Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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linear ones, they fix the normalizations of the

currents. The chiral currents F

L

= 1=2ðÞ(F



F

5

)

and F

R

= 1=2ðÞ(F



þF

5

) commute with each

other, so that the local current algebra decomposes

into two independent pieces.

The Dirac -functions in eqns [4] require that F

and

be interpreted as (unbounded) operator-valued

distributions; while the fixed-time condition suggests

these should make mathematical sense as

three-dimensional distributions, with x

held constant.

Such distributions may be modeled on the test-function

space D of real-valued, compactly supported, C

functions on the spacelike hyperplane R

. For functions

, f

2D, one has formally the ‘‘smeared currents’’

that are expected to be bona fide (unbounded)

operators in Hilbert space; suppressing x

ðf

Þ¼

ðxÞF

ðx

; xÞ



ðxÞF

ðx

; xÞ

½5

Equations [4] then become

ðf

Þ; F

ðf



¼F

ðf

Þ; F

ðf



¼i

ðc

abd

ðf

Þ; F

ðf



¼ i

ðc

abd

½6

Let g(x)beaC

map from R

to the Lie algebra G of

chiral SU(3)  SU(3), equal to zero outside a compact

set. The set of all such G-valued functions forms an

infinite-dimensional Lie algebra under the pointwise

bracket, [g, g

](x) = [g(x), g

(x)]. Let us call this Lie

algebra map

, G), where the subscript 0 indicates

the condition of compact support when that is

applicable (on compact manifolds, we omit the sub-

script). Expanding g(x) with respect to a fixed basis of

G, we straightforwardly identify the map g with the

two octets of test functions f

and f

. Then, defining

F(g) =

) þ

), eqns [6] are inter-

preted (for fixed x

)asarepresentationF of

map

, G).

Integrating out the spatial variables entirely using

eqns [1] leads to a representation at x

of G by the

charges F

and F

. The Adler–Weisberger sum rule

was first derived (in 1965) from the commutation

relations of these charges, together with the assump-

tion of a partially conserved axial-vector current

(PCAC). It connected nucleon -decay coupling with

pion–nucleon scattering cross sections, agreeing well

with experiment. Various low-energy theorems

followed, also in accord with experiment. Shortly

thereafter, Adler was able to eliminate the PCAC

assumption, and derived a further sum rule going

beyond an experimental test of the algebra of

charges to test the actual local current algebra.

Here, the prediction pertained to structure functions

in the deep inelastic scattering of neutrinos. This

was elaborated by Bjorken to inelastic electron

scattering. On the theoretical side, the study of the

chiral curre nt in perturbation theory led into the

theory of anomalies. All these ideas were highly

influential in subsequent theoretical work (Treiman

et al. 1985, Mickelsson 1989).

It is a natural idea to try to extend eqns [4] or [6],

which elegantly express the combined ideas of

locality and symmetry, to an equal-time commutator

algebra that would also include the space compo-

nents of the local currents F

, k = 1, 2, 3. One may

write without difficulty the commutators of the

charges in [1] with these space components:

½F

ðx

Þ; F

ðx

; xÞ ¼ ½F

ðx

Þ; F

ðx

; xÞ

¼ i

abd

ðx

; xÞ

½F

ðx

Þ; F

ðx

; xÞ ¼ ½F

ðx

Þ; F

ðx

; xÞ

¼ i

abd

ðx

; xÞ

½7

But the commutator of the local time compone nt

with the local space component of the current

cannot be merely the obvious extrapolation from

eqns [4] and [7], that is, it cannot be

½F

ðx

; xÞ; F

ðy

; yÞ

= y

= i

ð3Þ

ðx  yÞ

abd

ðx

; x Þ

and so forth . Under very general conditions, for a

relativistic theory based on local quantum fields or

local observables, additional ‘‘Schwinger terms’’ are

required on the right-hand side s of such commu-

tators (Renner 1968).

Well-known difficulties in specifying the Schwinger

terms are associated with the fact that operator-

valued distributions are singular when regarded as if

they were functions of spacetime points. Thus, the

product of two distributions at a point is often

singular or undefined. When the currents forming a

local current algebra are written as normal-ordered

products of field operator distributions and their

derivatives, the Schwinger terms in their commuta-

tion relations may be calculated, for example, by

‘‘splitting points’’ in the arguments of the underlying

fields, and subsequently letting the separation tend

toward zero. The general form of a Schwinger term

typically involves the derivative of a -function times

an operator. This may be a multiple of the identity

(i.e., a c-number) or not, depending on the underlying

Current Algebra 675

field-theoretic model. Furthermore, when the number

of spacetime dimensions is greater than 1 þ 1, the

c-number Schwinger terms turn out to be infinite.

Hence, we do not obtain this way a bona- fide

infinite-dimensional, equal-time commutator algebra

comprising all the components of the local currents.

Sugawara, Kac–Moody, and

Virasoro Algebras

Since equations such as [4] and [6] are not explicitly

dependent on how the currents are constructed from

underlying canonical fields, one has the possibility

of writing a theory entirely in terms of self-adjoint

currents as the dynamical variables, bypassing the

field operators entirely, and expressing a Hamilto-

nian operator directly in terms of such local

currents. This is in the spirit of approaches to

quantum field theory based on local algebras of

observables. It suggests consideration of relativistic

current algebras with finite c-number or operator

Schwinger terms in s þ 1 dimensions, s  1.

The Sugawara model, which is of this type, turned

out to be one of the most influential of those

proposed in the late 1960s and early 1970s.

Henceforth, let G be a compact Lie group, and G

its Lie algebra; let F

, a = 1, ..., dim G, be a basis for

G, with [F

, F

] = i

abd

. The Sugawara current

algebra, at the fixed time x

= y

(which, from here

on, we suppress in the notation), is given by

½J

ðxÞ; J

ðyÞ ¼ i

ð3Þ

ðx  yÞ

abd

ðxÞ

½J

ðxÞ; J

ðyÞ ¼ i

ð3Þ

ðx  yÞ

abd

ðxÞ

þ ic



ð3Þ

ðx  yÞI

½J

ðxÞ; J

‘

ðyÞ ¼ 0 ðk;‘ ¼ 1; 2; 3Þ

½8

where J



= (J

, J

), k = 1, 2, 3, is again a 4-vector, c is a

finite constant, and I is the identity operator. The time

components in eqns [8] behave like the local currents in

eqns [4]. The Schwinger term is a c-number, while

setting the commutators of the space components to

zero is the simplest choice consistent with the Jacobi

identity. The Sugawara Hamiltonian is given in terms of

the local currents by the formal expression:

H ¼

ðxÞ

k¼1

ðxÞ

½9

where the pointwise products of the currents require

interpretation in the particular representation. This

Hamiltonian leads to current conservation equations

for the J



Related to the Sugawara current algebra, with s = 1

and the spatial dimension compactified, are affine

Kac–Moody and Virasoro algebras (Goddard and

Olive 1986, Kac 1990). Consider the infinite-dimen-

sional Lie alge bra map(S

, G) of smooth functions

from the circle to G under the pointwise bracket. This

is also called a loop algebra. Referring to the basis F

define T

(m)

for integer m to be the Fourier function

 ! F

exp [im]. The pointwise bracket in

map(S

, G) gives [T

(m)

, T

(n)

] = i

abd

(mþn)

for these

generators. The corresponding (untwisted) affine

Kac–Moody algebra is a (uniquely defined, nontri-

vial) one-dimensional central extension of this loop

algebra – that is, the new generator commutes with all

elements of the Lie algebra and, in an irreducible

representation, must be a mul tiple of the identity.

In such a representation, the new bracket can be

written as

½T

ðmÞ

; T

ðnÞ

¼i

abd

ðmþnÞ

þ km



m;n

I ½10

where k is a constant. Here, T

(m = 0)

is again a

representation of G. Self-adjointness of the local

currents in the representation imposes the condition

(m)



= T

(m)

Now the compactly supported C

(tangent)

vector fields on a C

manifold M form a natural

Lie algebra under the Lie bracket, denoted by

vect

(M). In local Euclidean coordinates, for g

, g

vect

(M), one can write this bracket as

½g

; g

¼g

rg

 g

rg

½11

As the affine Kac–Moody algebras are central

extensions of the alge bra of G-valued functions on

, so are Virasoro algebras central extensions of the

algebra of vector fields on S

.LetL

(m)

denote

the (complexified) vec tor field described by

exp [ im](1=i)@=@,forintegerm. These genera-

tors then satisfy [L

(m)

, L

(n)

] = (m  n)L

(mþn)

Adjoining to the Lie algebra of vector fields a

new central element (commuting with all the

(m)

), the Virasoro bracket in an irreducible

representation is given by the formula

½L

ðmÞ

; L

ðnÞ

¼ðm  nÞL

ðmþnÞ

þ c

ðm þ 1Þmðm  1Þ



m;n

I ½12

where the numerical coefficient c is called the

Virasoro central charge; self-adjointness of the

currents imposes L

(m)



= L

(m)

. It is straight forward

to veri fy that eqn [12] satisfies the Jacobi identity.

The special form of the central term in the Virasoro

current alge bra results from the Gelfand–Fuks

cohomology on the algebra of vector fields.

676 Current Algebra

The Kac–Moody and Virasoro algebras, both

modeled on S

, may be combined to form a na tural

semidirect sum of Lie algebras, with the additional

bracket

½T

ðmÞ

; L

ðnÞ

¼mT

ðmþnÞ

½13

Roughly speaking, the Kac–Moody generators cor-

respond to Fourier transforms of charge densities on

, whereas the Virasoro generators correspond to

Fourier transforms of infinitesimal motions in S

The central extensions provide the finite, c-number

Schwinger terms. These structures have important

application to light cone current algebra, confor-

mally invariant quantum field theories in (1 þ 1)-

dimensional spacetime, the quantum theory of

strings, exactly solva ble models in statistical

mechanics, and many other domai ns.

Of greatest physical importance, both in quantum

field theory and statistical mechanics, are those

irreducible, self-adjoint representations of the Virasoro

algebra known as highest weight representations,

where the spectrum of the operator L

(m = 0)

is bounded

below. In these applications, one represents a pair of

Virasoro algebras by mutually commuting sets of

operators L

(m)

and



(m)

. In the quantum theory, for

example, one takes the total energy H /



(0)

þ L

(0)

and the total momentum P /



(0)

 L

(0)

. In a highest

weight representation, there is a unique eigenstate of

(0)

having the lowest eigenvalue h; for this ‘‘vacuum’’

jhi, L

(m)

jhi= 0, m > 0.

Friedan, Qiu, and Shenker showed in 1984 that

highest weight representations are characterized by a

class of specific, non-negative values of the central

charge c and, correspondingly, of h:eitherc  1(and

h  0) or c = 1  6(‘ þ 2)

1

(‘ þ 3)

1

, ‘ = 1, 2, 3, ...

(and h assumes a corresponding, specified set of values

for each value of ‘). In a beautiful application to the

study of the critical behavior of well-known statistical

systems, in which the generator of dilations is

proportional to



(0)

þ L

(0)

, they discovered a direct

correspondence with permitted values of the central

charge; thus, c = 1=2fortheIsingmodel,c = 7=10 for

the tricritical Ising model, c = 4=5 for the three-state

Potts model, and c = 6=7 for the tricritical three-state

Potts model.

Current Algebras and Groups

Local current algebras may be exponentiated to

obtain corresponding infinite-dimensional topologi-

cal groups (Pressley and Segal 1986, Mickelsson

1989, Kac 1990). Let G be a Lie group whose Lie

algebra is G. The algebra map

(M, G), consisting of

smooth, compactly supported G-valued functions on

M under the pointwise bracket, exponentiates to the

local current group Map

(M, G), consisting of

smooth maps from M to G that are the identity

outside a compact set in M, under the pointwise

group operation. When M is taken to be the four-

dimensional spacetime manifold (rather than a

spacelike hyperplane), the local current group

modeled on M is mathematically a gauge group for

nonabelian gauge field theory.

Likewise, the algebra vect

(M) exponentiates to

the group Diff

(M) of compactly supported C

diffeomorphisms of M (under composition). The

Kac–Moody and Virasor o algebras exponentiate to

central extensions of the loop group Map(S

, G) and

the diffeomorphism group Diff(S

), respectively. The

semidirect sums of the Lie algebras are the infinite-

simal generators of semidirect products of the

groups.

Under appropriate technical conditions, self-

adjoint representations of current algebras generate

(and may be obtained from) continuous unitary

representations of the corresponding groups. The

needed technical conditions have to do with the

existence of a dense set of analytic vectors belonging

to a common, dense invar iant domain of essential

self-adjointness for the currents.

Nonrelativistic Current Algebra

In nonrelativistic local current algebra, Schwinger

terms do not appear. In 1968, Das hen and Sharp

defined (at fixed time t, suppressed in the present

notation) a mass density (x) = m



(x) (x) and a

momentum density J(x) = (h=2i){



(x)r (x) 



(x)] (x)}, where is a second -quantized cano-

nical field; here we keep h in the notation. The

resulting equal-time algebra is the semidirect sum:

½ðxÞ;ðyÞ ¼0

½ðxÞ; J

ðyÞ ¼ ih

½

ð3Þ

ðx  yÞðxÞ

½J

ðxÞ; J

‘

ðyÞ ¼ih

½

ð3Þ

ðx  yÞJ

‘

ðyÞ





‘

½

ð3Þ

ðx  yÞJ

ðxÞ



½14

Since this current algebra is independent of whether

obeys commutation or anticommutation relations,

the information as to particle statistics (Bose or

Fermi) is not encoded in the Lie algebra itself but in

the choice of its representation (up to unitary

equivalence). Again interpreting  and J

as operator-

valued distributions, define (f ) =

xf(x)(x)

and J( g) =

x 

k = 1

(x)J

(x), where f and the

Current Algebra 677

components g

of the vector field g belong to the

function-space D. Then the Lie algebra becomes

½ðf

Þ;ðf

Þ ¼ 0

½ðf Þ; JðgÞ ¼ ihð g rf Þ

½Jðg

Þ; Jðg

Þ ¼ ihJð½g

; g

Þ

½15

Equations [15] are a representation by self-adjoint

operators of the semidirect sum of the abelian Lie

algebra D with vect

). The corresponding group

is the natural semidirect product of the space D

(regarded as an abelian topological group under

addition) with Diff

The construction generalizes to a general manifold

M or manifold with boundary (in place of R

), and

to a general set of charge densities that generate the

local Lie algebra map

(M, G). When M = S

, we have

the Kac–Moody and Virasoro algebras with central

charge zero. However, L

(0)

in the nonrelativistic

(1 þ 1)-dimensional quantum theories is propor-

tional to the total momentum P,andthusis

unbounded above and below.

The continuous unitary representations of

Diff

(M), or its semidirect product with a local

current group at fixed time, thus describe nonrela-

tivistic quantum systems (Albeverio et al. 1999,

Goldin 2004). The unitary representation V(),  2

Diff

(M), satisfies V(

) = exp [i(r=h)J( g)], where

r 2 R and 

is the one -parameter flow in Diff

(M)

generated by the vector field g. Such a representa-

tion may be described very generally by means of a

measure  on a configuration space , quasi-invariant

under a group action of Diff

(M)on, together

with a unitary 1-cocycle  on Diff

(M)  .The

Hilbert space for the representation is

H= L

d

(, W), which is the space of measurable

functions (),  2 ,takingvaluesinaninner

product space W, and square integrable with respect

to . The unitary representation V is given by

½VðÞðÞ¼



ðÞðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

d



d

ðÞ

½16

where  denotes the group action Diff

(M)   !

; 



is the measure on  transformed by  (which,

by the quasi-invariance of , is absolutely contin-

uous with respect to ); d



=d is the Radon–

Nikodym deriva tive; and 



():W!Wis a system

of unitary operators in W obeying the cocycle

equation





ðÞ¼



ðÞ



ð

Þ½17

Equations [16] and [17] hold outside sets of

-measure zero in . Given the quasi-invariant

measure  on , one may always choose W = C

and 



()  1 to obtain a unitary group representa-

tion on complex-valued wave functions; but inequi-

valent cocycles describe unitarily inequivalent

representations.

The configuration space 

(N)

, N = 1, 2, 3, ...,

consists of N-point subsets of R

, and 

(N)

is the

(local) Lebesgue meas ure on 

(N)

. The correspond-

ing diffeomorphism group and local current algebra

representations describe N identical quantum parti-

cles in s-dimensional space. When   1, we have

bosonic exchange symmetry. Inequivalent cocycles

on 

(N)

are obtained (for s  2) by inducing

(generalizing Mackey’s method) from inequivalent

unitary representation s of the fundamental group



[

(N)

]. For s  3, this fundamental group is the

symmetric group S

of particle permutations; the

odd representation of S

, N  2, gives fermionic

exchange symmetry, while the higher-dimensional

representations are associated with particles satisfy-

ing the parastatistics of Greenberg and Messiah.

When s = 2, however, 

[

(N)

] is the braid group

. Goldin, Menikoff, and Sharp obtained induced

representations of the current algebra describing the

intermediate statistics proposed by Leinaas and

Myrheim for identical particles in 2-space. Such

excitations, subsequently termed ‘‘anyons’’ by Wilc-

zek and characterized as charge-flux tube compo-

sites, are im portant constructs in the theory of

surface phenomena such as the quantum Hall effect,

and anyonic statistics has also been applied to the

study of high-T

superconductivity. Current algebra

representations induced by higher-dimensional

representations of B

describe the statistics of

‘‘plektons.’’ Similarly, current algebra in nonsimply

connected space describes the Aharonov–Bohm

effect.

Let



(h) =

xh(x)



(x) denote the smeared

creation field. Let the indexed set of representations



, J

, N = 0, 1, 2, ..., satisfying the current algebra

[15], act in Hilbert spaces H

, where



(h):H

Nþ1

, (h):H

Nþ1

, (h)jH

= 0, so that



and intertwine the N-particle diffeomorphism

group representations. Let (f ) and J(g) a ct on

N = 0

, so that (f )

= 

(f )

, J(g)

(g)

. Then conditions for a Fock space hier-

archy are specified by commutator brackets of the

fields with the currents:

½ðf Þ;



ðhÞ ¼



ð

N¼1

ðf ÞhÞ

½JðgÞ;



ðhÞ ¼



ðJ

N¼1

ðgÞhÞ

½18

The local creation and annihilation fields for anyons

in R

, obeying [18], satisfy q-commutati on relations,

where q is the relative phase change associated with

a single countercloc kwise exchange of two anyons,

678 Current Algebra

and the q-commutator [A, B]

= AB  qBA. These

relations generalize the canonical commutation

(q = 1) and anticommutation (q = 1) relations of

quantum field theory.

When  is the configuration space of infin ite but

locally finite subsets of R

, nonrelativistic current

algebra describes the physics of infinite gases in

continuum classical or quantum statistical

mechanics. Here, the most important kinds of

measures  are Poisson measures (associated with

gases of noninteracting particles at fixed average

density) or Gibbsian measures (associated with

translation-invariant two-body interactions). These

measures describe equilibrium states and correlation

functions in the classical case, and specify the

current algebra representations in the quantum

theory.

The group of volume-preserving diffeomorphisms

was taken by Arnold as the symmetry group of an

ideal, classical, incompressible fluid, and Marsden

and Weinstein described the hydrodynamics of such

a fluid using the Lie–Poisson bracket associated with

the nonrelativistic current algebra of divergenceless

vector fields. The idea of using this algebra to study

quantized fluid motion, included in the program

proposed by Rasetti and Regge, formed the basis of

the subsequent study of quantized vortex structures

in superfluids from the point of view of geometric

quantization on coadjoint orbits of the diffeomorph-

ism group. This leads to quantum configuration

spaces whose elements are no longer sets of points –

for example, spaces of vortex filaments in R

,or

ribbons and tubes in R

See also: Algebraic Approach to Quantum Field Theory;

Electroweak Theory; Quantum Chromodynamics;

Solitons and Kac–Moody Lie Algebras; Symmetries in

Quantum Field Theory: Algebraic Aspects; Toda Lattices;

Two-Dimensional Conformal Field Theory and Vertex

Operator Algebras.

Further Reading

Adler SL and Dashen RF (1968) Current Algebras and Applica-

tions to Particle Physics. New York: Benjamin.

Albeverio S, Kondratiev YuG, and Ro¨ ckner M (1999) Diffeo-

morphism groups and current algebras: configuration space

analysis in quantum theory. Reviews in Mathematical Physics

11: 1–23.

Arnol’d VI and Khesin BA (1998) Topological Methods in

Hydrodynamics. Applied Mathematical Sciences, vol. 125.

Berlin: Springer.

Gell-Mann M and Ne’eman Y (2000) The Eightfold Way (1964)

(reissued). Cambridge, MA: Perseus Publishing.

Goddard P and Olive D (1986) Kac–Moody and Virasoro

algebras in relation to quantum physics. International Journal

of Modern Physics A 1: 303–414.

Goldin GA (1996) Quantum vortex configurations. Acta Physica

Polonica B 27: 2341–2355.

Goldin GA (2004) Lectures on diffeomorphism groups in

quantum physics. In: Govaerts J, Hounkonnou N, and

Msezane AZ (eds.) Contempo rary Problems i n Mathematical

Physics: Proceedings of the Third International Workshop,

pp. 3–93. Singapore: World Scientific.

Goldin GA and Sharp DH (1991) The diffeomorphism group

approach to anyons. International Journal of Modern Physics

B 5: 2625–2640.

Ismagilov RS (1996) Representations of Infinite-Dimensional

Groups. Translations of Mathematical Monographs, vol. 152.

Providence, RI: American Mathematical Society.

Kac V (1990) Infinite Dimensional Lie Algebras. Cambridge:

Cambridge University Press.

Marsden J and Weinstein A (1983) Coadjoint orbits, vortices, and

Clebsch variables for incompressible fluids. Physica D

7: 305–323.

Mickelsson J (1989) Current Algebras and Groups. New York:

Plenum.

Ottesen JT (1995) Infinite Dimensional Groups and Algebras in

Quantum Physics. Berlin: Springer.

Pressley A and Segal G (1986) Loop Groups. Oxford: Oxford

University Press.

Renner B (1968) Current Algebras and Their Applications.

Oxford: Pergamon.

Sharp DH and Wightman AS (eds.) (1974) Local Currents and

Their Applications. New York: Elsevier.

Treiman SB, Jackiw R, Zumino B, and Witten E (1985) Current

Algebra and Anomalies. Singapore: World Scientific.

Current Algebra 679

Deformation Quantization

A C Hirshfeld, Universita

t Dortmund, Dortmund,

Germany

Introduction

Deformation quantization is an alternative way

of looking at quantum mechanics. Some of its

techniques were introduced by the pioneers of

quantum mechanics, but it was first proposed as

an autonomous theory in a paper in Annals of

Physics (

Bayen et al. 1978). More recent reviews

treat modern developments (HH I 2001,

Dito and

Sternheimer 2002, Zachos 2002).

Deformation quantization concentrates on the cen-

tral physical concepts of quantum theory: the algebra

of observables and their dynamical evolution. Because

it deals exclusively with functions of phase-space

variables, its conceptual break with classical mechanics

is less severe than in other approaches. It formulates the

correspondence principle very precisely which played

such an important role in the historical development.

Although this article deals mainly with nonrelati-

vistic bosonic systems, deformation quantization is

much more general. For inclusion of fermions and

the Dirac equation see (

Hirshfeld et al. 2002b). The

fermionic degrees of freedom may, in special cases, be

obtained from the bosonic ones by supersymmetric

extension (

Hirshfeld et al. 2004). For applications to

field theory, see

Hirshfeld et al. (2002). For the

relation to Hopf algebras see

Hirshfeld et al. (2003),

and to geometric algebra, see

Hirshfeld et al. (2005).

The observables of a physical system, such as the

Hamilton function, are smooth real-valued functions

on phase space. Physical quantities of the system at

some time, such as the energy, are calculated by

evaluating the Hamilton function at the point

= (q

, p

) in phase space that characterizes the

state of the system at this time (we assume for the

moment, a one-particle system). The mathematical

expression for this operation is

E ¼

Hðq; pÞ

ð2Þ

ðq  q

; p  p

Þdq dp ½1

where 

(2)

is the two-dimensional Dirac delta

function. The observables of the dynamical system

are functions on the phase space, the states of the

system are positive functionals on the observables

(here the Dirac delta functions), and we obtain the

value of the observable in a definite state by the

operation shown in eqn [1].

In general, functions on a manifold are multiplied

by each other in a pointwise manner, that is, given

two functions f and g, their product fg is the

function

ðfgÞðxÞ¼f ðxÞgðxÞ½2

In the context of classical mechanics, the observa-

bles build a commutative algebra, called the com-

mutative ‘‘classical algebra of observables.’’

In Hamiltonian mechanics there is another way to

combine two functions on phase space in such a way

that the result is again a function on the phase space,

namely by using the Poisson bracket

ff ; ggðq; pÞ¼

i¼1







q;p

¼ f @

 @



g ½3

in an abbreviated notation.

The notation can be further abbreviated by using x

to represent points of the phase-space manifold,

x = (x

, ..., x

), and introducing the Poiss on tensor



, where the indices i, j run from 1 to 2n.In

canonical coordinates 

is represented by the matrix

 ¼

0 I



½4

where I

is the n  n identity matrix. Then eqn [3]

becomes

ff ; ggðxÞ¼

f ðxÞ@

gðxÞ½5

where @

= @=@x

For a general observable,

f ¼ff ; Hg½6

Because  transforms like a tensor with respect

to coordinate transformations, eqn [5] may also be

written in noncanonical coordinates. In this case

the c omponents of  need not be constants, and

may depend on the point of the manifold at which

they are evaluated. But in Hamiltonian mechanics,

 is still required to be invertible. A manifold

equipped with a Poisson tensor of this kind is

called a symplectic manifold. In general, the tensor

 is no longer required to be invertible, but it

nevertheless suffices to define Poisson brackets via

eqn [5], and these brackets are required to have

the properties

1. {f , g} = {g, f },

2. {f , gh} = {f , g}h þ g{f , h}, and

3. {f ,{g, h}} þ {g,{h, f }} þ {h,{f , g}} = 0.

Property (1) implies that the Poisson bracket is

antisymmetric, property (2) is referred to as the Leibnitz

rule, and property (3) is called the Jacobi identity. The

Poisson bracket used in Hamiltonian mechanics satis-

fies all these properties, but we now abstract these

properties from the concrete prescription of eqn [3],and

a Poisson manifold (M, ) is defined as a smooth

manifold M equippedwithaPoissontensor, whose

components are no longer necessarily constant, such

that the bracket defined by eqn [5] has the above

properties. It turns out that such manifolds provide a

better context for treating dynamical systems with

symmetries. In fact, they are essential for treating gauge-

fieldtheories,whichgovernthefundamentalinterac-

tions of elementary particles.

Quantum Mechanics and Star Products

The essential difference between classical and

quantum mechanics is Heisenbe rg’s uncertainty

relation, which implies that in the latter, states can

no longer be represented as points in phase space.

The uncertainty is a consequence of the noncommu-

tativity of the quantum mechanical observables.

That is, the commutative classical algebra of

observables must be replaced by a noncommutative

quantum algebra of observables.

In the conventional approach to quantum

mechanics, this noncommutativity is implemented

by representing the quantum mechanical observables

by linear operators in Hilbert space. Physical

quantities are then represented by eigenvalues of

these operators, and physical states are related to the

operator eigen functions. Although these entities are

somehow related to their classical counterparts, to

which they are supposed to reduce in an appropriate

limit, the precise relationship has remained obscure,

one hundred years after the beginnings of quantum

mechanics. Textbooks refer to the correspondence

principle, which guided the pioneers of the subject.

Attempts to give this idea a precise formulation by

postulating a specific relation between the classical

Poisson brackets of observables and the commu-

tators of the corresponding quantum mechanical

operators, as undertaken, for example, by Dirac and

von Neumann, encountered insurmountable diffi-

culties, as pointed out by

Groenewold in 1946 in an

unjustly neglected paper (Groenewold 1948). In the

same paper Groenewold also wrote down the first

explicit representation of a ‘‘star product’’ (see eqn

[11]), without however realizing the potential of this

concept for overcoming the difficulties that he

wanted to resolve.

In the deformation quantization approach, there

is no such break when going from the classical

system to the corresponding quantum system; we

describe the quantum system by using the same

entities that are used to describe t he classical

system. The observables of the s ystem are described

by the same functions on phase space as their

classical counterparts. Uncertainty is realized by

describing physical states as distributions on phase

space that are not sharply localized, in contrast to

the Dirac delta functions which occur in the

classical case. When we evaluate an observable in

some definite state according to the quantum

analog of eqn [1] (see eqn [24]), values of the

observable in a whole region contribute to the

number that is obtained, which is thus an average

value of the observable in the given state. Non-

commutativity is incorporated by introducing a

noncommutative product for functions on phase

space, so that we get a new noncommutative

quantum algebra of observables. The systematic

work on deformation quantization stems from

Gerstenhaber’s seminal paper, where he introduced

the concept of a star product of smooth functions

on a manifold (

Gerstenhaber 1964).

For applications to quantum mechanics, we

consider smooth complex-valued functions on a

Poisson manifold. A star product f  g of two such

functions is a new smooth function, which, in

general, is described by an infinite power series:

f  g ¼ fg þðihÞC

ðf ; gÞþOðh

n¼0

ðihÞ

ðf ; gÞ½7

The first term in the series is the pointwise product

given in eqn [2], and (ih) is the deformation

parameter, which is assumed to be varying con-

tinuously. If h is identified with Planck’s constant,

then what varies is really the magnitude of the

2 Deformation Quantization

action of the dynamical system considered in units

of h: the classical limit holds for systems with large

action. In this limit, which we express here as h ! 0,

the star product reduces to the usual product. In

general, the coefficients C

will be such that the new

product is noncommutative, and we consider the

noncommutative algebra formed from the functions

with this new multiplication law as a deformation of

the original commutative algebra, which uses point-

wise multiplication of the functions.

The expressions C

(f , g) denote functions made

up of the derivatives of the functions f and g.Itis

obvious that without further restrictions of these

coefficients, the star product is too arbitrary to be of

any use. Gerstenhaber’s discovery was that the

simple requirement that the new product be asso-

ciative imposes such strong requirements on the

coefficients C

that they are essentially unique in

the most important cases (up to an equivalence

relation, as discussed below). Formally, Gerstenhaber

required that the coefficients satisfy the following

properties:

jþk = n

(f , g), h) =

jþk = n

(f , C

(g, h)),

2. C

(f , g) = fg,and

3. C

(f , g)  C

(g, f ) = {f , g}.

Property (1) guarantees that the star product is

associative: (f g) h=f (g h). Property (2) means

that in the limit h ! 0, the star product f  g agrees

with the pointwise product fg. Property (3) has at least

two aspects: (i) mathematically, it anchors the new

product to the given structure of the Poisson manifold

and (ii) physically, it provides the connection between

the classical and quantum behavior of the dynamical

system. Define a commutator by using the new

product:

½f ; g



¼ f  g  g  f ½8

Property (3) may then be written as

lim

h!0

ih

½f ; g



¼ff ; gg½9

Equation [9] is the correct form of the correspon-

dence principle. In general, the quantity on the left-

hand side of eqn [9] reduces to the Poisson bracket

only in t he classical limit. The source of the

mathematical difficulties that previous attempts to

formulate the correspondence principle encoun-

tered was related to t rying to enforce equality

between the Poisson bracket and the corresponding

expression involving the quantum mechanical com-

mutator. Equation [9] shows that such a relation in

general only holds up to corrections of higher order

in h.

For physical applications we usually require the

star product to be Hermitean:

f  g =

g 

f, where

denotes the complex conjugate of f. The star

products considered in this article have this

property.

For a given Poisson manifold, it is not clear a

priori if a star product for the smooth funct ions on

the manifold actually exists, that is, whether it is at

all possible to find coefficients C

that satisfy the

above list of properties. Even if we find such

coefficients, it it still not clear that the series they

define through eqn [7] yields a smooth function.

Mathematicians have worked hard to answer these

questions in the general case. For flat Euclidian

spaces, M = R

, a specific star product has long

been known. In this case, the components of the

Poisson tensor 

can be taken to be constants. The

coefficient C

can then be chosen antisymmetric,

so that

ðf ; gÞ¼



ð@

f Þð@

gÞ¼

ff ; gg½10

by property (3) above. The higher-order coefficients

may be obtained by exponentiation of C

. This

procedure yields the Moyal star product (

Moyal

1949):

f 

g ¼ f exp



ih







g ½11

In canonical coordinates, eqn [11] becomes

ðf 

gÞðq; pÞ

¼ f ðq; pÞexp



ih

ð@

 @



gðq; pÞ½12

m;n¼0

ih



mþn

ð1Þ

m!n!

ð@

f Þð@

gÞ½13

We now come to the question of uniqueness of the

star product on a given Poisson manifold. Two star

products  and 

are said to be ‘‘c-equivalent’’ if

there exists an invertible transition operator

T ¼ 1 þ hT

þ¼

n¼0

h

½14

where the T

are differential operators that satisfy

f 

g ¼ T

1

ððTf ÞðTgÞÞ ½15

It is known that for M = R

all admissible star

products are c-equivalent to the Moyal product. The

concept of c -equivalence is a mathematical one

(c stands for cohomology (

Gerstenhaber 1964)); it

does not by itsel f imply any kind of physical

equivalence, as shown below.

Deformation Quantization 3

Another expression for the Moyal product is a

kind of Fourier representation:

ðf 

gÞðq; pÞ

h



f ðq

; p

Þgðq

; p

exp

ih

ðpðq

 q

Þþqðp

 p



þðq

 q



½16

Equation [16] has an interesting geometrical inter-

pretation. Denote points in phase space by vectors,

for example, in two dimensions:

r ¼



; r



; r



½17

Now, consider the triangle in phase space spanned

by the vectors r  r

and r  r

. Its area (symplectic

volume) is

Aðr; r

; r

ðr r

Þ^ðr r

½pðq

q

Þþqðp

p

Þþðq

q

Þ ½18

which is proportional to the exponent in eqn [16].

Hence, we may rewrite eqn [16] as

ðf  gÞðrÞ

f ðr

Þgðr

Þexp

h

Aðr; r

; r



½19

Deformation Quantization

The properties of the star product are well adapted

for describing the noncommutative qua ntum algebra

of observables. We have already discussed the

associativity and the incorporation of the classical

and semiclassical limits. Note that the characteristic

nonlocality feature of quantum mechanics is also

explicit. In the expression for the Moyal product

given in eqn [13], the star product of the functions f

and g at the point x = (q, p) involves not only the

values of the functions f and g at this point, but also

all higher derivatives of these functions at x. But for

a smooth function, knowledge of all the derivatives

at a given point is equivalent to the knowledge of

the function on the entire space. In the integral

expression of eqn [16], we also see that knowledge

of the functions f and g on the whole phase space is

necessary to determine the value of the star product

at the point x.

The c-equivalent star products correspond to differ-

ent quantization schemes. Having chosen a quantiza-

tion scheme, the quantities of interest for the quantum

system may be calculated. It turns out that different

quantization schemes lead to different spectra for the

observables. The choice of a specific quantization

scheme can only be motivated by further physical

requirements. In the simple example we discuss below,

the classical system is completely specified by its

Hamilton function. In more general cases, one may

have to decide what constitutes a sufficiently large set

of good observables for a complete specification of the

system (

Bayen et al. 1978).

A state is characterize d by its energy E;theset

of all possible values for the energy is called the

spectrum of the system. The states are described

by dist ributions on phase space called projectors.

The state corresponding to the ener gy E is

denoted by 

(q, p). These distributions are

normalized:

2h



ðq; pÞdq dp ¼ 1 ½20

and idempotent:

ð

 

Þðq; pÞ¼

E;E



ðq; pÞ½21

The fact that the Hamilton function takes the value

E when the system is in the state corresponding to

this energy is expressed by the equation

ðH  

Þðq; pÞ¼E 

ðq; pÞ½22

Equation [22] corresponds to the time-independent

Schro¨ dinger equation, and is sometimes called the

‘‘-genvalue equation.’’ The spectral decomposition

of the Hamilton function is given by

Hðq; pÞ¼

E

ðq; pÞ½23

where the summation sign may indicate an integra-

tion if the spectrum is continuous. The quantum

mechanical version of eqn [1] is

E ¼

2h

ðH  

Þðq; pÞdq dp

2h

Hðq; pÞ

ðq; pÞdq dp ½24

where the last expression may be obtained by using

eqn [16] for the star product.

The time-evolution function for a time-indepen-

dent Hamilton function is denoted by Exp(Ht), and

the fact that the Hamilton function is the generator

of the time evolution of the system is expressed by

ih

ExpðHt Þ¼H  ExpðHtÞ½25

4 Deformation Quantization

This equation corresponds to the time-dependent

Schro¨ dinger equation. It is solve d by the star

exponential:

ExpðHtÞ¼

n¼0

it

h



ðHÞ

½26

where (H  )

= H  H H

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

n times

. Because each state

of definite energy E has a time evolution exp (iEt=h),

the complete time-evolution function may be written

in the form

ExpðHt Þ¼



iEt=h

½27

This expression is called the ‘‘Fourier–Dirichlet

expansion’’ for the time-evolution function.

Questions concerning the existence and unique-

ness of the star exponential as a C

function and the

nature of the spectrum and the projectors again

require careful mathematical analysis. The problem

of finding general conditions on the Hamilton

function H which ensure a reasonable physical

spectrum is analogous to the problem of showing,

in the conventional approach, that the symmetric

operator

H is self-adjoint and finding its spectral

projections.

The Simple Harmonic Oscillator

As an example of the above procedure, we treat the

simple one-dimensional harmonic oscillator charac-

terized by the classical Hamilton function

Hðq; pÞ¼

½28

In terms of the holomorphic variable s

a ¼

ﬃﬃﬃﬃﬃﬃﬃ

q þ i



;



a ¼

ﬃﬃﬃﬃﬃﬃﬃ

q  i



½29

the Hamilton function becomes

H ¼ !a



a ½30

Our aim is to calculate the time-evolution function.

We first choose a quantization scheme characterized

by the normal star product

f 

g ¼ f e

h



g ½31

we then have



a 

a ¼ a



a; a 



a ¼ a



a þ h ½32

so that



a½



¼ h ½33

Equation [25] for this case is

ih

Exp

ðHt Þ¼ðH þ h!a@

ÞExp

ðHt Þ½34

with the solution

Exp

ðHt Þ¼e

a



a=h

exp e

i!t



a=h



½35

By expanding the last exponential in eqn [35],we

obtain the Fourier–Dirichlet expansion

Exp

ðHtÞ¼e

a



a=h

n¼0

h



in!t

½36

From here, we can read off the energy eigenvalues

and the projectors describing the states by compar-

ing coefficients in eqns [27] and [36]:



ðNÞ

¼ e

a



a=h

½37



ðNÞ

h





h







ðNÞ



½38

¼ nh! ½39

Note that the spectrum obtained in eqn [39] does

not include the zero- point energy. The projector

onto the ground state 

(N)

satisfies

a 



ðNÞ

¼ 0 ½40

The spectral decomposition of the Hamilton func-

tion (eqn [23]) is in this case

H ¼

n¼0

nh!

h

a



a=h





¼ !a



a ½41

We now consider the Moyal quantization scheme.

If we write eqn [12] in terms of holomorphic

coordinates, we obtain

f 

g ¼ f exp

h

ð@



 @





g ½42

Here, we have

a 



a ¼ a



a þ

h

;



a 

a ¼ a



a 

h

½43

and again



a½



¼ h ½44

The value of the commutator of two phase-space

variables is fixed by property (3) of the star product,

Deformation Quantization 5