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arXiv:hep-th/9206092v1 24 Jun 1992

FINITE AND INFINITE SYMMETRIES IN

(2+1)-DIMENSIONAL FIELD THEORY*

R. Jackiw

Center for Theoretical Physics

Laboratory for Nuclear Science

and Department of Physics

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139U.S.A.

and

So-Young Pi

Physics Department

Boston University

350 Commonwealth Avenue

Boston, Massachusetts 02115

Dedicated to Franco Iachello on his 50th Birthday

Recent Problems in Mathematical Physics, Salamanca, Spain, June 1992; XIX International

Colloquium on Group Theoretical Methods in Physics, Salamanca, Spain, July 1992, Con-

densed Matter and High Energy Physics, Cagliari, Italy, September 1992.

Typeset in TEX by Roger L. GilsonCTP#2110

BU HEP 92-21

June 1992

* This work is supported in part by funds provided by the U. S. Department of Energy

(D.O.E.) under contract #DE-AC02-76ER03069 (RJ), and #DE-AC02-89ER40509(S-YP).

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ABSTRACT

These days, Franco Iachello is the eminent practitioner applying classical and finite groups

to physics. In this he is following a tradition at Yale, established by the late Feza Gursey,

and succeeding Gursey in the Gibbs chair; Gursey in turn, had Pauli as a mentor. Iachello’s

striking achievement has been to find an actual realization of arcane supersymmetry within

mundane adjacent even-odd nuclei. Thus far this is the only physical use of supersymmetry,

and its fans surely must be surprised at the venue. Here we describe the role of SO(2,1)

conformal symmetry in non-relativistic Chern–Simons theory: how it acts, how it controls

the nature of solutions, how it expands to an infinite group on the manifold of static solutions

thereby rendering the static problem completely integrable. Since Iachello has also used the

SO(2,1) group in various contexts, this essay is presented to him on the occasion of his fiftieth

birthday.

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I. INTRODUCTION

We shall discuss finite- and infinite-dimensional conformal symmetries of field theories

with non-relativistic kinematics. Such field theories also describe the second quantization of

non-relativistic particle mechanics. Particle mechanics, with its second order in time dynam-

ics, has the structure of a relativistic field theory in one time and zero space dimensions, and

a relativistic field theory in any dimension can enjoy conformal symmetry. Thus there are

family relationships between the conformal symmetries of non-relativistic field theory, non-

relativistic particle mechanics and relativistic field theory, and our first task is to describe

these interrelations.

A conformal transformation in (D + 1)-dimensional relativistic field theory changes the

independent variables, viz. the space-time coordinates xµof the fields (fields are dependent

variables), and infinitesimally reads

δfxµ= −fµ(x)(1.1)

where fµis a conformal Killing vector, i.e. fµsatisfies the conformal Killing equation.

∂µfν+ ∂νfµ=

2

D + 1gµν∂αfα

(1.2)

Here gµν is the Minkowski metric tensor with signature (1,−1,−1,...) and D is the spatial

dimensionality.

As is well-known, Eq. (1.2) has the finite number of1

2(D+2)(D+3) solutions for D > 1,

and conformal transformations form an SO(2,D + 1) group. The solutions to (1.2) comprise

D + 1 space-time translationsfµ(x) = aµ,aµconstant (1.3a)

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2D(D + 1) space-time rotationsfµ(x) = ωµνxν,ωµν= −ωνµ

(1.3b)

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a single scale transformationfµ(x) = axµ,a constant(1.3c)

D + 1 special conformal transformationsfµ(x) = 2c · xxµ− cµx2,cµconstant

(1.3d)

The finite versions of these are, respectively,

xµ→ xµ+ aµ

(1.4a)

xµ→ Λµνxµ,ΛµαΛνβgµν= gαβ

(1.4b)

xµ→ eaxµ

(1.4c)

xµ→

xµ− cµx2

1 − 2c · x + c2x2

(1.4d)

The last, the finite special conformal transformation, can also be seen as an inversion, xµ→

xµ/x2, followed by a translation and another inversion, i.e. a translation in the inverted

coordinate.

At D = 1 there exists an infinite number of solutions to (1.2) corresponding to arbitrary

redefinition of x±=

1

√2(x0±x1) and forming an infinite parameter group. Infinitesimally we

have

δfx±= −f±(x±) ,f±arbitrary(1.5)

while the finite version reads

x±→ X±(x±) ,X±arbitrary(1.6)

A linear conformal transformation on a space-time multiplet of Lorentz covariant rela-

tivistic fields ϕ, i.e. on the dependent variables, can be taken as

δfϕ = fα∂αϕ + ∂αfβ

?

∆

D + 1gαβ+1

2Σαβ

?

ϕ(1.7)

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Here Σαβis the spin-matrix, acting on the space-time components of ϕ and ∆ is a constant,

called the scale-dimension of ϕ. When the Lagrange density for ϕ possesses a conventional

relativistic kinetic term — quadratic in derivatives for Bose fields, linear for Fermi fields —

the kinetic action is invariant against conformal transformations (1.7) provided

∆ =D − 1

2

bosons (1.8a)

∆ =D

2

fermions (1.8b)

[These values for ∆ correspond to the dimensionality of a field in units of inverse length when

¯ h and c are scaled to unity.] Also the Bose field monomial

LI= ϕ2(D+1

D−1)

(1.9)

leads to an invariant action?dD+1xLI.

At D = 1, Bose fields become dimensionless, see (1.8a), and the conformally monomial

(1.9) cannot be formed. Nevertheless, there exists a non-trivial conformally invariant theory

— the completely integrable Liouville theory,

LLiouville=1

2∂µϕ∂µϕ −µ2

β2eβϕ

(1.10)

whose action is invariant provided the single-component scalar field ϕ is transformed according

to an inhomogenous generalization of (1.7),

δfϕ = fα∂αϕ +1

β∂αfα

(1.11a)

or equivalently

δfeβϕ= ∂α

?fαeβϕ?

(1.11b)

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