Three dimensional view of the SYK/AdS duality
JHE
Three dimensional view of the SYK/AdS duality
Sumit R. Das 0 1 3
Antal Jevicki 0 1 2
Kenta Suzuki 0 1 2
0 182 Hope Street, Providence, RI 02912 , U.S.A
1 Lexington , KY 40506 , U.S.A
2 Department of Physics, Brown University
3 Department of Physics and Astronomy, University of Kentucky , USA
4 V (y) with eigenvalue
We show that the spectrum of the SYK model can be interpreted as that of a 3D scalar coupled to gravity. The scalar has a mass which is at the BreitenholerFreedman bound of AdS2, and subject to a delta function potential at the center of the interval along the third direction. This, through KaluzaKlein procedure on AdS2 the spectrum reproducing the bilocal propagator at strong coupling. Furthermore, the leading 1=J correction calculated in this picture reproduces the known correction to the poles of the SYK propagator, providing credence to a conjecture that the bulk dual of this model can be interpreted as a three dimensional theory.
AdSCFT Correspondence; Gaugegravity correspondence; 1/N Expansion

HJEP09(217)
1 Introduction
2
3 3D interpretation
thus providing an example of the butter y e ect [40{49].
Like vector models, the SYK model is solvable at large N . Vector models, in general,
at large N can be expressed in terms of bilocal elds, and it was proposed in [50] that
these bilocal elds in fact provide a bulk construction of the dual higher spin theory [51],
with the pair of coordinates in the bilocal combining to provide the coordinates of the
emergent AdS spacetime.
The simplest proposal of [50] was implemented nontrivially in three dimensions giving
an understanding of bulk higher spin elds [52{54]. In the one dimensional SYK case [10, 11]
such bulk mapping is realized in its simplest form, with the bilocal times mapped to
AdS2 spacetime, thus providing an elementary example (in addition to the c = 1 matrix
model [55]) of how a Large N quantum mechanical model grows an additional dimension.
{ 1 {
Nevertheless, and despite great interest, the precise bulk dual of the SYK model is
still ununderstood. It has been conjectured in [56{59] that the gravity sector of this model
is the JackiwTeitelboim model [60, 61] of dilatongravity with a negative cosmological
constant, studied in [62], while [63] provides strong evidence that it is actually Liouville
theory. (See also [64{69]) It is also known that the matter sector contains an in nite tower
of particles [8{10]. Recently, couplings of these particles have been computed by calculating
six point functions in the SYK model [
70
].
In this paper, we provide a three dimensional interpretation of the bulk theory. The
zero temperature SYK model corresponds to a background AdS2
I, where I = S1=Z2 is a
nite interval whose size needs to be suitably chosen. There is a single scalar eld coupled to
derivative of the eld at the other end.1 The background can be thought of as coming from
the nearhorizon geometry of an extremal charged black hole which reduces the gravity
sector to JackiwTeitelboim model with the metric in the third direction becoming the
dilaton of the latter model [57]. The strong coupling limit of the SYK model corresponds
to a trivial metric in the third direction, while at nite coupling this acquires a dependence
on the AdS2 spatial coordinate. With a suitable choice of the size of the interval L and
the strength of the delta function potential V we show that at strong coupling, (i) the
spectrum of the KaluzaKlein (KK) modes of the scalar is precisely the spectrum of the
SYK model and (ii) the two point function2 with both points at the center of the interval
is in precise agreement with the strong coupling bilocal propagator, using the simplest
identi cation of the AdS coordinates proposed in [50]. For
nite coupling, we adopt the
proposal of [57, 58], and show that to order 1=J , the poles of the propagator shift in a
manner consistent with the explicit results in [9].
In section 2, we review relevant aspects of the bilocal formulation of the model. In
section 3, we discuss the three dimensional interpretation. Section 4 contains some concluding
remarks.
2
Overview of SYK
In this section, we will give a brief review of the Large N formalism and results along [10,
11]. The SachdevYeKitaev model [5] is a quantum mechanical many body system with
alltoall interactions on fermionic N sites (N
1), represented by the Hamiltonian
H =
Jijkl i j k l ;
(2.1)
1
N
X
4! i;j;k;l=1
{ 2 {
1We thank Edward Witten for a clari cation on this point.
2Note that this two point function is not the same as the standard AdS2 propagator. We thank Juan
Maldacena for discussions about this point.
where i are Majorana fermions, which satisfy f i; j g = ij . The coupling constant Jijkl
are random with a Gaussian distribution with width J . The generalization to analogous
qpoint interacting model is straightforward [5, 9]. After the disorder averaging for the
random coupling Jijkl, there is only one e ective coupling J in the e ective action. The
model is usually treated by replica method. One does not expect a spin glass state in this
model [7] so that we can restrict to the replica diagonal subspace [10]. The Large N theory
is simply represented through a (replica diagonal) bilocal collective eld:
where we have suppressed the replica index. The corresponding pathintegral is
1
N
X
N i=1
where Scol is the collective action:
N
2
+
Tr log
J 2N Z
2q
dt1dt2
q(t1; t2) :
Here the trace term comes from a Jacobian factor due to the change of pathintegral
variable, and the trace is taken over the bilocal time. One also has an appropriate order
O(N 0) measure . This action being of order N gives a systematic G = 1=N expansion,
while the measure
found as in [
72
] begins to contribute at oneloop level (in 1=N ). Other
formulations can be employed using two bilocal elds. These can be seen to reduce to Scol
after elimination.
In the above action, the rst linear term represents a conformal breaking term, while
the other terms respect conformal symmetry. In the IR limit with strong coupling jtjJ
1,
the collective action is reduced to the critical action
which exhibits an emergent reparametrization symmetry
Sc[ ] =
Tr log
N
2
J 2N Z
2q
dt1dt2
q(t1; t2) ;
(t1; t2) !
f (t1; t2) = f 0(t1)f 0(t2)
(f (t1); f (t2)) ;
1
q
with an arbitrary function f (t). This symmetry is responsible for the appearance of zero
modes in the strict IR critical theory. This problem was addressed in [10] with analog of the
quantization of extended systems with symmetry modes [73]. The above symmetry mode
representing time reparametrization can be elevated to a dynamical variable introduced
according to [
74
] through the FaddeevPopov method, leading to a Schwarzian action for
this variable [11] proposed by Kitaev, and established rst at quadratic level in [9]:
S[f ] =
N
12 B1 , with B1 representing the strength of the rst order
correction, established in numerical studies by Maldacena and Stanford [9]. The details of
the nonlinear evaluation are give in [11].
For the rest of this paper we proceed with q = 4. Fluctuations around the critical IR
background can be studied by expanding the bilocal eld as [10]
tj . With a simple coordinate transformation
the bilocal eld (t1; t2)
in a complete orthonormal basis
can be then considered as a eld in two dimensions (t; z). Expand the uctuation eld as
t =
(t1 + t2) ;
z =
(t1
t2) ;
u ;!(t; z) = sgn(z) ei!t Z (j!zj) ;
kernel [8]. Then, the quadratic action can be written as
S(2) =
3J
32p
X
;!
N ~ ;! g~( )
1 ~ ;! ;
where the normalization factor N is
and the kernel is given by
After a eld rede nition [10] the e ective action can be written as
N
=
8<(2 ) 1
:2
1 sin
g~( ) =
for
for
= 3=2 + 2n
= ir ;
2
3
cot
2
:
e
Sm =
1 3J Z
(t; z) hge(pDB)
1
i
(t; z) ;
{ 4 {
featuring the Bessel operator
The operator DB is in fact closely related to the laplacian on AdS2,
DB
rAdS2
= pzDB
1
p
z
1
4
where t and z in (2.10) are the Poincare coordinates in AdS2. This, therefore, realizes
the naive form of the proposal of [50]. However the action for (t; z) is nonpolynomial in
To understand the implications of this, consider the bilocal propagator, rst evaluated
in [8]. From the above e ective action, one has that the poles are determined as solutions
of g~( ) = 1, they represent a sequence denoted by pm as
=
tan
;
2m + 1 < pm < 2m + 2
(m = 0; 1; 2;
)
(2.20)
Therefore, the bilocal propagator is written as residues of
where z>(z<) is the greater (smaller) number among z and z0. The residue function is
derivatives.
2pm
Since that pm are zeros of ge( )
1
g~( )
1
e
g( )
=pm
1
=
3p2m
1, near each pole pm, we can approximate as
where fm can be determined from residue of 1=(g( )
1) at
= pm. Explicitly evaluating
these residues, the inverse kernel is written as an exact expansion
m=1 [p2m + (3=2)2][ pm
14 , (m > 0) in AdS2:
The e ective action near a pole labelled by m is that of a scalar eld with mass, M m2 =
g d2x
where the metric g
is given by g
= diag( 1=z2; 1=z2). It is clear from the above analysis
that a spectrum of a sequence of 2D scalars, with growing conformal dimensions is being
packed into a single bilocal eld. In other words the bilocal representation e ectively
packs an in nite product of AdS Laplacians with growing masses. An illustration of how
this can happen is given in the appendix A, relating to the scheme of Ostrogradsky. It
is this feature which leads to the suggestion that the theory should be represented by an
enlarged number of elds, or equivalently by an extra KaluzaKlein dimension.
For
nite coupling, the poles of the propagator is shifted. This has been calculated
by [9] in a 1=J expansion.
2
(pm)2 fm ;
e
6 p3m
{ 5 {
2
1
p2m
1
4
2m ;
:
:
(2.18)
(2.19)
(2.22)
(2.23)
(2.24)
(2.25)
3D interpretation
ary terms)
where 0 is a constant, and
by AdS2 with a metric
and a dilaton
According to [57] and [58], the bulk dual of the SYK model involves JackiwTeitelboim
theory of two dimensional dilaton gravity, whose action is given by (upto usual
boundSJT =
1
is a dilaton eld. The zero temperature background is given
where a is a parameter which scales as 1=J . In the following we will choose, without loss
of generality, 0 = 1.
This action can be thought as arising from a higher dimensional system which has
extremal black holes, and the AdS2 is the near horizon geometry [57]. The three dimensional
metric, with the dilaton being the third direction, is given by
ds2 =
1
z2
dt2 + dz2
+
1 +
dy2 :
a 2
z
This is in fact the nearhorizon geometry of a charged extremal BTZ black hole.
3.1
KaluzaKlein decomposition
We will now show that the in nite sequence of poles in the previous section from the
KaluzaKlein tower of a single scalar in a three dimensional metric (3.4) where the direction y is
an interval
L < y < L. The action of the scalar is
S =
where V (y) = V (y), with the constant V and the size L to be determined. This is
similar to HoravaWitten compacti cation on S1=Z2 [75] with an additional delta function
potential.3 The scalar satis es Dirichlet boundary conditions at the ends of the interval.
We now proceed to decompose the 3D theory into 2 dimensional modes. Using Fourier
transform for the t coordinate:
one can rewrite the action (3.5) in the form of
3See also [76, 77]. We are grateful to Cheng Peng for bringing this to our attention.
(t; z; y) =
e i!t
!(z; y) ;
{ 6 {
where D0 is the aindependent part and D1 is linear in a:
z2 + !
2
D1 =
a
z
2
m20
z2
V (y) ;
1
:
Here, we neglected higher order contributions of a. The eigenfunctions of D0 can be clearly
written in the form
!(z; y) =
!(z) fk(y) :
Then fk(y) is an eigenfunction of the Schrodinger operator
k2. This is a well known Schrodinger problem: the eigenfunctions and the eigenvalues are
HJEP09(217)
presented in detail in appendix B.
After solving this part, the kernels are reduced to
2
m20 + p2m
z2
;
D1 =
a
z
2
m20
2
qm
z2
where pm are the solutions of
(2=V )k = tan(kL)
while qm are the expectation values of
V (y) operator respect to fpm . If we choose
V = 3 and L = 2 the solutions of (3.11) agree precisely with the strong coupling spectrum
of the SYK model given by g~( ) = 1, as is clear from (2.14) and (2.16). This is our main
observation.
For these values of V , L, the propagator G is determined by the Green's equation of
D. We now use the perturbation theory to evaluate it. This will then be compared with
the corresponding propagator of the bilocal SYK theory.
3.2
Evaluation of G(0)
We start by determining the leading, zeroth order G(0) propagator obeying
D0 G(!0;)!0 (z; y; z0; y0) =
(z
z0) (y
y0) (! + !0) :
We rst separate the scaling part of the propagator by G(0) = p
Expanding in a basis of eigenfunctions fk(y),
z Ge(0) and multiplying z2.
e
G(0)(z; y; !; z0; y0; !0) =
X fk(y)fk0 (y0)Ge!;k;!0;k0 (z; z0)
(0)
k;k0
The Green's function Ge(!0;)k;!0;k0 (z; z0) is clearly proportional to (k
k0) and satis es the
equation
2 i G(0)
0
e!;k;!0;k0 (z; z0) =
z 32 (z
z0) (! + !0) (k
k0) : (3.14)
where we have de ned
2
0
k2 + m02 + 1=4:
{ 7 {
(3.8)
(3.9)
; (3.10)
(3.11)
(3.12)
(3.13)
(3.15)
e
Then, substituting this expansion into the Green's equation (3.12) and using eqs. (C.4)
and (C.2), one can x the coe cient g(0). Finally, the integral form of the propagator is
1=4, which is the BF bound of AdS2, we have
02 = p2m, and the equation which determine pm, (3.11) is precisely the equation which
determines the spectrum of the SYK theory found in [8, 10]. With this choice, the real
space zeroth order propagator in three dimensions is
G(0)(t; z; y; t0; z0; y0) =
m=0
jzz0j 21 X1 fpm (y)fpm (y0)
Z d!
2
e i!(t t0)Z d
N
Z (j!zj) Z (j!z0j) :
2
p2m
We now show that the above propagator with y = y0 = 0 is in exact agreement with
the bilocal propagator of the SYK model. The Green's function with these end points is
The operator which appears in (3.14) is the Bessel operator. Thus the Green's function
can be expanded in the complete orthonormal basis. For this, we use the same basis form
Z as in the SYK evaluation:4
and the integral over the continuous values can be now performed
exactly as in the calculation of the SYK bilocal propagator [10]. Closing the contour for
the continuous integral in Re( )! 1, one
of this contour. (1):
= 2n + 3=2, (n = 0; 1; 2;
nds that there are two types of poles inside
), and (2):
= pm, (m = 0; 1; 2;
).
The contributions of the former type of poles precisely cancel with the contribution from
the discrete sum over n. Details of the evaluation which explicitly shows the cancelation
4This represents a modi ed set of wavefunctions with boundary conditions at z ! 1 in contrast to the
standard AdS wavefunctions.
{ 8 {
are presented in appendix D. Therefore, the nal remaining contribution is just written as
residues of
Altogether we have shown that y = 0 mode 3D propagator is in precise agreement with
the q = 4 SYK bilocal propagator at large J given in eq. (2.21). The propagator is a sum
of nonstandard propagators in AdS2. While it vanishes on the boundary, the boundary
conditions at the horizon are di erent from that of the standard propagator in AdS.
In this section, we study the rst order eigenvalue shift due to D1 by treating this operator
as a perturbation onto the D0 operator. The result will con rm the duality a = 1=J , where
a is de ned in the dilaton background (3.3) and J is the coupling constant in the SYK
model.
Since the t and y directions are trivial, let us start with the kernels already solved for
these two directions given in eq. (3.10). The eigenfunction of D0 operator is
and using the orthogonality condition (C.3), its matrix element in the
space is found as
= a
dz jzj 2 Z 0 (j!zj)D1 jzj 2 Z (j!zj)
1
0
m02
aj!j
qm2 +
dz
Z 1
0
z
Z 0 (j!zj) h
dz
Z 0 (j!zj)Z (j!zj)
z2
J +1(j!zj)
J
1(j!zj)i :
(3.28)
{ 9 {
1
jzj 2 Z (j!zj) ;
N h 2
(m02 + p2m + 14 )i
; 0 :
Now following the rst order perturbation theory, we are going to determine the rst
order eigenvalue shift. Using the Bessel equation, the action of D1 on the D0
eigenfunction (3.23) is found as
1
D1 jzj 2 Z (j!zj) =
a
1
jzj 2
"
For the derivative term, we use the Bessel function identity (for example, see 8.472 of [78])
to obtain
J
1(x)
x
Z (j!zj)
j!j J +1(j!zj)
h
J
1(j!zj)i :
Therefore, now the matrix element is determined by integrals
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
For the continuous mode ( = ir), the integrals might be hard to evaluate. In the following,
we restrict ourself to the real discrete mode
= 3=2 + 2n. In such case,
= 0. Therefore,
the linear combination of the Bessel function is reduced to a single Bessel function as
Z (x) = J (x). Since
sin 2
(
( + )2
we have now found the matrix element for the discrete mode is given by
Now, we compare this result with the 1=J rst order eigenvalue shift of the SYK model,
which is for the zero mode found in [9] as
2aj!j sin 2
(
0
1) "
2
Next, let us focus on the zero mode ( = 0 = 3=2) eigenvalue. In the above formula,
taking the bare mass to the BF bound: m20 =
eigenvalue shift is found as
1=4 as before, the zero mode rst order
k(2; !) = 1
;
(zero temperature)
(3.32)
where
K
2:852 for q = 4. The !dependence of our result (3.31) thus agrees with that
of the SYK model. Furthermore, this comparison con rms the duality a = 1=J .
Finally, we can now complete our comparison by showing agreement for the m = 0
mode contribution to the propagator. We include the rst O(a) order shift for the pole as
=
+
6
aj!j 2 + q02
+ O(a2) :
For the zero mode part (m = 0) of the onshell propagator in eq. (D.6), the leading order is
O(1=a). This contribution comes from the coe cient factor of the Bessel function, which
was responsible for the double pole at
= 3=2. For other p0 setting them to 3=2, we obtain
the leading order contribution from the zero mode as
G(z0e)ro mode(t; z; 0; t0; z0; 0) =
4a (2 + q0)
This agrees with the order O(J ) contribution of the SYK bilocal propagator of
Maldacena/Stanford [9].
2
aj!j (2 + q02) :
K j!j +
2 J
3
2
9
(3.30)
(3.31)
(3.33)
(3.34)
In this paper we have provided a three dimensional perspective of the bulk dual of the
SYK model. At strong coupling we showed that the spectrum and the propagator of the
bilocal eld can be exactly reproduced by that of a scalar eld living in AdS2
with a delta function potential at the center. The metric on the interval in the third
direction is the dilaton of JackiwTeitelboim theory, which is a constant at strong coupling.
We also calculated the leading 1=J correction to the propagator which comes from the
corresponding term in the metric in the third direction, and showed that form of the poles
S1=Z2
of the propagator are consistent with the results of the SYK model [9].
We would like to emphasize that there are two aspects of this 3d perspective. The rst
concerns the agreement of the strong coupling spectrum and the form of the leading
nite
J correction. This agreement may very well follow from more general considerations [79].5
The second aspect is that the exact largeJ propagator agrees, and the form of the leading
enhanced correction for large but nite J agrees as well. We believe that this second aspect
is rather nontrivial and intruiging and the implications are yet to be fully understood.
This three dimensional view is a good way of repackaging the in nite tower of states
of the SYK model. Our analysis was done at the linearized level and the 3D gravity is
only used to x the background, as we did not treat them dynamically.6 Demonstrating
full duality at the nonlinear level is an open problem. In particular it would be interesting
if the three point function of bilocals [
70
] has a related 3d interpretation.
Acknowledgments
We acknowledge useful conversations with Robert de Melo Koch, Animik Ghosh, Juan
Maldacena, Cheng Peng, Al Shapere, Edward Witten and Junggi Yoon on the topics of
this paper. We also thank Wenbo Fu, Alexei Kitaev, Grigory Tarnopolsky and Jacobus
Verbaarschot for relevant discussions on the SYK model. This work is supported by the
Department of Energy under contract DESC0010010. The work of SRD is partially
supported by the National Science Foundation grant NSFPHY1521045. AJ would like to
thank the Galileo Galilei Institute for Theoretical Physics (GGI) for the hospitality and
INFN for partial support during the completion of this work, within the program \New
Developments in AdS3/CFT2 Holography". We also learned of possibly related work by
Marika Taylor [80].
A
Actions nonpolynomial in derivatives
To illustrate how an action which is nonpolynomial in derivatives can arise let us start
with the example of N decoupled elds
5We thank the referee for bringing this paper to our attention.
(A.1)
N
X
n=1
L =
'nDn'n :
One can then introduce elds
' =
which represents a transformation preserving the determinant:
Integrating 's out, one eventually ends up with the e ective Lagrangian
N
Y
n=1
Dn = Db +
Dbn :
N 1
Y
n=1
L' = '
PN
QnN=1 Dn
n1< <nN 1 Dn1
DnN 1
!
' :
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(B.1)
(B.2)
HJEP09(217)
n
N 1
X
DnN 1
Dnp
Dnp 1
;
;
Here all the poles are contained in the higherorder laplacian, as in eq. (2.16). The
opposite procedure of going from this e ective action with the N th order Laplacian to the
rst one, requires introducing N
1 extra elds, which would correspond to the scheme introduced by Ostrogradsky.
B
Schrodinger equation
In this appendix, we consider the equation of f (y), which is the Schrodinger equation:
h
where E is an eigenvalue of the equation. Since we con ned the eld in
L < y < L,
we have boundary conditions: f ( L) = 0. The continuation conditions at y = 0 are
f (+0) = f ( 0) and the other can be derived by integrating the Schrodinger equation (B.1)
over ( "; ") and taking limit " ! 0 as
f 0(+0)
f 0( 0) = V f (0) :
Since the potential of the Schrodinger equation is even function, the wave function is either
odd or even function of y.
(i) Odd: for odd parity case, a solution satisfying the boundary conditions at y =
where k2 = E. For odd parity solution, to satisfy the boundary condition f (+0) =
f ( 0), we need f ( 0) = 0. This implies that
k =
n
L
;
(n = 1; 2; 3;
)
(B.4)
Finally, let us prove the orthogonality of the parity even wave function (B.5):
Using the solution (B.5) and evaluating the integral in the lefthand side, one obtains
Z L
L
dy fm(y)fm0 (y) =
m;m0 :
B2 sin(L(k
k
k0
k0))
sin(L(k + k0))
k + k0
:
constant is xed as A = 1=pL.
Then, the continuity condition (B.2) is automatically satis ed. The normalization
(ii) Even: for even parity case, a solution satisfying the boundary conditions at y =
L
is given by
f (y) =
(B sin(k(y
L))
(0 < y < L)
B sin(k(y + L))
( L < y < 0)
where k2 = E. The evenness of the parity guarantees f ( 0) = f (+0). So, we only
need to impose the condition (B.2) on this solution. This condition gives an equation
k = tan(kL) :
2
V
s
Now we set L =
=2 and V = 3, then we have
(2=3)k = tan( k=2), which is
precisely the same transcendental equation determining poles of the q = 4 SYK
bilocal propagator (2.20). We denote the solutions of
(2=3)k = tan( k=2) by pm,
(2m + 1 < pm < 2m + 2), (m = 0; 1; 2;
). The normalization constant is xed as
B =
2k
2kL
sin(2kL)
:
(B.3)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
Now let's assume k 6= k0. Then, the integral result can be rearranged to the form of
k2
B2
k02 cos(Lk) cos(Lk0) h k0 tan(Lk)
k tan(Lk0) i = 0 ;
(B.10)
where the nal equality is due to the relation tan(Lk) =
2k=3. Next, we consider k = k0
case. In this case, due to the delta function identity, the result (B.9) is reduced to
B2 L
sin(2Lk)
2k
k;k0 = k;k0 ;
where for the equality we used eq. (B.7). Therefore, now we have proven the
orthogonality (B.8).
C
Completeness condition of Z
In this appendix, we summarize some properties of the Bessel function Z , which are used
to determine the zeroth order propagator (3.18). The linear combination of the Bessel
functions is de ned by [8]
Z (x) = J (x) +
From this orthogonality condition, one can x the normalization for the completeness
condition of Z . Namely, dividing each Z by pN , nally we nd the completeness
condition as
Z (jxj) Z (jx0j) = x (x
x0) :
D
Evaluation of the contour integral
In this appendix, we give a detail evaluation of the continuous and the discrete sums
appearing in eq. (3.19). As we de ned before, the integral symbol d is a shorthand
notation of a combination of summation over
= 3=2+2n, (n = 0; 1; 2;
) and integration
of
= ir, (r > 0). Namely,
which satis es the Bessel equation
Z (C.1) is given by
2 Z (j!zj) :
In [8], the orthogonality condition of the linear combination of the Bessel function
with
N
2
p2m
Z d Z (j!zj) Z (j!z0j) = I1 + I2 ;
I1
I2
1
X
n=0
2
2
;
0 2 sinh( r) r2 + p2m Zir(j!zj) Zir(j!z0j) :
Now, one can notice that the second term exactly cancels with the contribution from I1.
One can also repeat the above discussion for z0 > z case. Therefore, combining these two
cases the total contribution is now
I1 + I2 =
2 sin( pm)
J pm (j!jz>) +
Jpm (j!jz>) Jpm (j!jz<) ;
(D.5)
where z>(z<) is the greater (smaller) number among z and z0. Then, the propagator is
reduced to
Let us evaluate the continuous sum I2 rst. Using the symmetry of the integrand, one can
rewrite the integral as
I2 =
i Z i1
2
d
i1 sin(
We evaluate this integral by a contour integral on the complex
plane by closing the
contour in the Re( )> 0 half of the complex plane if z > z0. Inside of this contour, we
have two types of the poles. (i) at
= pm coming from the coe cient factor. (ii) at
= 3=2 + 2n, (n = 0; 1; 2;
) coming from
, where
= 1. After evaluating residues
at these poles, one obtains
I2 =
h
J pm (j!zj) +
pm Jpm (j!zj)i Jpm (j!z0j)
This agrees with the result given in eq. (3.22).
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