# Phase space holography with no strings attached

D. V. Khveshchenko  Department of Physics and Astronomy, University of North
Carolina, Chapel Hill, NC 27599

###### Abstract

This note discusses the Wigner function representation from the standpoint of
establishing a holography-like correspondence between the descriptions of a
generic quantum system in the phase space (’bulk’) picture versus its
spacetime (’boundary’) counterpart. Under certain circumstances the former
might reduce to the classical dynamics of a local metric-like variable while
the latter takes on the form of some bosonized collective field hydrodynamics.
This generic pseudo-holographic duality neither relies on any particular
symmetry of the system in question, nor does it require any connection to an
underlying ’string theory’, as in the various ’ad hoc’ scenarios of applied
holography.

Generalized holography

Among all the remarkable theoretical advances under the hashtag
${\\#}holography$ the one of the greatest interest to condensed matter physics
is the ongoing quest for generalizing the original, highly constrained (i.e.,
Lorentz- and maximally super-symmetric, etc.), string-theoretical holographic
conjecture to as broad as possible (i.e., non-Lorentz-, non-isotropic-, non-
super-symmetric, etc.) variety of quantum many-body systems ads .

In that regard, the so-called ’bottom-up’ holography has been portraying
itself as a powerful technique offering solutions to the traditionally hard
condensed matter problems. This innovative approach (sometimes referred to as
the anti-de-Sitter/condensed matter theory, or $AdS/CMT$, correspondence)
borrows its formal structure and mathematical apparatus (often ’ad verbatim’,
for the lack of an alternative) from the very specific, carefully crafted, and
highly symmetric examples of such duality known under the acronym $AdS/CFT$
and conjectured in the original context of fundamental string theory and its
various reductions ads .

So far, however, all the attempts of putting applied holography on a firm
foundation \- either along the lines of the geometrized renormalization group
(RG) flow or entanglement dynamics in tensor networks, or by using artificial
thermodynamic (Fisher-Ruppeiner, Fubini-Study, etc.) metrics, or else - have
remained consistently inconclusive.

Nonetheless, instead of striving to deliver a solid proof of principle, the
field of $AdS/CMT$ has, by and large, stayed the course reminiscent of
hacker’s code cracking: that is, trying to guess some higher-dimensional
enhanced gravity-like theory (often on the sole basis of technical
convenience) and then rely on the persuasive power of visual agreement between
some pre-selected experimental data plots and the (for the most part,
numerical) calculations based on the above $AdS/CMT$ ’dictionary’. A great
many number of the customarily verbose and look-alike accounts of such
pursuits can be readily found in the applied holographic literature from the
year $2007$ onwards holo . Judging by the factual outcome of this massive
attack, though, the code does not appear to have been cracked yet.

Such fundamental shortcomings notwithstanding, the still continuing occasional
exercises in the holographic phenomenology utilize a handful of the popular
bulk geometries, especially focusing on the legacy solutions of the
prototypical Einstein-Maxwell-dilaton theory. Other than their relative
simplicity and sheer availability (alongside, possibly, some lingering
anthropic factor) a proper justification of such choices does not appear to
have been an essential part of the holographic agenda, regardless of whether
the results were meant to be applied to the lattice Hubbard-like models,
superconducting cuprates, low-density $2DEG$, Dirac/Weyl materials such as
graphene, or else.

Lately, though, a gradual decline in such ’orthodox’ applications of the
$AdS/CMT$ machinery has been giving way to advanced hydrodynamics of strongly
coupled quantum matter and general out-of-equilibrium physics (eigenstate
thermalization, many-body (de)localization, chaos spreading, operator growth,
etc.). Correspondingly, instead of the once ubiquitous renditions of esoteric
black holes, nowadays a slide show on the topic of $AdS/CMT$ is more likely to
feature the images of water flows, rapids, whirlpools, and other familiar
hydrodynamic patterns video .

Of course, hydrodynamics, while suggesting some intriguing holographic
connections, has long been discussed outside of any holographic context.
Therefore, the renewed appreciation and novel applications (thanks to a number
of recent experimental advances) of the theory as Earthly as hydrodynamics
alone do not provide an answer to the question that should have (but does not
seem to 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 theoretical approach known as
collective field theory das  and its more recent developments that might be
capable of providing a much-needed justification for the ’stringy
hydrodynamics’ (especially, in those non-relativistic and/or rotationally-non-
invariant settings that are typical to the condensed matter applications but
do not normally occur in 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 to reformulate a quantum theory
of interest in terms of some intrinsically geometric bosonic variables.

Notably, while demonstrating some features reminiscent of the desired
holographic correspondence, this approach shows that the pertinent space-time
metrics may not be chosen at will and can often be quite different 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 $w({\bf x},{\bf p},t)$ defined in the
$2d+1$-dimensional phase space (plus time). The space of such functions is
equipped with the Moyal product

$f({\bf x},{\bf p})\star g({\bf x},{\bf p})=f({\bf x},{\bf p})e^{i{\hbar\over
2}({\overleftarrow{\partial}_{\bf x}}{\overrightarrow{\partial}_{\bf
p}}-{\overleftarrow{\partial}_{\bf p}}{\overrightarrow{\partial}_{\bf
x}})}g({\bf x},{\bf p})$ (1)

which introduces the symplectic structure through the Moyal bracket

${\\{}f,g{\\}}_{MB}=f\star g-g\star f$ (2)

The Wigner’s description is well suited for taking into account the underlying
theory’s invariance under the phase space volume-preserving diffeomorphisms,
including its natural time evolution that can be thought of as a canonical
transformation governed by the Liouville theorem.

Furthermore, when quantizing the system via the method of functional
integration, the function $w({\bf x},{\bf p},t)$ becomes a constrained field
variable implementing a coadjoint orbit’s quantization of the phase space
volume-preserving diffeomorphisms a la Kirillov-Kostant.

In this procedure, an orbit’s element
${\hat{g}}|\Psi_{0}><\Psi_{0}|{\hat{g}}^{-1}$ is constructed by acting with an
element $\hat{g}$ of the infinite-dimensional group of diffeomorphisms on the
projector to a chosen (e.g., ground) state $|\Psi_{0}>$. In the partition
function $Z=\int Dw\exp(-S(w))$ the integration then runs over the functions
satisfying the constraints

$w\star w=w~{}~{}~{}~{}~{}~{}~{}Trw=1$ (3)

and governed by the action

$S(w)=\int
dxdpdt(i\int^{1}_{0}dsw{\\{}\partial_{\tau}w,\partial_{s}w{\\}}_{MB}-wH)$ (4)

where the first term represents the Berry phase where the integral over the
auxiliary variable $s$ depends only on the boundary value $w({\bf x},{\bf
p},t,s=1)=w({\bf x},{\bf p},t)$ at $s=1$. This way, one arrives at the
formally exact geometrized description of the non-linear $\sigma$-model type
das .

The equation of motion derived from Eq.(4)

${\dot{w}}+{\\{}w,H{\\}}_{MB}=0$ (5)

reproduces the standard kinetic equation when the Moyal bracket is
approximated, to lowest order in the powers of $\hbar$, by the Poisson one
(hereafter the dot and prime stand for the time and space derivatives,
respectively)

${\dot{w}}+w^{\prime}\partial_{\bf p}H_{1}-H_{1}^{\prime}\partial_{\bf
p}w=St[w]$ (6)

where the one-body Hamiltonian $H_{1}$ may include an external potential. For
example, in the so-called ’non-critical $2d$ string theory’ where the spatial
coordinate originates from the eigenvalues of $N\times N$ matrices it happens
to be the inverted oscillator ($V\sim-x^{2}$) das . Also, the $n\geq 2$-body
terms $H_{n}$ in the Hamiltonian involving higher powers of $w$ are bundled
into the collision integral in the right hand side.

In the case of fermions, the semiclassical vacuum configuration corresponding
to the uniform Fermi sea is described by the expression

$w_{0}({\bf x},{\bf p},t)=\theta(\mu({\bf x},t)-\epsilon_{\bf p})$ (7)

where the local chemical potential $\mu({\bf x},t)$ denotes a sharp boundary
between the occupied and vacant momentum eigenstates with the dispersion
$\epsilon_{\bf p}$.

Facilitating further progress with the $2d+1$-dimensional ’bulk’ theory (4)
requires a convenient parametrization of the bounding momentum. Previously, a
similar task was tackled in the early works on multi-dimensional bosonization
where this goal was achieved by distinguishing between the Fermi momentum $\bf
p_{F}$ tracing the fiducial Fermi surface (FS) and the normal to the FS
(’radial’) degree of freedom describing fluctuations of the momentum
distribution bos .

For instance, in the much studied $d=2$ case the simplified action for the
vector $\bf k_{F}$ reads

$S({\bf p_{F}})=\int dxdt(i\int^{1}_{0}ds{\bf p_{F}}\partial_{\tau}{\bf
p_{F}}\times\partial_{s}{\bf p_{F}}-H({\bf p_{F}}))$ (8)

where the Hamiltonian $H$ is cast in terms of the local density $\rho={\bf
p_{F}}\times\partial_{t}{\bf p_{F}}/2$.

In what follows, we focus on the case of $d=1$ where the fluctuating FS can be
described in terms of $M\geq 1$ pairs of the Fermi momenta
$p^{(n)}_{\pm}(x,t)$ bounding the occupied states ($M>1$ accounts for the
possibility of ’folds’ in the presence of shock waves and other FS
singularities das ).

In particular, the $1d$ configuration (7) reads

$w_{0}(x,p,t)=\sum_{\pm}\sum_{\alpha=1}^{M}(\pm)\theta(p^{(\alpha)}_{\pm}(x,t)-p)$
(9)

while its small perturbation

$\delta w(x,p,t)={\hbar}\delta(x-x_{cl})\delta(p-p_{cl})$ (10)

is strongly peaked at the classical phase space trajectory
($x_{cl}(x_{0},p_{0},t)$, $p_{cl}(x_{0},p_{0},t)$) where the initial data
$x_{0}$ and $p_{0}$ are determined by the current values ($x,p$) at a later
time $t$.

It is worth mentioning, though, that while being capable of faithfully
reproducing the long-distance, late-time asymptotics of the response
functions, in its practical (hence, approximate) form the $d>1$-dimensional
bosonization technique is not well suited for computing the Lindhard-type
$2p_{F}$-singularities, just as it may not be sufficient for the single-
particle propagators bos .

In $d=1$, despite several decades of studies there has been a recent surge of
renewed interest in the out-of-equilibrium dynamics of quantum interacting
bosons and fermions. Many of those studies focus on the integrable and non-
ergodic systems which are governed by the generalized Gibbs ensembles (GGE)
and may not comply with the more generic eigenstate thermalization hypothesis
(ETH) hydro .

It is worth noting that in $1d$ Eq.(2) represents a classical analog of the
infinite-dimensional quantum algebra $W_{\infty}$ composed of the operators
${\hat{W}}_{mn}=({\hat{x}})^{m}({\hat{p}})^{n}$ with the commutation relations

$\displaystyle[{\hat{W}}_{mn},{\hat{W}}_{rs}]=\sum_{k=1}{(-\hbar)^{k}\over
k{\\!}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (11)
$\displaystyle({n{!}r{!}\over(n-k){!}(r-k){!}}-{m{!}s{!}\over(m-k){!}(s-k){!}}){\hat{W}}_{m+r-k,n+s-k}$

where the r.h.s. reduces to $(ms-nr){\hat{W}}_{m+r-1,n+s-1}$ in the limit
$\hbar\to 0$ limit, thereby encompassing the algebra $SL(2,R)$. This algebra
has been extensively studied in the context of Quantum Hall Effect (QHE) and
the various reincarnations of (effectively) non-commutative spacetimes.

An abstract Hilbert space can be readily equipped with a geometric structure
that has long been elucidated alongside the more familiar Berry phase.
However, the even (Fubini-Study metric) \- as opposed to the odd (Berry
curvature) - component of the same rank-$2$ tensor has been receiving less
attention.

Such a phase space metric can be naturally introduced in the context of
special coherent \- (de)localized neither in the coordinate, nor momentum
space - states $|p,x,0>=e^{i{\hat{P}}x-i{\hat{Q}}p}|0>$ which minimize, both,
the coordinate and momentum uncertainties.

Allowing, for the sake of generality, some coordinate-momentum cross-
correlations the corresponding Wigner function reads

$\displaystyle w_{coh}(x,p,0)=\int dye^{ipy}<\Psi|x+y/2><x-y/2|\Psi>$
$\displaystyle={\hbar\over D^{1/2}}\exp(-{\sigma_{p}\delta
x^{2}+\sigma_{x}\delta p^{2}+2\sigma_{xp}\delta x\delta
p\over{2D}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (12)

where $\delta x=x-x_{0}$, $\delta p=p-p_{0}$,
$D=\sigma_{x}\sigma_{p}-\sigma^{2}_{xp}$, and the parameters
$\sigma_{x},\sigma_{p},\sigma_{xp}$ control the Gaussian coordinate and/or
momentum variations.

The above suggests a naturally defined Fubini-type metric on the phase space
qm

$ds^{2}=(<\partial_{\mu}\Psi|\partial_{\nu}\Psi>-<\Psi|\partial_{\mu}\Psi><\partial_{\nu}\Psi|\Psi>)\delta_{\mu}\delta_{\nu}$
(13)

where $\delta_{\mu}=(dx,dp)$. Taken at its face value, this formula
establishes some form of superficial correspondence between single-particle
quantum mechanics and $2d$ metrics that can be viewed as solutions of certain
classical gravity.

Further generalizing Eq.(13) to include energy fluctuations one arrives at the
(Euclidean) $3d$ metric

$\displaystyle
ds^{2}=<(\Delta{\hat{H}})^{2}>dt^{2}+<(\Delta{\hat{x}})^{2}>dp^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+<(\Delta{\hat{p}})^{2}>dx^{2}+2<\Delta{\hat{x}}\Delta{\hat{p}}>dxdp~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(14)

given by the uncertainties of the conjugate variables ($x\leftrightarrow p$,
$t\leftrightarrow H$). Also, considering the metric (14) to be the expectation
value $ds^{2}=<\Psi|d{\hat{s}}^{2}|\Psi>$ of the operator-valued interval
$d{\hat{s}}^{2}$ paves the way for promoting the bulk (phase space)-to-
boundary (spacetime) relationship to the quantum level.

As the operators’ uncertainties depend on the choice of the state $|\Psi>$, so
does the dual metric (14). Heuristically, one might expect that for the
single-particle dispersion governed by the dynamical exponent $z$
($\epsilon_{\bf p}\sim{\bf p}^{z}$) the above variances behave as follows

$\displaystyle<(\Delta p)^{2}>\sim<(\Delta x)^{2}>^{-1}\sim p^{2}$
$\displaystyle<(\Delta x)(\Delta p)>\sim<(\Delta p)>^{1/2}<(\Delta
p)^{-1/2}\sim p^{0}$ $\displaystyle<(\Delta H)^{2}>\sim<(\Delta
p)^{2}>^{z}\sim p^{2z}$ (15)

so that the metric (14) conforms to the so-called Lifshitz variety (the
coefficients $A,B,C,D$ are constants)

$ds^{2}=Ap^{2z}dt^{2}+Bp^{2}dx^{2}+C{dp^{2}\over p^{2}}+2Ddxdp$ (16)

which has been often invoked in the various applications of $AdS/CMT$ ads .

Non-linear hydrodynamics

The formally exact representation (4) of the phase space dynamics provides a
basis for further simplifications, thus giving rise to (semi)classical
hydrodynamic equations for the various moments of the Wigner distribution

$w_{n}({\bf x},t)=\int d{\bf p}w({\bf x},{\bf p},t){\bf p}^{n}$ (17)

Among those moments are such standard hydrodynamic variables as the local mass
$\rho$ ($n=0$), momentum $Q$ ($n=1$), and energy $\epsilon$ ($n=z$) densities,
respectively. This transition from the entire Wigner function to the first few
of its moments can be thought of as a dimensional reduction from the
$2d+1$-dimensional bulk (phase space) to its $d+1$-dimensional boundary
hypersurface (spacetime) which, in practice, amounts to mere integration over
the $d$-dimensional momentum.

Correspondingly, the mass $J_{\rho}$, momentum $J_{Q}$, and energy
$J_{\epsilon}$ currents are given by the general expression

${\bf J}_{n}=\int d{\bf p}{\partial\epsilon_{p}\over\partial{\bf p}}w({\bf
x},{\bf p},t){\bf p}^{n}$ (18)

for $n=0,1,2$.

In the case of $d=1$ the lowest moments of the Wigner function (17) correspond
to the aforementioned bounding Fermi momenta $p_{\pm}=\int dpw(1\pm sgnp)/2$
or, equivalently, the local density and material velocity

$\rho={1\over 2\pi}(p_{+}-p_{-})~{}~{}~{}~{}~{}~{}v={1\over 2m}(p_{+}+p_{-})$
(19)

Note that limiting the momentum values to the interval $0<p<\infty$, similar
to the holographic radial variable, is dictated by the chiral nature of the
excitations carrying sign-definite momenta.

These variables have the Poisson bracket

${\\{}\rho(x),v(y){\\}}=\partial_{x}\delta(x-y)$ (20)

Then taking the various moments of Eq.(5) one arrives at the hydrodynamic
equations of motion which include the continuity equation

${\dot{\rho}}+(\rho v)^{\prime}=0$ (21)

and the inviscid Navier-Stokes (a.k.a. Euler/Burgers/Hopf) one

${\dot{v}}+vv^{\prime}=-{P^{\prime}\over\rho}-\kappa{V^{\prime}\over\rho}$
(22)

where the ’quantum pressure’ $P(\rho)$ (internal stress tensor of the $1d$
quantum fluid) is a system-specific function of the local density, while the
last term with the dispersion curvature
$\kappa={\partial^{2}_{p}\epsilon_{p}}$ represents the force exerted by the
external potential (if any).

It is well known that the non-linear hydrodynamic equations (21,22) can be
derived even from the free particle Schroedinger equation qm . Specifically,
by applying the Madelung parametrization of the wave function
$\Psi(x,t)={\sqrt{\rho}(x,t)}e^{iS(x,t)}$ and separating out the real and
imaginary parts one arrives at the coupled continuity and Navier-Stokes
equations, respectively, where

$v={1\over
2im}({\Psi^{*}\over\Psi})^{\prime}~{}~{}~{}~{}~{}~{}\rho=|\Psi|^{2}$ (23)

In the r.h.s. of (22) the pressure

$P={\hbar^{2}\over 8m}{(\rho^{\prime})^{2}\over\rho}$ (24)

contributes towards the overall energy density $\epsilon=\rho v^{2}/2+P$
which, in general, might be neither polynomial, nor separable as a sum of two
chiral components $P_{\pm}(p_{\pm})$.

It appears, however, that the pressure gradient term couples excitations with
opposite chiralities (left/right moving) at the level of operators with
dimensions of four or greater. Moreover, even if present, the non-chiral
corrections do not affect the states which are composed exclusively of the
chiral excitations with $p_{+}=0$ or $p_{-}=0$.

Thus, an arbitrary single-particle state $\Psi(x,t)=(2\pi i\hbar
t/m)^{-1/2}\int dye^{im(x-y)^{2}/2\hbar t}\Psi_{0}(y)$ of the free Hamiltonian
of mass $m$ with the initial data $\Psi_{0}(x)$ provides a valid solution to
the hydrodynamic equations (21,22) with the pressure (24). Correspondingly,
the pair of functions $\rho(x,t)$ and $v(x,t)$ determines a certain dual
metric, as explained below.

Solvable hierarchies

For certain choices of the Hamiltonian $H$ Eq.(22) appears to belong to the
infinite KdV (Korteweg–de Vries) hierarchy of integrable $1d$ systems hydro .
Such Hamiltonians $H^{\pm}_{k}$ are related to the Gelfand-Dickey polynomials
and form an infinite set of integrals of motion in involution
($[H^{\pm}_{n},H^{\pm}_{m}]=0$). In the asymptotically free regime of large
momenta (high energies) the $k^{th}$ member of this family describes small-
amplitude excitations with the dispersion exhibiting the dynamical exponent
$z=2k-1$.

In particular, the generic $1d$ Luttinger liquid (LL) behavior is associated
with the quadratic Hamiltonian $H=\sum_{\pm}H^{\pm}_{1}$ given by the standard
Sugawara construction

$H_{1}={1\over 2}\sum_{\pm}p^{2}_{\pm}={1\over 2}(p_{+}^{2}+p_{-}^{2})$ (25)

which gives rise to the equation of motion

${\dot{p}_{\pm}}\mp p_{\mp}^{\prime}=0$ (26)

whose solutions given by the (anti)holomorphic functions $p_{\pm}(x_{\pm})$
describe two decoupled chiral ($x_{\pm}=x\pm t$) pseudo-relativistic ($z=1$)
excitations.

Corrections to the LL Hamiltonian (25) may come from, both, the Gaussian terms
of higher operator dimensions which represent non-linear terms in the
dispersion of the collective $\rho$\- and $v$-modes, as well as from the non-
Gaussian ones which are due to some intrinsic non-linearity of the $1d$
dispersion, as in the case of non-relativistic fermions at a finite density
hydro .

For example, the next ($2^{nd}$) member of the KdV family is given by the non-
Gaussian expression

$\displaystyle H_{2}=\sum_{\pm}({\pm}){1\over 3}p_{\pm}^{3}+{1\over
2}(p^{\prime}_{\pm})^{2}=$ $\displaystyle{1\over 2}\rho v^{2}+{\pi^{2}\over
6}\rho^{3}+{1\over 2}(v^{\prime})^{2}+{\pi^{2}\over 2}(\rho^{\prime})^{2}$
(27)

for which the chiral components of Eq.(22) still remain uncoupled

${\dot{p}_{\pm}}\pm 3p_{\pm}p_{\pm}^{\prime}+p_{\pm}^{\prime\prime\prime}=0$
(28)

In the asymptotic regime of high energies the linearized Eq.(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 solitonic excitations (’cnoidal’ waves) hydro

$v(x,t)\sim{1\over\cosh^{2}x_{\pm}}$ (29)

whose propagation is described by the dispersion $\epsilon_{p}\sim p^{5/3}$.
Compactifying the spatial coordinate into a finite-length circle would then
replace (29) with the elliptic Jacobi function.

In general, the non-Gaussian terms in the Hamiltonian are sensitive to the
microscopic details of the model and stem from, both, kinetic and potential
terms in the total energy. Specifically, in the case of hard-core bosons, such
as the Tonks-Girardeau limit of the Lieb-Liniger model, the Hamiltonian
includes the pressure term $P(\rho)\sim\rho^{3}$. By contrast, in the quantum
Toda chain the function $P(\rho)$ is non-polynomial. However, despite not
being dividable into a sum of two chiral terms, the latter can still fit into
the KdV Hamiltonian (27) as the non-chiral terms appear to be irrelevant at
low momenta hydro .

Likewise, the deviations from the LL regime associated with a finite
dispersion curvature and/or chiral interactions can be studied with the use of
a linear combination ${\hat{H}}_{1}+{\hat{H}}_{2}+\dots$ of Eqs.(25) and (27)
. This way, one can obtain non-linear corrections $\delta\epsilon_{p}\sim
p^{5/3}$ to the linear LL spectrum at small momenta hydro .

Moreover, Eq.(27) can be further modified by including irrelevant non-Gaussian
terms, such as $p^{4}_{\pm}$, without destroying its integrability. Indeed,
such extension results in yet another, Gardner, equation (a.k.a. mixed
KdV-m(modified)KdV, which two equations are related by virtue of the Miura
transformation $p_{\pm}\to p^{2}_{\pm}+p_{\pm}^{\prime}$).

As a hallmark of integrability, the higher-$k$ level members of the KdV
hierarchy possess the bi-Hamiltonian structure relating them as follows sh

$\partial_{x}{\delta H^{\pm}_{k+1}\over\delta p_{\pm}}={\cal
D}^{\pm}_{x}{\delta H^{\pm}_{k}\over\delta p_{\pm}}$ (30)

where the long derivative is

${\cal D}^{\pm}_{x}=2p_{\pm}\partial_{x}+\partial_{x}p_{\pm}+\partial^{3}_{x}$
(31)

The higher level-$k$ members of the KdV and mKdV families can also be morphed
into a two-parameter Gardner sequence of Hamiltonians. Furthermore, certain
solvable systems of $M\geq$ coupled non-linear equations were shown to be
associated with the higher-spin symmetry algebras $SL(M,R)$ (e.g., the
Boussinesq equations for $M=3$) hs .

Generic Hamiltonians $\sum_{k}\mu_{k}{H}^{\pm}_{k}$ which includes different
members of the integrable family can be used to derive zero entropy GGE
hydrodynamics. In particular, one can study crossovers between the LL and
higher level-$k$ regimes at varying momenta. In essence, this construction
provides a (formally exact) bosonization scheme that was fully exploited,
e.g., in the context of the solvable Calogero-Sutherland model hydro .

Under the time evolution governed by a superposition of the different $H_{n}$
a generic initial condition produces a collection of solitons with different
velocities and a continuum of decaying dispersive waves. Being more robust the
soliton excitations dominate in the late-time behavior and, in particular, the
system’s equilibration towards a steady GGE state described by the density
matrix ${\hat{\cal G}}=\exp(-\sum_{k}\mu^{\pm}_{k}{\hat{H}}^{\pm}_{k})$ where
the chemical potentials $\mu_{k}$ are to be determined by equating the
averages of the commuting charges $<{\hat{H}}^{\pm}_{k}>=Tr({\hat{\cal
G}}{\hat{H}}^{\pm}_{k})/tr{\hat{\cal G}}$ to their chosen values.

Along these lines, one can also study the von Neumann entropy $S=-Tr{\hat{\cal
G}}\ln{\hat{\cal G}}$. The Wigner function satisfying the constraints (3)
corresponds to a pure state of zero entropy and the presence of an infinite
number of conserved charges precludes standard thermalization. When the
constraint ceases to hold the state becomes mixed, thus resulting in a finite
entropy. The ensuing thermalization can be accounted for by introducing
viscous terms, such as $\eta\rho^{\prime\prime}$, in the r.h.s. of Eq.(22).

Dual bulk gravity

A deep relationship between classical gravity and hydrodynamics has long been
known as one particular take on the holographic paradigm, often referred to as
the ’fluid-gravity’ correspondence. The crux of the matter is observation of
the similarity between the asymptotic near-boundary behavior of the Einstein
equations for the bulk metric and the Navier-Stokes ones describing a dual
boundary fluid in one lesser dimension (besides, the complementary
hydrodynamic behavior near the event horizon can be similar to that at the
boundary). Albeit being truncated and, therefore, approximate such relations
can be systematically improved, thus enabling certain computational
simplifications. Whether or not this duality can be promoted to the quantum
level requires further analysis.

Remarkably, in the case of $d=1$ this correspondence becomes exact.
Specifically, the Einstein equations stemming from the action of $3d$ gravity
with a negative cosmological constant (here $l$ and $G$ are the AdS radius and
Newton’s constant, respectively)

$S={l\over 16\pi G}\int dxdtdp{\sqrt{g}}(R+2/l^{2})=0$ (32)

coincide with the equations describing two decoupled Chern-Simons (CS) models
with the combined action sh

$S={l\over 16\pi G}Tr\int
dxdpdt\epsilon^{\mu\nu\lambda}({\hat{A}}_{\mu}^{\pm}\partial_{\nu}{\hat{A}}_{\lambda}^{\pm}+{\hat{A}}_{\mu}^{\pm}{\hat{A}}_{\nu}^{\pm}{\hat{A}}_{\lambda}^{\pm})$
(33)

The chiral connections ${\hat{A}}_{\mu}^{\pm}$ are matrices that can be
expanded in the basis spanned by the generators ${\hat{L}}^{\pm}_{0,\pm 1}$ of
the algebra $SL(2,R)\times SL(2,R)=SO(2,2)$. They obey the commutation
relations
$[{\hat{L}}^{\pm}_{n},{\hat{L}}^{\pm}_{m}]=(n-m){\hat{L}}^{\pm}_{n+m}$ and are
normalized, $Tr{\hat{L}}^{\pm}_{n}{\hat{L}}^{\pm}_{m}={1\over
2}\delta_{n0}\delta_{m0}-\delta_{n1}\delta_{m,-1}$.

The topological action (33) then reduces to a pure boundary term while the
equation of motion becomes that of null curvature

$\partial_{\mu}{\hat{A}}_{\nu}^{\pm}+{\hat{A}}_{\mu}^{\pm}{\hat{A}}_{\nu}^{\pm}-(\mu\leftrightarrow\nu)=0$
(34)

Parameterizing its solutions in terms of an arbitrary group element
${\hat{\chi}}_{\pm}$ and functions $p_{\pm}(x_{\pm})$ and $\mu_{\pm}(x_{\pm})$

${\hat{A}}^{\pm}(x,p,t)={\hat{\chi}}^{-1}_{\pm}(p){\hat{L}}_{0}(\mu_{\pm}\pm
p_{\pm}dx){\hat{\chi}}_{\pm}(p)$ (35)

one finds this equation to be equivalent to

${\dot{p}}_{\pm}\mp\mu_{\pm}^{\prime}=0$ (36)

which, in turn, coincides with one of the above solvable equations, provided
that the chemical potentials $\mu_{\pm}$ conjugate to the variables $p_{\pm}$
are given by the derivatives

$\mu_{\pm}={\delta H^{\pm}\over\delta p_{\pm}}$ (37)

Choosing the Hamiltonian appropriately one can then reproduce the solvable
(m)KdV, Gardner, and other equations. In particular, the KdV family is
recovered for

$\displaystyle{\hat{A}}_{p}^{\pm}(x,p,t)={1\over
p}{\hat{L}}^{\pm}_{0}~{}~{}~{}~{}~{}~{}{\hat{A}}_{x}^{\pm}(x,p,t)=p{\hat{L}}^{\pm}_{1}-{p_{\pm}\over
p}{\hat{L}}^{\pm}_{-1}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle{\hat{A}}_{t}^{\pm}(x,p,t)={p\mu_{\pm}}{\hat{L}}^{\pm}_{1}-\mu_{\pm}^{\prime}{\hat{L}}^{\pm}_{0}+{{\mu_{\pm}^{\prime\prime}-\mu_{\pm}p_{\pm}}\over
2p}{\hat{L}}^{\pm}_{-1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (38)

which expressions are manifestly Lorentz-non-invariant for all $k>1$.

Moreover, the equation of motion (34) can be converted into that of the
gravity model (32) under the identification of the $3d$ metric

$g_{\mu\nu}={l^{2}\over
4}<(A_{\mu}^{+}-A_{\mu}^{-})(A_{\nu}^{+}-A_{\nu}^{-})>$ (39)

On the gravity side the different saddle points of the coherent states path
integral can be identified as globally distinguishable (but locally $AdS_{3}$)
classical solutions. In particular, it can be shown that the only minima of
the action (32) corresponding to the boundary Hamiltonian $H_{1}+H_{2}$ are
those with constant (negative) curvature. The two competing minima are the
thermal $AdS_{3}$ and BTZ (Banados-Teitelboim- Zanelli) black hole.

However, by introducing higher order terms $H_{k}$ with $k\geq 3$ one can
generate new KdV-charged black hole configurations sh . The corresponding
boundary theory is encoded in the boundary conditions for the connection (35),
by varying which one can explore a variety of the integrable $1d$ systems.

The standard LL with $k=1$ is reproduced by introducing the original Brown-
Henneaux boundary conditions with constant $\mu_{\pm}\sim p_{\pm}$, the
outer/inner horizons being located at $p_{>/<}=(p_{+}\pm p_{-})/2$. The dual
metric

$\displaystyle ds^{2}={dp^{2}\over
p^{2}}+(p^{2}-2(p_{+}+p_{-})+{p^{2}_{+}+p^{2}_{-}\over
p^{2}})dt^{2}+~{}~{}~{}~{}~{}$
$\displaystyle(p^{2}+2(p_{+}+p_{-})+{p^{2}_{+}+p^{2}_{-}\over
p^{2}})dx^{2}+(p_{+}p_{-})dxdt~{}~{}~{}~{}~{}$ (40)

describes a rotating BTZ black hole with the event horizon but no curvature
singularity.

For static, yet non-constant, $p_{\pm}(x)$ the corresponding boundary
solutions possess non-trivial global charges given by the chiral surface
integrals $H_{k}^{\pm}$ while their bulk counterparts can be regarded as black
holes with multi-graviton excitations (’soft hair’) sh .

The general solution can be obtained by acting on the ground state (e.g., BTZ
black hole) with elements of the asymptotic symmetry group commuting with the
Hamiltonian. This way one can construct various constant curvature, yet
locally AdS, spacetimes with anisotropic Lifshitz scaling and dynamical
exponent $z=2k-1$. This opens up the possibility of studying nonrelativistic
holography without the need of bulk geometries which are asymptotically
Lifshitz spacetimes. Furthermore, in the case of a higher-spin symmetry
$SL(M,R)$ the list of attainable gravitational backgrounds may include the
asymptotically Lobachevsky, Schroedinger, warped $AdS$, etc. spacetimes carol
.

Shocks and other abrupt perturbations are characterized by FS breakdowns and
emergence of folds where the spatial derivative $\rho^{\prime}$ diverges,
thereby requiring several pairs of the bounding momenta $p_{\pm}$. In the
presence of shocks the conventional spacetime hydrodynamics becomes
insufficient for describing long-time behavior, although the full-fledged
phase space hydrodynamics can avoid such problems.

In that regard, particularly interesting are the non-stationary configurations
representing particles released from a confining potential which gets suddenly
switched on/off quench . Such quenching profiles generically have spacelike
boundaries where the saddle point solutions of the collective field
hydrodynamics diverge at finite times and the semi-classical description
fails. Ascertaining the emergent spacetimes and their dynamics then requires a
detailed study of fluctuations around the pertinent saddle points.

Reductions and generalizations

Despite having been repeatedly stated and extensively analyzed at the level of
salient symmetries and concomitant algebraic properties, the general
gravity/fluid correspondence in dimensions $d>1$ has not yet been put to much
of a practical use fluid .

Specifically, such a relationship was shown to exist between the solutions of
classical $d+2$-dimensional gravity and their $d+1$-dimensional hydrodynamic
counterparts, whereby the former would be given by the metric

$ds^{2}={dp^{2}\over
f(p)p^{2}}+p^{2}(\Delta_{\mu\nu}-f(p)u_{\mu}u_{\nu})dx^{\mu}dx^{\nu}$ (41)

parameterized in terms of the spacetime-dependent covariant velocity
$u_{\mu}({\bf x},t)$ and local temperature $T({\bf x},t)$ fluid . The latter
satisfy the hydrodynamic equations on a fixed background, provided that
$f(p)=1-(4\pi T/p)^{d}$ and $\Delta_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$. Thus,
a given fluid profile can be associated with a certain asymptotically
$AdS_{d+2}$-like spacetime with a horizon located at $p_{h}=4\pi T$.

A still more general (asymptotically accurate) solution can be constructed
with the use of the metric ansatz

$g_{\mu\nu}({\bf x},p,t)={g^{(0)}_{\mu\nu}\over
p^{2}}+g^{(2)}_{\mu\nu}+{p^{2}\over 4}g^{(2)}g^{(0)}g^{(2)}_{\mu\nu}$ (42)

with arbitrary functions $g^{(0,2)}_{\mu\nu}({\bf x},t)$ fluid .

In contrast to the generic case of $d>1$, pure gravity in $d\leq 1$ does not
support any finite energy excitations. Therefore, the fluid/gravity
correspondence takes on a particularly simple form where the dual bulk theory
appears to be non-dynamical and fully determined by the gapless boundary
degrees of freedom (’boundary gravitons’).

Of course, such a scenario of ’holography light’ does not quite rise to the
level of genuine holographic correspondence where the bulk theory is supposed
to have a non-trivial quantum dynamics which gets quenched and turns classical
only in a certain (large $N$) limit. It should be noted, however, that,
barring a few exceptions, the customarily 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 any back-reaction on the
fixed background metric appear to be by far the most common approximations
routinely made in the absolute majority of all the $AdS/CMT$ calculations
performed so far ads ; holo .

Nonetheless, there are still important differences between the situations in
$d=0$ and $d=1$. As per the above discussion, the latter is described by the
LL action of two chiral $1d$ bosons
$\phi_{\pm}(x_{\pm})=\ln\Phi^{\prime}_{\pm}(x_{\pm})$

$S^{\pm}_{LL}=\int
dxdt{\Phi^{\prime\prime}_{\pm}({\dot{\Phi}}^{\prime}_{\pm}\mp\Phi^{\prime\prime}_{\pm})\over(\Phi^{\prime}_{\pm})^{2}}=\int
dxdt\phi^{\prime}_{\pm}({\dot{\phi}}_{\pm}\mp\phi^{\prime}_{\pm})$ (43)

This action can also be obtained from the more general Alekseev-Shatashvili
functional which performs path-integral quantization on the co-adjoint orbit
of the (double) Virasoro group. Alternatively, it can be identified with the
large central charge limit in the conformal Liouville model, thus relating the
latter to its namesake (Liouville) theorem governing the phase space dynamics
in the (semi)classical limit.

Besides, this action can be viewed as a complexity functional defining an
associated quantum-information type of geometry on the Virasoro group, its
lower bound being given by the length of a proper geodesic on the co-adjoint
orbit qc .

In turn, the extensively studied case of $d=0$ can be attained in the
$AdS_{3}$ theory by taking the limit of a vanishing length of the compactified
spatial dimension. The resulting $AdS_{2}$ bulk theory, as well as its JT
(Jackiw-Teitelboim) extension, support a pseudo-Goldstone time
reparametrization mode with the $1d$ boundary action given by the Schwarzian
derivative syk .

Equivalently, it can be cast in terms of the Liouville quantum mechanics on
the quotient $Diff(S^{1})/PSL(2,R)$ with the action

$S_{L}=\int dt({1\over 2}{\dot{\phi}}^{2}+\lambda e^{\phi})$ (44)

for $\phi(t)=\ln{\dot{\Phi}}(t)$ where $t\to\Phi(t)$ is a diffeomorphism of
the thermal circle. In the context of the space-less random SYK (Sachdev-Ye-
Kitaev) and non-random tensor models this orbit emerges as the result of
factoring out the subspace of zero modes reflective of the $SL(2,R)$ symmetry
of the conformal saddle-point solutions syk . The integrable $1d$ dynamics in
such models is spatially ultra-local and corresponds to $z=\infty$, thus being
reminiscent of the popular $AdS/CMT$ schemes ads .

Notably, in contrast to the marginal nature of the $1d$ LL theory where the
interaction remains important at all energy/temperature scales, in the
SYK/tensor models it is strongly relevant in the infrared, thus only affecting
the conformal mean-field solutions below a certain energy/temperature scale.
Also, the maximally chaotic $AdS_{2}/JT$ gravity can be dual not to a certain
quantum mechanical ($d=0$) theory but (as in the case of SYK) a random
ensemble thereof. For comparison, in $d=1$ neither the boundary theory (43)
saturates the chaos bound, nor is the bulk behavior dominated by pure gravity.

In practice, establishing the SYK-to-$AdS_{2}/JT$ duality involves matching
thermodynamic properties of the two systems, alongside their various
correlation functions. However, achieving this correspondence beyond the
lowest order (two-point) correlation functions requires one to introduce
additional ’matter’ fields in the bulk which represents a tower of higher-spin
operators with the anomalous dimensions that all scale comparably with $1/l$
syk .

Likewise, in the KdV-to-$AdS_{3}$ correspondence the entropy, free energy,
etc. can be matched as well, giving rise to the dependencies $S={\pi\over
4}\sum_{\pm}p_{\pm}^{d}\sim T^{1/z}$ and
$E=\sum_{\pm}<{\hat{H}}^{\pm}_{k}>\sim T^{1+1/z}$, provided that one chooses
$\mu_{\pm}\sim T$ in order for the metrics (39) to remain regular everywhere
in space. Notably, the KdV-charged black holes’ thermodynamics differs from
that of the usual BTZ ones sh .

Comparison between the pertinent microstates on both sides of the latter
correspondence relies on the fact that the $2d$ phase space can be spanned by
the overcomplete basis of coherent states
$|\Psi>=\exp(i\sum_{nm}c_{nm}{\hat{W}}_{nm})|0>$ while the boundary $1d$
theory operates in the Hilbert space spanned by the vectors $|\pm
n>=\prod_{n}{\hat{p}}_{\pm}^{n}|0>$. Employing this basis the correlators of a
bulk field $\hat{O}$ of mass $m$ and dimension $\Delta=(d+1)/2\pm lm$ can be
evaluated by the saddle point method, thereby resulting in the semiclassical
expression for the (real-time) two-point function

${G}_{OO}(x,p,t)\sim\exp(-\Delta\int[g_{pp}dp^{2}-g_{tt}dt^{2}+g_{xx}dx^{2}]^{1/2})$
(45)

where the line integral is taken over the $3d$ geodesic connecting the end
points.

Placing the end points of this correlation function on the boundary yields the
single-particle boundary propagator. Fourier transforming this expression in
the spacetime domain one then obtains

$G_{OO}(\omega,k)\sim\exp(-\int dp[g_{pp}({\Delta^{2}\over
l^{2}}-{\omega^{2}\over g_{tt}}+{k^{2}\over g_{xx}})]^{1/2})$ (46)

For instance, the BTZ bulk metric (40) yields the following propagator of
massless $3d$ bulk fermions with the dimension $\Delta_{\Psi}=1$ and spin
$1/2$ LL

$G^{\pm}_{\Psi\Psi}(\omega,k)=({\omega\pm k\over\omega\mp k})^{1/2}$ (47)

which gives rise to the power-law spacetime behavior of the boundary
propagator $G^{\pm}_{\Psi\Psi}(t,x)\sim 1/|x_{\pm}|^{2}$. These results should
of course be distinguished from the standard LL propagator
$G^{\pm}_{\psi\psi}(\omega,k)=1/({\omega\mp k})$ of free chiral fermions with
the dimension $\Delta_{\psi}=1/2$.

In contrast, using the metric build out of the solitonic solution (29) changes
the spacetime decay from algebraic to exponential,
$G^{\pm}_{\Psi\Psi}(t,x)\sim\exp(-\Delta|x_{\pm}|/l)$.

In turn, the two-particle (energy) excitations representing gapless boundary
gravitons remain propagating, thus featuring the ordinary ballistic pole

$G_{\epsilon\epsilon}(\omega,k)=\int dtdxe^{i(kx-\omega
t)}[G^{\pm}_{\psi\psi}(x,t)]^{2}\sim{k^{2}\over k^{2}-\omega^{2}}$ (48)

Alternatively, this energy correlation function can be obtained from the
correlator $<w(x,p,t)w(x^{\prime},p^{\prime},t^{\prime})>$ computed as the
path integral over the Wigner function.

The ballistic behavior (48) should be contrasted against the diffusive one
observed in, e.g., a chain of coupled SYK models,
$G_{\epsilon\epsilon}(\omega,k)\sim{k^{2}/(k^{2}+i\omega)}$, which would be
indicative of a (maximally) chaotic state syk .

Further possible generalizations of the collective field hydrodynamics include
incorporation of the momentum Berry curvature in Eq.(6), exploration of the
effects of viscosity, generic dispersion with $z\neq 1,2$, etc. It would also
be interesting to investigate a development of turbulence which has long been
known to harbor some important connections to quantum gravity.

Discussion

In this note the Wigner function representation of generic quantum systems was
discussed from the standpoint of pinpointing the possible origin of the
hypothetical generalized holographic correspondence. To that end, using the
Kirillov-Kostant procedure of phase space quantization via the coherent state
path integral and collective field hydrodynamics may seem rather promising.

Specifically, in line with the holographic lore, path integral quantization on
a co-adjoint orbit of the $W_{\infty}$ group of the volume-preserving
diffeomorphisms of the phase space exposes an intrinsic relationship between
the $2d+1$-dimensional ’bulk’ description and the $d+1$-dimensional ’boundary’
hydrodynamics. The quantum bulk dynamics is described by the action composed
of the $W_{\infty}$ generators while the corresponding boundary variables are
given by the moments (17) of the Wigner function. Systematically implementing
this program can then be thought of as ’deriving’ the sought-out holographic
duality.

Importantly, such a generic form of correspondence neither requires a
reference to some underlying ’string’ theory, nor does it impose any
particular symmetry conditions on the boundary system in question, while the
putative bulk description does not necessarily have a gravity-like appearance.

Nonetheless, in the case of $d=1$ the corresponding $3d$ bulk behavior can
indeed be cast in terms of the (doubled) Chern-Simons theory or, equivalently,
non-dynamical Einstein gravity with a negative cosmological constant.
Furthermore, if the boundary Hamiltonian belongs to the integrable (e.g., KdV)
family, the corresponding set of the bulk metrics may include the familiar
BTZ, as would be generally anticipated in line with the $AdS/CFT$ paradigm ads
.

Furthermore, in $d=1$ the phase space description can be viewed as a formally
exact (non-linear) bosonization of the boundary system. Many of such systems
appear to be integrable (hence, non-ergodic) and possess an infinite number of
locally conserved currents given by the various moments of the Wigner function
which obey the equations of zero-entropy generalized quantum hydrodynamics of
the GGE type.

In higher dimensions the Wigner function’s moments, too, serve as coefficients
in the series expansions of the would-be local bulk metrics over the powers of
the momentum $\bf p$. Although with the increasing spatial dimension the
hydrodynamic description becomes less accurate, it remains capable of
capturing the salient features of the quantum phase space dynamics governed by
the conservation laws.

To summarize, the use of the phase space approach brings out the intrinsic
correspondence between a formally exact $2d+1$-dimensional and a less accurate
(’coarse-grained’) $d+1$-dimensional hydrodynamic descriptions of a given
quantum system. In this general setting, neither the latter needs to be a
conformal field theory, nor does the former have to have the appearance of
gravity.

In those $d=0$ and $d=1$ cases where the bulk indeed appears to be amenable to
a gravitational description the gravity theory has no dynamics of its own and
is fully determined by the boundary degrees of freedom. Accordingly, the
viable bulk metrics can be mapped onto the solutions of the boundary
hydrodynamics.

In that regard, the holographic custom of picking out a particular metric and
claiming some sort of the Einstein-Maxwell-dilaton theory to be the proper
bulk dual of a certain strongly correlated system does not appear to be backed
by the above conclusions. Nevertheless, in some cases, including $d=0$ and
$d=1$, certain phenomenological 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 to little else but
hydrodynamics then the whole issue would become largely moot and void.

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