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

Подождите немного. Документ загружается.

two-dimensional Hilbert space, for instance, we can

consider the initial pure state hij =: [ cos , sin ] and

the time-continuous measurement of the diagonal

observable [59] on it. The solution of eqn [58] is

given by eqn [60]:

A ¼

0 1



½59

E½



¼

cos

 e

t=4g

cos  sin 

t=4g

cos  sin  sin



½60

The off-diagonal elements of this density matrix

go to zero, that is, the coherent superposition

represented by the initial pure state becomes an

incoherent mixture represented by the diagonal

density matrix



Apart from the phenomenon of decoherence, the

stochastic equations show remarkable similarity

with the classical equations of time-continuous

measurement. The heuristic form of eqn [56] is

eqn [61] of invariable interpretation with respect

to the classical equation [19]:

¼h



þ g

½61

Equation [57] describes what is called the time-

continuous collapse of the quantum state under

time-continuous quantum measurement of

A. For

concreteness, we assume discrete spectrum for

A and

consider the spectral expansion

A ¼



½62

The real values a



are nondegenerate eigenvalues.

The Hermitian projectors



form a complete

orthogonal set:





I ½63





¼ 





½64

In a single ideal measurement of

A, the outcome a is

one of the a



’s singled out at random. The

probability distribution of the measurement out-

come and the corresponding collapse of the state are

given by



¼h





½65









¼:





½66

respectively. Equations [56] and [57] of continuous

measurements are an obvious time-continuous

resolution of the ‘‘sudden’’ ideal quantum measure-

ment (eqns [65] and [66]) in a sense that they

reproduce it in the limit t !1. The states





are

stationary states of eqn [57]. It can be shown that

they are indeed approached with probability p



for

t !1 (Gisin 1984).

Related Contexts

In addition to the two particular examples as

in postselection and in time-continuous measure-

ment, respectively, presented above, the weak

measurement limit itself has further variants.

A most natural example is the usual thermodynamic

limit in standard statistical physics. Then weak

measurements concern a certain additive micro-

scopic observable (e.g., the spin) of each constituent

and the weak value represents the corresponding

additive macroscopic parameter (e.g., the magneti-

zation) in the infinite volume limit. This example

indicates that weak values have natural interpreta-

tion despite the apparent artificial conditions of

their definition. It is important that the weak value,

with or without postselection, plays the physical role

similar to that of the common mean h



. If,

between their pre- and postselection, the states



become weakly coupled with the state of another

quantum system via the observable

A, their average

influence will be as if

A took the weak value





Weak measurements also open a specific loophole to

circumvent quantum limitations related to the

irreversible disturbances that quantum measure-

ments cause to the measured state. Noncommuting

observables become simultaneously measurable in

the weak limit: simultaneous weak values of non-

commuting observables will exist.

Literally, weak measurement had been coined

in 1988 for quantum measurements with (pre- and)

postselection, and became the tool of a certain time-

symmetric statistical interpretation of quantum states.

Foundational applications target the paradoxical

problem of pre- and retrodiction in quantum theory.

In a broad sense, however, the very principle of weak

measurement encapsulates the trade between asymp-

totically weak precision and asymptotically large

statistics. Its relevance in different fields has not yet

been fully explored and a growing number of founda-

tional, theoretical, and experimental applications are

being considered in the literature – predominantly in

the context of quantum physics. Since specialized

monographs or textbooks on quantum weak measure-

ment are not yet available, the reader is mostly referred

to research articles, like the recent one by Aharonov

and Botero (2005),coveringmanytopicsofpostse-

lected quantum weak values.

282 Quantum Mechanics: Weak Measurements

Nomenclature

a measurement outcome



a arithmetic mean of measurement

outcomes

A Hermitian operator, quantum observable

A(X) real phase-space function

E[ ...] stochastic expectation value

hf j

Ajii matrix element

hf jii inner product

H Hilbert space

space of Lebesgue square-integrable

complex functions

p probability distribution

tr trace

U unitary operator

Wiener process



white noise process



h...i



postselected mean value

 density operator

(X) phase-space distribution

 direct product

y operator adjoint

j...i state vector

h...j adjoint state vector

h...i



mean value

Further Reading

Aharonov Y, Albert DZ, and Vaidman L (1988) How the result of

measurement of a component of the spin of a spin-1/2 particle

can turn out to be 100. Physical Review Letters 60: 1351–1354.

Aharonov Y and Botero A (2005) Quantum averages of weak

values. Physical Review A (in print).

Aharonov Y, Popescu S, Rohrlich D, and Vaidman L (1993)

Measurements, errors, and negative kinetic energy. Physical

Review A 48: 4084–4090.

Aharonov Y and Vaidman L (1990) Properties of a quantum

system during the time interval between two measurements.

Physical Review A 41: 11–20.

Belavkin VP (1989) A new equation for a continuous

nondemolition measurement. Physics Letters A 140:

355–358.

Dio´si L (1988) Continuous quantum measurement and Itoˆ-

formalism. Physics Letters A 129: 419–432.

Gisin N (1984) Quantum measurements and stochastic processes.

Physical Review Letters 52: 1657–1660.

Giulini D, Joos E, Kiefer C, Kupsch J, Stamatescu IO et al. (1996)

Decoherence and the Appearance of a Classical World in

Quantum Theory. Berlin: Springer.

Gough J and Sobolev A (2004) Stochastic Schro¨dinger equations

as limit of discrete filtering. Open Systems and Information

Dynamics 11: 1–21.

Kraus K (1983) States, Effects, and Operations: Fundamental

Notions of Quantum Theory. Berlin: Springer.

von Neumann J (1955) Mathematical Foundations of Quantum

Mechanics. Princeton: Princeton University Press.

Nielsen MA and Chuang IL (2000) Quantum Computation and

Quantum Information. Cambridge: Cambridge University

Press.

Stratonovich R (1968) Conditional Markov Processes and Their

Application to the Theory of Optimal Control. New York:

Elsevier.

Wiseman HM (2002) Weak values, quantum trajectories, and the

cavity-QED experiment on wave-particle correlation. Physical

Review A 65: 32111–32116.

Quantum n-Body Problem

R G Littlejohn, University of California at Berkeley,

Berkeley, CA, USA

Introduction

This article concerns the nonrelativistic quantum

mechanics of isolated systems of n particles inter-

acting by means of a scalar potential, what we shall

call the ‘‘quantum n-body problem.’’ Such systems

are described by the kinetic-plus-potential

Hamiltonian,

H ¼ T þ V ¼

¼1



þ VðR

; ...; R

Þ½1

where R



, P



,  = 1, ..., n are the positions and

momenta of the n particles in three-dimensional

space, m



are the masses, and V is the potential

energy. This Hamiltonian also occurs in the

‘‘classical n-body problem,’’ in which V is usually

assumed to consist of the sum of the pairwise

gravitational interactions of the particles. In this

article, we shall only assume that V (hence H)is

invariant under translations, proper rotations, par-

ity, and permutations of identical particles. The

Hamiltonian H is also invariant under time reversal.

This Hamiltonian describes the dynamics of isolated

atoms, molecules, and nuclei, with varying degrees

of approximation, including the case of molecules in

the Born–Oppenheimer approximation, in which V

is the Born–Oppenheimer potential. We shall ignore

the spin of the particles, and treat the wave function

 as a scalar. We assume that  is an eigenfunction

of H, H=E. In practice, the value of n typically

ranges from 2 to several hundred. Often the cases

n = 3andn = 4 are of special interest. In this article,

we shall assume that n  3, since n = 2 is the trivial

case of central-force motion. The quantum n-body

problem is not to be confused with the ‘‘quantum

Quantum n-Body Problem 283

many-body problem,’’ which usually refers to the

quantum mechanics of large numbers of identical

particles, such as the electrons in a solid.

Of particular interest is the ‘‘reduction’’ of the

Hamiltonian [1], that is, the elimination of those

degrees of freedom that can be eliminated due to the

continuous symmetries of translations and rotations.

A basic problem is to write down the reduced

Hamiltonian and to make its analytical and geome-

trical properties clear. In the following we shall

present this reduction in two stages, dealing first with

the translations and second with the proper rotations.

In each stage, we shall describe the reduction first in

coordinate language and then in geometrical lan-

guage. The discrete symmetries of parity, time

reversal, and permutation of identical particles are

handled by standard methods of group representation

theory, and will not be discussed here.

There has been considerable interest in mathema-

tical circles in recent years in the reduction of

dynamical systems with symmetry, and the quantum

n-body problem is one of the most important such

systems from a physical standpoint. As such, the

basic theory of the quantum n-body problem has

received considerable attention in the physical

literature going back to the birth of quantum

mechanics, and continues to be of great practical

importance. This article and the bibliography

attempt to bridge these two centers of interest.

Reduction by Translations: Coordinate

Description

We begin with a coordinate description of the

reduction of the system [1] by translations. The

coordinates (R

, ..., R

) are coordinates on the con-

figuration space of the system, called the ‘‘original

configuration space’’ or OCS. The OCS is R

.The

original system has 3n degrees of freedom. The

translation group acts on configuration space by



7!R



þ ,for = 1, ..., n, where  is a displace-

ment vector. It acts on wave functions by

(R

, ..., R

) 7!(R

 , ..., R

 ).

To reduce the system by translations, we perform

a linear coordinate transformation on the OCS,

taking us from the original vectors (R

, ..., R

)toa

new set of n vectors (r

, ..., r

n1

, R

), where R

is the center-of-mass position,

¼1



½2

where M =



is the total mass of the system, and

the other n  1 vectors of the new coordinate system,

, ..., r

n1

), are required to be translationally

invariant, that is, independent linear functions of the

relative particle positions R



 R



.Wedenotethe

momenta conjugate to (r

, ..., r

n1

, R

)by

, ..., p

n1

, P

), of which P

turns out to be the

total momentum of the system,

¼1



½3

Under such a coordinate transformation, the poten-

tial energy becomes simply a function of the n  1

relative vectors, V(r

, ..., r

n1

), whereas the kinetic

energy becomes

T ¼

n1

;¼1





 p



½4

where K



is a symmetric tensor (the ‘‘inverse mass

tensor’’).

The vectors (r

, ..., r

n1

) specify the positions of n

particles relative to their center of mass. As described

so far, these vectors need only be independent,

translationally invariant linear combinations of the

particle postitions. However, it is convenient to

choose them so that the inverse mass tensor becomes

proportional to the identity, K



= (1=M)



.An

elegant way of doing this is the method of Jacobi

vectors, which involves splitting the original set of

particles into two nonempty subsets, which are then

split into smaller subsets, etc., until only subsets of a

single particle remain. The process can be represented

by a tree growing downward, with the original n

particles as the root, and the ends of the branches at

the bottom each containing one particle. Then the

vectors (r

, ..., r

n1

) (the Jacobi vectors) are chosen

to be proportional to the differences between the

centers of mass of the two subsets at each splitting.

With the right constants of proportionality, the

kinetic energy becomes

T ¼

n1

¼0



½5

Henceforth, we shall assume that the vectors

, ..., r

n1

) are Jacobi vectors with conjugate

momenta (p

, ..., p

n1

The choice of Jacobi vectors is not unique. In the

first place, there is a discrete set of possible ways of

splitting the original set of n particles into subsets

(of forming trees), each of which leads to the same

form [5] of the kinetic energy. More generally, the

kinetic energy [5] is invariant under transformations



n1

¼1





½6

284 Quantum n-Body Problem

where Q



is an orthogonal matrix, Q 2 O(n  1).

Such transformations are called ‘‘kinematic rota-

tions.’’ The discrete choices of trees in forming the

Jacobi vectors are equivalent to a discrete set of

kinematic rotations Q



that map one standard

choice of Jacobi vectors into the others.

Since the momentum P

of the center of mass

commutes with H, the eigenfunctions  of H can be

chosen to have the form

ðR

; ...; R

¼ expðiR

 P

=hÞ ðr

; ...; r

n1

Þ½7

This causes to be an eigenfunction of the

‘‘translation-reduced Hamiltonian,’’ H

= E

where

n1

¼0



þ Vðr

; ...; r

n1

Þ½8

The kinetic energy of the center of mass,

=2M, has been discarded from both H

and

, which represent physically the energy of the

system about its center of mass.

Reduction by Translations: Geometrical

Description

The kinetic energy T in eqn [1] specifies a metric



jdR



on the OCS (=R

). The transla-

tion group (=R

) acts freely on the OCS, with an

action that is generated by P

. This action defines

an orthogonal decomposition of the OCS,

= R

 R

3n3

, where R

is the orbit of the origin

(the other orbits of the translation group action are

parallel spaces), and R

3n3

is the orthogonal subspace

(henceforth the ‘‘translation-reduced configuration

space’’ or TRCS for short). The TRCS is physically

the space of configurations relative to the center of

mass. The vectors (r

, ..., r

n1

) are coordinates on

the TRCS. The TRCS possesses a metric which is the

projection of the metric on the OCS onto the TRCS

by means of the translation group action. The metric

can be projected because translations preserve the

original metric (they are isometries). Jacobi vectors

are Euclidean coordinates on the TRCS with respect

to this metric.

The tree method of constructing Jacobi vectors

can be understood in terms of certain group actions

which take place as each subset of particles is split

into two further subsets. The group action in

question leaves the center of mass of the original

subset invariant, while moving the two new subsets

apart along a line. This motion in the configuration

space is orthogonal to all the other group actions

that are created in the process of splitt ing subsets of

particles, including the original action of the

translation group. Thus, each splitting of a subset

of particles generates a three-dimensional subspace

of the OCS, on which one of the r



are coordinates.

The conjugate momentum p



is the generator of the

group action moving the two new subsets apart. The

final result is that the OCS is decomposed into n

orthogonal, three-dimens ional subspaces, one of

which contains the action of the original translation

group, and the others of which represent the

decomposition of the TRCS into n  1, three-

dimensional orthogonal subspaces.

The TRCS can also be seen as a global section of a

flat, trivial, principal fiber bundle created by the

action of the translation group on the OCS.

Alternatively, the TRC S can be seen as the quotient

space, R

. The construction is fairly simple

because the translation group is Abelian.

The wave function can be seen as a member of

the Hilbert space of wave functions on the TRCS,

upon which the reduced Hamiltonian H

of eqn [8]

acts. Alternatively, it can be seen as the function

obtained by restricting  on the OCS to the TRCS,

where  has a dependence along the orbits of the

translation group given by exp (iR

 P

=h), that

is, by an irredu cible representation (irrep) of the

translation group.

Reduction by Rotations: Coordinate

Description

The Hamiltonian H

acts on wave functions

defined on the TRCS and has 3n  3 degree s of

freedom. Consider a coordinate transformation to

eliminate further degree s of freedom due to the

rotational invariance. This coordinate transforma-

tion takes us from the Jacobi vectors {r



,  = 1, ...,

n  1} to orientational and shape coordinates. Shape

coordinates are a set of 3n  6 coordinates



,  = 1, ...,3n  6} that specify the shape of the

n-particle system, that is, they are 3n  6 independent

functions of the interparticle distances (hence rota-

tionally invariant). We will call the space upon which

the q



are coordinates ‘‘shape space.’’ For example, in

the case of the three-body problem, shape space is the

space of all triangles.

As for orientational coordinates, to define them it

is necessary first to define a ‘‘body frame.’’ We

assume we are already given one frame, the ‘‘space

frame,’’ a fixed inertial frame. The body frame is a

3-frame attached in a conventional way to each shape

of the system of particles, which rotates with the

particles. The orientational coordinates, to be

Quantum n-Body Problem 285

denoted by {

, i = 1, 2, 3}, are three coordinates (e.g.,

Euler angles) specifying the SO(3) rotation that maps

the space frame into the body frame. We shall write

the new coordinates collectively as {

, q



There is a great deal of arbitrariness in the choice

of a body frame, since for a given shape a body frame

can be attached in many ways, the different choices

being related by proper rotations. The only require-

ment is that the body frame should change smoothly

as the shape changes. Popular choices for the body

frame are the principal axis and Eckart frames.

When the potential energy is transformed to the new

coordinates, it becomes a function only of the {q



that is, of the shape. The potential can be written as

V = V(q). V is a scalar field on shape space.

The transformation of the kinetic energy is more

complicated. When the (Euclidean) metric tensor on

the TRCS is transformed to orientational and shape

coordinates there results a (3n  3)  (3n  3) com-

ponent matrix which may be partitioned into blocks

according to the coordinates {

, q



}, that is, accord-

ing to 3n  3 = 3 þ (3n  6). This matrix cannot be

made diagonal or even block diagonal by any choice

of orientational or shape coordinates, or by any

choice of body frame.

The components of the metric tensor in the new

coordinates are conveniently expressed in terms of

three fields on shape space. The first is the moment-of-

inertia tensor E, which describes the 3  3 upper block

of the metric tensor. Its components are given by

¼ M

n1

¼1





 r

i

j



½9

The vectors and tensor s in this equation can be

referred either to the space frame or the body fram e,

but the body frame is more convenient because then

the components of the vectors r



are functions only

of the shape coordinates q



. Thus, the body frame

components E

of the moment-of-inertia tensor

define a field on shape space.

The second field is the ‘‘gauge potential’’ A



,an

object with 3(3n  6) components A



, i = 1, 2, 3,

 = 1, ...,3n  6, which describes the off-diagonal

blocks of the metric tensor. It is defined by



¼ E

1

n1

¼1









½10

in which all vectors are understood to be referred to

the body frame (so the partial derivatives make

sense). The gauge potential A



is responsible for the

‘‘falling cat’’ phenomenon, in which a flexible body

of zero angular momentum nevertheless manages to

rotate.

The third field is the (3n  6)  (3n  6) lower

block of the metric tensor on the TRCS, an object

with two shape indices. It is given by



¼ M

n1

¼1













 A



 E  A



½11

where again the vectors are referred to the body

frame. The notation suggests (correctly) that g



the metric tensor on shape space.

On transforming the wave function from the

Jacobi vectors to coordinates (

, q



), it is convenient

to introduce a Jacobian factor, (r

, ..., r

n1

) =

1=4

(

, q



), where D = (det E)(det g



). This

causes the new wave function  to have the

normalization

3n6

¼1



jj

½12

where dR is the Haar measure on the group SO(3).

The factor D depends only on the q



, not the 

Then the Schro¨ dinger equation can be written as

 = E

, where H

is a differential operator

involving @=@

and @=@q



The orientational derivatives @=@

in H

are

conveniently expressed in terms of the angular

momentum operator L. When acting on the original

wave function  on the OCS, the angular momen-

tum is

L ¼

¼1



 P



½13

When this is transformed to the coordinates

, ..., r

n1

, R

), it becomes L = L

þ L

where L

= R

 P

, and

n1

¼1



 p



½14

Physically, L

is the angular momentum of the

system about the center of mass.

We shall henceforth drop the ‘‘tr’’ on H

, E

, and

, thereby restricting attention to the energy and

angular momentum about the center of mass.

The angular momentum L, when acting on wave

functions (r

, ..., r

n1

) on the TRCS, is a vector of

differential operators involving @=@r



. When these

are transformed to orientational and shape coordi-

nates, the components of L become differential

operators involving only orientational derivatives,

@=@

. There are no shape derivatives, @=@q



, since

L generates rotations, that is, changes in orientation,

not shape. Thus, one can solve for the operators

@=@

in terms of the components of L. This is true

286 Quantum n-Body Problem

both for the space and the body components of L,

although the differential operators are not the same

in the two cases. The space components of L satisfy

the usual angular momentum commutat ion rela-

tions, [L

, L

] = ih

ijk

, while the body compone nts

of satisfy [L

, L

] = ih

ijk

(with a minus sign

relative to the space commutation relations).

Thus, the Hamiltonian can be expressed in

terms of L and the shape momentum operators,



= ih@=@q



. The result is

H ¼

L  E

1

 L þ

ðp



 L  A



Þg



ðp



 L  A



þ V

ðqÞþVðqÞ½15

where all vectors are referred to the body frame,

where g



is the contravariant metric tensor on

shape space, and where V

is given by

h

1=4





1=4





½16

looks like a potential (it is a function of only q),

hence the notation, but physically it belongs to the

kinetic energy. It is sometimes called an ‘‘extrapoten-

tial.’’ It arises from nonclassical commutators in the

transformation of the kinetic energy (hence the h

dependence). The first term of eqn [15] is the kinetic

energy of rotation, also called the ‘‘vertical’’ kinetic

energy, the next two terms are the remainder of the

kinetic energy, somewhat imprecisely thought of as

the kinetic energy of vibrations or changes in shape,

also called the ‘‘horizontal kinetic energy,’’ and the

final term is the (true) potential, discussed above.

Since the Hamiltonian commutes with the angular

momentum, [H, L] = 0,  can be chosen to be

simultaneous eigenfunctions of L

and L

(the lat ter

being the space component), as well as of energy.

Let 

be these eigenfunctions, where l and m are

the quantum numbers of L

and L

, respectively.

Then by the transformation properties of  under

rotations, we can write



ð

; q



Þ¼

þl

k¼l



ðq



ÞD

ð

Þ½17

where D is a standard rotation matrix and 

are

functions only of q



. In these equations we use the

phase and other standard conventions of the theory of

rotations. The wave function  is a function only of q



and can loosely be thought of as the wave function on

shape space. It is not a scalar like , ,or, but rather

has 2l þ 1 components indexed by k.

The Schro¨ dinger equation for  can be written

as H = E, where H has the same form as in

eqn [15], except that now the components of the

angular momentum L

are interpreted, no longer

as differential operators in 

, but as (2l þ 1)

(2l þ 1) matrices that act on the ‘‘spinor’’ . These

matrices are the transposes of the usual angular

momentum mat rices in angular momentum theory,

that is, (L

)

= hk

jki.

This is the final form of the Schro¨ dinger equation

after all reductions by all continuous symmetries

have been carried out. The fully reduced system has

3n  5 degrees of freedom (3n  6 for the shape

coordinates, and one for the ‘‘spinor’’ index k).

Reduction by Rotations: Geometrical

Description

The proper rotation group SO(3) acts on the OCS

by R



7!RR



, and on the TRCS by r



7!Rr



, where

R 2 SO(3). Rotations acting on the OCS do not

commute with translations, but the action preserves

the translation fibers, and thus can be projected onto

the TRCS.

The action of SO(3) on the TRCS is effective but

not free, that is, most orbits are diffeomorphic to

SO(3), but a subset of measure zero (the ‘‘singular’’

orbits) are diffeomorphic to S

or a single point.

Configurations of the n-particle system in which the

particles do not lie on a line (‘‘noncollinear shapes’’)

have SO(3) orbits, those in which the particles do lie

on a line but are not coincident have S

orbits, and

the n-body collision (a single shape) has an orbit that

is a single point. Thus, the action of SO(3) on the

TRCS foliates the TRCS into a (3n  6)-parameter

family of copies of SO(3), plus the singular orbits. If

we exclude the singular orbits, then the TRCS has the

structure of an SO(3) principal fiber bundle. In

general, the bundle is not trivial. Shape space may

be defined as the quotient space under the SO(3)

action. Omitting the singular shapes, shape space is

the base space of the bundle. The coordinates q



introduced above are coordinates on shape space.

The singular shapes and orbits are physically acces-

sible, and there are important questions regarding the

behavior of the system in their neighborhood.

The definition of a body frame is equivalent to the

choice of a section of the fiber bundle, generally

only locally defined over some region of shape

space. A configuration (a point in the TRC S) on the

section defines an orientat ion of the n-particle

system for the given shap e, which serves as a

reference orientation to which others can be

referred. We think of the reference orientation as

one in which the space and body frames coincide; in

other orientations of the same shape, the body frame

has been rotated with the body to a new orientation.

The choice of the section (body frame) allows us to

Quantum n-Body Problem 287

impose coordinates on each (nonsingular) rotation

fiber, that is, we label points on the fiber by the

rotation that takes us from the section to the actual

configuration in question. This is why a choice of

body frame is necessary before defining orienta-

tional coordinates. Sections are only defined locally.

Popular choices of body frame, such as the principal

axis frame, imply multivalued sections, unless

branch cuts are introdu ced. Orientational coordi-

nates are simply coordinates on the group manifold

SO(3), transferred to the nonsingular rotation fibers,

with the group identity element mapped onto the

point where the fiber intersects the section.

The metric tensor determines much of the geome-

try of the reduction by rotations. Since the metric on

the TRCS is SO(3)-invariant, horizontal subspaces in

the SO(3) fiber bundle (the TRCS minus the singular

orbits) can be defined as the spaces orthogonal to the

fibers (hence orthogonal to the vertical subspaces).

This is a standard construction in Kaluza–Klein

theories, which reappears here. Thus, the bundle has

a connection, induced by the metric.

The moment-of-inertia tensor is the metric tensor

restricted to a fiber, evaluated in a basis of left-

(body frame) or right-invariant (space frame) vector

fields on SO(3), which are transported to the fibers

to create a basis of vertical vector fields.

The coordin ate description of the connection is

the gauge potential A



, in which the  index refers

to shape coordinates q



, and the components of the

3-vector A refer to the standard set of left- or right-

invariant vector fields on SO(3). The coordinate

representative of the curvature 2-form is conveni-

ently denoted by B



, defined by













 A



 A



½18

where it is understood that body frame components

are used. Direct calculation shows that it is nonzero,

hence the fiber bundle is not flat, for any value of

n  3. The curvature form B



appears in the

classical equation of motion and in the quantum

commutation relations.

The field B



satisfies differential equations on

shape space that have the form of Yang–Mills field

equations. It is interesting that the sources of this

field are singularities of the monopole type, located

on the singular shapes. In the case n = 3, the source

is a single monopole located at the three-body

collision, which is similar to a Dirac monopole in

electromagnetic theory.

The (3n  6)-dimensional horizontal subspaces of

the TRCS are annihilated by three differential forms,

whose values on a velocity vector of the system are

the components of the classical angular momentum L

(body or space components, depending on the basis

of forms). Thus, horizontal motions are those for

which L = 0, and horizontal lifts of curves in shape

space are motions of the system with vanishing

angular momentum. Since angular momentum is

conserved, such motions are generated by the

classical equations of motion and are physically

allowed. For loops in shape space, the holonomy

generated by the horizontal lift is physically the

rotation that a flexible body experiences when it is

carried under conditions of vanishing angular

momentum from an initial shape, through intermedi-

ate shapes and back to the initial shape. An example

is the rotation generated by the ‘‘falling cat.’’

Since the metric on the TRCS is SO(3)-invariant,

it may be projected onto shape space, which there-

fore is a Riemannian manifold in its own right. The

projected metric is ds

= g







. This metric is

not flat (the Riemann curvature tensor is nonzero

for all values n  3). Geodesics in shape space have

horizontal lifts that are free particle motions (V = 0)

of zero angular momentum. Conversely, such

motions project onto geodesics on shape space.

A popular choice of body frame in molecular

physics is the Eckart frame, which has advantages

for the description of small vibrations and other

purposes. The section defining the Eckart frame is a

flat vector subspace of the TRCS of dimension 3n  6

that is orthogonal (horizontal) to a particular fiber

(over an equilibrium shape) at a particular

orientation.

The geometrical meaning of eqn [17] is that

rotations act on a set of wave functions  that span

an irrep of SO(3) by multiplication by the represen-

tative element of the group. In standard physics

notation, l indexes the irrep, and m indexes the basis

vectors spanning the irrep. Thus, the values of these

wave functions at any point on the fiber are known

once their values are given at a reference point. A

convenient choice for the reference point is the point

on the section, and the wave functions 

are simply

the values of the 

on this reference point (with a

change of notation, m ! k). Thus, the wave func-

tions 

are properly not ‘‘wave functions on shape

space,’’ but rather wave functions on the section.

Shapespaceinthecasen = 3 is homeomorphic to

the region x

 0ofR

,andinthecasen = 4toR

A convenient tool for understanding the structure

of shape space is by its foliation under the action of

the kinematic rotations, eqn [ 5]. The kinematic

rotations commute with ordinary rotations, and

hence have an action on shape space. This action

preserves the eigenvalues of the moment-of-inertia

tensor.

288 Quantum n-Body Problem

Concluding Remarks

The quantum n-body problem provides an interesting

example in which nonabelian gauge theories find

application in nonrelativistic quantum mechanics. The

fields E, A



,andg



, and fields derived from them such

as the curvature tensor B



and the Riemann curvature

tensor derived from g



, satisfy a complex set of

differential equations on shape space that can be

derived by considering the vanishing of the Riemann

tensor on the TRCS. The resulting field equations are

useful in perturbation theory, for example, in the study

of small vibrations of a molecule. This means of

constructing field equations on the base space of a

bundle is standard in Kaluza–Klein theories, which are

an important line of thinking in modern attempts to

understand gauge field theories in particle physics.

The rotations generated by flexible bodies of vanish-

ing angular momentum (the ‘‘falling cat’’) are an

example of a ‘ ‘geometric phase,’ ’ that is, a nonabelian

generalization of ‘‘Berry’s phase.’’ It is interesting how

the associated gauge potential A



in this problem plays

a role in the dynamics of the n-particle system.

The Hamiltonian [15] is the starting point for

numerous practical calculations, for example, the

numerical evaluation of energy levels, cross-sections

and reaction rates in molecular physics. One can

compute, for example, chemical reaction rates for

molecular processes in atmospheric or astrophysical

contexts, where experiments would be difficult or

expensive. The numerical analysis of the Hamiltonian

[15] usually requires the introduction of a basis set and

the processing of large matrices. Current techniques

for basis set selection are not very satisfactory, and this

is an area where research into wavelets and numerical

analysis could have an impact.

See also: Bosons and Fermions in External Fields;

Gravitational N-Body Problem (Classical); Integrable

Systems: Overview.

Further Reading

Abraham R and Marsden JE (1978) Foundations of Mechanics,

2nd edn. Reading, MA: Benjamin/Cummings.

Aquilanti V and Cavalli S (1986) Coordinates for molecular

dynamics: orthogonal local frames. Journal of Chemical

Physics 85: 1355–1361.

Berry MV (1984) Quantal phase factors accompanying adiabatic

changes. Proceedings of the Royal Society of London. Series A

392: 45–57.

Coquereaux R (1988) Riemannian Geometry, Fiber bundles,

Kaluza–Klein Theories and All That. World Scientific Lecture

Notes in Physics, vol. 16. Singapore: World Scientific.

Ezra GS (1982) Symmetry Properties of Molecules. New York:

Springer.

Iwai T (1987) A gauge theory for the quantum planar three-body

problem. Journal of Mathematical Physics 28: 964–974.

Kupperman A (1993) A new look at symmetrized hyperspherical

coordinates. Advances in Molecular Vibrations and Collision

Dynamics vol. 2B: 117–186.

Littlejohn RG and Reinsch M (1997) Gauge fields in the

separation of rotations and internal motions in the n-body

problem. Reviews of Modern Physics 69: 213–275.

Mead CA (1992) The geometric phase in molecular systems.

Reviews of Modern Physics 64: 51–85.

Mitchell KA and Littlejohn RG (2000) Kinematic orbits and the

structure of the internal space for systems of five or more

bodies. Journal of Physics A 33: 1395–1416.

Naber GL (1997) Topology, Geometry and Gauge Fields.

New York: Springer.

Wilson EB Jr., Decius JC, and Cross PC (1955) Molecular

Vibrations: the Theory of Infrared and Raman Vibrational

Spectra. New York: McGraw-Hill.

Quantum Phase Transitions

S Sachdev, Yale University, New Haven, CT, USA

Introduction

The study of second-order phase transitions at

nonzero temperatures has a long and distinguished

history in statistical mechanics. Many key physical

phenomena, such as the loss of ferromagnetism

in iron at the Curie temperature or the critical

endpoint of CO

, are now understood in precise

quantitative detail. This understanding began in the

work of Onsager, and is based upon what may now

be called the Landau–Ginzburg –Wilson theory.

The content of this sophisticated theory may be sum-

marized in a few basic principles: (1) The collective

thermal fluctuations near second-order transitions

can be accurately described by simple classical

models, that is, quantum-me chanical effects can be

entirely neglecte d. (2) The classical models identify

an ‘‘order parameter,’’ a collective variable which

has to be treated on par with other thermodynamic

variables, and whose correlations exhibit distinct

behavior in the phases on either side of the

transition. (3) The thermal fluctuations of the

order parameter near the transition are controlled

by a continuum field theory whose structure is

usually completly dictated by simple symmetry

considerations.

Quantum Phase Transitions 289

This article will not consider such nonzero

temperature phase transitions, but will instead

describe second-order phase transitions at the

absolute zero of temperature. Such transitions are

driven by quantum fluctuations mandated by the

Heisenberg uncertainty principle: one can imagine

moving across the quantum critical point by

effectively ‘‘tuning the value of Planck’s constant,

h.’’ Clearly, quantum mechanics plays a central role

at such transitions, unlike the situation at nonzero

temperatures. The reader may object that absolute

zero is an idealization not realized by any experi-

mental system; hence, the study of quantum phase

transitions is a subject only of academic interest. As

we will illustrate below, knowledge of the zero-

temperature quantum critical points of a system is

often the key to understanding its finite-temperature

properties, and in some cases the influence of a zero-

temperature critical point can be detected at

temperatures as high as ambient room temperature.

We will begin in the following section by

introducing some simple lattice models which

exhibit quantum phase transitions. Next the theory

of the critical point in these models is based upon

a natural extension of the Landau–Ginzburg–Wilson

(LGW) method, and this will be presented. This

section will also describe the consequences of a zero-

temperature critical point on the nonzero tempera-

ture prop erties. Finally, we will consi der more

complex models in which quantum interference

effects play a more subtle role, and which cannot

be described in the LGW framework: such quantum

critical points are likely to play a central role in

understanding many of the correlated electron

systems of current interest.

Simple Models

Quantum Ising Chain

This is a simple model of N qubits, labeled by the

index j = 1, ..., N. On each ‘‘site’’ j there are two

qubit quantum states "

and #

(in practice, these

could be two magnetic states of an ion at site j in a

crystal). The Hilbert space therefore consists of 2

states, each consisting of a tensor product of the

states on each site. We introduce the Pauli spin

operators, ^



, on each site j, with  = x, y, z:

^



; ^

0 i

i 0



^

0 1



½1

These operators clearly act on the two states of the

qubit on site j, and the Pauli operators on different

sites commute.

The quantum Ising chain is defined by the simp le

Hamiltonian

¼J

N 1

j¼1

^

jþ1

 gJ

j¼1

^

½2

where J > 0 sets the energy scale, and g  0isa

dimensionless coupling constant. In the thermody-

namic limit (N !1), the ground state of H

exhibits

a second-order quantum phase transition as g is

tuned across a critical value g = g

(for the specific

case of H

it is known that g

= 1), as we will now

illustrate.

First, consider the ground state of H

for g  1.

At g = 0, there are two degenerate ‘‘ferromagnetically

ordered’’ ground states

j¼1

; +

j¼1

½3

Each of these states breaks a discrete ‘‘Ising’’

symmetry of the Hamiltonian rotations of all

spins by 180



about the x-axis. These states are

more succinctly characterized by defining the

ferromagnetic moment, N

,by

¼*

^

¼+

^

½4

At g = 0 we clearly have N

= 1. A key point is

that in the thermodynamic limit, this simple picture

of the ground state survives for a finite range of

small g (indeed, for all g < g

), but with 0 < N

< 1.

The quantum tunneling between the two ferromag-

netic ground states is exponentially small in N (and

so can be neglected in the thermodynamic limit),

and so the ground state remains 2-fold degenerate

and the discrete Ising symmetry remains broken.

The change in the wave functions of these states

from eqn [3] can be easily determined by perturba-

tion theory in g: these small g quantum fluctuations

reduce the value of N

from unity but do not cause

the ferromagnetism to disappear.

Now consider the ground state of H

for g  1.

At g = 1 there is a single nondegenerate ground

state which fully preserves all symmetries of H

)i¼2

N=2

j¼1

"ji

þ#ji



½5

It is easy to verify that this state has no ferromagnetic

moment N



)



^



)i= 0. Further, perturbation

theory in 1=g shows that these features of the ground

state are preserved for a finite range of large g values

290 Quantum Phase Transitions

(indeed, for all g > g

). One can visualize this ground

state as one in which strong quantum fluctuations

have destroyed the ferromagnetism, with the local

magnetic moments quantum tunneling between ‘‘up’’

and ‘‘down’’ on a timescale of order h=J.

Given the very distinct signatures of the small g

and large g ground states, it is clear that the ground

state cannot evolve smoothly as a function of g.

These must be at least one point of nonanalyticity as

a function of g: for H

it is known that there is only

a single nonanalytic point, and this is at the location

of a second-order quantum phase transition at

g = g

= 1.

The character of the excitations above the ground

state also undergoes a qualitative change across the

quantum critical point. In both the g < g

and g > g

phases, these excitations can be described in the

Landau quasiparticle scheme, that is, as super-

positions of nearly independent particle-like

excitations; a single well-isolated quasiparticle has

an infinite lifetime at low excitation energies.

However, the physical nature of the quasiparticles

is very different in the two phases. In the ferromag-

netic phase, with g < g

, the quasiparticles are

domain walls between regions of opposite

magnetization:

j; j þ 1

k¼1

‘¼jþ1

‘

½6

This is the exact wave function of a stationary

quasiparticle excitation between sites j and j þ 1at

g = 0; for small nonzero g the quasiparticle acquires

a ‘‘cloud’’ of further spin-flips and also becomes

mobile. However its qualitative interpretation as a

domain wall between the two degenerate ground

states remains valid for all g < g

. In contrast, for

g > g

, there is no ferromagnetism, and the non-

degenerate paramagnetic state has a distinct quasi-

particle excitation:

¼ 2

N=2

#



k6¼j



þ#



½7

This is a stationary ‘‘flipped spin’’ quasiparticle at

site j, with its wave function exact at g = 1.Again,

this quasiparticle is mobile and applicable for all

g > g

, but there is no smooth connection between

eqns [7] and [6].

Coupled Dimer Antiferromagnet

This model also involves qubits, but they are now

placed on the sites, j, of a two-dimensional square

lattice. Models in this class describe the magnetic

excitations of many experimentally important spin

gap compounds.

The Hamiltonian of the dimer antiferromag net is

illustrated in Figure 1 and is given by

¼J

hjki2A

^

þ ^

^

þ ^

^



hjki2B

^

þ ^

^

þ ^

^



½8

where J > 0 is the exchange constant, g  1 is the

dimensionless coupling, and the set of nearest-

neighbor links A and B are defined in Figure 1.An

important property of H

is that it is now invariant

under the full O(3) group of spin rotations under

which the ^



transform as ordinary vectors (in

contrast to the Z

symmetry group of H

). In

analogy with H

, we will find that H

undergoes a

quantum phase trans ition from a paramagnetic

phase which preserves all symmetries of the

Hamiltonian at large g, to an antiferromagnetic

phase which breaks the O(3) symmetry at small g.

This transition occurs at a critical value g = g

and the best current numerical estimate is

1=g

= 0.52337(3).

As in the previous section, we can establish the

existence of such a quantum phase transition by

contrasting the disparate physical properties at large

g with those at g  1. At g = 1 the exact ground

state of H

spin gap

hjki2A

ﬃﬃﬃ

#



½9

and is illustrated in Figure 2. This state is non-

degenerate and invariant under spin rotations, and

so is a paramagnet: the qubits are paired into spin

singlet valence bonds across all the A links.

The excitations above the ground state are

created by breaking a valence bond, so that the

pair of spins form a spin triplet with total spin

S = 1 – this is illustrated in Figure 3. It costs a large

energy to create this excitation, and at finite g the

Figure 1 The coupled dimer antiferromagnet. Qubits (i.e.,

S = 1=2 spins) are placed on the sites, the A links are shown as

full lines, and the B links as dashed lines.

Quantum Phase Transitions 291