Gravity Dual for Reggeon Field Theory and Non-linear Quantum Finance
aa r X i v : . [ h e p - t h ] J un UCB-PTH-09/20
Gravity Dual for Reggeon Field Theoryand Non-linear Quantum Finance
Yu Nakayama
Berkeley Center for Theoretical Physics,University of California, Berkeley, CA 94720, USA
Abstract
We study scale invariant but not necessarily conformal invariant deformationsof non-relativistic conformal field theories from the dual gravity viewpoint. Wepresent the corresponding metric that solves the Einstein equation coupled with amassive vector field. We find that, within the class of metric we study, when weassume the Galilean invariance, the scale invariant deformation always preserves thenon-relativistic conformal invariance. We discuss applications to scaling regime ofReggeon field theory and non-linear quantum finance. These theories possess scaleinvariance but may or may not break the conformal invariance, depending on theunderlying symmetry assumptions.
Introduction
The invasion of gauge/gravity correspondence into other areas of theoretical physicsis quite rapid these days. We have proliferating numbers of gauge/gravity correspon-dence beyond the classical AdS/CFT correspondence, first in AdS/QCD (like quark-gluonplasma applications) and then in AdS/CMP (like cold atom systems and high T c super-conductors). In these examples, the gravity dual approach has given birth to completelynovel models of the conventional physics. In particular, relatively new advancement in thegauge/gravity correspondence is to jump off from the island of the relativistic dispersionrelation and dive into the vast ocean of non-relativistic systems [1][2]. In this paper, weattempt to expand the realm of the gauge/gravity correspondence further by studyingyet another application to non-relativistic systems— Reggeon field theory and quantumfinance.In the free field theory limit, the both systems are reduced to the free Schr¨odingerequation, so they have the right symmetry that could fit in the non-relativistic conformalinvariant theories to begin with, albeit they are free. Once the interaction has beenintroduced, they are driven into the strongly coupled regime in the long distance (and overthe long time). Nobody has achieved non-perturbative understanding of their stronglycoupled regime, and our hope is that the gravitational dual approach may shed a newlight on the dynamics.Technically speaking, we have a completely orthogonal motivation for this study. Inthe relativistic conformal field theories, there is a long standing question whether the scaleinvariance suggests conformal invariance or not. Group theoretically, there is no reasonwhy the scale invariance indicates the conformal invariance because the former is merelya subgroup of the latter. Surprisingly, however, in the relativistic conformal field theoriesliving in higher than two space-time dimension, there are no known examples of unitaryscale invariant field theories but not conformal invariant. In two space-time dimension,under certain mild assumptions such as unitarity and discreteness of spectrum, it hasbeen proved that the scale invariance indeed implies the conformal invariance [3]. Whenwe remove a certain assumption such as unitarity, there are known examples of scaleinvariant but non-conformal field theories. See [4] for a recent construction of a non-trivially interacting example, which may be important in time-dependent string theories.It is of interest to ask the same question in the non-relativistic case. It is easy to1onstruct scale invariant but non-conformal field theories, as we will encounter in thispaper, once the Galilean invariance is broken. A part of the argument simply comes fromthe group structure because the non-relativistic conformal invariance demands i [ K, P i ] = G i , where K is the non-relativistic conformal transformation, so once the Galilean boost G i is broken, one cannot preserve K as long as momentum P i is conserved (see appendixA and B for more discussions). The remaining task is to find a scale invariant butGalilean non-invariant theory. An example is the Lifshitz-like field theory whose gravitycounterpart has been studied in [5]. We will see more examples from the gravity viewpointin the main part of the paper.The central question is, therefore, whether Galilean and scale invariant field theoriesare automatically non-relativistic conformal invariant or not. We cannot give a full ac-count of this statement nor will we give any counterexamples in this paper. We willsee, however, within the dual gravity background, the deformations of the non-relativisticconformal field theory that preserve the Galilean invariance as well as scale invariance donot break the conformal invariance, either.For applications to Reggeon field theory and non-linear quantum finance in the stronglycoupled scaling regime, we notice that they possess the scale invariance but may or maynot break the conformal invariance, depending on the underlying symmetry assumptions.Accordingly, we choose the gravity background in order to reproduce qualitative features(e.g. symmetries) of the theories we would like to study. In addition, we see that theReggeon field theory with a simple cubic interaction shows a non-trivial renormalizationgroup flow of the dynamical critical exponent. The tree level dynamical critical exponent Z = 2 will be modified through the interaction and the subsequent renormalization. Therenormalization of the dynamical critical exponent (also known as the Hurst exponent)might be of relevance in quantum finance, too.The organization of the paper is as follows. In section 2, we study various scale invari-ant but non-conformal deformations of the gravity solutions studied in literatures. Weshow that once Galilean invariance is imposed, the conformal invariance cannot be brokenwithin the class of models we consider. We also compute the correlation functions in thedeformed background. In section 3, we discuss the applications to Reggeon field theoryand non-linear quantum finance. We have included a short review of the supersymmet-ric quantum mechanical formulation of the Black-Scholes-Merton model in section 3.2 to2otivate the non-linear extension in section 3.3. We give further discussions and specu-lations in section 4. In appendix A, we study the criterion when the scale invariance andGalilean invariance indicates the conformal invariance from the field theory viewpoint.Appendix B summarizes the non-relativistic conformal algebra. In appendix C, we collectsome formulae for confluent Hypergeometric function used in the main text. In this section, we investigate scale invariant deformations of the gravity backgroundproposed in [1][2][5]. A particular class of the metric satisfies the Einstein equation withmassive vector field as a source of energy-momentum tensor. We study the correlationfunctions of the dual field theories by using the AdS/CFT technique.
Our starting point is the phenomenological gravity background for non-relativistic con-formal field theories presented in [1][2]: ds d +2 = − dt z + − dtdζ + dx i + dz z , (2.1)where i = 1 , · · · , d . In later applications, we set d = 2 for (physical) Reggeon field theoryand d = 1 for (one-factor) quantum finance. Originally, it was proposed to describe thecold atoms at criticality, or unitary fermion system for d = 3 [1].The metric (2.1) has the following isometries • Translations in x i and t . • Rotations of x i . • Galilean boost:( ζ , x i ) → (cid:0) ζ − v i x i + v t, x i − v i t (cid:1) . • particle number (translation in ζ ). 3 Scale transformation with dynamical critical exponent Z = 2:( t, ζ , x i , z ) → ( λ t, ζ , λx i , λz ). • Non-relativistic special conformal transformation:( t, ζ , x i , z ) → (cid:16) t ηt , ζ − η x i x i + z ηt , x i ηt , z ηt (cid:17) .We will compactify ζ as ζ ≃ ζ + 2 πR .We would like to study scale invariant but not necessarily conformal invariant defor-mations of the metric (2.1). By assuming the scale invariance and translation invariance,possible metric deformations are restricted in the following three classes: • Class 1: The deformations that break the spatial rotation: δds = (cid:8) dx i dtz , dx i dzz , dx i dζz (cid:9) .It also breaks the spatial parity invariance x i → − x i . This kind of spatial anisotropymight be relevant in cosmology or condensed matter physics. Our applications insection 3, however, will preserve this symmetry, so we will not discuss the conse-quence of these deformations further in this paper. • Class 2: The deformations that break the Galilean boost: δds = { dζ } . Thedeformation also breaks the non-relativistic conformal invariance. The resultingcontinuous symmetry is same as that of the Lifshitz-like field theory (except for thepresence of additional U (1) particle number conservation). • Class 3: The deformations that preserve the Galilean boost: δds = (cid:8) dzdζz , dzdtz (cid:9) .Actually, the deformation preserve the full non-relativistic conformal invariance aswe will see shortly.We first begin with how Class 3 deformations preserve the full non-relativistic confor-mal invariance. The key point is that the deformed metric is locally diffeomorphic to theoriginal non-relativistic conformal invariant metric (2.1). To see this, we introduce thecoordinate transformation t → t + α z , ζ → ζ + β log z (2.2) Notice that the deformation a i dx i dzz actually does not break the spatial rotation, so strictly speakingit is in class 3. By defining a new coordinate x ′ i = x i − a i z , we can see the spatial rotation is realized ina disguised way.
4o the metric (2.1). The resulting metric in the new coordinate is ds d +2 = − dt z − (4 α + 2 β ) dtdzz + − dtdζ + dx i + (1 − α − αβ ) dz z − α dzdζz . (2.3)We see that up to an overall scaling, the deformation by dtdzz and dzdζz can be undoneby the simple coordinate change. The global structure is slightly changed by the rescal-ing of variables. Conversely, under certain reasonable assumptions, the non-relativisticconformal background embedded in 2-dimension higher is locally unique as was shown in[6]. To see this we simply set α = 0 and rewrite the metric as ds d +2 = 2 ( − a ) dt z − adtdzz + − dtdζ + dx i + dz z , (2.4)where we have rescaled ( t, ζ ) → ( √ − a t, ζ √ − a ) so that ζ ≃ ζ + 2 π √ − a R . In thiscoordinate, the non-relativistic conformal transformation are realized in a slightly differentmanner than in the metric (2.1):( t, ζ , x i , z ) → (cid:18) t ηt , ζ − a log(1 + ηt ) − η x i x i + z ηt , x i ηt , z ηt (cid:19) . (2.5)In the limit a → a →
1, our metric is locally diffeomorphic to
AdS d +2 space (by the simplecoordinate transformation ζ → ζ + log z ). The latter space was studied in [7]. One canregard the metric (2.4) as a one-parameter interpolation between the metric studied in[1][2] and that in [7], where the both claim that the each model is dual to a non-relativisticconformal field theory. They are rather continuously connected.We see that some components of the curvature tensor are singular near z → a = 0 background just because they are locally diffeomorphic. A test particle will feelthe infinite tidal force near the conformal boundary. One particular component of theRiemann tensor, for instance, scales as R ζtzt ∝ a (1 − a ) /z , and it appears more singularthan in the case a = 0. Note that all these are just a consequence of the scale invarianceexcept for a proportionality constant, so they are not unexpected. Again one can removethis apparently more singular behavior by the coordinate transformation.We now move on to the equation of motion that has a solution (2.4). We first notethat the Einstein tensor for the metric (2.4) is given by the sum of “vacuum energy” ∝ g µν T tt : T µν = − ˜Λ g µν − ˜ Eδ µ δ ν g (2.6)To realize this form of the energy-momentum tensor, let us consider the following Einsteinaction coupled with a massive vector field (Proca action): S = Z d d x i dζ dtdz √− g (cid:18) R − Λ − F µν F µν − m A µ A µ (cid:19) , (2.7)where F µν = ∂ µ A ν − ∂ ν A µ . One can show that A = A µ dx µ = − √ − a z dt solves the Procaequation as well as the Einstein equation, providedΛ = −
12 ( d + 1)( d + 2) , m = 2( d + 2) . (2.8)In particular, for a = 1, the metric (2.4) is the solution of the vacuum Einstein equationwith cosmological constant. In this case, it is not necessary to introduce the massivevector field.Finally, we would like to briefly mention Class 2 deformation given by dζ . For sim-plicity, we turn off Class 3 deformations because they are locally cancelled by a coordinatetransformation. ds d +2 = − dt z + − dtdζ + dx i + dz z + cdζ . (2.9)Class 2 deformation breaks Galilean invariance as well as non-relativistic conformal invari-ance, and the Einstein tensor becomes truly anisotropic. One can no longer decomposethe Einstein tensor into the vacuum energy and the dust contribution unlike in Class 3deformation. Explicitly, the Einstein tensor takes the following form: G tt = 6 − d ( d + 1)(1 + 2 c )(1 + 2 c ) z , G tζ = − c + ( d + 1)( d + 2)(1 + 2 c )2(1 + 2 c ) z ,G ζζ = c − c + ( d + 2)( d + 3)(1 + 2 c )(1 + 2 c ) , G ii = 4 c + ( d + 1)( d + 2)(1 + 2 c )2(1 + 2 c ) z ,G zz = − c + ( d + 1)( d + 2)(1 + 2 c )2(1 + 2 c ) z , (2.10) It is also possible to replace the massive vector field with a particular scalar electro-dynamics in thebroken phase as in [2]. ζ direction and the pressure is anisotropic.It can be shown that the massive vector field minimally coupled to gravity as in (2.7)cannot support the energy-momentum tensor (2.10) on shell. It is, nevertheless, possibleto introduce further matters/interactions in the action to support the energy-momentumtensor. Since it is not particularly illuminating to do this without specifying the under-lying gravitational theory such as the string theory, we briefly mention one bottom-upconstruction.Possible independent tensor structures of the energy-momentum tensor constructedfrom A = a t dtz + a ζ dζ are given by g µν , A µ A ν , F [ µα F αν ] and F [ µα F αβ F βρ F ρν ] up to the fourthorder without higher derivatives. Other combinations are not independent or simplyvanish upon symmetrization. One can now construct the desired energy momentum tensorby an appropriate linear combination: T µν = ˜Λ g µν + g A µ A ν + g F [ µα F αν ] + g F [ µα F βγ F γδ F δν ] . (2.11)We can show that (2.11) is able to support the energy momentum tensor (2.10) by tuning a t and a ζ (and the coupling constants g i ). Note that we have five independent componentsof the Einstein equation, so the number of unknowns is sufficient. The equations of motionfor A should be fine-tuned as well so that the given a t and a ζ are solutions, which is alwayspossible in principle. So far, in our discussion, we have assumed that the scale invariant deformation preservesthe U (1) particle number. Once we relax the condition, there is no reason to retain ζ direction any more. Indeed, the non-relativistic gravity background that does not haveadditional U (1) isometry has been discussed in [5]. The metric takes the following form ds d +1 = − dt z + dx i + dz z . (2.12)The energy momentum tensor is supported by the massive vector field as in (2.7) without ζ direction, where m = d , Λ = − d +1) and A = dt √ z .The metric (2.12) is invariant under 7 Translations in ( t, x i ). • Rotations of x i . • Scale invariance: ( t, x i , z ) → ( λ t, λx i , λz ) so the dynamical critical exponent Z = 2.The metric is not invariant under the Galilean boost.Possible scale invariant deformations are restricted: • Class 1’: The deformations that preserve the same symmetry as (2.12) given by δds = (cid:8) dtdzz , dx i dzz (cid:9) . They are rather trivial because it can be undone by thecoordinate transformation ( t, x i ) → ( t + αz , x i + β i z ). • Class 2’: The deformations that break the rotational invariance as well as timereversal or spatial parity: δds = (cid:8) dtdx i z (cid:9) . This is the genuine deformation of thetheory. Our later applications, however, do not assume such a further symmetrybreaking of rotational invariance or parity, so we will not pursue this direction inthis paper.Conversely, as studied in [6], the d + 1 dimensional metric that has the same symmetryas the d dimensional Lifshitz field theory is locally unique. Having obtained the gravity background with the right symmetry, we can compute thecorrelation functions of the corresponding boundary theories by using the same holo-graphic technique employed in AdS/CFT correspondence. For this purpose, firstly, weconsider the scalar field propagating in the background (2.4) with the minimal action: S = − Z d d x i dζ dtdz √− g ( g µν ∂ µ φ † ∂ ν φ + m φ † φ ) . (2.13)As discussed before, ζ direction is compactified, and we focus on the mode with ζ KKmomentum i∂ ζ φ M = M φ M . With this assumption the field φ is dual to the boundaryoperator O which has a definite particle number M .After Fourier transforming in space-time directions, the equation of motion takes theform ∂ z φ − z ( d + 1 − iM a ) ∂ z φ + (cid:18) M w − k − m z − ( d + 2) iM az (cid:19) φ = 0 , (2.14)8here m = m + (2 − a ) M . A solution relevant to us is φ = z d/ − iMa K ν ( pz ) , (2.15)where p = √ k − M w and ν = q m − M a + ( d +2) .The two-point function of primary operators in the field theory is fixed by the non-relativistic invariance [8] up to an overall constant and it is given by h O ( k, w ) O † ( − k, − w ) i ∼ ( k − M w ) ν , (2.16)which we can explicitly confirm by the holographic computation. In the coordinate space, it is h O ( x, t ) O † (0 , i ∼ θ ( t ) 1 | ǫ t | ∆ e − iMx | t | , (2.17)where we have introduced UV cut-off ǫ and used the same regularization employed in [2].The scaling dimension ∆ is related to ν as ∆ = d +22 ± ν , where − signature is only possiblefor 0 < ν < θ ( t ) in (2.17). This causal structurewill be relevant in the application to quantum finance in section 3. As we will see, onceGalilean invariance is broken, this specific causal structure could be lost.Let us now consider the Galilean violating Class 2 deformation (2.9). Again for sim-plicity we turn off Class 3 deformation. The equation of motion for the scalar field as in(2.13) becomes ∂ z φ − z ( d + 1) ∂ z φ + (cid:18) c c w z − k + 2 M w c − m z (cid:19) φ = 0 , (2.18)where m = m + M c . The solution of the scalar equation of motion relevant to ourstudy is given by φ = e √ cwz i √ c +1 z d +1+ ν × More precisely, the Galilean invariance is sufficient to fix the form of the two-point function. This is not a trivial result even if our metric is locally diffeomorphic to the one studied in [1][2]. Recallthat the coordinate transformation in the bulk may induce a similar transformation in the boundarytheory. For instance, in the usual relativistic AdS/CFT, the correlation functions in the global AdSspace are different from those in the Poincar´e patch, related by the global conformal transformation. Thecomputation here shows that our coordinate change does not induce such a global transformation in theboundary theory. (1 + 2 c ) k + 2( − M + i p c (2 c + 1)(1 + ν )) w i p c (2 c + 1) w , ν ; − √ cwz i √ c ! , (2.19)where U ( a, b ; x ) is the confluent Hypergeometric function (we refer appendix C for details),and we have introduced ν = q m + ( d +2) as before (with a = 0). Note that in thisexpression, c → G ( z, w, k ) satisfies the same equation of motion (2.18)with the boundary condition G ( ǫ, w, k ) = 1. Subsequently, the boundary two-point func-tion is computed as h O ( k, w ) O † ( − k, − w ) i = G ( ǫ, k, w ) √ gg zz ∂ z G ( ǫ, k, w )= C w ν Γ( − ν )Γ (cid:18) (1+2 c ) k +2( − M + i √ c (2 c +1)(1+ ν )) w i √ c (2 c +1) w (cid:19) Γ( ν )Γ (cid:18) (1+2 c ) k +2( − M + i √ c (2 c +1)(1 − ν )) w i √ c (2 c +1) w (cid:19) + ultra local terms , (2.20)where we have omitted the ultra local contribution (= contact terms) in the boundarypropagator. C is a normalization constant independent of w and k .It is interesting to note that the boundary two-point function is same as that forthe Lifshitz-type background if one sets M = 0 up to a straightforward rescaling ofparameters. The latter has been computed as [5] h O ( k, w ) O † ( − k, − w ) i = C w ν Γ( − ν )Γ (cid:16) k +2 i (1+ ν ) w iw (cid:17) Γ( ν )Γ (cid:16) k +2 i (1 − ν ) w iw (cid:17) + ultra local terms . (2.21)A priori, this is not expected because the breaking of the Galilean invariance could intro-duce any function of f ( k /w ) in the boundary two-point function. The role of non-zero M simply shifts the momentum k → k − Mw c . Z = 2 and Galilean invariance We can generalized our construction by relaxing the dynamical critical exponent Z = 2.Although the non-relativistic conformal symmetry is broken, we are still able to preserve It is important to note that the limits M → c → c → M = 0 sector would be completely frozen as shown in [9]. Z 6 = 2 is given by ds d +2 = − dt z Z + − dζ dt + dx i + dz z . (2.22)Without breaking the Galilean invariance, one can introduce an analogue of Class 3deformation by δds = (cid:8) dζdzz , dtdzz Z +1 (cid:9) . Note that ζ now possesses a non-trivial scaling ζ → r −Z ζ .The two-point function of primary operators in this background is fixed by the Galileaninvariance. In the momentum space, it is given by h O ( w, k ) O † ( − w, − k ) i ∝ ( k − M w ) ν . (2.23)At first sight, this seems to be inconsistent with the dynamical critical exponent Z 6 = 2.However, one should be reminded that the particle number M also scales under the scalingsymmetry when Z 6 = 2: i [ D, M ] = (2 − Z ) M so that (2.23) transforms with weight 2 ν under the dilatation. Note that because of this non-trivial scaling, ζ direction cannotbe compactified. It may be rather interpreted as d + 1 dimensional field theory withadditional coordinate ζ .With this difficulty, we have to abandon the Galilean invariance to compactify ζ di-rection. The general metric ansatz would be ds = − dt z Z + − dζ dtz Z + dx i + dz z + cdζ , (2.24)where ζ is invariant under the dilation so that we can compactify ζ ∼ ζ + 2 R .In order to study the correlation functions, we introduce a minimally coupled scalarfield φ as in (2.13). The equation of motion is given by ∂ z φ − z ( d + 1) ∂ z φ + (cid:18) cz Z− c w − k + 2 M w c z Z− − m z (cid:19) φ = 0 , (2.25)where m = m + 2 M c . As in Z = 2, by setting M = 0, we obtain the same scalarequation of motion as in the Lifshitz type background studied in [5] (up to rescaling ofvariables).In order to compute the boundary two-point functions, we have to solve the equation(2.25) to obtain the bulk-boundary propagator. Unfortunately, we have been unable to11nd simple analytical solutions except for Z = 4 and c = 0. In this special case, we findthat the field profile is φ = e − i √ Mw √ z z d +22 + ν U √ k i √ M w + 12 (1 + ν ) , ν ; i √ M wz ! , (2.26)where ν = q m + ( d +2) . Accordingly, the two-point function is computed as h O ( k, w ) O † ( − k, − w ) i = C w ν Γ( − ν )Γ (cid:16) √ k i √ Mw + ν (cid:17) Γ( ν )Γ (cid:16) √ k i √ Mw + − ν (cid:17) + ultra local terms . (2.27)A similar observation has been made in [10]. Given the gravity solution, we discuss possible applications to the strongly coupled dualsystem in this section. We pick up two particular subjects, i.e. the scaling regime ofthe Reggeon field theory and the non-linear quantum finance. While our gravity solutionmight have more conventional applications such as in condensed matter physics, we chooserather “non-standard” applications so as to expand the realm of the “gauge/gravity”correspondence as discussed in Introduction. We hope our discussions will stimulate thefurther study on these yet unexplored subjects.
Non-relativistic conformal invariance was first discovered as a symmetry of the free Sch¨odingerequation [11][12]. The symmetry itself, however, could appear as an effective actionof some quasi-particles much like the relativistic dispersion relation and the emergent“Poincar´e invariance” that can be realized in condensed matter systems.One of such realizations is the physics of Reggeon in the high energy scatteringregime. It has been shown that in the large s limit with fixed momentum transfer t , the Reggeon exchange contributing to the partial wave expansion of the scattering ampli-tude is described by the so-called Reggeon field theory (see e.g. [13][14] for reviews). TheReggeon field theory is a (1 + 2) dimensional field theory living in t ∼ log s and transverse We denote the Mandelstam variables by boldface such as s and t . ⊥ = ( x , x ) plane. The “energy” E of the Reggeon field theory is identified with thecomplex angular momentum 1 − J of the underlying relativistic field theory (QCD).The non-interacting Reggeon field theory has a (1 + 2) dimensional dispersion relation E = 1 − J = α ′ k , (3.1)where α ′ is the Regge slope. The corresponding field theory can be formulated by the“Schr¨odinger” action S = Z dtdx (cid:0) iψ † ∂ t ψ − α ′ ( ∂ i ψ † ∂ i ψ ) − V ( ψ ) (cid:1) . (3.2)Note that the free part without the potential V ( ψ ) has the full symmetry of the non-relativistic conformal invariance.We would like to understand the IR limit of the Reggeon field theory, where the theoryis supposed to show scaling invariance. Our hope is that with the non-trivial potential V ( ψ ), the IR limit may show non-trivial strongly coupled fixed point that could be studiedby the gravity dual discussed in section 2. As we have seen in section 2, the structure ofthe gravity dual is rather robust, so we hope that the prediction from the gravity dual isuniversal.The properties of the IR fixed point theory depend on the symmetry assumption aboutthe potential V ( ψ ). In the literature [15], it has been suggested that, when the Reggeonis identified with the Pomeron, which has the vacuum quantum number, the lowest orderinteraction is given by the cubic coupling with a non-Hermitian coefficient: V P ( ψ ) = ir ( ψ ψ † + ψ † ψ ) , (3.3)which describes the three Pomeron interaction with intensity r . The non-trivial zero ofthe beta function for r has been found within the ǫ = 4 − d expansion. Note that thecubic coupling breaks particle number conservation and it naturally generates a non-zeromass term V mass = ψ † ψ as quantum corrections, so in order to obtain a non-trivial scalingregime in IR, we have to fine-tune the bare mass parameter (much like the Wilson-Fisherfixed point of λφ theory in 4 − ǫ dimension).Alternatively, one could imagine a hypothetical world where the Reggeon has a con-served particle number. In our real word, the Pomeron cannot have such a conservedquantum number, but we might expect that higher Reggeon might be described by such a13onserved quasi-particle in a certain parameter regime. In that case, the lowest interactionwould be V P ( ψ ) = λ | ψ | . (3.4)The interaction preserves the non-relativistic conformal invariance once we assume theexistence of a non-trivial fixed point. See appendix A for more discussions on the con-formal invariance at the one-loop order. The gravity dual that has the non-relativisticconformal invariance has been discussed in section 2.1. The Reggeon propagator is givenby G ( k, E ) = C ( E − α ′ k ) ν , (3.5)whose form is fixed by the Galilean invariance.Now let us go back to the originally proposed cubic Pomeron interaction. Since thenon-relativistic conformal invariance as well as the particle number conservation is broken,we expect the gravity dual is close to that studied in section 2.2, if any. Can we understandthe physics of the Reggeon field theory from the Lifshitz type gravity background or its1-dimensional higher counterpart (2.9)?The key ingredient to answer this question is to understand the dispersion relationfrom the predicted two-point function (2.20) from the gravity. We can easily see thatin the physical parameter region, there are no poles in (2.20) or (2.21) by noting Γ( z )has only single poles at negative integers (including zero), and Γ( z ) does not have zeroanywhere in z plane. Thus we see that the gravity computation would not predict anyquasi-particle excitation for the Reggeon field theory.One might wonder whether the prediction makes sense as a Reggeon field theory. Wecould circumvent the problem by doing a specific analytic continuation in (2.20) to use anegative value of − ≤ c ≤
0. Then, the Gamma function yields poles at w = α ′ eff k ,where the effective Regge slope is given by α ′ eff = 1 + 2 c M − i p c (2 c + 1)(1 + ν ) − iN p c (2 c + 1) (3.6)with a negative integer N = 0 , − , − , · · · . Accordingly, the two-point function near the14ole scales as G ( k, E ) = C E − ν +1 E − α ′ eff k . (3.7)The form is also predicted from the ǫ expansion of the Reggeon field theory (by assuming Z = 2: see [14] and references therein). We note that Γ function yields infinitely manypoles with different effective Regge slope α ′ eff . The residues of the poles have alternatingsignatures, so the higher poles may not be physical. m = 0 gives the largest Regge slope.From the gravity viewpoint, the reason why we have to take negative c is not obvious.The ζ direction is a closed time-like curve. On the other hand, the Reggeon field theory byitself is non-unitary because the interaction is anti-Hermitian, so one might not excludethe possibility that the gravity dual could be a non-unitary theory as indicated by thenegative c .Finally, we would like to study the deviation from Z = 2. While the tree level actionof the Reggeon field theory is compatible with the Z = 2 scaling, there is no protection ofthe dynamical critical exponent Z from the renormalization once we break the Galileaninvariance (and particle number conservation). Indeed, the two-loop ǫ expansion predictsthat the dynamical critical exponent would be [17][18]2 Z = 1 + ǫ
24 + ǫ (cid:18) (cid:19) . (3.8)Similarly, to the first order in ǫ , the two-point function scales as G ( k, E ) = C E γ ( E − α ′ k E ξ ) , (3.9)where Z = − ξ = 1 .
74, and the anomalous dimension γ is given by γ = − ǫ − ǫ (cid:18) (cid:19) (3.10)In particular, γ = − .
32 for d = ǫ = 2.Since the relation between the underlying relativistic field theory and the effectiveReggeon field theory is quite non-trivial, the unitarity bound (e.g. Froissart bound and thecondition that the elastic cross section is less than the total cross section: σ el ≤ σ tot ) of the It is interesting to note that the quasi-particle spectrum is impossible in non-trivial relativistic con-formal field theories; see e.g. [16] for recent applications to (un)particle physics. The non-relativisticscale invariance can accommodate such a quasi-particle spectrum density in the two-point function. − γ ≤ Z ≤ Z ≤ Z = 1. The other bound − γ ≤ Z is equivalent to thecondition for the conformal dimension: ∆ ≥ d −
2. When the theory is conformallyinvariant, the condition is weaker than the unitarity bound ∆ ≥ d . In this way, theunitarity bound for the underlying theory is related to the physical requirement of thedual gravitational theory.The corresponding dual gravity has been described in section 2.4. Unfortunately,the gravity solution cannot predict the value of Z ≥
1. Only we could do is to studythe correlation functions with a given dynamical critical exponent Z . In particular, itwould be interesting to study the causal structure and dispersion relation of the two-point function gravity as we have done in Z = 2 above. We defer this study because theanalytic expression for the gravity two-point function is unavailable, and we would haveto resort to the numerical computation. In later section 3.4 we will come back to theissue of non-trivial dynamical critical exponent and its relation to fractal geometry in thecontext of non-linear quantum finance. Before we discuss the application of our holographic computation to the non-linear quan-tum finance, we briefly review the classical Black-Scholes-Merton model of option pricing[19][20] from the (supersymmetric) path integral viewpoint. Since the computation isreduced to a linear differential equation i.e. “Schr¨odinger equation”, we call it as linearquantum finance.Our starting point is the generalized Langevin equation: ∂X ( t ) ∂t = ∂W ( X ) ∂X + ση ( t ) , (3.11) We refer [21][22][23] for reviews of quantum physics approach to finance with the usage of pathintegral. η ( t ) is a Gaussian random noise that satisfies h η ( t ) η (0) i = δ ( t ) . (3.12)In the Black-Scholes-Merton model, the “superpotential” W ( X ) is simply given by alinear function W ( X ) = µX , where X ( t ) is the logarithm of the risky asset price S ( t )(i.e. S ( t ) = e X ( t ) ), but for a while we keep the superpotential more general.Suppose we would like to compute the expectation value of a certain quantity F ( X )averaged over the random paths satisfying (3.11). By introducing the Gaussian measurefor the random noise η ( t ) to realize (3.12), one can express it in the form of the pathintegral as h F ( X ) i = Z [ D η ][ D X ] F ( X ) J ( X ) δ (cid:18) ∂X∂t − ∂W∂X − ση (cid:19) exp (cid:18) − Z dt η (cid:19) . (3.13)The Jacobian appearing in the measure is the usual Fadeev-Popov factor associated withthe delta functional constraint to impose (3.11), and it can be exponentiated to a form ofaction by introducing Grassmann fields ¯ ψ ( t ) and ψ ( t ) as J ( X ) = Z [ D ψ ][ D ¯ ψ ] exp (cid:18) − σ Z dt ¯ ψ∂ t ψ − ∂ W∂X ¯ ψψ (cid:19) . (3.14)On the other hand, the Gaussian noise η can be removed in the path integral by usingthe delta functional constraint, so the Bosonic action becomes S bos = Z dt σ ( ∂ t X − ∂ X W ) . (3.15)We see that the total action is nothing but that of the supersymmetric quantum mechanicsin the Euclidean signature [24] upon partial integration of the cross term ∂ t X∂ X W = ∂ t W in the Bosonic action.The Black-Scholes-Merton model specifies the superpotential W = µX and discussesthe geometric Brownian motion. Since the superpotential is linear, the fermionic partcompletely decouples, so we can safely neglect the Jacobian factor (3.14) in the following.To interpret the parameters in the action, we note that the no-arbitrage assumptionrequires µ = r − σ , where r is the risk free interest rate (say, bank interest rate), and σ is the volatility of the risky asset (say, stock).Let us now compute a present value of a European call option as the simplest appli-cation of the Black-Scholes-Merton model. For this, we simply compute the (discounted)17xpectation value for C ( X ∗ , t ∗ ) = ( e X ∗ − E ) θ ( e X ∗ − E ) , (3.16)where E is the exercise price of the call option, and X ∗ is the value of the asset atthe maturity time t ∗ . We also impose the boundary condition X ( t ) = X = log S .Explicitly, the discounted European option price is given by C ( X , t )= e − r ( t ∗ − t ) Z ∞ log E dX ∗ Z X ( t ∗ )= X ∗ X ( t )= X [ D X ]( e X ∗ − E ) exp (cid:18) − Z dt σ ( ∂ t X − µ ) (cid:19) . (3.17)The Gaussian path integral for X ( t ) is readily performed as C ( X , t ) = Z ∞ log E dX ∗ ( e X ∗ − E ) G ( X − X ∗ , t − t ∗ ) , (3.18)where the propagator G ( X − X ∗ , t − t ∗ ) is just given by G ( X − X ∗ , t − t ∗ ) = e − r ( t ∗ − t ) p πσ ( t ∗ − t ) exp (cid:18) − ( X − X ∗ + µ ( t ∗ − t )) σ ( t ∗ − t ) (cid:19) , (3.19)which is nothing but a free particle propagator (with a constant external vector potential)corresponding to the Bosonic action (3.15). One may easily perform the remaining integral(3.18) to obtain the celebrated Black-Scholes formula for the European call option: C ( X , t ) = S N ( d ) − Ee − r ( t ∗ − t ) N ( d ) , (3.20)where d = log( S /E ) + ( r + σ )( t ∗ − t ) σ √ t ∗ − t d = d − σ √ t ∗ − t (3.21)and N ( x ) = √ π R x −∞ e − t dt . A European option is a contract that entitles one to buy a share of stock at time t ∗ at a fixed price E . He will gain S ( t ∗ ) − E when the stock price S ( t ∗ ) = e X ∗ is higher than E . On the other hand, hedoes not lose anything when the stock price is lower than E because he does not have to buy the stock.This is the origin of the step function in (3.16). Because of the risk free interest rate r , the value of the option is naturally deflated by the factor e − r ( t ∗ − t ) .
18y construction, we can regard C ( X , t ) as the “wavefunction”. Indeed, it satisfiesthe “Schr¨odinger equation” (or Kolmogorov backward equation): ∂C∂t = rC − µ ∂C∂X − σ ∂ C∂X . (3.22)Unconventional terms rC and µ ∂C∂X can be absorbed by the change of variables C ( X, t ) → e rt C ( X − µt, t ) to directly compare with the Wick rotated Schr¨odinger equation. Notethat with this change of variable, the propagator used in the Black-Scholes-Merton modelis simply the Wick rotated version of the Schr¨odinger propagator obtained in the gravityapproach in (2.17). The equation (3.22) is known as the Black-Scholes equation in finance.The propagator can be obtained either in the first quantized formulation presentedhere or in the second quantized formulation by quantizing the Schr¨odinger action, bothof which are equally valid in the linear quantum finance. We use the second quantizedmethod in the following subsections to discuss the non-linear quantum finance. As a first order approximation to the real market, the Black-Scholes-Merton model hasbeen quite successful both theoretically and experimentally. Theoretically, the Black-Scholes-Merton model has an advantage that it possesses equivalent Martingale measure,which fits with the efficient market hypothesis with no arbitrage. There are several problems, however: • The fluctuation in the real market is not Gaussian unlike (3.12). It does not decayas fast as Gaussian (fat tail problem), and the history shows that the catastrophicloss (gain) is more likely than the Gaussian model. • Furthermore, people argue that there is a slight non-trivial time dependence in themarket performance. The non-equal time auto-correlation is slightly positive whenthe time scale is small (i.e. the companies winning continues to win for a while:winner has a momentum). On the other hand, the correlation is slightly negativewhen the time scale is large (i.e. no company cannot consistently dominate the It roughly means that nobody cannot consistently beat the market no matter how smart he is. There is great evidence on this point; see e.g. [23][25][26] and references therein. Also some people believe the usage of market cycle. • The Gaussian hypothesis neglects possible non-trivial higher-point correlations (orcumulant) in the market. The non-trivial higher point correlation functions clearlyindicate the underlying non-linear nature of the market. People “hope” that byusing non-linear analysis, they may be able to beat the market because the non-linearity might break the efficient market hypothesis.There are many models to account for these issues, but we would like to focus on aparticular attempt suggested in [22] to use a non-linear quantum field theory because wemay be able to tackle the problem by using our gravitational dual realization. The ideais inspired by the appearance of the effective Schr¨odinger equation in the Black-Scholes-Merton model and its analogy to the Reggeon field theory. We take the second quantizedapproach and regard the particle propagator as h φ ( x , t ) φ † ( x, t ) i = Z [ D φ ] φ ( x .t ) φ † ( x, t ) exp( − S [ φ ]) , (3.23)where S [ φ ] = R dtdx (cid:16) − φ † ∂ t φ + σ ∂ x φ † ∂ x φ (cid:17) is the free Euclidean Sch¨odinger action.The proposal is that we replace the free Sch¨odinger action by the interacting action S [ φ ] = Z dtdx (cid:18) − φ † ∂ t φ + σ ∂ x φ † ∂ x φ + V ( φ ) (cid:19) . (3.24)In the short range, where δX and δt are small (i.e. in the UV limit), it is reduced to theoriginal Black-Scholes-Merton model, while when δX or δt are large (i.e. in the IR limit),the non-linearity will be of importance and the theory may show a non-trivial scalingregime as in the Reggeon field theory. The formulation is quite analogous to the Reggeonfield theory except that our “space time” is (1 + 1) dimension (or d = 1) rather than(1 + 2) dimension (or d = 2) in Reggeon field theory.The prescription is simply to replace the free particle propagator in the option pric-ing formula by the interacting propagator obtained from the non-linear action (3.24).We keep the form of the Schr¨odinger action by performing the replacement G ( X, t ) → e rt G ( X − µt, t ) at the end of the computation to incorporate the drift and deflation becausethe Schr¨odinger action possess a manifest enhanced symmetry (non-relativistic conformalsymmetry) and it is a good starting point to discuss its gravitational dual. This observation is controversial once we subtracted the underlying uptrend of the whole market.See e.g. [25] and references therein.
20e focus on the IR limit of the non-linear theory (3.24) so that we may expect tolearn the structure of the correlation functions from the gravity dual with the scalingsymmetry. As in the Reggeon field theory, the IR structure will depend on the symmetryassumption of the potential V ( φ ). The potential could depend on higher derivatives,while these higher derivative terms are always perturbatively irrelevant around the freefield fixed point.Let us first consider the case where the potential V ( φ ) preserves the Galilean invari-ance. Slightly weaker assumption that the theory is invariant under the particle numberleads to the same constraint on the potential as long as we discard any higher derivativeinteractions. The gravity dual corresponding to this assumption is the non-relativisticconformal invariant background studied in section 2.1. The non-relativistic conformalinvariance fixes the form of the propagator. In our context, the propagator should read G ( x, t ) = θ ( t ) e − rt | πσ t | ∆ e − ( x − µt )22 σ t (3.25)from section 2.4. The only non-trivial parameter is the scaling dimension ∆. Note thatthe exponential damping is not alleviated and still be present due to the non-relativisticconformal invariance. The option-pricing formula would be slightly modified by replacing √ t factor by | t | ∆ .Now, in more general situations, we do not impose the Galilean invariance (e.g. brokenby higher derivative interactions), or we even do not expect particle number conservation(e.g. by introducing a cubic coupling as in Reggeon field theory). The gravity dual forsuch deformations has been studied in section 2.1 and 2.2.We have studied the corresponding propagator in section 2.4. The most importantfeature of the two-point function studied there is its causal structure (see section 3.1).They do not have any quasi-particle dispersion relation as long as c >
0. The Greenfunction carries information both forward in time and backward in time. This may ormay not be a problem for the quantum financial interpretation because we would like toestimate the future option price from the available data at present. Analytic continuationto c < The particle number conservation may be important to understand the conservation of probabilityin quantum mechanics, while the situation and its necessity is less obvious in the quantum finance. ∼ θ ( t ) (cid:18) x − µtt (cid:19) − ν +1) e − rt q πσ eff t e − ( x − µt )22 σ eff t (3.26)for large t with x/t fixed as long as the poles are well separated. Here σ eff dependson the pole location, which is related to the effective Regge slopes in the gravity dualdescription of the Reggeon field theory (c.f. (3.6)). Thus, the option price behaves as if itwere composed of multiple securities with different effective volatilities. The appearanceof the multiple effective volatilities is of great significance in the real market and has beeninvestigated as a first step to improve the Black-Scholes-Merton model (e.g. GARCHmodel). It may also be related to the multi-fractal analysis of the market pursued in [26].On the other hand, at a specific parameter region of the scale invariant but non-conformal gravity background such as Lifshitz-type background, long distance (fixed time)propagators can be power-like without any exponential tail by tuning the conformal di-mension (see [5] for a detailed demonstration). Explicitly, the two-point function of op-erator O whose scaling dimension is 4 has been shown to behave as h O ( t, x ) O † (0 , i = C| x | (3.27)in | x | → ∞ limit. To obtain this result, we cannot use the formula (2.21) because theseparation between the analytic part and the non-analytic part becomes more complicatedthan the derivation of (2.21) when the scaling dimension is an integer. Similarly, whenever ν is a positive integer, the power-like behavior can appear. This also holds in the M = 0sector of the deformed theory studied in section 2.1. The emergent power-like decay isquite promising in the application of non-linear quantum finance to understand the fattail problem mentioned above.Finally, once we break the Galilean invariance or particle number conservation, thereis no reason why the dynamical critical exponent Z remains to be 2. Indeed, the non-trivial dynamical critical exponent is proposed in finance as well. As we have discussed insection 3.1, this is also expected in ǫ expansion of Reggeon field theory with no particlenumber conservation. We will further investigate this issue in the next subsection.22 .4 Deviation from Z = 2 : Hurst exponent We have seen that in Reggeon field theory with cubic non-Hermitian interaction, the ǫ expansion around d = 4 leads to a non-trivial dynamical critical exponent Z 6 = 2. This isnot totally unexpected because the violation of the non-relativistic conformal invarianceinduced by the violation of the Galilean invariance does not protect the dynamical criticalexponent.The concept of dynamical critical exponent was first introduced in finance by Mandel-brot (see [26] and references therein), and it is called a Hurst exponent. Let us considera random variable Q ( t ). When the random variable shows a scaling auto-correlationfunction h| Q ( t + T ) − Q ( t ) | q i = C q T q H (3.28)for q > − C q , we say that the random variable Q ( t ) has a Hurst exponent H . Here in order to define H , we have assumed that the underlying distribution for Q ( t )has stationary increment so that (3.28) does not depend on t . Note that H could dependon q due to the anomalous dimension of composite operators when the theory is highlynon-linear.For example, let us consider the case Q ( t ) = B ( t ) = R t η ( s ) ds with the Gaussianrandom noise η ( t ) satisfying (3.12). B ( t ) is the Brownian motion, and it is easy to see H = . Similarly fractional Brownian motion Q ( t ) = B H ( t ) = Z −∞ (cid:16) ( t − s ) H− − ( − s ) H− (cid:17) η ( s ) ds + Z t ( t − s ) H− η ( s ) ds (3.29)has a non-trivial Hurst exponent H 6 = which appeared in its definition. The Brownianmotion with H = is Martingale, while the fractional Brownian motion is not Martingaleand it shows a memory effect. The Hurst exponent is related to the fractal dimension ofthe sample path of the random process by d = 2 − H .Schematically one could write (3.28) as δX = ( δt ) H . If we compare this definitionwith our discussions of the non-linear quantum finance, we can identify H = Z from thedispersion, or scaling relation E = k Z , where Z is the dynamical critical exponent. Here,it is important to carefully subtract the drift contribution in order to satisfy the stationarycondition. The failure to do this in market data leads to a spurious Hurst exponent. Werecall that the Gaussian assumption leads to H = and this is the value mostly studied23n previous sections in particular with the non-relativistic conformal invariance ( Z = 2),which is manifest after the change of variables C ( X, t ) → e rt C ( X − µt, t ). Our non-linearaction (3.24) has the obvious time-translation invariance. Historical analysis of marketdata suggests that H is slightly above , but not significantly larger [25].An interpretation of the Hurst exponent in finance is that when < H <
1, thesecurity is persistent in memory, and when 0 < H < it is anti-persistent. The argumentis simple [26]. Let us compute the auto-correlation function C ( τ ) = h ( Q ( t ) − Q ( t − τ ))( Q ( t + τ ) − Q ( t )) ih ( Q ( t + τ ) − Q ( t )) i = 2 H− − , (3.30)where we have used (3.28) with q = 2, i.e. h ( Q ( t + τ ) − Q ( t )) i = C τ H . Since C ( τ )governs the persistence of memory, it leads to the above interpretation. The bound H < Z >
1, has been observed in the gravity before (c.f. the discussion in section 3.1 andits relation to the unitarity bound.) The Reggeon field theory in ǫ expansion predicts H = 0 .
57 for d = 2 and H = 0 .
64 for d = 1, which is consistent with the observation thatthe market is usually persistent in memory.The gravitational dual theory by itself cannot tell which exponent should be realizedin nature or in market. The analogue of the holographic running might be importantfrom the “UV” critical exponent Z = 2 ( H = ) to that for the “IR” exponent. Byintroducing two types of massive vector fields with different mass, which is related to Z ,we may obtain the geometry interpolating different dynamical critical exponents. On theother hand, the experimental determination of the Hurst exponent from the market datais a challenging subject. See [25][26] for some related discussions. We just mention thatbecause the non-trivial Hurst exponent typically indicates the violation of the Martingaleor the efficient market hypothesis, we expect the Hurst exponent or non-linear fixedpoint is rather unstable and cannot last forever. This is not necessarily the case especially when the distribution does not have stationary increment.See e.g. [27]. A similar argument could be stated in field theories. Non-local but free field theories suchas S = R dwd d kφ † ( w H − k ) ν φ known as generalized free field theories give non-trivial dynamical criticalexponent, but they are rather trivial. In such theories, higher cumulant does not contain any non-trivialinformation at all. Discussions
In this paper, we proposed novel applications of non-relativistic gauge/gravity correspon-dence. One is the Reggeon field theory and another is the non-linear quantum finance.The relation between the two has been suggested some time ago [22], and we have mademore concrete proposals in this paper. We have seen that the strongly coupled regimeof the both theories is beyond the scope of the perturbative field theories, and the dualgravity computation is promising. We have proposed the two-point functions from thegravity computation.The quantum finance is base on the relation between stochastic systems and quantumfield theories. We believe that the stochastic process in gauge/gravity correspondence isworth studying further. Recently, an effective Langevin equation for heavy charged parti-cles in the quark-gluon plasma has been computed by using the AdS/CFT correspondence[28][29][30], where they have pointed out its relation to black hole Hawking radiation. In-troducing external potential in their setup, we can study the geometric Brownian motionof stochastic strings, which may be relevant in quantum finance. The random noise pre-dicted in AdS/CFT is not white, so when it turns out to be solvable (possibly by studyingthe dual gravity regime), it might become a novel non-Gaussian market model.On the theoretical side, it would be interesting to find a way to convert their effectiveLangevin equations into the form of non-relativistic field theories as we have presentedin section 3.1. In this way, we may be able to find a clue to derive our proposed phe-nomenological gravity background from the string theory. All in all, there is no concreteexamples of non-relativistic gauge/gravity correspondence whose gauge side is identifiedwith the gravity side (and vice versa), so this approach could be a major breakthrough ifsuccessful [31].We would like to conclude this paper with a few words about our philosophy of ap-plying gauge/gravity correspondence to non-linear finance. Much like in the cold atomsor unitary fermion systems, there is currently no theoretical foundation that the gravitydescription is suitable for describing the financial market except for general symmetryarguments and its validity in the free theory limit. From the phenomenological modelbuilding perspective, there is nothing wrong about proposing a “solvable” model fromcompletely different perspective. Indeed, it might be better suited here than the con-densed matter application because we do not know the fundamental principle of market25nlike in the condensed matter systems. On the other hand, the Reggeon field theory does have its origin in QCD, so it would be wonderful to derive the phenomenologicalnon-relativistic background discussed in this paper from the AdS/QCD correspondencewhose theoretical foundation is much firmer.
Acknowledgements
The author would like to thank Soo-Jong Rey for suggesting AdS/Finance correspondence.The work was supported in part by the National Science Foundation under Grant No.PHY05-55662 and the UC Berkeley Center for Theoretical Physics.
A Scale invariance vs conformal invariance in Galileaninvariant theories
In this appendix, we discuss the energy-momentum tensor of Galilean invariant field the-ories to study the condition when the scale invariance indicates the conformal invariance.The non-relativistic energy-momentum tensor with the conformal invariance has beenstudied in [32]. We assume the translational invariance in time and space, which meanswe have a conserved energy-momentum tensor ∂ t T i + ∂ j T ji = 0 ∂ t T + ∂ i T i = 0 (A.1)corresponding to H = Z d d xT , P i = Z d d xT i . (A.2)The spatial rotational invariance demands that the energy-momentum tensor be symmet-ric T ij = T ji (note, however, T i = T i in general).We also assume that the theory is invariant under the Galilean boost by demandingthat the U (1) particle number density is related to the energy-momentum tensor m ˙ ρ = − ∂ i T i . (A.3)26hen the Galilean boost density G i = tT i − mx i ρ satisfies ∂ t G i + ∂ j ( tT ij − x i T j ) = 0 . (A.4)The corresponding conserved charge is G i = R d d x G i Now suppose that the energy-momentum tensor satisfies the following condition2 T − T ij δ ij = ∂ t S + ∂ j A j , (A.5)then one can show that the dilatation density D = tT − x i T i − S D = R d d x D . Thus, the condition (A.5)is the requirement of scale invariance. Note that one can always redefine 2 T → T + ∂ j A j to remove A j , so only the non-trivial condition is the existence of S . Furthermore,if S is a total divergence such that S = ∂ i σ i , then we can improve the energy-momentumtensor as T → T + ∂ j ∂ t σ j + ∂ i A i so that the right hand side of (A.5) is zero.The condition that the right hand side of (A.5) can be improved to be zero is ananalogue of the traceless condition for the energy-momentum tensor in relativistic fieldtheories [3]. As in the relativistic case, (A.5) indicates, if the right hand side vanishes, anadditional conserved density K = t T − tx i T i + m x ρ , (A.7)which generates non-relativistic conformal transformation whose charge is K = R d d x K .In order to show that (A.7) is conserved, the Galilean invariance (A.3) and the tracecondition on the energy-momentum tensor are crucial.The discussion here states that the scale invariance and Galilean invariance do notnecessarily imply the conformal invariance in non-relativistic field theories, much like inrelativistic field theories. The criterion of the conformal invariance is whether we couldimprove the energy-momentum tensor so that the trace condition 2 T − T ij δ ij = 0 issatisfied. A non-trivial possibility of S which is not a divergence of another current isthe obstruction. As far as we know, however, there is no known physical example ofnon-trivial S in literatures. 27et us do a no-relativistic version of the exercise done in [3]. Consider the non-relativistic action S = Z dtd d x (cid:18) iφ † a ∂ t φ a − m ∂ i φ † a ∂ i φ a − λ abcd φ a φ b φ † c φ † d (cid:19) (A.8)in d = 2 − ǫ dimension. The reality of the action demands λ abcd = λ ∗ cdab . To the first orderin ǫ , the candidate for S and A j is given by S = iξ ab φ † a φ b A j = ξ ab m (cid:16) φ † a ∂ j φ b + φ a ∂ j φ † b (cid:17) , (A.9)with (real) antisymmetric ξ ab . At one-loop, the scale invariance demands − ǫλ abcd + 116 π λ abef λ efcd + ξ mc λ abmd + ξ md λ abmc − ξ am λ mbcd − ξ bm λ macd = 0 (A.10)while the conformal invariance demands − ǫλ abcd + 116 π λ abef λ efcd = 0 . (A.11)The latter is in principle stronger than the former because of the additional unknowns ξ ab .We can show, however, that (A.10) implies (A.11) by contracting (A.10) with ξ ma λ cdmb + ξ mb λ cdma − ξ cm λ mdab − ξ dm λ mcab . Thus, the scale invariance and Galilean invariance doessuggest the conformal invariance in λ | φ | non-linear Schr¨odinger theory.Let us finish this appendix with another peculiar example in (1 + 0) dimension. Con-sider the non-relativistic Liouville-like Lagrangian L = iφ † ∂ t φ − µe | φ | = − ρ∂ t θ − µe ρ (A.12)where we have introduced the polar coordinate φ = √ ρe iθ . The theory is scale invariantunder ρ → ρ + λ . Note that if it were defined in (1 + d ) dimension ( d >
0) with additionalkinetic terms, the theory would be Galilean invariant (but not scale invariant). TheHamiltonian T = µe ρ = − ∂ t θ is a time derivative, so the dilatation charge D = tµe ρ + θ is conserved. However, since θ is not a derivative of something else, we cannot constructa conserved non-relativistic conformal charge. This example might suggest a possibilityto construct a counterexample, but we have not come up with any in higher dimensionswith Galilean invariance. 28 Non-relativistic conformal algebra
We summarize the non-relativistic conformal algebra for d = 2. For d = 1, there is noangular momentum J , and no indices for P and G in the following commutation relations. i [ J, P ] = − iP , i [ J, ¯ P ] = i ¯ P , i [ J, G ] = − iG , i [ J, ¯ G ] = i ¯ G ,i [ H, G ] =
P , i [ H, ¯ G ] = ¯ P , i [ K, P ] = − G , i [ K, ¯ P ] = − ¯ G ,i [ D, P ] = − P , i [ D, ¯ P ] = − ¯ P , i [ D, G ] =
G , i [ D, ¯ G ] = ¯ G ,i [ D, H ] = − H , i [ H, K ] =
D , i [ D, K ] = 2
K , i [ P, ¯ G ] = 2 M , (B.1)where J = J is U (1) angular momentum, P = P + iP and ¯ P = P − iP are spatialmomenta, H is the Hamiltonian, G = G + iG and ¯ G = G − iG are Galilean boost, D is the dilatation, K is the special conformal transformation, and M is the number densityoperator which is the center of the non-relativistic conformal algebra.The representation theory of the non-relativistic conformal algebra relevant for thenon-relativistic conformal field theories can be found in [33][34]. C Confluent Hypergeometric function
The confluent Hypergeometric function U ( a, b ; x ) is a solution of Kummer’s equation x d Udx + ( b − x ) dUdx − ax = 0 (C.1)with the series expansion U ( a, b ; x ) = x − b (cid:20) Γ( − b )Γ( a ) + ( − − a + b )Γ( − b )Γ( a ) x + · · · (cid:21) + Γ(1 − b )Γ(1 + a − b ) − a Γ( − b )Γ(1 + a − b ) x + · · · . (C.2)Alternatively it has an integral representation U ( a, b ; x ) = 1Γ( a ) Z ∞ dte − xt t a − (1 + t ) b − a − . (C.3)It satisfies U ( a, b ; x ) = z − b U (1 + a − b, − b ; x ) . (C.4)29 eferences [1] D. T. Son, Phys. Rev. D , 046003 (2008) [arXiv:0804.3972 [hep-th]].[2] K. Balasubramanian and J. McGreevy, Phys. Rev. Lett. , 061601 (2008)[arXiv:0804.4053 [hep-th]].[3] J. Polchinski, Nucl. Phys. B , 226 (1988).[4] C. M. Ho and Y. Nakayama, JHEP , 109 (2008) [arXiv:0804.3635 [hep-th]].[5] S. Kachru, X. Liu and M. Mulligan, Phys. Rev. D , 106005 (2008) [arXiv:0808.1725[hep-th]].[6] S. Schafer-Nameki, M. Yamazaki and K. Yoshida, arXiv:0903.4245 [hep-th].[7] W. D. Goldberger, JHEP , 069 (2009) [arXiv:0806.2867 [hep-th]].[8] M. Henkel, J. Statist. Phys. , 1023 (1994) [arXiv:hep-th/9310081].[9] Y. Nakayama and S. J. Rey, arXiv:0905.2940 [hep-th].[10] S. Sekhar Pal, arXiv:0808.3232 [hep-th].[11] C. R. Hagen, Phys. Rev. D , 377 (1972).[12] U. Niederer, Helv. Phys. Acta (1972) 802.[13] H. D. I. Abarbanel, J. B. Bronzan, R. L. Sugar and A. R. White, Phys. Rept. ,119 (1975).[14] M. Moshe, Phys. Rept. , 255 (1978).[15] V. N. Gribov, Sov. Phys. JETP (1968) 414 [Zh. Eksp. Teor. Fiz. (1967) 654].[16] H. Georgi, Phys. Rev. Lett. , 221601 (2007) [arXiv:hep-ph/0703260].[17] M. Baker, Phys. Lett. B , 158 (1974).[18] J. B. Bronzan and J. W. Dash, Phys. Rev. D , 4208 (1974) [Phys. Lett. B , 496(1974 ERRAT,D12,1850.1975)]. 3019] F. Black, M. Scholes, Journal of Political Economy 81 (3): 637-654 (1973).[20] R. Merton, Bell Journal of Economics and Management Science 4 (1): 141-183 (1973).[21] E. Baaquie, “Quantum finance : path integrals and Hamiltonians for options andinterest rate” Cambridge University Press, 2004.[22] J. Dash, “Quantitative finance and risk management : a physicist’s approach” WorldScientific Pub., 2004.[23] H. Kleinert, “Path integrals in quantum mechanics, statistics, polymer physics, andfinancial markets” World Scientific, 2006.[24] G. Parisi and N. Sourlas, Nucl. Phys. B , 321 (1982).[25] A. Lo and A. MacKinlay, “A Non-Random Walk Down Wall Street”, PrincetonUniversity Press, 1999.[26] B. Mandelbrot, “Fractals and Scaling in Finance”, Springer Verlag, 1997.[27] J. McCauley, K. Bassler, and G. Gunaratne , Physica A37, 202, 2008.[28] J. de Boer, V. E. Hubeny, M. Rangamani and M. Shigemori, arXiv:0812.5112 [hep-th].[29] D. T. Son and D. Teaney, arXiv:0901.2338 [hep-th].[30] G. C. Giecold, E. Iancu and A. H. Mueller, arXiv:0903.1840 [hep-th].[31] Work in progress.[32] R. Jackiw and S. Y. Pi, Phys. Rev. D , 3500 (1990) [Erratum-ibid. D , 3929(1993)].[33] V. K. Dobrev, H. D. Doebner and C. Mrugalla, Reports on mathematical physics,39, 201 (1997)[34] Y. Nakayama, JHEP0810