Phase space holography with no strings attached
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Phase space holography with no strings attached
D. V. Khveshchenko
Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599
This note discusses the Wigner function representation from the standpoint of establishing aholography-like correspondence between the descriptions of a generic quantum system in the phasespace (’bulk’) picture versus its spacetime (’boundary’) counterpart. Under certain circumstancesthe former might reduce to the classical dynamics of a local metric-like variable while the latter takeson the form of some bosonized collective field hydrodynamics. This generic pseudo-holographic du-ality neither relies on any particular symmetry of the system in question, nor does it require anyconnection to an underlying ’string theory’, as in the various ’ad hoc’ scenarios of applied holography.
Generalized holography
Among all the remarkable theoretical advances underthe hashtag holography the one of the greatest interestto condensed matter physics is the ongoing quest for gen-eralizing the original, highly constrained (i.e., Lorentz-and maximally super-symmetric, etc.), string-theoreticalholographic conjecture to as broad as possible (i.e., non-Lorentz-, non-isotropic-, non-super-symmetric, etc.) va-riety of quantum many-body systems .In that regard, the so-called ’bottom-up’ holographyhas been portraying itself as a powerful technique offer-ing solutions to the traditionally hard condensed mat-ter problems. This innovative approach (sometimes re-ferred to as the anti-de-Sitter/condensed matter theory,or AdS/CM T , correspondence) borrows its formal struc-ture and mathematical apparatus (often ’ad verbatim’,for the lack of an alternative) from the very specific,carefully crafted, and highly symmetric examples of suchduality known under the acronym
AdS/CF T and conjec-tured in the original context of fundamental string theoryand its various reductions .So far, however, all the attempts of putting appliedholography on a firm foundation - either along the lines ofthe geometrized renormalization group (RG) flow or en-tanglement dynamics in tensor networks, or by using ar-tificial thermodynamic (Fisher-Ruppeiner, Fubini-Study,etc.) metrics, or else - have remained consistently incon-clusive.Nonetheless, instead of striving to deliver a solid proofof principle, the field of AdS/CM T has, by and large,stayed the course reminiscent of hacker’s code crack-ing: that is, trying to guess some higher-dimensional en-hanced gravity-like theory (often on the sole basis of tech-nical convenience) and then rely on the persuasive powerof visual agreement between some pre-selected experi-mental data plots and the (for the most part, numerical)calculations based on the above
AdS/CM T ’dictionary’.A great many number of the customarily verbose andlook-alike accounts of such pursuits can be readily foundin the applied holographic literature from the year 2007onwards . Judging by the factual outcome of this mas-sive attack, though, the code does not appear to havebeen cracked yet.Such fundamental shortcomings notwithstanding, the still continuing occasional exercises in the holographicphenomenology utilize a handful of the popular bulk ge-ometries, especially focusing on the legacy solutions ofthe prototypical Einstein-Maxwell-dilaton theory. Otherthan their relative simplicity and sheer availability(alongside, possibly, some lingering anthropic factor) aproper justification of such choices does not appear tohave been an essential part of the holographic agenda,regardless of whether the results were meant to be ap-plied to the lattice Hubbard-like models, superconduct-ing cuprates, low-density 2 DEG , Dirac/Weyl materialssuch as graphene, or else.Lately, though, a gradual decline in such ’orthodox’applications of the
AdS/CM T machinery has been giv-ing way to advanced hydrodynamics of strongly coupledquantum matter and general out-of-equilibrium physics(eigenstate thermalization, many-body (de)localization,chaos spreading, operator growth, etc.). Correspond-ingly, instead of the once ubiquitous renditions of eso-teric black holes, nowadays a slide show on the topic of
AdS/CM T is more likely to feature the images of wa-ter flows, rapids, whirlpools, and other familiar hydrody-namic patterns .Of course, hydrodynamics, while suggesting some in-triguing holographic connections, has long been discussedoutside of any holographic context. Therefore, the re-newed appreciation and novel applications (thanks to anumber of recent experimental advances) of the theory asEarthly as hydrodynamics alone do not provide an an-swer to the question that should have (but does not seemto have been) long dominated the holographic discourse,that is: ’So why, on Earth, strings?!’.The goal of this note is to recall a decades-old the-oretical approach known as collective field theory andits more recent developments that might be capable ofproviding a much-needed justification for the ’stringy hy-drodynamics’ (especially, in those non-relativistic and/orrotationally-non-invariant settings that are typical to thecondensed matter applications but do not normally occurin the original string-theoretical context). Alternatively,this approach can be viewed as a variant of the long-pursued idea of (non-linear) ’bosonization’ that aims toreformulate a quantum theory of interest in terms of someintrinsically geometric bosonic variables.Notably, while demonstrating some features remi-niscent of the desired holographic correspondence, thisapproach shows that the pertinent space-time metricsmay not be chosen at will and can often be quitedifferent from the routinely utilized ones. Phase space quantization
A systematic description of many-body dynamics in d spatial dimensions calls for the use of the Wigner function w ( x , p , t ) defined in the 2 d + 1-dimensional phase space(plus time). The space of such functions is equipped withthe Moyal product f ( x , p ) ⋆ g ( x , p ) = f ( x , p ) e i ¯ h ( ←− ∂ x −→ ∂ p − ←− ∂ p −→ ∂ x ) g ( x , p )(1)which introduces the symplectic structure through theMoyal bracket { f, g } MB = f ⋆ g − g ⋆ f (2)The Wigner’s description is well suited for taking into ac-count the underlying theory’s invariance under the phasespace volume-preserving diffeomorphisms, including itsnatural time evolution that can be thought of as a canon-ical transformation governed by the Liouville theorem.Furthermore, when quantizing the system via themethod of functional integration, the function w ( x , p , t )becomes a constrained field variable implementing acoadjoint orbit’s quantization of the phase space volume-preserving diffeomorphisms a la Kirillov-Kostant.In this procedure, an orbit’s element ˆ g | Ψ >< Ψ | ˆ g − is constructed by acting with an element ˆ g of the infinite-dimensional group of diffeomorphisms on the projector toa chosen (e.g., ground) state | Ψ > . In the partition func-tion Z = R Dw exp( − S ( w )) the integration then runsover the functions satisfying the constraints w ⋆ w = w T rw = 1 (3)and governed by the action S ( w ) = Z dxdpdt ( i Z dsw { ∂ τ w, ∂ s w } MB − wH ) (4)where the first term represents the Berry phase where theintegral over the auxiliary variable s depends only on theboundary value w ( x , p , t, s = 1) = w ( x , p , t ) at s = 1.This way, one arrives at the formally exact geometrizeddescription of the non-linear σ -model type .The equation of motion derived from Eq.(4)˙ w + { w, H } MB = 0 (5)reproduces the standard kinetic equation when the Moyalbracket is approximated, to lowest order in the powers of¯ h , by the Poisson one (hereafter the dot and prime standfor the time and space derivatives, respectively)˙ w + w ′ ∂ p H − H ′ ∂ p w = St [ w ] (6) where the one-body Hamiltonian H may include an ex-ternal potential. For example, in the so-called ’non-critical 2 d string theory’ where the spatial coordinateoriginates from the eigenvalues of N × N matrices it hap-pens to be the inverted oscillator ( V ∼ − x ) . Also,the n ≥ H n in the Hamiltonian involvinghigher powers of w are bundled into the collision integralin the right hand side.In the case of fermions, the semiclassical vacuum con-figuration corresponding to the uniform Fermi sea is de-scribed by the expression w ( x , p , t ) = θ ( µ ( x , t ) − ǫ p ) (7)where the local chemical potential µ ( x , t ) denotes a sharpboundary between the occupied and vacant momentumeigenstates with the dispersion ǫ p .Facilitating further progress with the 2 d + 1-dimensional ’bulk’ theory (4) requires a convenientparametrization of the bounding momentum. Previously,a similar task was tackled in the early works on multi-dimensional bosonization where this goal was achieved bydistinguishing between the Fermi momentum p F tracingthe fiducial Fermi surface (FS) and the normal to the FS(’radial’) degree of freedom describing fluctuations of themomentum distribution .For instance, in the much studied d = 2 case the sim-plified action for the vector k F reads S ( p F ) = Z dxdt ( i Z ds p F ∂ τ p F × ∂ s p F − H ( p F )) (8)where the Hamiltonian H is cast in terms of the localdensity ρ = p F × ∂ t p F / d = 1 where thefluctuating FS can be described in terms of M ≥ p ( n ) ± ( x, t ) bounding the occupiedstates ( M > ).In particular, the 1 d configuration (7) reads w ( x, p, t ) = X ± M X α =1 ( ± ) θ ( p ( α ) ± ( x, t ) − p ) (9)while its small perturbation δw ( x, p, t ) = ¯ hδ ( x − x cl ) δ ( p − p cl ) (10)is strongly peaked at the classical phase space trajectory( x cl ( x , p , t ), p cl ( x , p , t )) where the initial data x and p are determined by the current values ( x, p ) at a latertime t .It is worth mentioning, though, that while being ca-pable of faithfully reproducing the long-distance, late-time asymptotics of the response functions, in its prac-tical (hence, approximate) form the d > p F -singularities, just as it may notbe sufficient for the single-particle propagators .In d = 1, despite several decades of studies there hasbeen a recent surge of renewed interest in the out-of-equilibrium dynamics of quantum interacting bosons andfermions. Many of those studies focus on the integrableand non-ergodic systems which are governed by the gen-eralized Gibbs ensembles (GGE) and may not complywith the more generic eigenstate thermalization hypoth-esis (ETH) .It is worth noting that in 1 d Eq.(2) represents a clas-sical analog of the infinite-dimensional quantum algebra W ∞ composed of the operators ˆ W mn = (ˆ x ) m (ˆ p ) n withthe commutation relations[ ˆ W mn , ˆ W rs ] = X k =1 ( − ¯ h ) k k (11)( n ! r !( n − k )!( r − k )! − m ! s !( m − k )!( s − k )! ) ˆ W m + r − k,n + s − k where the r.h.s. reduces to ( ms − nr ) ˆ W m + r − ,n + s − inthe limit ¯ h → SL (2 , R ). This algebra has been extensively studiedin the context of Quantum Hall Effect (QHE) and thevarious reincarnations of (effectively) non-commutativespacetimes.An abstract Hilbert space can be readily equippedwith a geometric structure that has long been eluci-dated alongside the more familiar Berry phase. However,the even (Fubini-Study metric) - as opposed to the odd(Berry curvature) - component of the same rank-2 tensorhas been receiving less attention.Such a phase space metric can be naturally introducedin the context of special coherent - (de)localized nei-ther in the coordinate, nor momentum space - states | p, x, > = e i ˆ P x − i ˆ Qp | > which minimize, both, the co-ordinate and momentum uncertainties.Allowing, for the sake of generality, some coordinate-momentum cross-correlations the corresponding Wignerfunction reads w coh ( x, p,
0) = Z dye ipy < Ψ | x + y/ >< x − y/ | Ψ > = ¯ hD / exp( − σ p δx + σ x δp + 2 σ xp δxδp D ) (12)where δx = x − x , δp = p − p , D = σ x σ p − σ xp , and theparameters σ x , σ p , σ xp control the Gaussian coordinateand/or momentum variations.The above suggests a naturally defined Fubini-typemetric on the phase space ds = ( < ∂ µ Ψ | ∂ ν Ψ > − < Ψ | ∂ µ Ψ >< ∂ ν Ψ | Ψ > ) δ µ δ ν (13)where δ µ = ( dx, dp ). Taken at its face value, this for-mula establishes some form of superficial correspondencebetween single-particle quantum mechanics and 2 d met-rics that can be viewed as solutions of certain classicalgravity. Further generalizing Eq.(13) to include energy fluctu-ations one arrives at the (Euclidean) 3 d metric ds = < (∆ ˆ H ) > dt + < (∆ˆ x ) > dp + < (∆ˆ p ) > dx + 2 < ∆ˆ x ∆ˆ p > dxdp (14)given by the uncertainties of the conjugate variables( x ↔ p , t ↔ H ). Also, considering the metric (14)to be the expectation value ds = < Ψ | d ˆ s | Ψ > of theoperator-valued interval d ˆ s paves the way for promotingthe bulk (phase space)-to-boundary (spacetime) relation-ship to the quantum level.As the operators’ uncertainties depend on the choiceof the state | Ψ > , so does the dual metric (14). Heuris-tically, one might expect that for the single-particle dis-persion governed by the dynamical exponent z ( ǫ p ∼ p z )the above variances behave as follows < (∆ p ) > ∼ < (∆ x ) > − ∼ p < (∆ x )(∆ p ) > ∼ < (∆ p ) > / < (∆ p ) − / ∼ p < (∆ H ) > ∼ < (∆ p ) > z ∼ p z (15)so that the metric (14) conforms to the so-called Lifshitzvariety (the coefficients A, B, C, D are constants) ds = Ap z dt + Bp dx + C dp p + 2 Ddxdp (16)which has been often invoked in the various applicationsof
AdS/CM T . Non-linear hydrodynamics
The formally exact representation (4) of the phasespace dynamics provides a basis for further simplifica-tions, thus giving rise to (semi)classical hydrodynamicequations for the various moments of the Wigner distri-bution w n ( x , t ) = Z d p w ( x , p , t ) p n (17)Among those moments are such standard hydrodynamicvariables as the local mass ρ ( n = 0), momentum Q ( n = 1), and energy ǫ ( n = z ) densities, respectively.This transition from the entire Wigner function to thefirst few of its moments can be thought of as a dimen-sional reduction from the 2 d + 1-dimensional bulk (phasespace) to its d + 1-dimensional boundary hypersurface(spacetime) which, in practice, amounts to mere integra-tion over the d -dimensional momentum.Correspondingly, the mass J ρ , momentum J Q , and en-ergy J ǫ currents are given by the general expression J n = Z d p ∂ǫ p ∂ p w ( x , p , t ) p n (18)for n = 0 , , d = 1 the lowest moments of the Wignerfunction (17) correspond to the aforementioned boundingFermi momenta p ± = R dpw (1 ± sgnp ) / ρ = 12 π ( p + − p − ) v = 12 m ( p + + p − ) (19)Note that limiting the momentum values to the interval0 < p < ∞ , similar to the holographic radial variable, isdictated by the chiral nature of the excitations carryingsign-definite momenta.These variables have the Poisson bracket { ρ ( x ) , v ( y ) } = ∂ x δ ( x − y ) (20)Then taking the various moments of Eq.(5) one arrivesat the hydrodynamic equations of motion which includethe continuity equation˙ ρ + ( ρv ) ′ = 0 (21)and the inviscid Navier-Stokes (a.k.a. Eu-ler/Burgers/Hopf) one˙ v + vv ′ = − P ′ ρ − κ V ′ ρ (22)where the ’quantum pressure’ P ( ρ ) (internal stress tensorof the 1 d quantum fluid) is a system-specific function ofthe local density, while the last term with the dispersioncurvature κ = ∂ p ǫ p represents the force exerted by theexternal potential (if any).It is well known that the non-linear hydrodynamicequations (21,22) can be derived even from the freeparticle Schroedinger equation . Specifically, by apply-ing the Madelung parametrization of the wave functionΨ( x, t ) = √ ρ ( x, t ) e iS ( x,t ) and separating out the real andimaginary parts one arrives at the coupled continuity andNavier-Stokes equations, respectively, where v = 12 im ( Ψ ∗ Ψ ) ′ ρ = | Ψ | (23)In the r.h.s. of (22) the pressure P = ¯ h m ( ρ ′ ) ρ (24)contributes towards the overall energy density ǫ = ρv / P which, in general, might be neither polyno-mial, nor separable as a sum of two chiral components P ± ( p ± ).It appears, however, that the pressure gradient termcouples excitations with opposite chiralities (left/rightmoving) at the level of operators with dimensions of fouror greater. Moreover, even if present, the non-chiral cor-rections do not affect the states which are composed ex-clusively of the chiral excitations with p + = 0 or p − = 0.Thus, an arbitrary single-particle state Ψ( x, t ) =(2 πi ¯ ht/m ) − / R dye im ( x − y ) / ht Ψ ( y ) of the free Hamil-tonian of mass m with the initial data Ψ ( x ) providesa valid solution to the hydrodynamic equations (21,22) with the pressure (24). Correspondingly, the pair offunctions ρ ( x, t ) and v ( x, t ) determines a certain dualmetric, as explained below. Solvable hierarchies
For certain choices of the Hamiltonian H Eq.(22) ap-pears to belong to the infinite KdV (Korteweg–de Vries)hierarchy of integrable 1 d systems . Such Hamiltonians H ± k are related to the Gelfand-Dickey polynomials andform an infinite set of integrals of motion in involution([ H ± n , H ± m ] = 0). In the asymptotically free regime oflarge momenta (high energies) the k th member of thisfamily describes small-amplitude excitations with the dis-persion exhibiting the dynamical exponent z = 2 k − d Luttinger liquid (LL)behavior is associated with the quadratic Hamiltonian H = P ± H ± given by the standard Sugawara construc-tion H = 12 X ± p ± = 12 ( p + p − ) (25)which gives rise to the equation of motion˙ p ± ∓ p ′∓ = 0 (26)whose solutions given by the (anti)holomorphic func-tions p ± ( x ± ) describe two decoupled chiral ( x ± = x ± t )pseudo-relativistic ( z = 1) excitations.Corrections to the LL Hamiltonian (25) may comefrom, both, the Gaussian terms of higher operator di-mensions which represent non-linear terms in the disper-sion of the collective ρ - and v -modes, as well as fromthe non-Gaussian ones which are due to some intrinsicnon-linearity of the 1 d dispersion, as in the case of non-relativistic fermions at a finite density .For example, the next (2 nd ) member of the KdV familyis given by the non-Gaussian expression H = X ± ( ± ) 13 p ± + 12 ( p ′± ) =12 ρv + π ρ + 12 ( v ′ ) + π ρ ′ ) (27)for which the chiral components of Eq.(22) still remainuncoupled ˙ p ± ± p ± p ′± + p ′′′± = 0 (28)In the asymptotic regime of high energies the linearizedEq.(28) describes small waves with the expressly Lorentz-non-invariant cubic dispersion ( z = 3).In the opposite, low-energy and essentially non-perturbative, limit Eq.(28) permits non-linear solitonicexcitations (’cnoidal’ waves) v ( x, t ) ∼ x ± (29)whose propagation is described by the dispersion ǫ p ∼ p / . Compactifying the spatial coordinate into a finite-length circle would then replace (29) with the ellipticJacobi function.In general, the non-Gaussian terms in the Hamiltonianare sensitive to the microscopic details of the model andstem from, both, kinetic and potential terms in the totalenergy. Specifically, in the case of hard-core bosons, suchas the Tonks-Girardeau limit of the Lieb-Liniger model,the Hamiltonian includes the pressure term P ( ρ ) ∼ ρ .By contrast, in the quantum Toda chain the function P ( ρ ) is non-polynomial. However, despite not being di-vidable into a sum of two chiral terms, the latter canstill fit into the KdV Hamiltonian (27) as the non-chiralterms appear to be irrelevant at low momenta .Likewise, the deviations from the LL regime associatedwith a finite dispersion curvature and/or chiral interac-tions can be studied with the use of a linear combinationˆ H + ˆ H + . . . of Eqs.(25) and (27). This way, one canobtain non-linear corrections δǫ p ∼ p / to the linear LLspectrum at small momenta .Moreover, Eq.(27) can be further modified by includ-ing irrelevant non-Gaussian terms, such as p ± , withoutdestroying its integrability. Indeed, such extension re-sults in yet another, Gardner, equation (a.k.a. mixedKdV-m(modified)KdV, which two equations are relatedby virtue of the Miura transformation p ± → p ± + p ′± ).As a hallmark of integrability, the higher- k level mem-bers of the KdV hierarchy possess the bi-Hamiltonianstructure relating them as follows ∂ x δH ± k +1 δp ± = D ± x δH ± k δp ± (30)where the long derivative is D ± x = 2 p ± ∂ x + ∂ x p ± + ∂ x (31)The higher level- k members of the KdV and mKdV fam-ilies can also be morphed into a two-parameter Gardnersequence of Hamiltonians. Furthermore, certain solvablesystems of M ≥ coupled non-linear equations were shownto be associated with the higher-spin symmetry algebras SL ( M, R ) (e.g., the Boussinesq equations for M = 3) .Generic Hamiltonians P k µ k H ± k which includes differ-ent members of the integrable family can be used to de-rive zero entropy GGE hydrodynamics. In particular, onecan study crossovers between the LL and higher level- k regimes at varying momenta. In essence, this construc-tion provides a (formally exact) bosonization scheme thatwas fully exploited, e.g., in the context of the solvableCalogero-Sutherland model .Under the time evolution governed by a superpositionof the different H n a generic initial condition produces acollection of solitons with different velocities and a con-tinuum of decaying dispersive waves. Being more robustthe soliton excitations dominate in the late-time behav-ior and, in particular, the system’s equilibration towardsa steady GGE state described by the density matrix ˆ G = exp( − P k µ ± k ˆ H ± k ) where the chemical potentials µ k are to be determined by equating the averages of the com-muting charges < ˆ H ± k > = T r ( ˆ G ˆ H ± k ) /tr ˆ G to their chosenvalues.Along these lines, one can also study the von Neu-mann entropy S = − T r ˆ G ln ˆ G . The Wigner functionsatisfying the constraints (3) corresponds to a pure stateof zero entropy and the presence of an infinite numberof conserved charges precludes standard thermalization.When the constraint ceases to hold the state becomesmixed, thus resulting in a finite entropy. The ensuingthermalization can be accounted for by introducingviscous terms, such as ηρ ′′ , in the r.h.s. of Eq.(22). Dual bulk gravity
A deep relationship between classical gravity and hy-drodynamics has long been known as one particular takeon the holographic paradigm, often referred to as the’fluid-gravity’ correspondence. The crux of the matteris observation of the similarity between the asymptoticnear-boundary behavior of the Einstein equations for thebulk metric and the Navier-Stokes ones describing a dualboundary fluid in one lesser dimension (besides, the com-plementary hydrodynamic behavior near the event hori-zon can be similar to that at the boundary). Albeit beingtruncated and, therefore, approximate such relations canbe systematically improved, thus enabling certain com-putational simplifications. Whether or not this dualitycan be promoted to the quantum level requires furtheranalysis.Remarkably, in the case of d = 1 this correspondencebecomes exact. Specifically, the Einstein equations stem-ming from the action of 3 d gravity with a negative cos-mological constant (here l and G are the AdS radius andNewton’s constant, respectively) S = l πG Z dxdtdp √ g ( R + 2 /l ) = 0 (32)coincide with the equations describing two decoupledChern-Simons (CS) models with the combined action S = l πG T r Z dxdpdtǫ µνλ ( ˆ A ± µ ∂ ν ˆ A ± λ + ˆ A ± µ ˆ A ± ν ˆ A ± λ ) (33)The chiral connections ˆ A ± µ are matrices that can be ex-panded in the basis spanned by the generators ˆ L ± , ± ofthe algebra SL (2 , R ) × SL (2 , R ) = SO (2 , L ± n , ˆ L ± m ] = ( n − m ) ˆ L ± n + m andare normalized, T r ˆ L ± n ˆ L ± m = δ n δ m − δ n δ m, − .The topological action (33) then reduces to a pureboundary term while the equation of motion becomesthat of null curvature ∂ µ ˆ A ± ν + ˆ A ± µ ˆ A ± ν − ( µ ↔ ν ) = 0 (34)Parameterizing its solutions in terms of an arbitrarygroup element ˆ χ ± and functions p ± ( x ± ) and µ ± ( x ± )ˆ A ± ( x, p, t ) = ˆ χ − ± ( p ) ˆ L ( µ ± ± p ± dx ) ˆ χ ± ( p ) (35)one finds this equation to be equivalent to˙ p ± ∓ µ ′± = 0 (36)which, in turn, coincides with one of the above solvableequations, provided that the chemical potentials µ ± con-jugate to the variables p ± are given by the derivatives µ ± = δH ± δp ± (37)Choosing the Hamiltonian appropriately one can then re-produce the solvable (m)KdV, Gardner, and other equa-tions. In particular, the KdV family is recovered forˆ A ± p ( x, p, t ) = 1 p ˆ L ± ˆ A ± x ( x, p, t ) = p ˆ L ± − p ± p ˆ L ±− ˆ A ± t ( x, p, t ) = pµ ± ˆ L ± − µ ′± ˆ L ± + µ ′′± − µ ± p ± p ˆ L ±− (38)which expressions are manifestly Lorentz-non-invariantfor all k > d metric g µν = l < ( A + µ − A − µ )( A + ν − A − ν ) > (39)On the gravity side the different saddle points of the co-herent states path integral can be identified as globallydistinguishable (but locally AdS ) classical solutions. Inparticular, it can be shown that the only minima of theaction (32) corresponding to the boundary Hamiltonian H + H are those with constant (negative) curvature.The two competing minima are the thermal AdS andBTZ (Banados-Teitelboim- Zanelli) black hole.However, by introducing higher order terms H k with k ≥ . The corresponding boundary theory isencoded in the boundary conditions for the connection(35), by varying which one can explore a variety of theintegrable 1 d systems.The standard LL with k = 1 is reproduced by introduc-ing the original Brown-Henneaux boundary conditionswith constant µ ± ∼ p ± , the outer/inner horizons beinglocated at p >/< = ( p + ± p − ) /
2. The dual metric ds = dp p + ( p − p + + p − ) + p + p − p ) dt +( p + 2( p + + p − ) + p + p − p ) dx + ( p + p − ) dxdt (40)describes a rotating BTZ black hole with the event hori-zon but no curvature singularity.For static, yet non-constant, p ± ( x ) the correspond-ing boundary solutions possess non-trivial global chargesgiven by the chiral surface integrals H ± k while their bulkcounterparts can be regarded as black holes with multi-graviton excitations (’soft hair’) . The general solution can be obtained by acting on theground state (e.g., BTZ black hole) with elements of theasymptotic symmetry group commuting with the Hamil-tonian. This way one can construct various constantcurvature, yet locally AdS, spacetimes with anisotropicLifshitz scaling and dynamical exponent z = 2 k − SL ( M, R ) the listof attainable gravitational backgrounds may include theasymptotically Lobachevsky, Schroedinger, warped
AdS ,etc. spacetimes .Shocks and other abrupt perturbations are character-ized by FS breakdowns and emergence of folds wherethe spatial derivative ρ ′ diverges, thereby requiring sev-eral pairs of the bounding momenta p ± . In the presenceof shocks the conventional spacetime hydrodynamics be-comes insufficient for describing long-time behavior, al-though the full-fledged phase space hydrodynamics canavoid such problems.In that regard, particularly interesting are the non-stationary configurations representing particles releasedfrom a confining potential which gets suddenly switchedon/off . Such quenching profiles generically havespacelike boundaries where the saddle point solutions ofthe collective field hydrodynamics diverge at finite timesand the semi-classical description fails. Ascertaining theemergent spacetimes and their dynamics then requiresa detailed study of fluctuations around the pertinentsaddle points. Reductions and generalizations
Despite having been repeatedly stated and extensivelyanalyzed at the level of salient symmetries and concomi-tant algebraic properties, the general gravity/fluid corre-spondence in dimensions d > .Specifically, such a relationship was shown to exist be-tween the solutions of classical d + 2-dimensional gravityand their d + 1-dimensional hydrodynamic counterparts,whereby the former would be given by the metric ds = dp f ( p ) p + p (∆ µν − f ( p ) u µ u ν ) dx µ dx ν (41)parameterized in terms of the spacetime-dependent co-variant velocity u µ ( x , t ) and local temperature T ( x , t ) .The latter satisfy the hydrodynamic equations on a fixedbackground, provided that f ( p ) = 1 − (4 πT /p ) d and∆ µν = g µν + u µ u ν . Thus, a given fluid profile can be asso-ciated with a certain asymptotically AdS d +2 -like space-time with a horizon located at p h = 4 πT .A still more general (asymptotically accurate) solutioncan be constructed with the use of the metric ansatz g µν ( x , p, t ) = g (0) µν p + g (2) µν + p g (2) g (0) g (2) µν (42)with arbitrary functions g (0 , µν ( x , t ) .In contrast to the generic case of d >
1, pure gravityin d ≤ N ) limit. It shouldbe noted, however, that, barring a few exceptions, thecustomarily assumed ’classicality’ of the bulk geometry(regardless of whether or not the 1 /N - and/or ’stringy’corrections are important) and a complete neglect of anyback-reaction on the fixed background metric appear tobe by far the most common approximations routinelymade in the absolute majority of all the AdS/CM T cal-culations performed so far .Nonetheless, there are still important differences be-tween the situations in d = 0 and d = 1. As per theabove discussion, the latter is described by the LL actionof two chiral 1 d bosons φ ± ( x ± ) = ln Φ ′± ( x ± ) S ± LL = Z dxdt Φ ′′± ( ˙Φ ′± ∓ Φ ′′± )(Φ ′± ) = Z dxdtφ ′± ( ˙ φ ± ∓ φ ′± )(43)This action can also be obtained from the more generalAlekseev-Shatashvili functional which performs path-integral quantization on the co-adjoint orbit of the (dou-ble) Virasoro group. Alternatively, it can be identifiedwith the large central charge limit in the conformal Li-ouville model, thus relating the latter to its namesake(Liouville) theorem governing the phase space dynamicsin the (semi)classical limit.Besides, this action can be viewed as a complexityfunctional defining an associated quantum-informationtype of geometry on the Virasoro group, its lower boundbeing given by the length of a proper geodesic on theco-adjoint orbit .In turn, the extensively studied case of d = 0 can beattained in the AdS theory by taking the limit of a van-ishing length of the compactified spatial dimension. Theresulting AdS bulk theory, as well as its JT (Jackiw-Teitelboim) extension, support a pseudo-Goldstone timereparametrization mode with the 1 d boundary actiongiven by the Schwarzian derivative .Equivalently, it can be cast in terms of theLiouville quantum mechanics on the quotient Dif f ( S ) /P SL (2 , R ) with the action S L = Z dt ( 12 ˙ φ + λe φ ) (44)for φ ( t ) = ln ˙Φ( t ) where t → Φ( t ) is a diffeomorphismof the thermal circle. In the context of the space-less random SYK (Sachdev-Ye-Kitaev) and non-random ten-sor models this orbit emerges as the result of factoringout the subspace of zero modes reflective of the SL (2 , R )symmetry of the conformal saddle-point solutions . Theintegrable 1 d dynamics in such models is spatially ultra-local and corresponds to z = ∞ , thus being reminiscentof the popular AdS/CM T schemes .Notably, in contrast to the marginal nature of the 1 d LL theory where the interaction remains important atall energy/temperature scales, in the SYK/tensor mod-els it is strongly relevant in the infrared, thus only affect-ing the conformal mean-field solutions below a certainenergy/temperature scale. Also, the maximally chaotic
AdS /JT gravity can be dual not to a certain quantummechanical ( d = 0) theory but (as in the case of SYK) arandom ensemble thereof. For comparison, in d = 1 nei-ther the boundary theory (43) saturates the chaos bound,nor is the bulk behavior dominated by pure gravity.In practice, establishing the SYK-to- AdS /JT dualityinvolves matching thermodynamic properties of the twosystems, alongside their various correlation functions.However, achieving this correspondence beyond the low-est order (two-point) correlation functions requires one tointroduce additional ’matter’ fields in the bulk which rep-resents a tower of higher-spin operators with the anoma-lous dimensions that all scale comparably with 1 /l .Likewise, in the KdV-to- AdS correspondence the en-tropy, free energy, etc. can be matched as well, giv-ing rise to the dependencies S = π P ± p d ± ∼ T /z and E = P ± < ˆ H ± k > ∼ T /z , provided that one chooses µ ± ∼ T in order for the metrics (39) to remain regulareverywhere in space. Notably, the KdV-charged blackholes’ thermodynamics differs from that of the usual BTZones .Comparison between the pertinent microstates on bothsides of the latter correspondence relies on the fact thatthe 2 d phase space can be spanned by the overcompletebasis of coherent states | Ψ > = exp( i P nm c nm ˆ W nm ) | > while the boundary 1 d theory operates in the Hilbertspace spanned by the vectors | ± n > = Q n ˆ p n ± | > . Em-ploying this basis the correlators of a bulk field ˆ O of mass m and dimension ∆ = ( d + 1) / ± lm can be evaluated bythe saddle point method, thereby resulting in the semi-classical expression for the (real-time) two-point function G OO ( x, p, t ) ∼ exp( − ∆ Z [ g pp dp − g tt dt + g xx dx ] / )(45)where the line integral is taken over the 3 d geodesic con-necting the end points.Placing the end points of this correlation function onthe boundary yields the single-particle boundary propa-gator. Fourier transforming this expression in the space-time domain one then obtains G OO ( ω, k ) ∼ exp( − Z dp [ g pp ( ∆ l − ω g tt + k g xx )] / ) (46)For instance, the BTZ bulk metric (40) yields the fol-lowing propagator of massless 3 d bulk fermions with thedimension ∆ Ψ = 1 and spin 1 / G ± ΨΨ ( ω, k ) = ( ω ± kω ∓ k ) / (47)which gives rise to the power-law spacetime behavior ofthe boundary propagator G ± ΨΨ ( t, x ) ∼ / | x ± | . Theseresults should of course be distinguished from the stan-dard LL propagator G ± ψψ ( ω, k ) = 1 / ( ω ∓ k ) of free chiralfermions with the dimension ∆ ψ = 1 / G ± ΨΨ ( t, x ) ∼ exp( − ∆ | x ± | /l ).In turn, the two-particle (energy) excitations repre-senting gapless boundary gravitons remain propagating,thus featuring the ordinary ballistic pole G ǫǫ ( ω, k ) = Z dtdxe i ( kx − ωt ) [ G ± ψψ ( x, t )] ∼ k k − ω (48)Alternatively, this energy correlation function can be ob-tained from the correlator < w ( x, p, t ) w ( x ′ , p ′ , t ′ ) > com-puted as the path integral over the Wigner function.The ballistic behavior (48) should be contrastedagainst the diffusive one observed in, e.g., a chain of cou-pled SYK models, G ǫǫ ( ω, k ) ∼ k / ( k + iω ), which wouldbe indicative of a (maximally) chaotic state .Further possible generalizations of the collective fieldhydrodynamics include incorporation of the momentumBerry curvature in Eq.(6), exploration of the effectsof viscosity, generic dispersion with z = 1 ,
2, etc. Itwould also be interesting to investigate a development ofturbulence which has long been known to harbor someimportant connections to quantum gravity.
Discussion
In this note the Wigner function representation ofgeneric quantum systems was discussed from the stand-point of pinpointing the possible origin of the hypothet-ical generalized holographic correspondence. To thatend, using the Kirillov-Kostant procedure of phase spacequantization via the coherent state path integral and col-lective field hydrodynamics may seem rather promising.Specifically, in line with the holographic lore, path inte-gral quantization on a co-adjoint orbit of the W ∞ groupof the volume-preserving diffeomorphisms of the phasespace exposes an intrinsic relationship between the 2 d +1-dimensional ’bulk’ description and the d + 1-dimensional’boundary’ hydrodynamics. The quantum bulk dynamicsis described by the action composed of the W ∞ genera-tors while the corresponding boundary variables are givenby the moments (17) of the Wigner function. Systemati-cally implementing this program can then be thought ofas ’deriving’ the sought-out holographic duality.Importantly, such a generic form of correspondenceneither requires a reference to some underlying ’string’ theory, nor does it impose any particular symmetry con-ditions on the boundary system in question, while theputative bulk description does not necessarily have agravity-like appearance.Nonetheless, in the case of d = 1 the corresponding3 d bulk behavior can indeed be cast in terms of the(doubled) Chern-Simons theory or, equivalently, non-dynamical Einstein gravity with a negative cosmologi-cal constant. Furthermore, if the boundary Hamiltonianbelongs to the integrable (e.g., KdV) family, the corre-sponding set of the bulk metrics may include the familiarBTZ, as would be generally anticipated in line with the AdS/CF T paradigm .Furthermore, in d = 1 the phase space description canbe viewed as a formally exact (non-linear) bosonizationof the boundary system. Many of such systems appearto be integrable (hence, non-ergodic) and possess an in-finite number of locally conserved currents given by thevarious moments of the Wigner function which obey theequations of zero-entropy generalized quantum hydrody-namics of the GGE type.In higher dimensions the Wigner function’s moments,too, serve as coefficients in the series expansions of thewould-be local bulk metrics over the powers of the mo-mentum p . Although with the increasing spatial dimen-sion the hydrodynamic description becomes less accurate,it remains capable of capturing the salient features of thequantum phase space dynamics governed by the conser-vation laws.To summarize, the use of the phase space approachbrings out the intrinsic correspondence between a for-mally exact 2 d + 1-dimensional and a less accurate(’coarse-grained’) d + 1-dimensional hydrodynamic de-scriptions of a given quantum system. In this generalsetting, neither the latter needs to be a conformal fieldtheory, nor does the former have to have the appearanceof gravity.In those d = 0 and d = 1 cases where the bulk indeedappears to be amenable to a gravitational descriptionthe gravity theory has no dynamics of its own and isfully determined by the boundary degrees of freedom.Accordingly, the viable bulk metrics can be mapped ontothe solutions of the boundary hydrodynamics.In that regard, the holographic custom of picking out aparticular metric and claiming some sort of the Einstein-Maxwell-dilaton theory to be the proper bulk dual of acertain strongly correlated system does not appear to bebacked by the above conclusions. Nevertheless, in somecases, including d = 0 and d = 1, certain phenomenolog-ical predictions may indeed turn out to be right - albeit,quite possibly, for the wrong reason. On the other hand,if the essentials of practical holography were to amount tolittle else but hydrodynamics then the whole issue wouldbecome largely moot and void. S. A. Hartnoll, Class. Quant. Grav. , 224002 (2009); C.P. Herzog, J.Phys. A42 , 723105 (2010); S.Sachdev, An-nual Review of Cond. Matt. Phys. , 9 (2012); J. Zaanenet al, ’Holographic Duality in Condensed Matter Physics’,Cambridge University Press, 2015; M. Ammon and J. Erd-menger, ’Gauge/Gravity Duality’, Cambridge UniversityPress, 2015; S.A. Hartnoll, A.Lucas, and S. Sachdev, ’Holo-graphic Quantum Matter’, MIT Press, 2018. A. Dhar, G. Mandal, and S.R. Wadia, arxiv:hep-th/9210120;arXiv:hep-th/9204028; arXiv:hep-th/9207011;arXiv:hep-th/9309028; arxiv:hep-th/9212027;S. R. Das et al, arXiv:hep-th/9111021;arXiv:hep-th/9112052;arxiv:hep-th/0401067;arxiv:hep-th/0503002; S. Iso, D. Karabali, B.Sakita arxiv:hep-th/9202012; S.R. Das and L.H. Santos, arxiv:hep-th/0702145; N. Banerjeea,S.Duttab, arXiv:1112.5345; S. R. Das and S.D. Mathur arxiv:hep-th/9507141; A. Jevicki,arxiv:hep-th/9309115.pdf. https://arxiv.org/search/?query=holographic+cond-mat https://projects.ift.uam-csic.es/holotube/video-gallery/ A. Luther, Phys. Rev. B 19, 320 (1979); F.D.M.Haldane,Varenna lectures, Proceedings (1992); cond-mat/0505529;D.V.Khveshchenko and P.C.E.Stamp, Phys.Rev.Lett.71,2118 (1993); Phys.Rev.B49, 21 5227 (1994); A. C. Netoand E. Fradkin, Phys.Rev.Lett 72, 1393 (1994); Phys.Rev. B 49, 10877 (1994); A.Hougton and J.B.Marston,Phys.Rev.B48,7790 (1993); A.Hougton, H.J.Kwon,and J.B.Marston, Phys.Rev.Lett. 73, 284 (1994);Phys.Rev.B50, 1351 (1994); Advances in Physics 49,141 (2000); D.V.Khveshchenko, R.Hlubina, and T.M.Rice,Phys.Rev.B48, 10766 (1993); L.B.Ioffe, D.Lidsky, andB.L.Altshuler, Phys.Rev.Lett.73, 472 (1994); C.Castellani,C.Di Castro, and W.Metzner, Phys.Rev.Lett.72, 316(1994); J.Polchinski, Nucl.Phys.B422, 617 (1994);D.V.Khveshchenko, Phys. Rev. B49, 16893 (1994);arXiv:cond-mat/9404094; Phys.Rev. B, 52, 4833(1995). A. M. J. Schakel, arxiv:cond-mat/9607164; A. G.Abanov and P. B. Wiegmann, arxiv:cond-mat/0504041;E. Bettelheim, A. G. Abanov, and P. Wiegmann,arxiv:nlin/0605006; arXiv:0810.5327; arXiv:0804.2272arxiv: cond-mat/0607453; P.B.Wiegmann,arXiv:1112.0810; arXiv:1211.5132; E. Bettelheim andP. Wiegmann, arXiv:1104.1854; E. Bettelheim, Y.Kaplan,and P. Wiegmann, arXiv:1103.4236; arXiv:1011.1993;M. Laskin, T. Can, and P. Wiegmann, arXiv:1412.8716;M. Laskin et al, arXiv:1602.04802; S. Klevtsov et al,arXiv:1510.06720; M. Pustilnik and K. A. Matveev,arXiv:1507.05639; arXiv:1706.07004; arXiv:1606.05553;B. Doyon, H. Spohn, T. Yoshimura, arXiv:1704.04409;B. Doyon and T. Yoshimura, arXiv:1611.08225; M.Fagotti, arXiv:1708.05383; B. Doyon, arXiv:1712.05687;arXiv:1912.08496; P. Ruggiero et al, arxiv.org:1910.00570;Zhe-Yu Shi,1, Chao Gao, and Hui Zhai, arXiv:2011.01415;D. S. Dean et al, arXiv:1902.02594; arXiv:2012.13958; H.Spohn, arXiv:2101.06528. J. R. Klauder, arxiv: quant-ph/0112010; C. Anastopou-los, arxiv:quant-ph/0312025; T. Curtright and D. Fair-lie arxiv:math-ph/0207008; arxiv: math-ph/0303003; R. Carroll, arxiv:quant-ph/0401082; D.H. Delphenich arxiv:gr-qc/0211065; T. Curtright arxiv: hep-th/0307121; C.Anastopoulos and N. Savvidou, arxiv:quant-ph/0304049;T. L. Curtright and C. K. Zachos, arxiv: 1104.5269; V.V. Dodonov and A. V. Dodonov, arXiv:1504.00862; M.Geiller, C. Goeller, N. Merino, arXiv:2011.09873. A. Achucarro and P.K. Townsend, Phys. Lett. B180,89 (1986); E. Witten, Commun. Math. Phys. 121, 351(1989); J.D. Brown and M. Henneaux, Commun. Math.Phys. 104, 207 (1986); O. Coussaert, M.Henneaux,and P. van Driel arxiv:gr-qc/9506019; R.E. Goldsteinand D.M. Petrich, Phys. Rev. Lett. 67 (3203) 1991;H.Afshar et al, arXiv:1512.08233; arXiv:1611.09783;arXiv:1603.04824; M. Henneauxa, W. Merbisa and A.Ranjbara arXiv:1912.09465; S. Li, N. Toumbas, J.Troost, arXiv:1903.06501; A. P´erez, D. Tempo, R. Tron-coso, arXiv:1605.04490; A.Perez arXiv:1711.02646; D.Grumiller1, W. Merbis, arXiv:1906.10694; E Ojed.aaand A.Pereza arXiv:1906.11226; A. Dymarsky andS.Sugishitaa arXiv:2002.08368; J.Matulich, S. Pro-hazka, and J. Salzer, arXiv:1903.09165; E. Ojeda andA. Perez, arXiv:1906.11226; J.Cotler and K. Jensen,arXiv:2006.08648. M.Henneaux and S.-J. Rey arXiv:1008.4579; G. Com-pere, and W. Song arXiv:1306.0014; M. Gutperle and YiLi arXiv:1412.7085; M. Beccaria et al, arXiv:1504.06555;Grumiller, R. Troncoso arXiv:1607.05360; E. Ojeda and A.Perez, arXiv:2009.07829. P.M. Petropoulos, arXiv:1406.2328; P. M. Petropoulos,K.Siampos, arXiv:1510.06456; J. Gath, A. Mukhopad-hyay, A. C. Petkou, arXiv:1506.04813; L. Ciambelli, et al,arXiv:1802.05286; arXiv:1802.06809; D. Hansen, J. Har-tong, and N. A. Obers3, arXiv:2001.10277; J. de Boerand D. Engelhardt, arXiv:1604.05327; M. Rangamani et al,arXiv:0811.2049; A. Campoleoni et al, arXiv:1812.04019. M. Kulkarni, G. Manda, and T. Morita, arXiv:1806.09343;S. R. Das, S. Hampton, S. Liu, arXiv:1910.00123;arXiv:1903.07682; S. R. Das, S. Hampton, S. Liu,arXiv:1903.07682; arXiv:1910.00123. S. Bhattacharyya et al, arXiv:0712.2456; S. Bhat-tacharyya, arXiv:0803.2526; N. Ambrosetti, J. Charbon-neau, and S. Weinfurtner, arXiv:0810.2631; N. Banerjeeet al, arXiv:0809.2596; S. Bhattacharyya, S. Minwalla,S. R. Wadia, arXiv:0810.1545; M. Haack and A. Yarom,arXiv:0806.4602; S. Bhattacharyya et al, arXiv:0809.4272;S. Bhattacharyya et al, arXiv:0712.2456; M. Ranga-mani, arXiv:0905.4352; V. E. Hubeny, S. Minwalla,M. Rangamani, arXiv:1107.5780; N. Banerjeea, S.Dutta,arxiv.org/pdf/1112.5345; Yanyan Bu and M.Lublinsky,arXiv:1409.3095; A. Campoleoni et al, arXiv:1812.04019;L. Ciambelli et al, arXiv:2006.10083. G. Barnich1, H. A. Gonz´alez, and P. Salgado-Rebolledo, arXiv:1707.08887; P. Caputa, J. M. Magan,arXiv:1807.04422; J. M. Magan, arXiv:1805.05839; J.Ergmenter et al, arxiv.org/pdf/2004.03619; A.Jain,arXiv:2008.03994; W. Merbis and M. Riegler,arXiv:1912.08207. S. Sachdev and J. Ye, Phys. Rev. Lett. 70(1993) 3339; S. Sachdev, ibid 105, 151602 (2010);Phys.Rev.X5, 041025 (2015); 2 A. Kitaev, KITPseminars, 2015; arXiv:1711.08169; A. Kitaev and S.J. Suh, arXiv:1711.08467;1808.07032; E. Wit-ten, arXiv:1610.09758, R. Gurau, arXiv:1611.04032;1705.08581; I. R. Klebanov and G. Tarnopolsky, PhysRev D95, 046004 (2017), S. Giombi, I. R. Klebanov,and G. Tarnopolsky, ibid 96, 106014 (2017); J. Malda-cena, S.H. Shenker and D. Stanford, JHEP 08 (2016)106; J. Maldacena and D. Stanford, Phys. Rev. D94, 106002 (2016); J. Maldacena, D. Stanford and Z.Yang, arXiv:1606.01857; D. Stanford and E. Witten,JHEP10(2017)008; J. Polchinski and V. Rosenhaus,JHEP 1604, 001 (2016); D. J. Gross and V. Rosenhaus,JHEP 1705 (2017) 092, ibid 1712 (2017) 148; G. Sarosi,arXiv:1711.08482]; H. W. Lin, J.Maldacena, and Y. Zhao,arXiv:1904.12820; Y.Gu, X-L. Qi, and D. Stanford, J.High Energ. Phys. (2017) 2017: 125; Y. Gu, A. Lucas,and X-L. Qi, SciPost Phys. 2, 018 (2017); J. High Energ.Phys. (2017) 2017: 120; Y. Gu and A.Kitaev, J. HighEnerg. Phys. (2019) 2019: 75; 11 D. Bagrets, A. Altlandand A. Kamenev, Nucl. Phys. B 911, 191 (2016); ibid921, 727 (2017); T. G. Mertens, G. J. Turiaci and H. L. Verlinde, JHEP08(2017)136; T. G. Mertens, JHEP1805 (2018) 036; Z. Yang, arXiv:1809.08647; A. Jevicki,K. Suzuki and J. Yoon, JHEP 1607, 007 (2016); A.Jevicki, K. Suzuki, arXiv:1608.07567; S. R. Das, A.Jevicki, K. Suzuki, JHEP09(2017)017; S. R. Das et al,JHEP07(2018)184; ibid 02(2018)162; G. Mandal, P. Nayakand S. R. Wadia, JHEP 1711 (2017) 046; A. Gaikwad etal, arXiv:1802.07746; T. G. Mertens and G. J. Turiaci,arXiv:1904.05228; D.V.Khveshchenko, arXiv:1905.04381.15