ЖЭТФ, 2019, том 156, вып. 4 (10), стр. 707-710

QUANTIZATION OF HYDRODYNAMICS: ROTATING

SUPERFLUID, AND GRAVITATIONAL ANOMALY

P. Wiegmann^***

Kadanoff Center for Theoretical Physics, University of Chicago

60637, Chicago, IL, USA

Received May 27, 2019,

revised version June 1, 2019

Accepted for publication June 2, 2019

Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

DOI: 10.1134/S0044451019100122

some, not immediately obvious, consequences of quan-

tization. The guidance for the quantization comes from

The problem of quantization of hydrodynamics be-

the intersection between quantum chiral hydrodynam-

yond linear approximation is commonly considered in-

ics and quantum two-dimensional gravity.

tractable. Nevertheless, nature confronts us with beau-

Classical ideal flows are characterized by the Hamil-

tiful quantum non-linear ideal fluids with experimen-

tonian and the Poisson structure

tally accessible precise quantization. Among them, two

∫

ρ_A

quantum fluids stand out: superfluid helium and elec-

H =

u²dV,

(2)

tronic fluid in the fractional quantum Hall state. In

{ω(r), ω(r^′)} = ρ-1A (∇_r × ∇_r′ )ω(r)δ(r - r^′).

both cases, the precise quantization of vortex circula-

tion in superfluid helium and the precise quantization

It is well known, that the Poisson structure is the

of electric transport in FQHE leave no doubts of the

Lie-Poisson algebra of area preserving diffeomorphisms

quantum nature of these fluids.

SDiff. Hence flows of an ideal fluid are the actions of

The fundamental aspects of quantization of fluid

area-preserving diffeomorphism SDiff, and should be

dynamics most clearly appear in ideal flows. These are

studied from a geometric standpoint, see, e. g., [4].

incompressible flows ∇· u = 0 of homogeneous inviscid

Formally the quantization amounts supplanting the

fluids. The problem of quantization is further special-

Poisson brackets by the commutator

ized in chiral two-dimensional ideal flows which we con-

sider in the paper. These are 2D flows with extensive

{, }→

[ , ]

vorticity: the mean vorticity

iℏ

∫

and identifying the Hilbert space with a representation

2Ω =

ω dV, ω(r) = ∇×u,

(1)

space of SDiff. The latter, however, is not well under-

stood. The difficulties appear in the regularization of

remains finite, as V → ∞. The chiral flows are distin-

short-distance divergencies. A regularization must be

guished by the holomorphic character of the quantum

consistent with fundamental local symmetries of the

states.

theory. The local “symmetry” of fluids is the relabeling

symmetry, or equivalently the invariance with respect

Two most perfect quantum fluids, rotating super-

fluid helium [1], and FQHE fall to the class of ideal

to re-parametrization of flows. We had shown that

chiral flows (see [2, 3] for the correspondence between

the relabeling symmetry alone determines the universal

FQHE and superfluid hydrodynamics). In the paper

quantum corrections to the Euler equation. Relabeling

we had shown how to quantize chiral flows and describe

are diffeomorphisms in the manifold of Lagrangian co-

ordinates. Invariance with respect to diffeomorphisms

^* E-mail: wiegmann@gmail.com, wiegmann@uchicago.edu

is also a guiding principle of quantum gravity. In the

^** Also at IITP RAS, 127994, Moscow, Russian Federation

paper, we described the correspondence between chiral

707

P. Wiegmann

ЖЭТФ, том 156, вып. 4 (10), 2019

flows and 2D gravity, and explained how this correspon-

If we assume that vorticity is a smooth function and is

dence yields to a unique short-distance regularization.

positive ω > 0, then we had shown that

The result of the regularization is quantum corrections

(

)

ℏΩ

to the Euler equation expressed in terms of the gravi-

T_zz =

∂2z log ω -

(∂_z

log ω)²

(5)

12π

tational anomaly.

We start by the notion of quantum stress. Consider

Then the quantum Helmholtz equation reads

a traceless part of the momentum flux tensor

(

)

ℏΩ

ρ_AD_tω =

∇R×∇ω,

(6)

Π^′ij = ρ_A u_iu_j -

u²δij

48π

where

In the quantum case, the velocity u_i is an operator.

We will be interested in expectation value of Π^′ij . The

R = -ω-1Δlogω.

(7)

cumulant of the bilinear product of velocities is the

The reader may recognize that T_zz is the Schwarzian

(minus) quantum stress

of a Riemann surface with the metric

(

)

-T^′ij = ρ_A

〈〈u_iu_j〉〉 -

〈〈u²〉〉δij

ds² = ω|dz|²,

(8)

The quantum stress corrects the Euler equation as

and that R in (7) is the curvature of that surface. These

relations are not accidental. We argued that the chiral

ρ_AD_tu_i + ∇_ip = ∇^jT^′ij.

(3)

flow may be understood as evolving Riemann surface.

In this equation, all entries are assumed to be expec-

The surface which hosts the fluid is a complex manifold

tation values u_i → 〈u_i〉, hence the equation could be

equipped with a closed vorticity 2-form,

treated as classical. Also

ω_ijdxⁱ ∧ dxⁱ,

ω_ij = ∂_iu_j - ∂_ju_i.

D_t = ∂_t + 〈u〉 · ∇

Because vorticity of the chiral flow does not change the

is the material derivative with respect to the expecta-

sign

tion value of velocity and ρ_A is the mass density of the

ω=

ϵ^ijω_ij > 0,

fluid. The divergency of the quantum stress vanishes,

it yields to the corrections to hydrodynamics.

the chiral flow gives the host surface a Kähler structure

We write the Euler equation in the Helmholtz form

with the Kähler form

by taking the curl of (3)

ω dz ∧ dz,

ρ_AD_tω = ϵ^ik∂_k∇^jT^′ij.

(4)

and the Riemannian metric (8). The Kähler form,

We see that unless the rhs of (4) vanishes, the ma-

the volume element of the surface is the vorticity in

jor property of classical hydrodynamics, the Helmholtz

the fluid volume. Adopting the language of quantum

law does not hold for the expectation value of vortic-

gravity we identify the manifold of Lagrangian coordi-

ity. We recall that the Helmholtz law D_tω = 0 states

nates with a target space and the host surface with a

that vorticity is frozen in the flow. Departure from the

world-sheet.

Helmholtz law is the major result of the paper. De-

The coordinates of the tangent space of the surface

spite it the Kelvin theorem (conservation of vorticity

appear as Clebsch variables. We recall that Clebsch

of a fluid parcel) is intact. The Kelvin theorem holds

variables parameterize vorticity as

when the divergence of the stress has no circulation

along a liquid contour

ω = ∇λ¹ × ∇λ².

(9)

∮

∇^jT_ijdxⁱ = 0

It follows that the intersection of level lines of λ¹ and

λ² are position of vortices. Hence, vorticity

and it, indeed, vanishes.

ω = det∥∂_iλ^a∥,

In the paper, we expressed the stress in terms of

the expectation values of the vorticity. We present the

is the Jacobian of the map

result in complex coordinates

(x¹, x²) → (λ¹, λ²)

T^′ijdxⁱdx^j =

[T_zz(dz)² + T_zz(dz)²].

708

ЖЭТФ, том 156, вып. 4 (10), 2019

Quantization of hydrodynamics. . .

∫

and

ω_k = e-ik·rω(r)dV,

e^aidxⁱ = dλ^a

∑

(10)

ω_k = Γ

e^- 2

kz^†i e-2 kzi ,

are the vielbeins.

i≤Nv

In these terms the diffeomorphisms in the space of

Clebsch variables which leave vorticity unchanged ap-

where Γ is a circulation of each vortex and z_i is a com-

pear as relabeling of vortices. This is relabeling symme-

plex coordinate of a vortex. The operators acts in the

try or a diffeomorphism invariance of hydrodynamics.

Bargmann state [6,7] of holomorphic polynomials of z_i

In the literature the relabeling symmetry usually refers

and the conjugate coordinate z^†i obey the Heisenberg

to fluid atoms. In our approach, it is relabeling of vor-

algebra

tices. We want to keep this major symmetry intact

[z_i, z_j ] = (πn_A)-1δ_ij ,

in quantization. This amounts that the short distance

where

cut-off must be kept uniform in Clebsch coordinates

n_A = (Γ/h)ρ_A

and corresponds to the interval ds of the auxiliary sur-

face. The relation between Clebsch and Eulerian coor-

is the number of helium atoms. Then it follows that

dinates

vorticity operator obeys the sine-algebra [8]

dλⁱ = dxⁱ/ℓ[ω],

(

)

k·k′

where

[ω_k, ω_k′ ] = ie 4πnA ekk′ ωk+k′ ,

ℓ[ω] = ω-1/2

(11)

⁽k×k^′)

e_kk′ = 2Γ sin

4πn_A

is a mean distance between vortices suggests that the

short distance cut-off should be ℓ[ω]. It is non-uniform.

This algebra is a finite-dimensional “approximation” of

It depends on the flow and on the position within the

the Lie algebra SDiff. The latter is obtained by sup-

flow.

planting the Poisson brackets by the Lie brackets. The

Based on this principle we were able to obtain the

sine-algebra depends on two deformation parameters ℏ

diffeomorphism-invariant regularization of the bilinear

and n_A. The limit ℏ → 0 and n_A → ∞ which keeps

of velocities u_iu_j. One way of doing this is to split

the mass density ρ_A = hn_A/Γ and is taken at a fixed k

points

brings back the Poisson structure (2). In order to ob-

(

ϵ⁾

tain the quantum corrections we need a different limit

u_i(r)u_j(r) → u_i r +

u_j r -

when k × k^′ increases with the same rate as n_A. A

proper execution of this limit yields the result for the

and to take into account that the short distance cut-off

quantum stress (5). The formula (5) essentially means

is the functional of the flow ϵ = ℓ[ω].

that the stress obeys the conformal Ward identity and

We illustrate this idea in terms of the path inte-

its moments are generators of the Virasoro algebra. Let

gral approach to quantization. In this approach, one

us represent the stress by the Laurent series about the

typically integrates over pathlines of fluid parcels. In-

origin (the fixed point of the rotation)

stead, we choose to integrate over vorticity and for this

purpose we need to know the measure on the space

∑

T_zz(z) = -

z-n-2L_n

of vorticity. In order to determine it, we invoke the

relation of the chiral flow to 2D quantum gravity de-

scribed above. Since vorticity is a metric we effectively

and set ω = 2Ω. Then L_n generate the Virasoro alge-

integrate over metrics. The measure on the space of

bra (with the central charge c = 1)

metrics has been established in quantum gravity [5].

It consists of the Fadeev-Popov determinant restor-

[L_n, L_m] = (n - m)Ln+m +

(n³ - n)δn+m,0.

ing the re-parametrization invariance of the surface. In

quantum gravity, it is a source of the Liouville action.

Summing up, in the paper we present a consistent

In quantum hydrodynamics, it is a source of quantum

scheme of quantization of chiral flows of the ideal 2D

stress.

fluid. The quantization is based on a geometric rela-

In order to prove that the measure on vorticity con-

tion of chiral flows to two-dimensional quantum gravity

figuration is the same as a measure of metrics, we em-

and is implemented by the gravitational anomaly. The

ploy the following procedure. We first assume that the

effect of the gravitational anomaly violates the major

flow consists of a large, albeit a finite number of vor-

property of classical hydrodynamics, the Helmholtz

tices. Then the vorticity operator reads

law: vortices are no longer frozen into the flow. We

709

P. Wiegmann

ЖЭТФ, том 156, вып. 4 (10), 2019

show that quantum stress generates the Virasoro al-

2. P. B. Wiegmann, Phys. Rev. B 88, 241305 (2013).

gebra, the centrally extended algebra of holomorphic

diffeomorphisms. The result follows as the limit of

3. P. Wiegmann, Zh. Eksp. Teor. Fiz. 144, 617 (2013)

[J. Exp. Theor. Phys. 117, 538 (2013)].

the finite-dimensional approximation of Lie algebra

of area-preserving diffeomorphisms SDiff yielding dif-

4. V. I. Arnold and B. A Khesin, Topological Methods

feomorphism invariant regularization of the advection

in Hydrodynamics, Vol. 125, Springer (1999).

term in Euler equation. The main applications of this

theory are rotating superfluid and electronic systems

5. A. M. Polyakov, Gauge Fields and Strings (Con-

in the magnetic field in the regime of a fractional Hall

temporary Concepts in Physics), Harwood Academic

effect.

Publishers, Switzerland (1987).

The full text of this paper is published in the English

6. S. M. Girvin, A. H. MacDonald, and P. M. Platzman,

version of JETP.

Phys. Rev. B 33, 2481 (1986).

7. V. Bargmann, Comm. on Pure and Appl. Math. 14,

187 (1961).

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