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Holographic description of quantum field theory Sung-Sik Lee Department of Physics & Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada (Dated: January 24,2010) Abstract 0 1 0 We propose that general D-dimensional quantum field theories are dual to (D + 1)-dimensional local 2 quantumtheorieswhichingeneralincludeobjectswithspintwoorhigher. Usingageneralprescription,we n a J construct a (D+1)-dimensional theory which isholographically dual tothe D-dimensional O(N) vector 4 model. Fromtheholographic theory, thephasetransitionandcriticalpropertiesofthemodelindimensions 2 ] D > 2aredescribed. h t - p e h [ 2 v 3 2 2 5 . 2 1 9 0 : v i X r a 1 I. INTRODUCTION Quantum field theory is a universal language that describes long wavelength fluctuations in quantum systems made of many degrees of freedom. Although strongly coupled quantum field theories commonly arise in nature, it is notoriously difficult to find a systematic way of under- standingstronglycoupledquantumfield theories. Theanti-deSitterspace/conformalfieldtheorycorrespondence[1–3]openedthedoortounder- standaclassofstronglycoupledquantumfieldtheories. Accordingtotheduality,certainstrongly coupled quantum field theories in D dimensions can be mapped into weakly coupled gravitational theories in (D + 1) dimensions in large N limits. Although the original correspondence hasbeenconjecturedbased onthesuperstringtheory,itispossiblethattheunderlyingprincipleis more general and a wider class of quantum field theories can be understood through holographic descriptions[4–7], whichmay havedifferent UV completionthanthestringtheory. In this paper, we provide a prescription to construct holographic theories for general quantum field theories. As a demonstration of the method, we explicitly construct a dual theory for the D-dimensional O(N) vector model, and reproduce the phase transition and critical behaviors of themodelusingtheholographicdescription. The paper is organized in the following way. In Sec. II, we will convey the main idea behind the holographic description by constructing a dual theory for a toy model. In Sec. III A, using the general idea presented in Sec. II, we will explicitlyconstruct a holographictheory dual to the D-dimensional O(N) vector model. In Sec. III B, we will consider a large N limit where the theorybecomesclassicalforO(N)singletfieldsinthebulk. InSec. IIIC,thephasetransitionand critical propertiesoftheO(N) modelwillbediscussedusingtheholographictheory. II. TOY-MODEL:0-DIMENSIONALSCALARTHEORY In this section, we will construct a holographic theory for one of the simplest models : 0- dimensional scalar theory. In zero dimension, the partition function is given by an ordinary inte- gration, Z[ ] = dΦe S[Φ]. (1) − J Z 2 Weconsideran actionS[Φ] = S [Φ]+S [Φ] with M J S [Φ] = M2Φ2, M ∞ S [Φ] = Φn. (2) n J J n=1 X Here S is the bare action with ‘mass’ M. S is a deformation with sources . The values of M n J J ’s are notnecessarily small. In thefollowing,wewillconsiderdeformationsuptoquarticorder n J : = 0 for n > 4. However, the following discussion can be straightforwardly generalized to n J moregeneral cases. For a given set of sources , quantum fluctuations are controlled by the bare mass M. One n J useful way of organizing quantum fluctuations is to separate high energy modes and low energy modes,andincludehighenergyfluctuationsthroughaneffectiveactionforthelowenergymodes. Although there is only one scalar variable in this case, this can be done through the Polchinski’s renormalizationgroupscheme[8]. First,an auxiliaryfield Φ˜ withmassµ isintroduced, Z[ ] = µ dΦdΦ˜ e (S[Φ]+µ2Φ˜2). (3) − J Z At this stage, Φ˜ is a pure auxiliary field without any physical significance. Then, we find a new basisφ and φ˜ ˜ Φ = φ+φ, Φ˜ = Aφ+Bφ˜, (4) insuchawaythatthe‘lowenergyfield’φhasamassM′ whichisslightlylargerthantheoriginal massM. Asaresult,quantumfluctuationsforφbecomeslightlysmallerthantheoriginalfieldΦ. The missing quantum fluctuations are compensated by the ‘high energy field’ φ˜with mass m′. If wechoosethemassofthelowenergy field φ as M′2 = M2e2αdz (5) with dz being an infinitesimally small parameter and α being a positive constant, we have to choose ′ ′ MM m M A = , B = , (6) − m′µ M′µ where e2αdz M2 m′2 = M2 = . (7) e2αdz 1 2αdz − 3 Note that m′2 is very large, proportional to 1/dz. This is because φ˜ carries away only infinites- ′ imally small quantum fluctuations of the original field Φ. Moreover, m is independent of the arbitrary massµ becauseφ˜isphysical. Interms ofthenew variables,thepartitionfunctioniswrittenas ′ ′ Z[ ] = Mm + MM dφdφ˜e (SJ[φ+φ˜]+M′2φ2+m′2φ˜2). (8) J M′ m′ − (cid:18) (cid:19)Z Ifwerescale thefields, φ e αdzφ, φ˜ e αdzφ˜, (9) − − → → the quadratic action for low energy field φ can be brought into the form which is the same as the originalbareaction, Z[ ] = m dφdφ˜e (Sj[φ+φ˜]+M2φ2+m2φ˜2), (10) − J Z where 4 S [φ+φ˜] = j (φ+φ˜)n, (11) j n n=1 X with j = e nαdz and m = m′e αdz. Note that j ’s become smaller than the original sources n n − − n J , which isamanifestationofreduced quantumfluctuationsfor thelowenergy field φ. Thenew n J actioncan beexpandedinpowerofthelowenergy field, S [φ+φ˜] = S [φ˜]+(j +2j φ˜+3j φ˜2 +4j φ˜3)φ j j 1 2 3 4 +(j +3j φ˜+6j φ˜2)φ2 +(j +4j φ˜)φ3 +j φ4. (12) 2 3 4 3 4 4 In the standard renormalization group (RG) procedure[8, 9], one integrates out the high energy field to obtain an effective action for the low energy field with renormalized coupling constants. Here we take an alternative view and interpret the high energy field φ˜ as fluctuating sources for the low energy field. This means that the sources for the low energy field can be regarded as dynamicalfields insteadoffixed couplingconstants. To makethis moreexplicit,we decouplethe highenergy field and thelowenergy field byintroducingHubbard-Stratonovichfields J andP , n n Z[J] = m dφdφ˜Π4n=1(dJndPn) e−(Sj′+M2φ2+m2φ˜2), (13) Z 4 where ′ ˜ S = S [φ] j j +iP J iP (j +2j φ˜+3j φ˜2 +4j φ˜3)+J φ 1 1 1 1 2 3 4 1 − +iP J iP (j +3j φ˜+6j φ˜2)+J φ2 2 2 2 2 3 4 2 − +iP J iP (j +4j φ˜)+J φ3 3 3 3 3 4 3 − +iP J iP j +J φ4. (14) 4 4 4 4 4 − Nowweintegrateoutφ˜toobtainaneffectiveactionforthesourcefields. Themassm2forthehigh energy field is proportional to 1/dz and only terms that are linear in dz contributeto theeffective action(for thederivation,seetheAppendixA), Z[ ] = dφΠ4 (dJ dP ) e (SJ[φ]+M2φ2+S(1)[J,P]), (15) J n=1 n n − Z where 4 S(1)[J,P] = i(J +nαdz )P n n n n −J J n=1 X αdz + (i ˜ +2P ˜ +3P ˜ +4P ˜)2 (16) 2M2 J1 1J2 2J3 3J4 with ˜ = ( +J )/2. n n n J J After repeating the steps from Eqs. (3) to (15) R times, one obtains a path integral for the partitionfunction Z[ ] = ΠR Π4 (DJ(k+1)DP(k))e S(R)[J(k),P(k)]Z[J(R+1)], (17) J k=1 n=1 n n − Z where R 4 S(R)[J(k),P(k)] = i(J(k+1) J(k) +nαdzJ(k))P(k) n − n n n Xk=1hXn=1 αdz + (iJ˜(k) +2P(k)J˜(k) +3P(k)J˜(k) +4P(k)J˜(k))2 (18) 2M2 1 1 2 2 3 3 4 i withJ˜(k) = (J(k+1) +J(k))/2and J(1) = . Thenon-trivialsolutionforEq. (17)isgivenby n n n n n J Z[ ] = Π4 (DJ DP ) e S[J,P], (19) J n=1 n n − Z where ∞ S[J,P] = dz i(∂ J +nαJ )P z n n n Z0 α h + (iJ +2P J +3P J +4P J )2 . (20) 2M2 1 1 2 2 3 3 4 i 5 Here DJ DP represent functional integrations over one dimensional fields J (z),P (z) which n n n n are defined on the semi-infiniteline [0, ). The boundary valueof J (z) is fixed by the coupling n ∞ constants of the original theory, J (0) = . P (z) is the conjugate field of J (z). The physical n n n n J meaning of P becomes clear from the equation of motionfor thecorresponding source field. By n takingderivativeofEq. (14) withrespect toJ , oneobtains n < φn >= i < P > . (21) n − Therefore, P describes physical fluctuations of the operator φn, and an expectation value of P n n alongtheimaginaryaxisgivesan expectationvalueoftheoperator. Thetheory givenby Eqs. (19) and (20), which is exactly dual to the originaltheory, is an one- dimensional local quantum theory. The emergent dimension z corresponds to logarithmicenergy scale[10]. The parameter α determines the rate the energy scale is changed. Despite the apparent similaritywiththestandardRGtheory,thereisanimportantdifference. IntheusualRGapproach, quantum fluctuations of high energy modes modify coupling constants for low energy modes and generated new terms in the effectiveaction. In the present approach, new terms are not generated and the structure of the bare theory is maintained at each level. In particular, the highest order coupling(J inthiscase) obeysthestrictconstraint, 4 ∂ J +4αJ = 0 (22) z 4 4 which is nothing but the classical scaling. This is because there is no dynamics for the conjugate field P for the highest order coupling. Instead, other couplings acquire non-trivialdynamics and 4 some of them become propagating modes in the bulk. Fluctuations of those dynamical coupling fieldsembodyquantumfluctuationsinthisapproach. Inquantumfieldtheory,thereisaredundancyas towhatenergy scaleoneshouldusetodefine theory. This makes the reparametrization of the RG flow to be a gauge symmetry. Because of thisredundancy, thepartitionfunctionin Eq. (19) does not depend on therate high energy modes are eliminated as far as all modes are eventually eliminated. Moreover, at each step of mode elimination, one could have chosen α differently. Therefore, α can be regarded as a function of z. If one interprets z as ‘time’, it is natural to identify α(z) as the ‘lapse function’, that is, α(z) = g (z), where g (z) is the metric. Then one can view Eqs. (19) and (20) as an one- zz zz dimensiopnal gravitational theory with matter fields J . This becomes more clear if we write the n Lagrangian as L = P ∂ J αH, (23) n z n − 6 whereH istheHamiltonian(thereasonwhyH isnotHermitianinthiscaseisthatwestartedfrom theEuclideanfieldtheory). However,thereisoneimportantdifferencefromtheusualgravitational theory. In theHamiltonianformalismofgravity[11], thelapsefunctionis aLagrangianmultiplier which imposes the constraint H = 0. However, in Eq. (19), α is not integrated over and the Hamiltonianconstraintisnotimposed. Thisisduetothepresenceoftheboundaryatz = 0which explicitly breaks the reparametrization symmetry. In particular, the ‘proper time’ from z = 0 to z = givenby ∞ ∞ l = α(z)dz (24) Z0 is a quantity of physical significance which measures the total warping factor. To reproduce the originalpartitionfunctioninEq. (1)fromEq. (19),onehastomakesurethatl = toincludeall ∞ modes in theinfrared limit. Therefore, l shouldbe fixed to be infinite. As a result,one shouldnot integrateoverallpossibleα(z)someofwhichgivedifferentl. Thisisthephysicalreasonwhythe Hamiltonian constraint is not imposed in the present theory. This theory is a gravitational theory with thefixed size along thez direction. Nonetheless,the partitionfunction does not depend on a specificchoiceofα(z)asfaraslisfixed. Ifonewantstomakethisgaugesymmetrymoreexplicit, one could integrate in the gauge degree of freedom by summing over different α(z) with fixed l. Forexample,wecan integrateoverκ parametrizingl-preserving fluctuationsofα as z1,z2 α(z) = α (z)+κ (δ(z z ) δ(z z )), (25) 0 z1,z2 − 1 − − 2 whereα (z) is a‘gaugefixed’lapsefunction. Integrationoverκ generates theconstraint 0 z1,z2 H(z ) = H(z ). (26) 1 2 However,thisisatrivialconstraintwhichisalreadyimplementedinEq. (19)withfixedα,because itis simplytheconservationof‘energy’. Althoughtherearemanyfieldsinthebulk,i.e. J ,P foreachn,thereisonlyonepropagating n n mode, and the remaining fields are non-dynamical in the sense they strictly obey constraints im- posedbytheirconjugatefields. Thisisnotsurprisingbecausewestartedwithonedynamicalfield Φ. There is a freedom in choosing one independent field. In this case, it is convenient to choose J asanindependentfieldbecausetheconjugatefieldP ismultipliedbyJ inEq. (20),whereJ 3 3 4 4 isnon-dynamicalduetoEq. (22). IntegratingoverP , oneobtains 3 M2 i S = ∞dz (∂ J +3αJ )2 (∂ J +3αJ )(3J P +2J P +iJ ) 32αJ2 z 3 3 − 4J z 3 3 3 2 2 1 1 Z0 4 4 h +i(∂ J +αJ )P +i(∂ J +2αJ )P . (27) z 1 1 1 z 2 2 2 i 7 P and P are Lagrangianmultiplierswhichimposetheconstraints, 1 2 J 2 (∂ J +αJ ) = (∂ J +3αJ ), z 1 1 z 3 3 2J 4 3J (∂ J +2αJ ) = 3(∂ J +3αJ ). (28) z 2 2 z 3 3 4J 4 Remarkably,theseconstraintshavealocalsolution,thatis,thefieldsJ andJ atascalez depend 1 2 onlyon theindependentfield J at thesamescale, 3 J3 J J = 3 + e 2αz 3 , 1 16J2 J2 − 2J 4 4 3J2 J = 3 + e 2αz. (29) 2 2 − 8J J 4 This locality is guaranteed because all source fields at a given scale are tied with one fluctuating field φ˜ at the same scale. Here we considered the case with Z symmetry where = 0 for odd 2 n J n. From thisonecan writedownthelocalaction fortheindependentfield J inthebulk, 3 M2 J4 e 2αz S = ∞dz (∂ J +3αJ )2 +∂ 3 + J2 − J2 . (30) 32αJ2 z 3 3 z 256J3 16J2 3 Z0 (cid:20) 4 (cid:26) 4 4 (cid:27)(cid:21) The first term is a bulk term and the second term is a boundary term. Since J (z = 0) = 0, 3 the boundary terms contribute only at the infrared limit z = . The theory for J is free in the 3 ∞ bulk,buttheboundarytermcontainsnon-trivialinteractions. Presumably,thistheoryisnoteasier to solve than the original theory due to the boundary interactions. However, the construction of the dual theory for the toy model illustrateshow one can construct dual theories for more general field theories. Now, rather than trying to analyze the theory (30), we will move on to apply the prescriptiontomorenon-trivialfield theory: D-dimensionalO(N)vectormodel. III. D-DIMENSIONALO(N)VECTORTHEORY A. Constructionofdualtheory WeconsideraD-dimensionalvectorfield theory, Z[ ] = DΦ e (SM[Φ]+SJ[Φ]), (31) a − J Z 8 where S [Φ] = dxdy Φ (x)G 1(x y)Φ (y), M a −M − a Z S [Φ] = dx Φ + Φ Φ + Φ Φ Φ + Φ Φ Φ Φ a a ab a b abc a b c abcd a b c d J J J J J Z h + ij∂ Φ ∂ Φ + ij Φ ∂ Φ ∂ Φ + ij Φ Φ ∂ Φ ∂ Φ . (32) Jab i a j b Jabc a i b j c Jabcd a b i c j d i Here dxand dy areintegrationsona D-dimensionalmanifold D. Here weuse D = RD M M forsimRplicity. ΦRa isO(N) vectorfield. G−M1(x) is theregularized kineticenergy with G 1(x) = dpp2K 1(p/M)eipx, (33) −M − Z where px p x . K 1(s) is an analytic function of s2, which remains to be order of 1 for s < 1 i i − ≡ and growssmoothlyfor s > 1,forexample, K 1(s) = es2. (34) − The mass scale M is a UV cut-off above which fluctuations of Φ are suppressed. S is a de- a J formation of the free theory. We consider sources which are fully symmetric in the flavor ab... J indices a,b,.... In general, the sources may depend on x. Although we can add more general deformations,wewillproceed withthisquarticaction(32)whichissufficienttoillustrategeneral features oftheholographicdescription. Tointegrateouthighenergy modes,weaddan auxiliaryvectorfield Φ˜ , a Z[ ] = [detG˜ ] N/2 DΦDΦ˜ e (SM[Φ]+SJ[Φ]+S˜[Φ˜]), (35) D − − J Z where S˜[Φ˜] = dxdy Φ˜ (x)G˜ 1(x y)Φ˜ (y). (36) a −D − a Z The form of the propagator G˜ for the auxiliary field does not affect the final answer. Then, we D find anewbasis φand φ˜, Φ (x) = φ (x)+φ˜ (x), a a a Φ˜ (x) = dy A(x,y)φ (y)+B(x,y)φ˜ (y) , (37) a a a Z (cid:16) (cid:17) whereAand B are chosento satisfy G 1 +ATG˜ 1A = G 1, −M −D −M′ G 1 +BTG˜ 1B = G˜′ 1, −M −D − G 1 +ATG˜ 1B = 0 (38) −M −D 9 sothatthelowenergyfieldφhasaslightlysmallerUVcut-offM′ = Me αdz andthehighenergy − field φ˜hasa propagatorG˜′ = −(GM′ −GM). Thenthepartitionfunctioncan bewritten as Z[ ] = [detG˜′−1detG−1′ detGM]N/2 DφDφ˜e−(SJ[φ+φ˜]+SM′[φ]+S˜′[φ˜]), (39) J M Z where S˜′ = dxdy φ˜ (x)G˜′ 1(x y)φ˜ (y). (40) a − a − Z A rescalingofxand thefields, x eαdzx, → φ e(2 d)αdz/2φ , a − a → φ˜ e(2 d)αdz/2φ˜ (41) a − a → bringsthekineticenergy forthelowenergy field totheoriginalformas Z[ ] = [detG˜ 1]N/2 DφDφ˜e (Sj[φ+φ˜]+SM[φ]+S˜[φ˜]), (42) − − J Z where S [φ] = dx j φ +j φ φ +j φ φ φ +j φ φ φ φ j a a ab a b abc a b c abcd a b c d Z h +jij∂ φ ∂ φ +jij φ ∂ φ ∂ φ +jij φ φ ∂ φ ∂ φ (43) ab i a j b abc a i b j c abcd a b i c j d i with ja(x) = e2+2Dαdz a(eαdzx), J j (x) = e2αdz (eαdzx), ab ab J jabc(x) = e6−2Dαdz abc(eαdzx), J j (x) = e(4 D)αdz (eαdzx), abcd − abcd J jij(x) = ij(eαdzx), ab Jab jij (x) = e2−2dαdz ij (eαdzx), abc Jabc jij (x) = e(2 d)αdz ij (eαdzx) (44) abcd − Jabcd and G˜ 1(x y) = e(2+d)αdzG˜′ 1(eαdz(x y)). (45) − − − − 10