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

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Weakly Coupled Oscillators

E M Izhikevich, The Neurosciences Institute,

San Diego, CA, USA

Y Kuramoto, Hokkaido University, Sapporo, Japan

Introduction

Practically any physical, chemical, or biological

system can exhibit rhythmic oscillatory activity, at

least when the conditions are right. Winfree (2001)

reviews the ubiquity of oscillations in nature,

ranging from autocatalytic chemical reactions to

pacemaker cells in the heart, to animal gates, and to

circadian rhythms. When coupled, even weakly,

oscillators interact via adjustment of their phases,

that is, their timing, often leading to synchroniza-

tion. In this chapter, we review the most important

concepts needed to study and understand the

dynamics of coupled oscillators.

From a mathematical point of view, an oscillator

is a dynamical system,

x ¼ f ðxÞ; x 2 R

½1

having a limit-cycle attractor – periodic orbit   R

Its period is the minimal T > 0suchthat

ðtÞ¼ðt þ TÞ for any t

and its frequency is =2=T.Letx(0) = x

2  be

an arbitrary point on the attractor, then the state of

the system, x(t), is uniquely defined by its phase

# 2 S

relative to x

,whereS

is the unit circle.

Throughout this article, we assume that the

periodic orbit  is exponentially stable, which

implies normal hyperbolicity. In this case, there is a

continuous transformation  : U ! S

defined in a

neighborhood U   such that #(t) =(x(t)) for any

trajectory in U, that is,  maps solutions of [1] to

solutions of

# ¼  ½2

Such a transformation removes the amplitude but

saves the phase of oscillation.

Accordingly, there is a continuous transformation

that maps solutions of the weakly coupled network

of n oscillators,

¼ f

ðx

Þþ"g

ðx

; ...; x

;"Þ;" 1 ½3

onto solutions of the phase system

¼ 

þ "h

ð#

; ...;#

;"Þ;#

2 S

½4

which is easier for studying the collective properties

of [3].

The oscillators are said to be frequency locked when

[4] has a stable periodic orbit #(t) = (#

(t), ..., #

(t))

on the n-torus T

,asinFigure 1a. The ‘‘rotation

vector’’ or ‘‘winding ratio’’ of the orbit is the set of

integers q

: q

: : q

such that #

makes q

rotations

while #

makes q

rotations, etc., as in the 2 : 3

frequency locking in Figure 1a. The oscillators

are entrained when they are 1 : 1: :1 frequency

locked. The oscillators are phase locked when there is

an (n  1)  n integer matrix K having linearly

independent rows such that K#(t) = const. For exam-

ple, the two oscillators in Figure 1b are phase locked

with K = (2, 3), while those in Figure 1c are not. The

oscillators are synchronized when they are entrained

and phase locked. Synchronization is in-phase when

(t) =  = #

(t) and out-of-phase otherwise. Two

oscillators are said to be synchronized antiphase when

(t)  #

(t) = . Frequency locking without phase

locking, as in Figure 1c, is called phase trappi ng. The

relationship between all these definitions is depicted

in Figure 2.

Phase Resetting

An exponentially stable periodic orbit is a normally

hyperbolic invariant manifold, hence its sufficiently

small neighborhood, U, is invariantly foliated by

(c)

(a)

(b)

Identify

Figure 1 A 2-torus and its representation on the square.

(Modified from Hoppensteadt and Izhikevich 1997.)

Frequency locking

Phase locking

phase

Antiphase

Synchronization

Entrainment

(1:1 frequency locking)

Figure 2 Various degrees of locking of oscillators. (Modified

from Izhikevich 2006.)

448 Weakly Coupled Oscillators

stable submanifolds (Guckenheimer 1975)illustrated

in Figure 3. The manifolds represent points having

equal phases and, for this reason, they are called

isochrons (from Greek ‘‘iso’’ meaning equal and

‘‘chronos’’ meaning time).

The geometry of isochrons determines how the

oscillators react to perturbations. For example, the

pulse in Figure 3, right, moves the trajectory from

one isochron to another, thereby changing its pha se.

The magnitude of the phase shift depends on the

amplitude and the exact timing of the stimulus

relative to the phase of oscillation #. Stimulating the

oscillator at different phases, one can measure the

phase transition curve (Winfree 2001)

new

¼ PTCð#

old

and the phase resetting curve

PRCð#Þ¼PTCð#Þ#

ðshift ¼ new phase  old phaseÞ

Positive (negative) values of the PRC correspond to

phase advances (delays). PRCs are convenient when

the phase shifts are small, so that they can be

magnified and clearly seen, as in Figure 4. PTCs are

convenient when the phase shifts are large and

comparable with the period of oscillation.

In Figure 5 we depict phase portraits of the

Andronov–Hopf oscillator receiving pulses of

magnitude 0.5 ( left) and 1. 5 ( right). Notice the

drastic difference between the corresponding PRCs

or PTCs. Winfree (2001) distinguishes two cases:

1. type 1 (weak) resetting results in continuous PRCs

and PTCs with mean slope 1, and

2. type 0 (strong) resetting results in discontinuous

PRCs and PTCs with mean slope 0.

–1.5 –1 –0.5 0 0.5 1 1.5

–2

–1.5

–1

–0.5

0.5

1.5

van der Pol oscillator

pulse

–1.5 –1 – 0.5

0 0.5 1 1.5

–1.5

–1

–0.5

0.5

1.5

Andronov–Hopf oscillator

Re z

Im z

θ = 5π /4

θ = 3π /4

θ = π /2

θ = π /4

θ = π

θ = 3π /2

θ = 7π /4

θ = 0

Figure 3 Isochrons of Andronov–Hopf oscillator (

z = (1 þi )z  zjzj

, z 2 C) and van der Pol oscillator (

x = x  x

 y ,

y = x ):

–0.2

0.2

Stimulus phase, θ

PRC(θ)

Re z (t )

Andronov–Hopf oscillator

PRC

2π

Stimulus phase, θ

(t )

van der Pol oscillator

PRC

2π

Figure 4 Examples of phase response curves (PRCs) of the

oscillators in Figure 3. PRC

(#) and PRC

(#) correspond to

horizontal (along the first variable) and vertical (along the second

variable) pulses with amplitudes 0.2. An example of oscillation is

plotted as a dotted curve in each subplot (not to scale).

2π 2π

−π

2π

Stimulus phase, θ Stimulus phase, θ

PTC(θ)

{θ

+ PRC(θ)} mod 2π

PTC(θ)

{θ

+ PRC(θ)} mod 2π

Phase transition

–1

2π

–π

Stimulus phase, θ Stimulus phase, θ

PRC(θ) PRC(θ)

Phase resetting

Type 1 (weak) resetting Type 0 (strong) resetting

= 0, 2π

= π

θ = π /2

= 3π /2

Figure 5 Types of phase resetting of the Andronov–Hopf

oscillator in Figure 3.

Weakly Coupled Oscillators 449

The discontinuity of type 0 PRC in Figure 5 is a

topological property that cannot be removed by

reallocating the initial point x

that corresponds to

zero phase. The discontinuity stems from the fact

that the shifted image of the limit cycle (dashed

circle) goes beyond the central equilibrium at which

the phase is not defined.

The stroboscopic mapping of S

to itself, called

Poincare

phase map,

kþ1

¼ PTCð#

Þ½5

describes the response of an oscillator to a T-periodic

pulse train. Here, #

denotes the phase of oscillation

when the kth input pulse arrives. Its fixed points

correspond to synchronized solutions, and its periodic

orbits correspond to phase-locked states.

Weak Coupling

Now consider dynamical systems of the form

x ¼ f ðxÞþ"sðtÞ½6

describing periodic oscillators,

x = f (x), forced by

a weak time-depended input "s(t), for example, from

other oscillators in a network. Let (x) denote the

phase of oscillation at point x 2 U, so that the map

 : U ! S

is constant along each isochron. This

mapping transforms [6] into the phase model

# ¼  þ "Qð#ÞsðtÞ

with function Q(#), illustrated in Figure 6, satisfying

three equivalent conditions:

1. Winfree: Q(#) is normalized PRC to infinitesimal

pulsed perturbations;

2. Kuramoto: Q(#) = grad (x); and

3. Malkin: Q is the solution to the adjoint problem

Q ¼fDf ððtÞÞg

Q ½7

with the normalization Q(t)  f ((t)) =for any t.

The function Q(#) can be found analytically in a

few simple cases:

1. a nonlinear phase oscillator

x = f (x) with x 2 S

and f > 0 has Q(#) ==f ((#));

2. a system near saddle-node on invariant circle

bifurcation has Q(#) proportional to 1  cos #;

and

3. a system near supercritical Andronov–Hopf

bifurcation has Q(#) proportional to sin(#  ),

where 2 S

is a constant phase shift.

Other interesting cases, including homoclinic,

relaxation, and bursting oscillators are considered

by Izhikevich (2006).

Treating s(t)in[6] as the input from the network,

we can transform weakly coupled oscillators

¼ f

ðx

Þþ"

j¼1

ðx

; x

zﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{

ðtÞ

; x

2 R

½8

to the phase model

¼ 

þ " Q

ð#

Þ

j¼1

ðx

ð#

Þ; x

ð#

ÞÞ

zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{

ðtÞ

½9

having the form [4] with h

= Q

, or the form

¼ 

þ "

j¼1

ð#

where h

= Q

. Introducing phase deviation vari-

ables #

=

t þ ’

, we transform this system into the

form

’

¼ "

j¼1

ð

t þ ’

; 

t þ ’

which can be averaged to

’

¼ "

j¼1

ð’

 ’

Þ½10

with the functions

ðÞ¼ lim

T!1

ð

t; 

t  Þdt ½11

–1

Andronov–Hopf oscillator

(θ)

Phase, θ

–1

2π

(θ)

Phase, θ

–1

2π

(θ)

van der Pol oscillato

–1

–2

Q (θ)

Figure 6 Solutions Q = (Q

, Q

) to the adjoint problem [7] for

oscillators in Figure 3.

450 Weakly Coupled Oscillators

describing the interaction between oscillators

(Ermentrout and Kopell 1984). To summarize, we

transformed weakly coupled system [8] into the

phase model [10] with H given by [11] and each Q

being the solut ion to the adjoint problem [7]. This

constitutes the Malkin theorem for weakly coupled

oscillators (Hoppens teadt and Izhikevi ch 1997,

theorem 9.2).

Existence of one equilibrium of the phase model

[10] implies the existence of the entire circular

family of equilibria, since translation of all ’

by a

constant phase shift does not change the phase

differences ’

 ’

and hence the form of [10]. This

family corresponds to a limit cycle of [8], on which

all oscillators have equal frequencies and constant

phase shifts, that is, they are synchronized, possibly

out of phase.

We say that two oscillators, i and j, have resonant

(or commensurable) frequencies when the ratio



=

is a rational number, for exampl e, it is p=q

for some integer p and q. They are nonr esonant

when the ratio is an irrational number. In this case,

the function H

defined above is constant regardless

of the details of the oscillatory dynamics or the

details of the coupling, that is, dynamics of two

coupled nonresonant oscillators is described by an

uncoupled phase model. Apparently, such oscillator s

do not interact; that is, the phase of one of them

cannot change the phase of the other one even on

the long timescale of order 1=".

Synchronization

Consider [8] with n = 2, describing two mutually

coupled oscillators. Let us introduce ‘‘slow’’ time

 = "t and rewrite the corresponding phase model

[10] in the form

’

¼ !

þ H

ð’

 ’

’

¼ !

þ H

ð’

 ’

where

= d=d and !

= H

(0) is the frequency

deviation from the natural oscillation, i = 1, 2. Let

 = ’

 ’

denote the phase difference between the

oscillators; then



¼ ! þ HðÞ½12

where

! ¼ !

 !

and HðÞ¼H

ðÞH

ðÞ

is the frequency misma tch and the antisymmetric

part of the coupling, respectively, illustrated in

Figure 7, dashed curves. A stable equilibrium of

[12] corresponds to a stable limit cycle of the phase

model.

All equilibria of [12] are solutions to H() = !,

and they are intersections of the horizontal line !

with the graph of H. They are stable if the slope

of the graph is negative at the intersection. If

oscillators are identical, then H() is an odd

function (i.e., H() = H()), and  = 0and

 =  are always equilibria, possibly unstable,

corresponding to the in-phase and antiphase syn-

chronized solutions. The in-phase synchronization

of gap–junction coupled oscillators in Figure 7 is

stable because the slope of H (dashed curves) is

negative at  = 0. The max and min values of the

function H determine the tolerance of the network

to the frequency mismatch !, since there are no

equilibria outside this range.

Now consider a network of n > 2 weakly coupled

oscillators [8]. To determine the existence and

stability of synchronized states in the network, we

need to study equilibria of the corresponding phase

model [10]. The vector  = (

, ..., 

)isan

equilibrium of [10] when

0 ¼ !

j6¼1

ð

 

Þðfor all iÞ½13

It is stable when all eigenvalues of the linearization

matrix (Jacobian) at  have negative real parts,

except one zero eigenvalue corresponding to the

eigenvector along the circular family of equilibria (

plus a phase shift is a solution of [13] too since the

phase shifts 

 

are not affected).

In general, determining the stability of equilibria

is a difficult problem. Ermentrout (1992) found a

simple sufficient condition. If

1. a

= H

(

 

)  0, and

2. the directed graph defined by the matrix a = (a

)

is connected, (i.e., each oscillator is influenced,

possibly indirectly, by every other oscillator),

then the equilibrium  is neutrally stable, and the

corresponding limit cycle x(t þ )of[8] is asympto-

tically stable.

Andronov–Hopf oscillator

Phase difference,

H(χ)

(χ)

–0.5

0.5

2π

van der Pol oscillator

Phase difference, χ

H(χ)

(χ)

Figure 7 Solid curves: functions H

() defined by [11]

corresponding to the gap–junction input g(x

, x

) = (x

 x

, 0).

Dashed curves: functions H() = H

()  H

().: Parameters

are as in Figure 3.

Weakly Coupled Oscillators 451

Another sufficient condition was found by

Hoppensteadt and Izhikevich (1997). If system [10]

satisfies

1. !

=  = !

= ! (identical frequencies)

2. H

() = H

() (pairwise odd coupling)

for all i and j, then the network dynamics converge to a

limit cycle. On the cycle, all oscillators have equal

frequencies 1 þ "! and constant phase deviations.

The proof follows from the observation that [10]

is a gradient system in the rotating coordinates

’ = ! þ  with the energy function

EðÞ¼

i¼1

j¼1

ð

 

where

ðÞ¼



ðsÞds

One can check that dE()=d = 

(

)

 0

along the trajectories of [12] with equality only at

equilibria.

Mean-Field Approximations

Let us represent the phase model [10] in the form

’

¼ !

j6¼i

ð’

 ’

where

= d=d,  = "t is the slow time, and

= H

(0) are random frequency deviations. Collec-

tive dynamics of this system can be analyzed

in the limit n !1. We illustrate the theory

using the special case, H() = sin , known as the

Kuramoto (1984) model:

’

¼ !

j¼1

sinð’

 ’

Þ;’

2½0; 2½14

where K > 0 is the coupling strength and the factor

1=n ensures that the model behaves well as n !1.

The complex-valued sum of all phases,

j¼1

i’

ðKuramoto synchronization indexÞ½15

describes the degree of synchronization in the

network. Apparently, the in-phase synchronized

state ’

=  = ’

corresponds to r = 1 with

being the population phase. In contrast, the inco-

herent state with all ’

having different values

randomly distributed on the unit circle, corresponds

to r  0. Intermediate values of r correspond to a

partially synchronized or coherent state, depicted in

Figure 8. Some phases are synchronized forming a

cluster, while others roam around the circle.

Multiplying both sides of [15] by e

i’

and

considering only the imaginary parts, we can rewrite

[14] in the equivalent form

’

¼ !

þ Kr sinð  ’

that emphasizes the mean-filed character of interac-

tions between the oscillators: they all are pulled into

the synchronized cluster (’

! ) with the effective

strength proportional to the cluster size r. This pull

is offset by the random frequency deviations !

that

pull away from the cluster.

Let us assume that !

’s are distributed randomly

around 0 with a symmetrical probability density

function g(!), for example, Gaussian. Kuramoto has

shown that in the limit n !1, the cluster size r

obeys the self-consistenc y equation

r ¼ rK

þ=2

=2

gðKr sin ’Þcos

’ d’ ½16

Notice that r = 0, corresponding to the incoherent

state, is always a solution of this equation. When the

coupling strength K is greater than a certain critical

value,

gð0Þ

an additional , nontrivial solution r > 0 appears,

which corre sponds to a partially synchronized

state. Expanding g in a Taylor series, one gets the

scaling r =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

16(K  K

)=(g

(0)K

)

. Thus, the

stronger the coupling K relative to the random

distribution of frequencies, the more oscillators syn-

chronize into a coherent cluster. The issue of stability

of incoherent and partially synchronized states is

discussed by Strogatz (2000). Other generalizations

of the Kuramoto model are reviewed by Acebron et al.

(2005). An extended version of this article with the

i ψ

Figure 8 Kuramoto synchronization index [15] describes the

degree of coherence in the network [14].

452 Weakly Coupled Oscillators

emphasis on computational neuroscience can be found

in the recent book by Izhikevich (2006).

See also: Bifurcations of Periodic Orbits; Dynamical

Systems and Thermodynamics; Hamiltonian Systems:

Stability and Instability Theory; Singularity and Bifurcation

Theory; Stability Theory and KAM; Synchronization of

Chaos.

Further Reading

Acebron JA, Bonilla LL, Vincente CJP, Ritort F, and Spigler R (2005)

The Kuramoto model: a simple paradigm for synchronization

phenomena. Reviews of Modern Physics 77: 137–186.

Ermentrout GB (1992) Stable periodic solutions to discrete and

continuum arrays of weakly coupled nonlinear oscillators. SIAM

Journal on Applied Mathematics 52: 1665–1687.

Ermentrout GB and Kopell N (1984) Frequency plateaus in a chain

of weakly coupled oscillators, I. SIAM Journal on Applied

Mathematics 15: 215–237.

Glass L and MacKey MC (1988) From Clocks to Chaos. Princeton:

Princeton University Press

Guckenheimer J (1975) Isochrons and phaseless sets. Journal of

Mathematical Biology 1: 259–273.

Hoppensteadt FC and Izhikevich EM (1997) Weakly Connected

Neural Networks. New York: Springer.

Izhikevich EM (1999) Weakly connected quasiperiodic oscillators,

FM interactions, and multiplexing in the brain. SIAM Journal

on Applied Mathematics 59: 2193–2223.

Izhikevich EM (2006) Dynamical Systems in Neuroscience: The

Geometry of Excitability and Bursting. Cambridge, MA: The

MIT Press.

Kuramoto Y (1984) Chemical Oscillations, Waves, and Turbulence.

New York: Springer.

Pikovsky A, Rosenblum M, and Kurths J (2001) Synchronization: A

Universal Concept in Nonlinear Science. Cambridge: Cambridge

University Press.

Strogatz SH (2000) From Kuramoto to Crawford: exploring the

onset of synchronization in populations of coupled oscillators.

Physica D 143: 1–20.

Winfree A (2001) The Geometry of Biological Time, 2nd edn.

New York: Springer.

Wheeler–De Witt Theory

J Maharana, Institute of Physics, Bhubaneswar, India

Introduction

It is recognize d that one of the outstanding problems

in modern physics is to formulate the quantum

theory of gravity, synthesizing the principles of

quantum mechanics and general theory of relativity.

The fundamental units for measuring time, length,

and energy, known as Planck time, Planck length,

and Planck energy, respectively, are defined to be

=(hG=c

)

1=2

=5.39 10

44

s, l

=(hG=c

)

1=2

1.61 10

33

cm, and m

=(hc =G)

1=2

=2.17 

5

g, in terms of the Newton’s constant, G,

velocity of light, c, and h=h=2,h being the

Planck’s constant. We may con clude, on dimen-

sional arguments, that quantum gravity effects will

play an important role when we consider physical

phenomena in the vicinity of these scales. Therefore,

when we probe very short distances, consider

collisions at Planckian energies, and envisage evolu-

tion of the universe in the Planck era, the quantum

gravity will come into play in a predominant

manner. The purpose of this article is to present an

overview of an approach to quantize Einstein’s

theory of gravity, pioneered by Wheeler and De

Witt almost four decades ago. We proceed to

recapitulate various prescriptions for quantizing

gravitation and then discuss simple derivation of

the Wheeler–De Witt (WDW) equation in general

relativity and some of its applications in the study of

quantum cosmology. There are, broadly speaking,

three different approaches to quantize gravity.

The general theory of relativity has been tested to

great degree of accuracy in the classical regime. The

geometrical description of spacetime plays a cardinal

role in Einstein’s theory. Therefore, the general

relativists emphasize the geometrical attributes of the

theory and the central role played by the spacetime

structure in their formulation of quantum theory.

It is natural to adopt a background-independent

approach. In contrast, the path followed by

quantum field theorists, where the prescription is

valid in the weak-field approximation, the theory is

quantized in a given background, usually the Min-

kowskian space. It is argued by the proponents of the

geometric approach, that the background metric

should emerge from the theory in a self-consistent

manner rather than being introduced by hand when

we quantize the theory. One of the earliest attempts

to quantize gravity was to follow the route of

canonical method. The canonical quantization

approach has many advantages. One of the impor-

tant features is that it is quite similar to the

prescriptions adopted in quantum field theory where

one uses notion of operators, commutation relations,

etc. Moreover, the subtleties encountered in quantiz-

ing gravity are transparent. Therefore, the canonical

procedure is preferred over the path-integral formula-

tion, although the latter has its own advantages too.

Another positive aspect of the canonical approach is

that the requirement of background-independent

Wheeler–De Witt Theory 453

formulation could be maintained to some extent.

Thus, there is room for exploring some of the

nonperturbative attributes of the theory. The relati-

vists favor canonical formulation, since some of the

geometrical features of general theory of relativity

could be incorporated here and be explored to see

how far the quantum theory captures such properties

of the classical theory. As we shall discuss in sequel,

some of the interesting issues of quantum cosmology

are addressed in this approach. However, there are

limitations and short comings in this formulation and

we refer the reader to the text books and review

articles for further reading and critical assessments of

canonical approach to quantize gravity.

The second approach is primarily the endeavor of

physicists who have devoted their research to

quantum field theory. Feynman’s seminal work on

quantization of gravity from this perspective has

profoundly influenced the subsequent developments.

The quantization of gravity is carried out in the

weak-field approximation such that the graviton is

identified as the fluctuation over the Minkowski

background metric. It is a massless spin-2 field as one

concludes from the properties of low-energy gravita-

tional interaction in the classical limit. Furthermore,

the gauge invariance associated with a spin-2 mass-

less field gets intimately related with invariance of

Einstein’s theory under general coordinate transfor-

mation. In this setup, the field-theoretic techniques

could be employed to quantize theory and to consider

perturbative expansions for the scattering amplitudes.

It is realized that low-energy amplitudes computed

from the massless spin-2 theory match with those

derived from the Einstein–Hilbert action in the weak-

field approximation. Furthermore, the theory is not

perturbatively renormalizable since the coupling

constant carries dimension. One of the most impor-

tant outcomes of the investigations from this per-

spective is the discovery, due to Feynman, that the

introduction of ghost fields is necessary in order to

maintain unitarity of the S-matrix when one goes

beyond the tree level. As is well known, this work has

profoundly influenced frontiers of research in physics

leading to quantization of Yang–Mills theory which,

in turn, paved way for electroweak theory and the

QCD. It is worthwhile to mention in passing that the

quantum phenomena associated with gravity in the

nonperturbative regime cannot be addressed in this

framework.

In recent years, superstring theory has been at the

center stage in order to provide a unified theory of

fundamental interactions. It is postulated that all

elementary constituents of matter and the carriers of

the interactions such as gauge bosons and graviton

are excitations of one-dimensional extended objects:

the string s. The superstring theories are perturba-

tively consistent in critical ten dimensions. The

closed-superstring spectrum contains a spin-2 mass-

less state which is identified to be the graviton. It is

well known that perturbative computation of pro-

cesses involving graviton turn out to be finite.

Moreover, the Einstein–Hilbert term appears natu-

rally when one derives the string effective action.

Therefore, it is expected that string theory will be

able to provide answers to questions related to

quantum gravity. Indeed, the theory has met with

success in resolving some important issues. We note

that cosmological scenario has been discussed in the

string theory framework and the WDW equation

has played an important role in study of quantum

string cosmology. We shall comment on this aspect

towards the end of this article.

The Canonical Structure of Einstein

Gravity

The Einstein–Hilbert action is

S ¼

16G

ﬃﬃﬃﬃﬃﬃﬃ

g

xðR  2Þ½1

where R is the Ricci scalar derived from the metric,



, and  is the cosmological constant. The field

equations are derived from the action by the

standard variati onal technique. Note that R involves

second derivative of the metric. If we have compac t

manifolds with boundary @M such that variations of

the metric vanish on the bounda ry and the normal

derivatives do not, it is necessary to add a surface

term to this action. The exact form of this term will

be discussed later. The Einstein’s theory of gravita-

tion is manifestly covariant. The associated action

[1] is invariant under genera l coordinate transforma-

tions: under x



0

(x),

0

ðx

Þ¼g



ðxÞ

0



0



½2

Therefore, we expect that the theory will be

endowed with constraints expressed in terms of the

canonical variables. One can implement general

coordinate transformations so that there are only

two pairs of canonical phase-space variables on a

spacelike hypersurface. In other words, from physi-

cal considerations, graviton has only two polariza-

tions whereas the metric has ten components.

Therefore, the two physical degrees of freedom can

be obtained using the freedom of choosing the

‘‘gauge’’ transformations in this context. It is

desirable to identify the constraints and analyze

their structure, most appropriately in Dirac’s

454 Wheeler–De Witt Theory

formalism, and to quantize the theory canonically as

the next step. This is the path we intend to follow in

order to arrive at the WDW equation.

The Classical Constraints

The Hamiltonian approach is most appropriate to

employ the constraint formalism due to Dirac. We

recall that the Lagrangian formulation is manifestly

covariant as is reflected in the field equations;

whereas the spacetime covariance is lost in the

passage to the Hamiltonian approach. Furthermore,

the spatial components of the metric are the

dynamical degrees of freedom. We adopt the

formalism introduced by Arnowitt, Deser, and

Misner (ADM) for the so-called 3 þ 1 split of the

hyperbolic Riemannian spacetime metric, g



. One

introduces the lapse function, N

, and the shift

function, N

. We suppress the factors of 1=16G,

etc., for the time being for the general discussions

and shall reintroduce them later. The family of

spacelike hypersurfaces, 

, are constructed, with

metric h

induced on it. Here t is a timelike

parameter, param etrize 

. The distance between

points on two neighbor ing hyp ersurface, 

and



tþdt

, with coordinates (t, x

)and(t þ dt, x

þ dx

respectively, is given by

¼ðN

þ h

ðN

dt þ dx

ÞðN

dt þ dx

¼ g







½3

The indices of tensors defined on 

are raised and

lowered by h

and its inverse h

. The relations

between the components of g



and N

, N

, h

can

be obtained easily,

¼ h

ðN

; g

¼ h

½4

The above relations can be inverted to give

¼ h

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

g

½5

The relations between spatial components, g

,ofg



and h

and some other useful relations are listed

below for later conveniences:

¼ h



ðN

ﬃﬃﬃﬃﬃﬃﬃ

g

¼ N

ﬃﬃﬃ

ðN

½6

Note that (N

, N

) are introduced to specify the

deformation of the hypersurface and therefore, the

evolution equations through the Hamiltonian will

not determine them; they are arbitrary functions.

Consequently, [4] implies that g

and g

will enter

the Hamiltonian as arbitrary functions. As alluded

to above, h

and their conjugate momenta 

are the

dynamical degrees of freedom. We may choose

, N

) = N



and h

as independen t variables

rather than (g

, g

) = g

0

and h

for convenience

and go back to the other set of variables through [4]

and [5] if we desire. Let 



be canonically conjugate

momenta to N



, then it is obvious that a Lagrangian

multiplier, 



, is necessary so that . term has to be

supplemented to the Hamiltonian due to the

arbitrariness of N



. We remind the reader that in

electrodynamics an analogous situation arises while

analyzing its canonical structure – local gauge

symmetry plays a crucial role there. It is obvious

that the generic form of the Hamiltonian is (w e shall

introduce 1=16G, etc., later)

H ¼

½h

;

þN

½h

;

þ:



½7

From the perspective of constraint analysis, it is

natural that 



0 appears as a first-class constraint

as they are multiplied by arbitrary functions. More-

over, this constraint must hold good under the

deformation of the surface which implies {



,H}

must vanish weakly leading to H



0. As a

consistency requirement, these must be first-class

constraints if N



are to be arbitrary functions. We

identify that 



0 and H



0 are the primary and

secondary constraints, respectively. Thus far, we

have discussed the case for pure gravity; the

presence of matter fields in the full action modifies

the treatment appropriately.

Let us analyze the structure of the constraints for

the Einstein–Hilbert action [1]. For a compact

manifold with boundary @M, we have to add the

surface term which takes the form:

8G

ﬃﬃﬃ

Here K stands for the trace of the extrinsic curvature

of the boundary 3-surface and h = det h

; note that

is the induced metric on the 3-surface. If we

include mat ter fields, the correspon ding action is to

be taken into account. Once we make the 3 þ 1 split

of the metric, the action assumes the following form:

S ¼

16G

x dtN

ﬃﬃﬃ

 K

 K

R  2



½8

where



þ D



½9

Wheeler–De Witt Theory 455

Here D

represents covariant derivative of N

with

the connections computed from h

and

R is

curvature of the 3-surface. The canonical momenta

are



ﬃﬃﬃ

16G

 h



½10

and we can invert this relation to get

¼

16G

ﬃﬃﬃ









The Hamiltonian form of action is give n by

x dt



 N



½11

Notice that [8] does not involve time derivatives of

and N

, their corresponding canonical momenta

vanish.



 0;

 0 ½12

as expected from our earlier discussions about the

role of N



. A straightforward constraint analysis

leads to the pair of constraints

¼2D



 0 ½13

16G

ﬃﬃﬃ









ﬃﬃﬃ

16G

R  0 ½14

We mention in passing that the above constraint

equations get modified in the presence of matter

fields in the theory. This is relevant. The WDW

equation plays an important role in quantum

cosmology to describe the evolution of the universe

in early epochs and the equation is studied in the

presence of a generic matter content, that is, a scalar

field with potential. The constraint equations [13]

and [14] modify to

¼H

þH

matter

 0 ½15

¼H

þH

matter

 0 ½16

The Algebra of Constraints

In order to compute the classical Poisson bracket

algebra of the constraints [13] and [14], we use the

canonical Poisson bracket relations for the phase-

space variables on 

ðxÞ; h

ðx

Þg ¼ 0 ½17

f

ðxÞ;

ðx

Þg ¼ 0 ½18

ðxÞ;

ðx

Þg ¼ 

ði



jÞ

 ðx; x

Þ½19

Thus, Poisson brackets among the constraints [13]

and [14] are

ðxÞ; H

ðx

Þg ¼ H

ðxÞ@

ð x; x

þH

ðx; x

Þ½20

ðxÞ; H

ðx

Þg ¼ H

ðxÞ@

ð x; x

Þ½21

ðxÞ; H

ðx

Þg ¼ h

ðxÞH

ðxÞ@

ðx ; x

 h

ðx

ÞH

ðx

Þ@

ðx; x

Þ½22

When we resort to canonical quantization, the

starting point is the Hamiltonian action in the first-

order formalism, where the canonical variables are

subjected to the constraints [13] and [14] in terms of

and H

satisfying the algebra given by [20]–[22].

One encounters a number of important issues while

proceeding to canonically quantize the theory. We

shall mention only a few of them in what follows. It

is important to address issues related to the role of

the constraints in the quantized theory and how to

deal with the Lagrange multipliers N

and N

simple proposal is to solve the constraints at the

classical level and identify the physical degrees of

freedom and quantize the theory subsequently.

There are four constraints (first class), H

, H

therefore, out of the 12 phase-space variables,

, 

), only eight are independent. We need to

supply four gauge conditions in order to render the

theory (classically) solvable. Thus, we are left with

four physical degrees of freedom in the Hamiltonian

phase space and we can quantize them. The

implementation of this idea is easier said than

done. One obstacle is that the constraint s cannot

be solved in a closed form in this formalism. If we

fix a gauge and quantize the theory, we obviously

break the gauge invariance. It is essential to show,

subsequently, that a ll physically observable quanti-

ties are independent of the gauge choice. Another

criticism of this formalism is that we already get rid

of some of the components of the metric. Therefore,

the spirit of the general theory of relativity, which is

based on the geometrical structure of spacetime, is

somewhat diluted. There are other suggestions

where h

and their conjugate momenta are elevated

to quantum status before supplying the gauge

conditions. The issues of gauge fixing and dealing

with the constraints are addressed at the quantum

level. We replace the canonical Poisson bracket

456 Wheeler–De Witt Theory

algebra by the canonical commutators and procee d

further. The momentum operat or assumes the form

^

¼ih



h

and the wave functional depends on h

that is, [h].

There are many technical problems related to the

properties of the states and we shall not deal with

them due to limitations of space. It is essential to

discuss the role of the constraints in the quantum

theory. We demand that the quantum constraints

annihilate the physical states (recall the Gauss law

constraint in gauge theories). However, the issue of

operator ordering is to be dealt with which in turn is

connected with the Hermiticity properties of the

quantum constraints. The Hamiltonian constraint

 0 (henceforth denoted as H and defined as the

Hamiltonian) is a product of the metric

and 

There is certain ambiguity in defining the constraint.

Therefore, one has to choose a convention.

The condition that the Hamiltonian,

, consisting

of gravitational and matter components, annihilates

the state is expressed as

 ¼ 0 ½23

When we ado pt coordinate representation for 

the above equation takes the form

16GG

ijkl



h



h





ﬃﬃﬃ

16G

R  2ÞþH

matter

½h;¼0 ½24

This is the celebrated WDW equation. Here we have

considered a simple case where matter Hamiltonian

density generically contains a single scalar field, ,

and therefore  is functional of 3-metric on 

and

. G

ijkl

is the De Witt metric in the superspace:

ijkl

ﬃﬃﬃ

ðh

hjl þ h

 h

Þ½25

Remarks The space of all 3-metrics and the scalar

field (h

, ), on 

, for the description of classical

evolutions is called the superspace (no connection

with the superspace of supersymmetry). Thus,

[h

, ] is a functional on superspace. Furthermore,

 carries no explicit dependence on t. This is a

consequence of the fact that ‘‘time’’ plays the role of

a parameter in the general theory of relativity, thus

the dynamical variables h

and  already provide the

evolutionary processes although t does not make its

appearance. As mentioned earlier, we always discuss

the case when 

is compact. Another point to note

is that the quantum momentum constraint,

,asan

operator annihilates the wave function which is a

statement of the quantum-mechanical invariance of

the theory under three-dimensional diffeomorph-

isms. However, the WDW equation conveys invar-

iance of the theory under reparametrization,

although careful analysis is necessary to prove this

point. Now we proceed to discuss the solutions of

the WDW equation.

WDW Equation and the Solutions

It is recognized that the WDW equation [24] is a

second-order hyperbolic functional differential equa-

tion and naturally it has enormous number of

solutions. Therefore, if we want the WDW equation

to have any predictive power, it is necessary to

introduce boundary conditions. One of the possible

choice is to specify the wave function on the

boundary of the superspace. Indeed, the central

issue of quantum co smology is about the choice of

various boundary conditions which has been an

important topic of debates. This point will be briefly

discussed later. Notice that the boundary condition

has to be introduced keeping in mind how the

universe is expected to behave as it evolves. There is

a proposition that the boundary condition for the

quantum evolution of the universe be given the

status of a physical law. Therefore, the role of the

wave functional, [h

(x), (x), B], its evolution, and

interpretation are central to the development of

quantum cosmology. Thus,  represents the ampli-

tude for the universe to hav e h

(x) on the 3-surface,

B, and matter field (x). It is argued that path-

integral formalism should be adopted as an alt er-

native to the canonical presc ription to solve for the

wave function, rather the transition amplitude,

satisfying the WDW equation. Her e the first step is

to define the Euclidean version of the gravitational

action keeping in mind the subtleties. As is well

known, we deal with propagator (or transition

amplitude) in the path-integral approach where the

functional integral is carried out over a set of

4-metrics and matter fields with Euclidean action

inside the integral acting as the weight factor. We

recall that while formulating quantum mechanics in

the path-integral approach, we sum over all possible

paths in the functional integral. However, in the

semiclassical approximation, the amplitude is domi-

nated by the action corresponding to the classical

path and we approximate the wave function as 

(i=h)S

and it gets modified appropriately in the

Euclidean formulation. In this background, we

briefly discuss how the wave function of the

universe is obtained in the path-integral formalism.

Wheeler–De Witt Theory 457