The group-theoretic approach to perfect uid equations with conformal symmetry Anton Galajinsky

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The group-theoretic approach to perfect ﬂuid

equations with conformal symmetry

Anton Galajinsky

Laboratory of Applied Mathematics and Theoretical Physics,

TUSUR, Lenin ave. 40, 634050 Tomsk, Russia

e-mail: a.galajinsky@tusur.ru

Abstract

The method of nonlinear realizations is a convenient tool for building dynamical realiza-

tions of a Lie group, which relies solely upon structure relations of the corresponding Lie

algebra. The goal of this work is to discuss advantages and limitations of the method,

which is here applied to construct perfect ﬂuid equations with conformal symmetry.

Four cases are studied in detail, which include the Schr¨odinger group, the `-conformal

Galilei group, the Lifshitz group, and the relativistic conformal group.

PACS: 11.30.-j, 02.20.Sv, 47.10.A, 47.10.ab

Keywords: perfect ﬂuid equations, conformal symmetry

arXiv:2210.14544v2 [hep-th] 9 Nov 2022

1. Introduction

Recently, there has been a considerable eﬀort to understand the hydrodynamic limit

of the AdS/CFT-correspondence [1, 2, 3].1The main point of the study was to establish

that the low-frequency behavior of an interacting ﬁeld theory at ﬁnite temperature could be

described by ﬂuid mechanics.

The conventional formulation of ﬂuid dynamics relies upon an expansion scheme in which

the eﬀects of viscosity and heat transfer are regarded as corrections to the perfect ﬂuid equa-

tions (see e.g. [2]). As is known, for a properly chosen equation of state the (non)relativistic

perfect ﬂuid equations exhibit conformal invariance (see e.g. [5, 6, 7]). It is then natural to

wonder whether such equations can be formulated on purely group-theoretic grounds.

A convenient tool for building dynamical realizations of a Lie group, which relies solely

upon structure relations of the corresponding Lie algebra, is provided by the method of

nonlinear realizations [8]. The scheme includes several steps. First of all, one introduces

coordinates and ﬁelds, whose number in general is equal to the number of generators of a

Lie algebra at hand. Then one builds a formal group-theoretic element g, which is a product

of exponentials of the type eiαT , where αis a coordinate (or a ﬁeld) and Tis a Lie algebra

generator. After that, one studies the left action of the group upon the group-theoretic

element gand establishes transformation laws of the coordinates and ﬁelds. Finally, one

computes the Maurer-Cartan one-forms g−1dg. By construction, they hold invariant under

the left action of the group upon the group theoretic element and, hence, provide convenient

building blocks to formulate invariant equations of motion. If desirable, they can also be

used to eliminate some of the ﬁelds from the consideration by imposing constraints. Within

the method of nonlinear realizations, imposing constraints is attributed to the inverse Higgs

phenomenon [9].

An alternative formulation of ﬂuid dynamics in terms of group-valued variables, which

relies upon Kirillov’s method of orbits [10], was proposed in [11]. The formalism is particu-

larly suitable for taking into account constituent particles, which carry nonabelian charges

or spin degrees of freedom, as well as for incorporating anomalies [12]–[16].

The goal of this work is to discuss advantages and limitations of the method of nonlinear

realizations, which is here applied to construct perfect ﬂuid equations with conformal sym-

metry. Four cases are studied in detail, which include the Schr¨odinger group, the `-conformal

Galilei group, the Lifshitz group, and the relativistic conformal group.

The organization of the work is as follows. In the next section, we brieﬂy outline key

features of the method of nonlinear realizations, which are used in later section to explore

ﬂuid equations with conformal symmetry. For simplicity of the presentation, the construction

is illustrated by the example of so(2,1) algebra, which was ﬁrst studied in [17]. In Sect. 3,

perfect ﬂuid equations with the Schr¨odinger symmetry are built in terms of the invariant

Maurer-Cartan one-forms associated with the Schr¨odinger group and an invariant derivative.

It is demonstrated that a proper equation of states, which links pressure to ﬂuid density,

1Literature on the subject is overwhelmingly large. For a review and further references see [4].

comes about quite naturally without the need to invoke more sophisticated arguments [5, 6].

In Sect. 4 and Sect. 5, a similar analysis is performed for the `-conformal Galilei group

and the Lifshitz group, respectively. To the best of our knowledge, the Lifshitz-invariant

equations in Sect. 5 are new. Sect. 6 is focused on the case of the relativistic conformal

group. In contrast to the nonrelativistic examples, the construction of relativistic ﬂuid

equations invariant under the conformal group in terms of the Maurer-Cartan invariants

alone turns to be problematic and extra arguments need to be invoked. In the concluding

Sect. 7, we summarize our results and discuss possible further developments.

Throughout the paper, summation over repeated indices is understood unless otherwise

is stated.

2. The method of nonlinear realizations

In this section, we outline key features of the method of nonlinear realizations [8], which

will be used below to explore ﬂuid equations with conformal symmetry. For simplicity of the

presentation, the construction will be illustrated by the example of so(2,1) algebra

[H, D] = iH, [H, K] = 2iD, [D, K] = iK, (1)

which was ﬁrst studied in [17]. Above His interpreted as the temporal translation generator,

Dlinks to dilatation, and Kis associated with the special conformal transformation.

In its essence, the method of nonlinear realizations is a tool to build dynamical realizations

of a Lie group, which is based solely upon structure relations of the corresponding Lie algebra.

As the ﬁrst step of the construction, one introduces coordinates and ﬁelds whose number in

general is equal to the number of generators of the Lie algebra. The choice is not unique

and it essentially depends on a dynamical realization one seeks for. For example, if one is

concerned with one-dimensional mechanics originating from (1), a temporal variable tand

a function u(t), which describes a particle dynamics, are needed. It seems natural to link

tto Hand u(t) to D. In order to treat all the generators on equal footing, one introduces

one more ﬁeld w(t), which is regarded as a partner of K. At this stage, it is not yet clear

whether u(t) or w(t) will be more suitable for describing a reasonable conformal mechanics

model and one anticipates an SO(2,1)-invariant constraint, which will link the ﬁelds to each

other.

Then one introduces the group-theoretic element

g=eitH eiu(t)Deiw(t)K.(2)

In general, the expressions in (1) are regarded as formal Lie brackets (rather than commu-

tators of operators) and the exponential entering (2) is treated as the exponential map of a

Lie algebra to a neighborhood of the unit group element. Yet, nothing prevents one from

assuming that a speciﬁc representation of the Lie algebra is chosen such that (1) represents

commutators of the generators and (2) is an operator acting upon a state. In the latter case,

the exponential entering (2) can be regarded as a formal Taylor series eA=P∞

n=0

n!, in

which case the well known Baker-Campbell-Hausdorﬀ formula

eiA T e−iA =T+

∞

n=1

n![A, [A, . . . [A, T ]. . . ]]

| {z }

ntimes

(3)

is applicable, where Aand Tare two arbitrary operators. The conventional property of the

exponential

eαAeβA =e(α+β)A,(4)

holds true as well, where α,βare constants and Ais an operator action upon a state in

the representation chosen. Eq. (3) is the cornerstone of the whole consideration to follow.

Note that the order in which the factors contribute to the group-theoretic element (2) can

be chosen at will and diﬀerent options are related by coordinate and ﬁeld redeﬁnitions.

At the second step of the construction, one analyzes the left action of the group upon

the group-theoretic element

g0=eiβH eiλDeiσK ·g=eit0Heiu0(t0)Deiw0(t0)K,(5)

where β,λ,σare real transformation parameters, and makes use of (3) so as to establish

transformation laws of the coordinates and ﬁelds. For most physical applications it suﬃces

to consider their inﬁnitesimal form.

When performing calculations, depending on a speciﬁc operator eLat hand, it might

prove helpful to insert the unit operator e−BeB, with Bto be speciﬁed below, either to the

left or to the right of eL. The following chain of relations

eiλDeitH =eiλDeitH e−iλDeiλD

| {z }

=eiλD (1 + itH +. . . )e−iλDeiλD =

(1 + it(H+ [iλD, H] + . . . ) + . . . )eiλD =ei(1+λ)tH eiλD (6)

gives the example of how eiλD ”passes through” eitH . The Baker-Campbell-Hausdorﬀ formula

was repeatedly applied in the second line of (6). At the same time, when computing eiσK eitH

with inﬁnitesimal σ, it proves convenient to place eitH e−itH to the left of the operator

eiσK eitH =eitH e−itH

| {z }

eiσK eitH =eitH e−itH (1 + iσK)eitH =

eitH 1+iσK−[itH, K] + 1

2![itH, [itH, K]]=ei(t+σt2)HeiσK e2iσtD,(7)

where we used the fact that eiσAeiσB =eiσB eiσA for inﬁnitesimal σ.

Taking into account the technicalities, from Eq. (5) one obtains the inﬁnitesimal SO(2,1)-

transformations acting upon the temporal variable and the ﬁelds u(t), w(t) (each transfor-

mation is separated by semicolon)

t0=t+β, u0(t0) = u(t), w0(t0) = w(t);

t0= (1 + λ)t, u0(t0) = u(t) + λ, w0(t0) = w(t);

t0=t+σt2, u0(t0) = u(t)+2σt, w0(t0) = w(t) + eu(t)σ. (8)

The third step of the method consists in computing the Maurer-Cartan one-forms2

g−1dg = iωHH+ iωDD+ iωKK, (9)

where

ωH=e−udt, ωD=du −2we−udt, ωK=dw −wdu +w2e−udt. (10)

By construction, they hold invariant under the group transformation (5), (8) and, hence,

provide convenient building blocks to formulate invariant equations of motion. If desirable,

they can also be used to eliminate some of the ﬁelds entering the group-theoretic element

from the consideration by imposing constraints.

In the last step of the construction, one speciﬁes a dynamical realization of the group at

hand by choosing judiciously a combination of the Maurer-Cartan invariants which results

in a reasonable set of second order diﬀerential equations. The latter are identiﬁed with the

equations of motion of a dynamical system. This item is not straightforward and requires

guesswork. For example, if one wishes to use (10) in order to formulate one-dimensional

SO(2,1)-invariant mechanics, one imposes the constraint ωD= 0, which links wto u

w=1

deu

dt ,(11)

and then postulates the equation of motion [17]

ωK−γ2ωH= 0,(12)

in which γis a (coupling) constant. Implementing the ﬁeld redeﬁnition u= ln ρ2, which is

meant to remove the du

dt 2-term from (12), one ﬁnally arrives at

d2ρ

dt2=γ2

ρ3,(13)

which is the conventional 1dconformal mechanics equation of motion [18].

To give another example, it is known that the Schwarzian derivative

S(ρ(t)) =

...

ρ(t)

˙ρ(t)−3

2¨ρ(t)

˙ρ(t)2

,(14)

where ρ(t) is a real function and the dot designates the derivative with respect to t, holds

invariant under the SL(2, R)-transformation acting upon the argument

ρ0(t) = aρ(t) + b

cρ(t) + d,(15)

2Given gin (2), the inverse element is g−1=e−iwK e−iuDe−itH , while the diﬀerential reads dg =

eitH (idtH)eiuD eiwK +eitH eiuD (iduD)eiwK +eitH eiuDeiwK (idwK).

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Thegroup-theoreticapproachtoperfectuidequationswithconformalsymmetryAntonGalajinskyLaboratoryofAppliedMathematicsandTheoreticalPhysics,TUSUR,Leninave.40,634050Tomsk,Russiae-mail:a.galajinsky@tusur.ruAbstractThemethodofnonlinearrealizationsisaconvenienttoolforbuildingdynamicalrealiza-tionsofaLiegroup...

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