Holographic realtime nonrelativistic correlators at zero and finite temperature
Edwin Barnes^{1}^{1}1Present address: CMTC, University of Maryland, College Park, MD 20742, barnes@umd.edu Diana Vaman^{2}^{2}2dv3h@virginia.edu, Chaolun Wu^{3}^{3}3cw2an@virginia.edu
Department of Physics, The University of Virginia
McCormick Rd, Charlottesville, VA 22904
We compute a variety of two and threepoint realtime correlation functions for a stronglycoupled nonrelativistic field theory. We focus on the theory conjectured to be dual to the Schrödingerinvariant gravitational spacetime introduced by Balasubramanian, McGreevy, and Son, but our methods apply to a large class of nonrelativistic theories. At zero temperature, we obtain timeordered, retarded, and Wightman nonrelativistic correlators for scalar operators of arbitrary conformal dimension directly in field theory by applying a certain lightlike Fourier transform to relativistic conformal correlators, and we verify that nonrelativistic AdS/CFT reproduces the results. We compute thermal two and threepoint realtime correlators for scalar operators dual to scalar fields in the black hole background which is the finite temperature generalization of the Schrödinger spacetime. This is done by first identifying thermal realtime bulktoboundary propagators which, when combined with Veltman’s circling rules, yield two and threepoint correlators. The twopoint correlators we obtain satisfy the KallenLehmann relations. We also give retarded and timeordered threepoint correlators.
Contents
 1 Introduction and Summary

2 Zerotemperature twopoint functions from CFT correlators
 2.1 Timeordered and reversetimeordered twopoint functions in position space
 2.2 Timeordered and reversetimeordered twopoint functions in momentum space
 2.3 Wightman twopoint functions in position space
 2.4 Wightman twopoint functions in momentum space
 2.5 Retarded and advanced twopoint functions in position space
 2.6 Retarded and advanced twopoint functions in momentum space
 3 Zerotemperature threepoint functions from CFT correlators
 4 Zerotemperature bulktoboundary propagators in momentum space
 5 Finitetemperature correlators
 A Zerotemperature timeordered threepoint functions in position space
 B Zerotemperature timeordered threepoint functions in momentum space
 C Zerotemperature Wightman threepoint functions in position space
 D Zerotemperature retarded and advanced threepoint functions in position space
 E metric fluctuation spectrum
1 Introduction and Summary
The AdS/CFT correspondence [1, 2, 3] has enjoyed much success in elucidating aspects of stronglyinteracting relativistic field theories over the past twelve years. However, there exists a broad array of stronglycoupled systems which admit a fieldtheoretic description but do not exhibit Lorentz invariance. Cold atomic gases in the socalled unitarity limit provide a prime example of this. The unitarity limit refers to a regime in which the system resides precisely at the crossover point between a gas of Cooper pairs and a BoseEinstein condensate. This regime arises when the interatomic potential is tuned such that the range of the potential vanishes while the scattering length diverges, thus removing all scales in the problem. The ability to realize this scaling limit experimentally by exploiting the phenomena of Feshbach resonances [4] has drawn wide attention to this system and emphasized the need for more progress on the theory front to balance the successes in the laboratory. Recent approaches to this problem that employ field theory techniques directly can be found in [5, 6, 7, 8].
The unitary Fermi gas is a natural candidate for testing the waters of nonrelativistic AdS/CFT as it possesses not just scale invariance but a larger group of symmetries known as the Schrödinger group, which can be thought of as the nonrelativistic analog of the conformal group. The fact that the Schrödinger group is not only a close cousin of conformal symmetry but that it can also be embedded in the conformal group was a key observation in formulating the first proposal for a nonrelativistic AdS/CFT correspondence [9, 10]. This proposal posited that the nonrelativistic (Schrödinger) correspondence works in much the same way as its relativistic counterpart, but with a particular planewavetype geometry (sourced by additional nontrivial background fields) replacing AdS as the dual gravitational spacetime at zero temperature.
The form of the Schrödinger spacetime suggests that we should think of the dual Schrödinger theory as being embedded within a parent theory living in one higher dimension. If we denote the time coordinate in the parent theory by , then the Schrödinger theory is obtained by switching to light cone coordinates , where is one of the spatial directions, and then projecting the spectrum onto a sector of fixed momentum. The coordinate plays the role of time in the Schrödinger theory. If is also compactified, then the momentum assumes a discrete set of values that are naturally interpreted as particle number for the nonrelativistic Schrödinger theory, and the procedure is referred to as discrete light cone quantization (DLCQ). At zero temperature, there are subtleties involved with this compactification since is lightlike, and one needs to consider large particle number in order to trust the gravity dual [11, 12, 13]. This idea of embedding the Schrödinger theory within a parent theory is closely related to the notion of Bargmann spacetimes [14, 15, 16], and it greatly facilitates the formulation of a nonrelativistic version of AdS/CFT.
The original conjecture of Refs. [9, 10] has since been placed on firmer ground by subsequent work which uplifted the dual gravitational background to a solution of IIB supergravity and extended it to a finite temperature black hole solution [11, 12, 13]. Furthermore, it was shown that at finite temperature, is no longer a lightlike direction, so that subtleties pertaining to its compactification no longer arise. The work of [11, 12, 13] also clarified that the parent theory related to the Schrödinger metric constructed in [9, 10] is an example of a well known class of noncommutative theories known as dipole theories [17, 18, 19]. The dipole theories are obtained by starting from an ordinary commutative field theory and replacing regular multiplication of fields with a noncommutative star product. The particular parent theory described by the Schrödinger metric is the dipole theory that results when super YangMills is deformed in this way, and the corresponding Schrödinger theory is just the DLCQ of this. In what follows, we will always refer to these theories simply as the parent and Schrödinger theories^{1}^{1}1Although Schrödinger invariance is broken at finite temperature, we will continue to use the term “Schrödinger theory” even in this case., and we will also use these terms to distinguish between their gravity duals, which are also related by DLCQ. These discoveries constitute an important milestone toward extending AdS/CFT to the realm of nonrelativistic theories and applying it to improve our understanding of systems like the unitary Fermi gas.
Some of the most important information that AdS/CFT provides about a field theory are its correlation functions. As for theories exhibiting conformal invariance, the two and threepoint correlation functions for Schrödingerinvariant theories also provide crucial checks of the conjectured correspondence since these correlators are largely determined by the symmetry group alone [20, 21]. In particular, the twopoint functions are completely fixed by the symmetry up to an overall constant. Unlike the conformallyinvariant case, however, it is not true that the functional form of the threepoint functions are fully determined. It was verified in [22, 23, 24] that AdS/CFT reproduces both two and threepoint scalar correlation functions.^{2}^{2}2Intriguingly, the authors of [22] also computed the threepoint function using a certain field theory model for a unitary Fermi gas and showed that, in addition, AdS/CFT correctly computes the piece of the threepoint function which is not determined by Schrödinger symmetry.
In this paper, we will focus on computing realtime correlation functions for scalar fields in Schrödinger field theories. Realtime correlation functions are of particular interest since these in turn yield quantities like conductivity and viscosity. However, realtime correlation functions in AdS/CFT pose more technical challenges relative to Euclidean ones since it is less clear how to formulate basic AdS/CFT recipes in Minkowski signature, where the bulk spacetime tends to be more complicated. Early attempts to deal with this problem yielded a casebycase treatment [25, 26], and only in the last two years have more systematic approaches appeared in the literature [27, 28, 29]. In the context of Schrödingerinvariant theories, the methods of [27, 28] were employed by Hoang and Leigh [24] to compute timeordered and Wightman twopoint functions at zero temperature. We will show that the approach given in [29] reproduces their findings in a simple way, and we further use it to compute a wide variety of realtime two and threepoint correlators at both zero and finite temperature. Our results for the zerotemperature twopoint functions are also consistent with the bulktobulk correlators computed in [30].
The complete functional form of the zerotemperature two and threepoint correlation functions can in fact be computed in real time without having to invoke AdS/CFT or any other method for solving the Schrödinger field theory due to the intimate relation between the Schrödinger and conformal groups. More specifically, we may obtain correlation functions of the Schrödinger theory directly from the correlation functions of a CFT by switching to light cone coordinates and Fouriertransforming with respect to , as was originally noticed some time ago in Ref. [21] and more recently exploited in the context of AdS/CFT in Refs. [22, 23]. This procedure is essentially equivalent to performing DLCQ, with the role of the Fourier transform being to project onto a sector of fixed momentum. However, as stressed in [21], it is important to note that the Fouriertransform trick must be performed keeping noncompact so as to avoid having to construct CFT correlators on a lightlike circle. We will still refer to this procedure as DLCQ since we are free to consider fixed particle number in the resulting expressions, keeping in mind the zerotemperature subtleties mentioned earlier.
We will apply the Fouriertransform technique to compute various zerotemperature correlators in real time. Although the authors of [22, 23] computed realtime correlators, they did not keep track of the type of correlator (e.g. timeordered, retarded, etc.), as their primary focus was on testing whether nonrelativistic AdS/CFT reproduces Schrödingerinvariant expressions. Since the prescriptions which distinguish between the different types of realtime correlators are well known in the case of a relativistic CFT (see [29] for a review), it is a straightforward task to perform DLCQ on these correlators to produce realtime Schrödinger correlators, and we will show that the type of correlator is preserved under DLCQ. In addition, we verify that the standard KallenLehmann relations are satisfied by the various realtime Schrödinger correlators.
We stress that the CFT correlators that we Fourier transform are not the correlators of the parent theory. In particular, note that the parent dipole theory is not Lorentzinvariant [17, 18, 19]. Therefore, the resulting Schrödingerinvariant correlators will not contain the same overall constants as those of the DLCQ of the dipole theory. Our aim in applying the Fourier transform to CFT correlators is to ascertain the form of the various realtime Schrödinger correlators at zero temperature; this provides us with an important check of the exact Schrödinger theory correlators we will compute from nonrelativistic AdS/CFT.
We compute the zerotemperature correlators in momentum space as well as position space. DLCQ applied to momentum space correlators simply amounts to performing a rotation in the plane spanned by and , the energy and momentum associated with and . In the case of threepoint functions, we express the timeordered and retarded correlators in a way that is very reminiscent of AdS/CFT, namely in terms of an integral over what are readily interpreted as three bulktoboundary propagators in the Schrödinger spacetime. This allows us to define in a natural way nonrelativistic Feynman and retarded bulktoboundary propagators which in turn are easily verified to reproduce the corresponding twopoint correlators that we compute directly in field theory. Wightman propagators are then defined in such a way that they reproduce Wightman twopoint functions. The exercise of constructing the different realtime bulktoboundary propagators and establishing their interrelations not only provides additional consistency checks of nonrelativistic AdS/CFT, but also helps set the stage for the computations of thermal correlation functions, which rely heavily on correctly identifying these propagators.
One of the main objectives of this work is to compute realtime twopoint and threepoint nonrelativistic correlation functions for scalar operators dual to minimally coupled scalars at finite temperature. The Schrödinger theory at finite temperature also possesses a nonzero chemical potential which breaks Schrödinger invariance, meaning that we can no longer receive guidance by applying the Fouriertransform trick to CFT correlators as in the zerotemperature case. With the confidence gained from crosschecking AdS/CFT with field theory at zero temperature, we therefore focus solely on AdS/CFT in the finite temperature case.
The first step is to compute realtime bulktoboundary propagators. This is done by starting with the Euclidean version of the metric which describes a black hole in the Schrödinger spacetime [11, 12, 13] and solving the scalar wave equation to obtain the Euclidean bulktoboundary propagator in momentum space. This is the Euclidean propagator in the gravity dual of the parent dipole theory. We analytically continue this to a retarded propagator and then perform DLCQ to get the retarded Schrödinger bulktoboundary propagator. In [29], it was shown that in the relativistic case, the other realtime thermal propagators can be derived from the retarded one using certain identities. These identities naturally extend to the parent dipole theory, where we can then apply DLCQ to show that similar relations between Schrödinger propagators also hold.
Once we have the realtime Schrödinger bulktoboundary propagators, we can assemble them into realtime two and threepoint correlators. This problem was solved in the case of thermal SYM in [29] by extending Veltman’s circling rules [31, 32, 33] to the gravitational theory on the black hole spacetime, and similar arguments should apply in the present context as well. We compute twopoint correlators in this way and confirm that the results satisfy standard KallenLehmann relations. The retarded threepoint was worked out explicitly in [29], and we borrow the result to immediately write down the threepoint function for the parent dipole theory. A simple application of DLCQ to the result in turn gives the retarded Schrödinger threepoint correlator. In addition, we apply the circling rules to construct the timeordered Schrödinger threepoint function.
Before proceeding with the calculations, we pause for a moment to briefly discuss the dimensionality of the theories we consider. We keep the number of spatial dimensions (denoted ) arbitrary as much as possible. There is certainly no obstacle to doing so in our discussion of DLCQ applied directly to CFT correlators at zero temperature. Furthermore, we may still consider the dimensional Schrödinger spacetime as a “bottomup” holographic model even though this spacetime can only be uplifted to a solution of IIB supergravity in the case . Leaving the dimension arbitrary here also facilitates comparison with our field theory results. We could adopt a similar philosophy toward the Schrödinger black hole spacetime, however it is considerably more difficult to solve the wave equation for values of not equal to 2, so we restrict ourselves to in this case.
We will consider various types of correlators and bulktoboundary propagators throughout the paper, and we distinguish between these with different fonts of the letter , as we will now clarify. Relativistic CFT correlators are denoted by , while nonrelativistic Schrödinger correlators are denoted by . The symbol specifies bulktoboundary propagators in the gravity dual of the parent theory. We also define bulktoboundary propagators for the gravity dual of the Schrödinger theory, and these we denote by . In addition, we use tildes to denote momentum space functions, the superscript to denote causal threepoint correlators, and additional decorations to distinguish between the different types of realtime functions. In particular, the subscripts , , and denote timeordered, retarded, and advanced functions respectively. Reversetimeordered functions are distinguished from timeordered functions by an additional bar (e.g. ) which should not be confused with complex conjugation, for which we use the symbol . Wightman twopoint functions have superscripts , while Wightman threepoint functions are specified by a subscript which shows the order of operator insertions. For example, is a Schrödinger Wightman threepoint function in momentum space proportional to , where the are arbitrary scalar operators in the Schrödinger theory.
The paper is organized as follows. In section 2, we compute all types of scalar realtime Schrödinger twopoint correlators at zero temperature in both position and momentum space by applying the Fouriertransform trick to relativistic CFT correlators. We also verify that the results satisfy the KallenLehmann relations. In section 3, we perform similar computations for realtime zerotemperature threepoint correlators. We confirm that standard relations among the threepoint functions hold, and we also express the results in a way that is very suggestive of an AdS/CFT calculation, allowing us to identify Schrödinger “bulktoboundary propagators”. In section 4, we construct realtime bulktoboundary propagators directly in the zerotemperature gravity dual and show that the results are consistent with the answers obtained using the Fouriertransform trick in sections 2 and 3. Finally in section 5, we apply the intuition developed in the zerotemperature case to construct thermal realtime bulktoboundary propagators for scalar fields. These are then employed to compute two and threepoint scalar correlators for the Schrödinger theory at finite temperature. The first four appendices each contain details about the application of the Fouriertransform trick to particular zerotemperature threepoint correlators considered in section 3. In appendix E, we construct an explicit example of a massive scalar in the effective 5d gravity theory which arises when a fluctuation in a particular offdiagonal metric component has nontrivial charges on the 5d internal space.
2 Zerotemperature twopoint functions from CFT correlators
In this section, we will obtain Schrödingerinvariant nonrelativistic zerotemperature twopoint functions in position space by starting with realtime relativistic conformal twopoint functions in light cone coordinates and Fouriertransforming with respect to one of these coordinates. We also obtain momentum space twopoint functions by translating the Fouriertransform trick to momentum space, where it becomes a simple redefinition of momenta. We will pay particular attention to how the different types of relativistic realtime functions (e.g. timeordered, retarded, etc.) map to the different types of nonrelativistic functions. We will see that the type of realtime correlator is preserved under the special Fourier transform. We also verify that standard KallenLehmann relations between the various realtime functions are satisfied by the nonrelativistic correlators we obtain.
2.1 Timeordered and reversetimeordered twopoint functions in position space
Consider first the timeordered relativistic conformal function:
(2.1) 
Here, , where is the relativistic time coordinate, and is one of the spatial coordinates. We use the mostly plus signature for Minkowski space. The coordinate vector represents the remaining spatial coordinates which coincide with the spatial coordinates of the nonrelativistic theory: . Following [21, 23], we can obtain a nonrelativistic twopoint function by performing a Fouriertransform with respect to the direction:
(2.2) 
This integral can be done by introducing a Schwinger parameter:
(2.3) 
The integral on yields a function:
(2.4) 
The function will evaluate to zero unless . Therefore, we find
(2.5) 
This has the form of a nonrelativistic timeordered twopoint function where the nonrelativistic time coordinate is . Hoang and Leigh [24] found the same result employing a very different approach [27, 28]. We have retained the in the final expression because it removes a potential singularity at . It is clear from (2.2) that this singularity should not be present.
If we instead start with the reversetimeordered relativistic twopoint function:
(2.6) 
the same calculation now gives
(2.7) 
This has the form of a nonrelativistic reversetimeordered function.
2.2 Timeordered and reversetimeordered twopoint functions in momentum space
To make it clear how the Fouriertransform trick translates to momentum space, we first consider a simple example. The Fourier transform of the conformal timeordered twopoint function with and is
(2.8) 
where and . Introducing the light cone coordinates , we can write this as
(2.9)  
We now define the nonrelativistic energy and mass in terms of the relativistic energy and momentum component :
(2.10) 
Applying the inverse of this map,
(2.11) 
we obtain
(2.12)  
This is the nonrelativistic timeordered propagator in momentum space with nonrelativistic energy
(2.13) 
It is easy to check that (2.12) is the Fourier transform of (2.5) with and :^{3}^{3}3Our conventions for both relativistic and nonrelativistic Fourier transforms are established in this subsection. In particular, note that we define our nonrelativistic Fourier transform with an additional overall factor of because the nonrelativistic time coordinate is .
(2.14) 
Obviously the same approach can be applied to any other momentumspace function since, from the point of view of the Fourier transform in (2.8) and (2.9), we are simply redefining momentumspace variables. Starting from the relativistic timeordered twopoint function with arbitrary and ,
(2.15)  
the map (2.11) gives the nonrelativistic timeordered function^{4}^{4}4The second line of (LABEL:nrfeynmanmomentumtwopt) can be obtained with the help of the identity , .:
The reversetimeordered twopoint function is just minus the complex conjugate of :
2.3 Wightman twopoint functions in position space
Next consider the following relativistic Wightman function:
(2.18) 
We can facilitate taking the Fourier transform by rewriting this as
(2.19) 
Before taking the Fourier transform, it is useful to first factor out a factor of from the denominator:
(2.20) 
One can then use the positivity of to introduce a Schwinger parameter as we did in the case of timeordered functions in the previous subsection. The integral then gives a function:
(2.21) 
Since the argument of the function does not contain , a step function involving does not arise in this case. The result is then
(2.22) 
This agrees with the result found in [24] using the methods of [27, 28]. The role of the in the exponent is to ensure that vanishes at , while the in the denominator controls the location of the branch cut for noninteger :^{5}^{5}5Throughout this paper, we adopt the convention that if no prescription is given explicitly.
(2.23) 
If we instead consider starting with the other relativistic Wightman function,
(2.24) 
the same steps lead to an integration as above, but with instead of appearing in the integrand, yielding
(2.25) 
Using (2.5), (2.22), and (2.25) it is easy to check that the relations
(2.26) 
are satisfied.^{6}^{6}6When , the in the exponents can be discarded relative to . When , , , and each vanish identically, and (2.26) is satisfied trivially.
2.4 Wightman twopoint functions in momentum space
Now consider the relativistic Wightman functions with and in momentum space:
(2.27) 
Under (2.11), these transform to
(2.28) 
For (), the function has support at positive (negative) values of , so that
(2.29) 
which is the Fourier transform of (2.22) for , , and
(2.30) 
To obtain the nonrelativistic Wightman functions for general and , we start with the Fourier transform of (2.18):
(2.31) 
The map (2.11) then gives
(2.32) 
We have used that
(2.33) 
to rewrite the argument of the step function before applying the map. The second relativistic Wightman function has the form
(2.34) 
Now observing that
(2.35) 
we see that the map (2.11) leads to
(2.36) 
So far, we have seen that the relativistic (reverse)timeordered twopoint function transforms into a nonrelativistic (reverse)timeordered twopoint function, and the relativistic Wightman functions map to nonrelativistic Wightman functions. It remains for us to check what happens to relativistic retarded and advanced functions under this mapping.
2.5 Retarded and advanced twopoint functions in position space
The relativistic conformal retarded twopoint correlator can be written in terms of Wightman functions:
(2.37) 
In terms of light cone coordinates, we therefore have
(2.38)  
Factoring out the in the denominators, assuming , and Fouriertransforming leads to
(2.39)  
Notice that we have chosen to write instead of in the denominators. Since we are assuming , we may rewrite , and the term can be eliminated by rescaling the in the denominators of the integrands. The in the factors appearing outside the integrals cannot be neglected even though because the in these factors control branch cuts, as we discussed in the previous subsection on Wightman functions.
Changing the integration variable to , we have
(2.40)  
The integrals can be evaluated using
(2.41) 
and we find
(2.42) 
This has the form of a nonrelativistic retarded twopoint function. This expression is valid for ; when , it is easy to check that . (This can be done by setting in (2.38) and Fouriertransforming.) An analogous calculation reveals that if we start with the relativistic advanced function, we obtain the nonrelativistic advanced function:
(2.43) 
A shortcut to obtaining this result is to write down the analog of (2.39) for and to notice that can be obtained from by sending and in the final answer. It is easy to check using (2.22) and (2.25) that standard relations between retarded/advanced and Wightman functions:
(2.44) 
hold for the nonrelativistic twopoint functions we have obtained.
We conclude that the type of realtime twopoint functions is preserved under the mapping which takes us from relativistic to nonrelativistic functions.
2.6 Retarded and advanced twopoint functions in momentum space
The Fourier transform of the relativistic retarded twopoint function (2.38) with and is given by
(2.45) 
Applying (2.11), we obtain
(2.46)  
This function has a pole at
(2.47) 
Since this pole lies in the lower half plane regardless of the value of or , we see that this function has the analytic properties expected of a retarded Green’s function. We may therefore redefine to absorb the positive coefficient which multiplies it in :
(2.48) 
The advanced function is the conjugate of this:
(2.49) 
Comparing these results with (2.12), we see that , and obey the following relation:
(2.50) 
On the other hand, we can obtain what appears to be a slightly different relation directly from (2.46):
(2.51) 
At first glance, it may seem surprising that these two relations hold simultaneously. However, their compatibility has a simple origin in the form of the imaginary part of the retarded twopoint correlator. Using (2.46), we can write this as . Note that only the imaginary part is affected by the presence of the step functions in (2.50) and (2.51). Since the function requires that and have the same sign, this is equivalent to , which agrees with (2.48). Also notice that we can write the relation in a third way:
(2.52) 
This form of the relation makes it apparent that there is no chemical potential at zero temperature. We will now see that the relations (2.50), (2.51), and (2.52) hold for general values of and .
The Fourier transform of (2.38) for general and is
(2.53) 
where was given in (2.15). This yields
Similar reasoning leads to the following advanced twopoint function:
It is not difficult to check that the relations (2.50), (2.51), and (2.52) hold for general and . Furthermore, the relation (2.50) provides a useful check of the consistency of our results since it must also hold in position space, and one can readily verify that (2.5), (2.42), and (2.43) satisfy (2.50). We will see later on in section 5 that the second relation, (2.51), generalizes to a relation that is also valid at finite temperature.
3 Zerotemperature threepoint functions from CFT correlators
In this section, we apply the Fouriertransform trick to relativistic threepoint functions to obtain Schrödingerinvariant nonrelativistic threepoint functions in position space. We do this for timeordered and Wightman functions and show that the usual identity relating these types of correlators is satisfied. In position space, retarded and advanced functions can be expressed in terms of the Wightman functions [36].
We compute nonrelativistic timeordered and retarded/advanced threepoint functions in momentum space as well. This can be done either by computing the Fourier transforms of the nonrelativistic position space threepoint functions, or by starting with relativistic conformal threepoint functions in momentum space and performing an appropriate redefinition of momenta as we did for twopoint functions in section 2. We will apply the former approach to obtain timeordered functions and the latter approach for retarded/advanced functions. Wightman threepoint functions are quite unwieldy in momentum space, so we do not include them here.
The final answers we obtain for the momentum space threepoint functions are of the form one would expect from an AdS/CFT calculation. In particular, we express the results as integrals of products of three functions which are naturally interpreted as bulktoboundary propagators for scalar fields in nonrelativistic AdS/CFT [9, 10]. In section 4, we reproduce these bulktoboundary propagators by applying DLCQ to propagators in the gravity dual of the dipole theory and show that they give rise to twopoint functions which are consistent with the nonrelativistic correlators we computed in section 2.
3.1 Timeordered threepoint functions in position space
The relativistic timeordered threepoint function in position space has the form
(3.1)  
We have introduced the notation . The constants are related to the conformal dimensions of the scalar fields:
(3.2) 
To obtain the nonrelativistic correlator , we must Fouriertransform with respect to the . The integrals can be computed with the help of Schwinger parameters, and the details can be found in appendix A. The result is
where
(3.4) 
and