2r0Þa2r0 ½2iðt1 t2Þ2r0 ; (1.13) G3ðt1; t; t2Þ ¼ ht1j’ðtÞjt2i ¼ hn ¼ 0jOyðt1Þ’ðtÞOðt2Þjn ¼ 0i ¼ hn ¼ 0j’ð0Þjn ¼ 0ii 2 2r 0þ ð2r0Þa2r0 ðt1 tÞðt t 2Þðt2 t1Þ2r0 : (1.14) The expressions (1.13) and (1.14) also arise from calcu-lations based on a scalar field in AdS2, at the boundary of the AdS2bulk.
In Sec. II, we extend the investigation to the quantum mechanical four-point function.
G4ðt1; t2; t3; t4Þ ¼ ht1j’ðt2Þ’ðt3Þjt4i ¼ hn ¼ 0jOyðt
1Þ’ðt2Þ’ðt3ÞOðt4Þjn ¼ 0i: (1.15) 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.
II. CORRELATION FUNCTION AND CONFORMAL BLOCK A. Four-point function inCFT1
To calculate G4in (1.15), insert complete sets of jni states between the operators. Also without loss of generality evalu-ate the sums at special values: t1¼ ia, t4¼ 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’ðtÞjn0i to hn ¼ 0j’ð0Þjn0 ¼ 0i. This was ac-complished by dAFF [2] with the SOð2; 1Þ Wigner-Eckart theorem. This procedure leads to [4]
G4ðt1; t2; t3; t4Þ ¼ jhn ¼ 0j’ð0Þjn ¼ 0ij22ð1 Þ 22þ2r0 2ð2r 0Þ ðt13t24Þ2ðt 14Þ2r02 X1 n¼0 1 ð2r0þ nÞ2ð1 nÞ xn n! ; tij ti tj; x t12t34 t13t24: (2.1)
(The scaling parameter a is set to unity.)
Remarkably, the sum may be evaluated in terms of the hypergeometric function2F1. The final expression for G4is G4ðt1; t2; t3; t4Þ ¼ jhn ¼ 0j’ð0Þjn ¼ 0ij2 1 22þ2r0 ð2r0Þ ðt13t24Þr0ðt 12t34Þþr0 xr0 2F1ð; ;2r0; xÞ: (2.2)
The polynomial in tijprovides conformal covariance, while the x dependence is conformally invariant. (In one dimen-sion 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 since2F1possesses a Mellin-Barnes representation.
2dAFF Collaboration Ref. [2].
R. JACKIW AND S.-Y. PI PHYSICAL REVIEW D 86, 045017 (2012)
2F1ð; ; 2r0; xÞ ¼ ð2r0Þ 2ðÞ Zi1 i1ds 2ð þ sÞðsÞ ð2r0þ sÞ ðxÞs: (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 inCFT1
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 eigen-functions 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. G4ðt1; t2; t3; t4Þ ¼ h’1ðt1Þ’2ðt2Þ’3ðt3Þ’4ðt4Þi ¼ 1 ðt12Þ1þ2ðt 34Þ3þ4ðt13Þ34ðt14Þ1234ðt24Þ12 FðxÞ ¼ pðt1; t2; t3; t4ÞFðxÞ: (2.4) The t-polynomial p carries the conformal transformation property of G4, while F is invariant. iis 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, 2¼ 3 ¼ , ’1 ¼ Oy, ’4 ¼ O, ’2;3¼ ’.)
The block decomposition states FðxÞ ¼X
i
biBiðxÞ; (2.5)
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.
C ¼1
2ðHK þ KHÞ D
2; (2.6)
CðpBÞ ¼ cðpBÞ: (2.7)
In (2.6) and (2.7), the individual generators are sums of the corresponding derivative operators
H ¼ H1þ H2; K ¼ K1þ K2; D ¼ D1þ D2 Hi¼ i @ @ti ; Di¼ i ti @ @ti þ i ; Ki¼ i t2 i @ @ti þ 2iti : (2.8) c is the eigenvalue. Thus the derivative operatorD corre-sponding to C, D t2 12 @2 @t1@t2þ 2t12 2 @ @t1 1 @ @t2 þ ð1þ 2Þ2 ð1þ 2Þ; (2.9) acts on pB as DðpBÞ ¼ pðx2ð1xÞB00 þð1þ1234Þx2B0þ1234xBÞ (2.10) (dash signifiesdxd ). The eigenvalue equation reads
x2ð1 xÞB00þ ð1 þ 12 34Þx2B0þ 1234xB ¼ cB; (2.11) and is solved by B ¼ x2F1ð 12; þ 34; 2; xÞ; (2.12a) c ¼ ð 1Þ: (2.12b)
In order to match this block to the four-point function (2.2) where 1 ¼ 4¼ r0, 2 ¼ 3¼ we must set ¼ r0, so that
B ¼ xr0
2F1ð; ;2r0; xÞ: (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 short-distance expansion for ’1ðt1Þ’2ðt2Þ. In our construction ’1ðt1Þ is replaced by Oyðt1Þ, which does not have an evident short distance expansion with ’2ðt2Þ. 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.
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 elabo-rate some of their further properties, which follow from (1.2) and (1.10).
A. Energy eigenstates
Since the action of H on jti is known from (1.10a), it is readily seen that3
3Energy eigenstates were defined by dAFF, Ref. [2].
jEi ¼ 2r0 E 1=2 ðaEÞr0 Z1 1 dt 2e iEtjti (3.1)
is an orthonormal energy eigenstate. The prefactor ensures normalization.
hEjE0i ¼ ðE E0Þ: (3.2) The SOð2; 1Þ generators act as
HjEi ¼ EjEi; (3.3a)
DjEi ¼ i E d dEþ 12 jEi; (3.3b) KjEi ¼ E d2 dE2 d dEþ ðr0 1=2Þ2 E jEi: (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 2r0ðaEÞ
r0 E1=2 e
iEt: (3.4)
B. (In)completeness of the jti states Combining (3.1) with (3.4) gives
jEi ¼ 22r0 E ðaEÞ2r0 Z1 1 dt 2jtihtjEi; (3.5a) or 22r0ðaHÞ 2r0 H jEi ¼ Z1 1 dt 2jtihtjEi: (3.5b) Since the energy eigenstates are complete, we arrive at an (in)complete relation for the jti states.
1 H aH 2 2r 0 ¼Z1 1 dt 2jtihtj: (3.6) C. State-operator correspondence
In the paper [1] it is shown that
jci eHajt ¼ 0i (3.7) satisfies Rjci¼r0jci; hence jci is proportional to jn¼0i. Naming the proportionality constantN , we have
jci ¼ N jn ¼ 0i;
jN j2 ¼ hcjci ¼ ht ¼ 0je2Hajt ¼ 0i; ¼Z1
0
dEe2Eajht ¼ 0jEij2:
(3.8a)
The matrix element (with a restored) is given by (3.4). Therefore jN j2¼Z1 0 dEe2Ea1 E aE 2 2r 0 ¼ ð2r0Þ 42r0 : (3.8b)
Then (3.7) and (3.8) show that eHajt ¼ 0i ¼ 1 22r0 1=2ð2r 0Þjn ¼ 0i; jt ¼ 0i ¼ 1 22r0 1=2ð2r 0ÞeHajn ¼ 0i: (3.9a)
Since H generates t evolution, a further consequence is4 jti ¼ eiHtjt ¼ 0i ¼1=2ð2r0Þ
22r0 e
ðaþitÞHjn ¼ 0i: (3.9b) This is an interesting alternative to (1.7) and (1.8).
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 Rjti ¼1 2 aH þK a jti ¼ i1 2½a þ t 2=ad dtþ r0t a jti: (3.10) Upon defining a new ‘‘time’’ ,
t ¼ a tan=2 (3.11)
[compare (1.7)] the expression in the last parenthesis of (3.10) may be rewritten as
ðcos=2Þ2r0 d
dððcos=2Þ
2r0jt ¼ a tan=2iÞ: Hence if we define new time states ji
ji ¼ ðcos=2Þ2r0jt ¼ a tan=2i; (3.12) it follows that R translates infinitesimally.
Rji ¼ i d
dji: (3.13)
Explicitly the state ji is given by ji ¼ ~NðÞ expðeiL
þÞjn ¼ 0i; (3.14a) ~
NðÞ ¼ ðcos=2Þ2r0Nðt ¼ a tan=2Þ;
¼ ½ð2r0Þ1=2eir0: (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.
4Nakayama, Ref [6].
R. JACKIW AND S.-Y. PI PHYSICAL REVIEW D 86, 045017 (2012)
In terms of the new variable, the two-point function becomes (see footnote 4)
G2ð0; Þ ¼ ð2r0Þ ½2ifsin½0
2 g2r0
: (3.15)
One may also consider evolution generated by1 2ðaH
K aÞ. 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.
IV. CONCLUSION
We have studied the four-point function and its confor-mal block for CFT1 conformal quantum mechanics. We used operators that are not primary ½OðtÞ and states that
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 four-point 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
We acknowledge conversations with S. Behbahani, C. Chamon, D. Harlow and L. Santos. This research is supported by the DOE under Grant Nos. DE-FG02-05ER41360 (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 propor-tionality sign.
[2] Scale-free quantum mechanics was investigated in re-sponse 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 G2ðt1; t2Þ at, t1¼ 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).
