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

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result rely on relativistic causality and the uncer-

tainty principle. The creation of the new state

happens over timescales of order 1/M. Before that

well-defined event, the colliding particles are

approaching at nearly the speed of light, and hence

cannot affect the distributions of each others’

partons. After the new particle is created, the

fragments of the hadrons recede from each other,

and the subsequent time development, when

summed over all possible final states that include

the heavy particle, is finite in perturbation theory as

a direct result of the unitarity of QCD.

Structure of Hadronic Final States

A wide range of semi-inclusive cross sections are

defined by measuring properties of final states that

depend only on the flow of energy, and which bring

QCD perturbation theory to the threshold of

nonperturbative dynamics. Schematically, for a

state N = jk

...k

i, we define S(N) =

s(

where s() is some smooth function of directions.

We generalize the e



annihilation case above, and

define a cross section in terms of a related, but

highly nonlocal, matrix element,

dðQÞ



x e

iQx







ð0Þ





sðÞEðÞS





ðxÞ





½25

where 

is a zeroth-order cross section, and where

E is an operator at spatial infinity, which measures

the energy flow of any state in direction : E()

...k

i= (1=Q)



( 

). This may seem a

little complicated, but like the total annihilation cross

section, the only dimensional scale on which it

depends is Q. The operator E can be defined in a

gauge-invariant manner, through the energy–momen-

tum tensor for example, and has a meaning indepen-

dent of partonic final states. At the same time, this

sort of cross section may be im plemented easily in

perturbation theory, and like the total annihilation

cross section, it is infrared safe. To see why, notice

that when a massless (k

= 0) particle decays into two

particles of momenta xk and (1 x)k (0  x  1), the

quantity S is unchanged, since the sum of the new

energies is the same as the old. This makes the

observable S(N) insensitive to processes at low

momentum transfer.

For the case of leptonic annihilation, the lowest-

order perturbative contribution to energy flow

requires no powers of 

, and consists of an

oppositely moving quark and antiquark pair. Any

measure of energy flow that includes these config-

urations will dominate over correlations that require



corrections. As a result, QCD predicts that in

most leptonic annihilation events, energy will flow

in two back-to-back collimat ed sets of particles,

known as ‘‘jets.’’ In this way, quarks and gluons are

observed clearly, albeit indirectly.

With varyi ng choices of S, many properties of

jets, such as their distributions in invariant mass,

and the probabilities and angular distributions of

multijet events, and even the energy dependence of

their particle multiplicities, can be computed in

QCD. This is in part because hadronization is

dominated by the production of light quarks,

whose production from the vacuum requires very

little momentum transfer. Paradoxically, the very

lightness of quarks is a boon to the use of

perturbative metho ds. All these considerations can

be extended to hadronic scattering, and jet and other

semi-inclusive properties of final states also com-

puted and compared to experiment.

Conclusions

QCD is an extremely broad field, and this article has

hardly scratched the surface. The relation of QCD-

like theories to supersymmetric and string theories,

and implications of the latter for confinement and

the computati on of higher-order perturbative ampli-

tudes, have been some of the most exciting devel-

opments of recent years. As another example, we

note that the reduction of the heavy-quark propa-

gator to a nonabelian phase, noted in our discussion

of confinement, is related to additional symmetries

of heavy quarks in QCD, with many consequences

for the analysis of their bound states. Of the

bibliography given below, one may mention the

four volumes of Shifman (2001, 2002), which

communicate in one place a sense of the sweep of

work in QCD.

Our confidence in QCD as the correct description of

the strong interactions is based on a wide variety of

experimental and observational results. At each stage in

the discovery, confirmation, and exploration of QCD,

the mathematical analysis of relativistic quantum field

theory entered new territory. As is the case for gravity or

electromagnetism, this period of exploration is far from

complete, and perhaps never will be.

See also: AdS/CFT Correspondence; Aharonov–Bohm

Effect; BRST Quantization; Current Algebra; Dirac

Operator and Dirac Field; Euclidean Field Theory;

Effective Field Theories; Electroweak Theory; Lattice

Gauge Theory; Operator Product Expansion in Quantum

Field Theory; Perturbation Theory and its Techniques;

Perturbative Renormalization Theory and BRST;

Quantum Field Theory: A Brief Introduction; Random

152 Quantum Chromodynamics

Matrix Theory in Physics; Renormalization: General

Theory; Scattering in Relativistic Quantum Field Theory:

Fundamental Concepts and Tools; Scattering,

Asymptotic Completeness and Bound States;

Seiberg–Witten Theory; Standard Model of Particle

Physics.

Further Reading

Bethke S (2004) 

at Zinnowitz, 2004. Nuclear Physics

Proceeding Supplements 135: 345–352.

Brodsky SJ and Lepage P (1989) Exclusive processes in quantum

chromodynamics. In: Mueller AH (ed.) Perturbative Quantum

Chromodynamics. Singapore: World Scientific.

Collins JC, Soper DE, and Sterman G (1989) Factorization. In:

Mueller AH (ed.) Perturbative Quantum Chromodynamics.

Singapore: World Scientific.

Dokshitzer Yu L and Kharzeev DE (2004) Gribov’s conception of

quantum chromodynamics. Annual Review of Nuclear and

Particle Science 54: 487–524.

Dokshitzer Yu L, Khoze V, Troian SI, and Mueller AH (1988)

QCD coherence in high-energy reactions. Reviews of Modern

Physics 60: 373.

Eidelman S et al. (2004) Review of particle physics. Physics

Letters B 592: 1–1109.

Ellis RK, Stirling WJ, and Webber BR (1996) QCD and Collider

Physics, Cambridge Monographs on Particle Physics, Nuclear

Physics and Cosmology, vol. 8. Cambridge: Cambridge

University Press.

Greensite J (2003) The confinement problem in lattice gauge

theory. Progress in Particle and Nuclear Physics 51: 1.

Mandelstam S (1976) Vortices and quark confinement in

nonabelian gauge theories. Physics Reports 23: 245–249.

Muta T (1986) Foundations of Quantum Chromodynamics.

Singapore: World Scientific.

Neubert H (1994) Heavy quark symmetry. Physics Reports 245:

259–396.

Polyakov AM (1977) Quark confinement and topology of gauge

groups. Nuclear Physics B 120: 429–458.

Schafer T and Shuryak EV (1998) Instantons and QCD. Reviews

of Modern Physics 70: 323–426.

Shifman M (ed.) (2001) At the Frontier of Particle Physics:

Handbook of QCD, vols. 1–3. River Edge, NJ: World Scientific.

Shifman M (ed.) (2002) At the Frontier of Particle Physics:

Hand book of QCD, vol. 4. River Edge, NJ: World Scientific.

Sterman G (1993) An Introduction to Quantum Field Theory.

Cambridge: Cambridge University Press.

’t Hooft G (1977) On the phase transition towards permanent

quark confinement (1978). Nuclear Physics B 138: 1.

’t Hooft G (ed.) (2005) Fifty Years of Yang–Mills Theories.

Hackensack: World Scientific.

Weinberg S (1977) The problem of mass. Transactions of the

New York Academy of Science 38: 185–201.

Wilson KG (1974) Confinement of quarks. Physical Review D

10: 2445–2459.

Quantum Cosmology

M Bojowald, The Pennsylvania State University,

University Park, PA, USA

Introduction

Classical gravity, through its attract ive nature, leads

to a high curvat ure in important situations. In

particular, this is realized in the very early universe

where in the backward evolution energy densities

are growing until the theory breaks down. Mathe-

matically, this point appears as a singularity where

curvature and physical quantities diverge and the

evolution breaks down. It is not possible to set up an

initial-value formulation at this place in order to

determine the further evolution.

In such a regime, quantum effects are expected to

play an important role and to modify the classical

behavior such as the attractive nature of gravity or the

underlying spacetime structure. Any candidate for

quantum gravity thus allows us to reanalyze the

singularity problem in a new light which implies the

tests of the characteristic properties of the respective

candidate. Moreover, close to the classical

singularity, in the very early universe, quantum

modifications will give rise to new equations of

motion which turn into Einstein’s equations only on

larger scales. The analysis of these equations of

motion leads to new classes of early universe

phenomenology.

The application of quantum theory to cosmology

presents a unique problem with not only mathema-

tical but also many conceptual and philosophical

ramifications. Since by definition there is only one

universe which contains everything accessible, there

is no place for an outsid e observer separate from the

quantum system. This eliminates the most straight-

forward interpretations of quantum mechanics and

requires more elabor ate, and sometimes also more

realistic, constructions such as decoherence. From

the mathematical point of view, this situation is

often expected to be mirrored by a new type of

theory which does not allow one to choose initial or

boundary conditions separa tely from the dynamical

laws. Initial or boundary conditions, after all, are

meant to specify the physical system prepared for

observations which is impossible in cosmology.

Since we observe only one universe, the expectation

goes, our theories should finally present us with only

Quantum Cosmology 153

one, unique solution without any freedom for

further conditions. This solution then contains all

the information about observations as well as

observers. Mathematically, this is an extremely

complicated problem which has received only scant

attention. Equations of motion for quantum cosmol-

ogy are usually of the type of partial differential or

difference equations such that new ingredients from

quantum gravity are needed to restrict the large

freedom of solutions.

Minisuperspace approximation

In most investigations, the problem of applying full

quantum gravity to cosmology is simplified by a

symmetry reduction to homogeneous or isotropic

geometries. Originally, the reduction was performed

at the classical level, leaving in the isotropic case

only one gravitational degree of freedom given by

the scale factor a. Together with homogeneous

matter fields, such as a scalar , there are then

only finitely many degrees of freedom which one can

quantize using quantum mechanics. The classical

Friedmann equation for the evolution of the scale

factor, depending on the spatial curvature k = 0or

1, is then quantized to the Wheeler–DeWitt

equation, commonly written as

‘

x

 ka



ða;Þ

¼

8G

matter

ðaÞ ða;Þ½1

for the wave function (a, ). The matter Hamilto-

nian

matter

(a), such as

matter

ðaÞ¼

h

3

@

þ a

VðÞ½2

is left unspecified here, and x parametrizes factor

ordering ambiguities (but not completely). The

Planck length ‘

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

8Gh

is defined in terms of

the gravitational constant G and the Planck

constant h.

The central conceptual issue then is the generality

of effects seen in such a symmetric model and its

relation to the full theory of quantum gravity. This

is completely open in the Wheeler–DeWitt form

since the full theory itself is not even known. On the

other hand, such relations are necessary to value any

potential physical statement about the origin and

early history of the universe. In this context,

symmetric situations thus present models, and the

degree to which they approximate full quantum

gravity remains mostly unknown. There are exam-

ples, for instance, of isotropic models in anisotropic

but still homogeneous models, where a minisuper-

space quantization does not agree at all with the

information obtained from the less symmetric

model. However, often those effects already have a

classical analog such as instability of the more

symmetric solutions. A wider investigation of the

reliability of models and when correction terms

from ignored degrees of freedom have to be included

has not been done yet.

With candidates for quantum gravity being

available, the current situation has changed to

some degree. It is then not only possible to reduce

classically and then simply use quantum

mechanics, but also perform at least some of the

reduction steps at the quantum level. The relation

to models is then much clearer, and consistency

conditions which arise in the full theory can be

made certain to be observed. Moreover, relations

between models and the full theory can be studied

to elucidate the degree of approximation. Even

though new techniques are now available, a

detailed investigation of the degree of approxima-

tion given by a minisuperspace model has not been

completed due to its complexity.

This program has mostly been developed in the

context of loop quantum gravity, where the specia-

lization to homogeneous models is known as loop

quantum cosmology. More specifically, symmetries

can be introduced at the level of states and basic

operators, where symmetric states of a model are

distributions in the full theory, and basic operators

are obtained by the dual action on those distribu-

tions. In such a way, the basic representation of

models is not assumed but derived from the full

theory where it is subject to much stronger

consistency conditions. This has implications even

in homogeneous models with finitely many degrees

of freedom, despite the fact that quantum mechanics

is usually based on a unique representation if the

Weyl operators e

isq

and e

itp

for the variables q and p

are represented weakly continuously in the real

parameters s and t.

The continuity condition, however, is not neces-

sary in general, and so inequivalent representations

are possible. In quantum cosmology this is indeed

realized, where the Wheeler–DeWitt representation

assumes that the conjugate to the scale factor,

corresponding to extrinsic curvature of an isotropic

slice, is represented through a continuous Weyl

operator, while the representation derived for loop

quantum cosmology shows that the resulting opera-

tor is not weakly continuous. Furthermore, the scale

factor has a continuous spectrum in the Wheeler–

DeWitt representation but a discrete spectrum in the

loop representation. Thus, the underlying geometry

154 Quantum Cosmology

of space is very different, and also evolution takes

a new form, now given by a difference equation of

the type

ðV

þ5

 V

þ3

Þe

þ4

ðÞ

ð2 þ k

ÞðV

þ1

 V

1



ðÞ

þðV

3

 V

5

Þe

ik

4

ðÞ

¼

G‘

matter

ðÞ



ðÞ½3

in terms of volume eigenvalues V



= (‘

jj=6)

3=2

For large  and smooth wave functions, one can see

that the difference equation reduces to the

Wheeler–DeWitt equation with jj/a

to leading

order in derivatives of . At small ,closetothe

classical singularity, however, both equations have

very different properties and lead to different

conclusions. Moreover, the prominent role of

difference equations leads to new mathematical

problems.

This difference equation is not simply obtained

through a discretization of [1], but derived from a

constraint operator constructed with methods from

full loop quantum gravity. It is, thus, to be regarded

as more fundamental, with [1] emerging in a

continuum limit. The structure of [3] depends on

the properties of the full theory such that its

qualitative analysis allows conclusions for full

quantum gravity.

Applications

Traditionally, quantum cosmology has focused on

three main conceptual issues:



the fate of classical singularities,



initial conditions and the ‘‘prediction’’ of inflation

(or other early universe scenarios), and



arrow of time and the emergence of a classical

world.

The first issue consists of several subproblems since

there are different aspects to a classical singularity.

Often, curvature or energy densities diverge and one

can expect quantum gravity to provide a natural

cutoff. More importantly, however, the classical

evolution breaks down at a singularity, and quan-

tum gravity, if it is to cure the singularity problem,

has to provide a well-defined evolution which does

not stop. Initial conditions are often seen in relation

to the singularity problem since early attempts tried

to replace the singularity by choosing appropriate

conditions for the wave function at a = 0. Different

proposals then lead to different solutions for the

wave function, whose dependence on the scalar 

can be used to determine its probability distribution

such as that for an inflaton. Since initial conditions

often provide special properties early on, the

combination of evolution and initial conditions has

been used to find a possible origin of an arrow

of time.

Singularities

While classical gravity is based on spacetime

geometry and thus metric tensors, this structure is

viewed as emergent only at large scales in canonical

quantum gravity. A gravitational system, such as a

whole universe, is instead described by a wave

function which, at best, yields expectation values for

a metric. The singularity problem thus takes a

different form since it is not metrics which need to

be continued as solutions to Einstein’s field equa-

tions but the wave function describing the quantum

system. In the strong curvature regime around a

classical singularity, one does not expect classical

geometry to be applicable, such that classical

singularities may just be a reflection of the break-

down of this picture, rather than a breakdown of

physical evolution. Nevertheless, the basic feature of

a singularity as presenting a boundary to the

evolution of a system equally applies to the quantum

equations. One can thus analyze this issue, using

new properties provided by the quantum evolution.

The singularity issue is not resolved in the

Wheeler–DeWitt formulation since energy densities,

with a being a multiplication operator, diverge and

the evolution does not continue anywhere beyond

the classical singularity at a = 0. In some cases one

can formally extend the evolution to negative a, but

this possibility is not generic and leaves open what

negative a means geometrically. This is different in

the loop quantization: here, the theory is based on

triad rather than metric variables. There is thus a

new sign factor corresponding to spatial orientation,

which implies the possibility of negative  in the

difference equation. The equation is then defined on

the full real line with the classical singularity  = 0

in the interior. Outside  = 0, we have positive

volume at both sides, and opposite orientations.

Using the difference equation, one can then see that

the evolution does not break down at  = 0,

showing that the quantum evolution is singularity

free.

For the example [3] shown here, one can follow

the evolution, for instance, backward in internal

time , starting from initial values for at large

positive . By successively solving for

4

, the wave

function at lower  is determined. This goes on in

this manner only until the coefficient V

3

 V

5

4

vanishes, which is the case if and only if  = 4.

Quantum Cosmology 155

The value

of the wave function exactly at the

classical singularity is thus not determined by initial

data, but one can easily see that it completely drops

out of the evolution. In fact, the wave function at all

negative  is uniquely determined by initial values at

positive . Equation [3] corresponds to one parti-

cular ordering, which in the Wheeler–DeWitt case is

usually parametrized by the parameter x (although

the particular ordering obtained from the continuum

limit of [3] is not contained in the special family

[1]). Other nonsingular orderings exist, such as that

after symmetrizing the constraint operator, in which

case the coefficients never become 0.

In more complicated systems, this behavior is

highly nontrivial but still known to be realized in a

similar manner. It is not automatic that the internal

time evolution does not continue since even in

isotropic models one can easily write difference

equations for which the evolution breaks down.

That the most natural orderings imply nonsingular

evolution can be taken as a support of the general

framework of loop quantum gravity. It should also

be noted that the mechanism described here,

providing essentially a new region beyond a classical

singularity, presents one mechanism for quantum

gravity to remove classical singularities, and so far

the only known one. Nevertheless, there is no claim

that the ingredients have to be realized in any

nonsingular scenario in the same manner. Different

scenarios can be imagined, depending on how

quantum evolution is understood and what the

interpretation of nonsingular behavior is. It is also

not claimed that the new region is semiclassical in

any sense when one looks at it at large volume. If

the initial values for the wave function describe a

semiclassical wave packet, its evolution beyond the

classical singularity can be deformed and develop

many peaks. What this means for the re-emergence

of a semiclassical spacetime has to be investigated in

particular models, and also in the context of

decoherence.

Initial Conditions

Traditional initial conditions in quantum cosmology

have been introduced by physical intuition. The

main mathematical problem, once such a condition

is specified in sufficient detail, then is to study well-

posedness, for instance, for the Wheeler–DeWitt

equation. Even formulating initial conditions

generally, and not just for isotropic models, is

complicated, and systematic investigations of the

well-posedness have rarely been undertaken. An

exception is the historically first such condition,

due to DeWitt, that the wave function vanishes at

parts of minisuperspace, such as a = 0 in the

isotropic case, corresponding to classical singulari-

ties. This condition, unfortunately, can easily be

seen to be ill posed in anisotropic models where in

general the only solution vanishes identically. In

other models, lim

a !0

(a) does not even exist.

Similar problems of the generality of conditions

arise in other scenarios. Most well known are the

no-boundary and tunneling proposal where initial

conditions are still imposed at a = 0, but with a

nonvanishing wave function there.

This issue is quite different for difference equa-

tions since at first the setup is less restrictive: there

are no continuity or differentiability conditions for a

solution. Moreover, oscillations that become arbi-

trarily rapid, which can be responsible for the

nonexistence of lim

a !0

(a), cannot be supported

on a discrete lattice. It can then easily happen that a

difference equation is well posed, while its con-

tinuum limit with an analogous initial condition is

ill posed. One example are the dynamical initial

conditions of loop quantum cosmology which arise

from the dynamical law in the following way: the

coefficients in [3] are not always nonzero but vanish

if and only if they are multiplied with the value of

the wave function at the classical singularity  = 0.

This value thus decouples and plays no role in the

evolution. The instance of the difference equation

that would determine

, for example, the equation

for  = 4 in the backward evolution, instead implies

a condition on the previous two values,

and

in the example. Since they have already been

determined in previous iteration steps, this translates

to a linear condition on the initial values chosen. We

thus have one example where indeed initial condi-

tions and the evolution follow from only one

dynamical law, which also extends to anisotropic

models. Without further conditions, the initial-value

problem is always well posed, but may not be

complete, in the sense that it results in a unique

solution up to norm. Most of the solutions,

however, will be rapidly oscillating. In order to

guarantee the existence of a continuum approxima-

tion, one has to add a condition that these

oscillations are suppressed in large volume regimes.

Such a condition can be very restrictive, such that

the issue of well-posedness appears in a new guise:

nonzero solutions do exist, but in some cases all of

them may be too strongly oscillating.

In simple cases, one can use generating function

techniques advantageously to study oscillating solu-

tions, at least if oscillations are of alternating nature

between two subsequent levels of the difference

equation. The idea is that a generating function

G(x) =

has a stronger pole at x = 1if

156 Quantum Cosmology

is alternating compared to a solution of constant

sign. Choosing initial conditions which reduce the

pole order thus implies solutions with suppressed

oscillations. As an example, we can look at the

difference equation

nþ1



n1

¼ 0 ½4

whose generating function is

GðxÞ¼

x þ

ð1 þ 2xð1  logð1  xÞÞÞ

ð1 þ xÞ

½5

The pole at x = 1 is removed for initial values

(2 log 2  1) which corresponds to nonoscil-

lating solutions. In this way, analytical expressions

can be used instead of numerical attempts which

would be sensitive to rounding errors. Similarly, the

issue of finding bounded solutions can be studied by

continued fraction methods. This illustrates how an

underlying discrete structure leads to new questions

and the application of new techniques compared to

the analysis of partial differential equations which

appear more commonly.

More General Models

Most of the time, homogeneous models have been

studied in quantum cosmology since even formulat-

ing the Wheeler–DeWitt equation in inhomogeneous

cases, the so-called midisuperspace models, is

complicated. Of particular interest among homo-

geneous models is the Bianchi IX model since it has

a complicated classical dynamics of chaotic beha-

vior. Moreover, through the Belinskii–Khalatnikov–

Lifschitz (BKL) picture, the Bianchi IX mixmaster

behavior is expected to play an important role even

for general inhomogeneous singularities. The classi-

cal chaos then indicates a very complicated

approach to classical singularities, with structure

on arbitrarily small scales.

On the other hand, the classical chaos relies on a

curvature potential with infinitely high walls, which

can be mapped to a chaotic billiard motion. The

walls arise from the classical divergence of curva-

ture, and so quantum effects have been expected to

change the picture, and shown to do so in several

cases.

Inhomogeneous models (e.g., the polarized

Gowdy models) have mostly been studied in cases

where one can reformulate the problem as that of a

massless free scalar on flat Minkowski space. The

scalar can then be quantized with familiar techni-

ques in a Fock space representation, and is related to

metric components of the original model in rather

complicated ways. Quantization can thus be per-

formed, but transforming back to the metric at the

operator level and drawing conclusions is quite

involved. The main issue of interest in the recent

literature has been the investigation of field theory

aspects of quantum gravity in a tractable model. In

particular, it turns out that self-adjoint Hamilto-

nians, and thus unitary evolution, do not exist in

general.

Loop quantizations of inhomogeneous models are

available even in cases where a reformulation such

as a field theory on flat space does not exist, or is

not being made use of to avoid special gauges. This

is quite valuable in order to see if specific features

exploited in reformulations lead to artifacts in the

results. So far, the dynamics has not been investi-

gated in detail, even though conclusions for the

singularity issue can already be drawn.

From a physical perspective, it is most important

to introduce inhomogeneities at a perturbative level

in order to study implications for cosmological

structure formation. On a homogeneous back-

ground, one can perform a mode decomposition of

metric and matter fields and quantize the homo-

geneous modes as well as amplitudes of higher

modes. Alternatively, one can first quantize the

inhomogeneous system and then introduce the mode

decomposition at the quantum level. This gives rise

to a system of infinitely many coupled equations of

infinitely many variables, which needs to be trun-

cated, for example, for numerical investigations. At

this level, one can then study the question to which

degree a given minisuperspace model presents a

good approximation to the full theory, and where

additional correction terms should be introduced. It

also allows one to develop concrete models of

decoherence, which requires a ‘‘bath’’ of many

weakly interacting degrees of freedom usually

thought of as being provided by inhomogeneities in

cosmology, and an understanding of the semiclassi-

cal limit.

Interpretations

Due to the complexity of full gravity, investigations

without symmetry assumptions or perturbative

approximations usually focus on conceptual issues.

As already discussed, cosmology presents a unique

situation for physics since there cannot be any

outside observer. While this fact has already

implications on the interpretation of observations

at the classical level, its full force is noticed only in

quantum cosmology. Since some traditional inter-

pretations of quantum mechanics require the role of

Quantum Cosmology 157

observers outside the quantum system, they do not

apply to quantum cosmology.

Sometimes, alternative interpretations such as

Bohm theory or many-world scenarios are cham-

pioned in this situation, but more conventional

relational pictures are most widely adopted. In

such an interpretation, the wave function yields

relational probabilities between degrees of free-

dom rather than absolute probabilities for mea-

surements done by an outside observer. This has

been used, for instance, to determine the prob-

ability of the right initial conditions for inflation,

but it is marred by unresolved interpretational

issues and still disputed. These problems can be

avoided by using effective equations, in analogy

to an effective action, which modify classical

equations on small scales. Since the new equa-

tions are still of classical type, that is, differential

equations in coordinate time, no interpretational

issues arise at least if one stays in semiclassical

regimes. In this manner, new inflationary scenar-

ios motivated from quantum cosmology have

been developed.

In general, a relational interpretation, though

preferable conceptually, leads to technical

complications since the situation is much more

involved and evolution is not easy to disentangle.

In cosmology, one often tries to single out one

degree of freedom as internal time with respect to

which evolution of other degrees of freedom is

measured. In homogeneous models, one can

simply take the volume as internal time, such as

a or  earlier, but in full no candidate is known.

Even in homogeneous models, the volume is not

suitable as internal time to describe a possible

recollapse. One can use extrinsic curvature

around such a point, but then one has to under-

stand what changing the internal time in quantum

cosmology implies, that is, whether evolution

pictures obtained in different internal time for-

mulations are equivalent to each other.

There are thus many open issues at different

levels, which, strictly speaking, do not apply only to

quantum cosmology but to all of physics. After all,

every physical system is part of the universe, and

thus a potential ingredient of quantum cosmology.

Obviously, physics works well in most situations

without taking into account its being part of one

universe. Similarly, much can be learned about a

quantum universe if only some degrees of freedom

of gravity are considered as in mini- or

midisuperspace models. In addition, complicated

interpretational issues, as important as they are for

a deep understanding of quantum physics, do not

prevent the development of physical applications in

quantum cosmology, just as they did not do so in

the early stages of quantum mechanics.

See also: Canonical General Relativity; Cosmology:

Mathematical Aspects; Loop Quantum Gravity; Quantum

Geometry and its Applications; Spacetime Topology,

Causal Structure and Singularities; Wheeler–De Witt

Theory.

Further Reading

Bojowald M (2001a) Absence of a singularity in loop

quantum cosmology. Physical Review Letters 86: 5227–5230

(gr-qc/0102069).

Bojowald M (2001b) Dynamical initial conditions in

quantum cosmology. Physical Review Letters 87: 121301

(gr-qc/0104072).

Bojowald M (2003) Initial conditions for a universe. General

Relativity and Gravitation 35: 1877–1883 (gr-qc/0305069).

Bojowald M (2005) Loop quantum cosmology. Living Reviews in

Relativity (to appear).

Bojowald M and Morales-Te´cotl HA (2004) Cosmological

applications of loop quantum gravity. In: Procee dings of

the Fift h Mexican School (DGFM): The Early Universe

and Observational Cosmology, Lecture Notes in Physics,

vol. 646, pp. 421–462. Berlin: Springer (gr-qc/0306008).

DeWitt BS (1967) Quantum theory of gravity. I. The canonical

theory. Physical Review 160: 1113–1148.

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

Decoherence and the Appearance of a Classical World in

Quantum Theory. Berlin: Springer.

Hartle JB (2003) What connects different interpretations of

quantum mechanics? quant-ph/0305089.

Hartle JB and Hawking SW (1983) Wave function of the

universe. Physical Review D 28: 2960–2975.

Kiefer C (2005) Quantum cosmology and the arrow of time. In:

Proceedings of the Conference DICE2004, Piombino, Italy,

September 2004 (gr-qc/0502016).

Kuchar

KV (1992) Time and interpretations of quantum gravity.

In: Kunstatter G, Vincent DE, and Williams JG (eds.)

Proceedings of the 4th Canadian Conference on General

Relativity and Relativistic Astrophysics. Singapore: World

Scientific.

McCabe G (2005) The structure and interpretation of cosmology:

Part II. The concept of creation in inflation and quantum

cosmology. Studies in History and Philosophy of Modern

Physics 36: 67–102 (gr-qc/0503029).

Vilenkin A (1984) Quantum creation of universes. Physical

Review D 30: 509–511.

Wiltshire DL (1996) An introduction to quantum cosmology. In:

Robson B, Visvanathan N, and Woolcock WS (eds.)

Cosmology: The Physics of the Universe, pp. 473–531.

Singapore: World Scientific (gr-qc/0101003).

158 Quantum Cosmology

Quantum Dynamical Semigroups

R Alicki, University of Gdan

sk, Gdan

sk, Poland

Introduction

With a given quantum system we associate a

Hilbert space H such that pure states of the system

are represented by normalized vectors in H or

equivalently by one-dimensional projections j ih j,

whereas mixed states are given by density matrices

 =

j, p

> 0,

= 1, that is, positive

trace-1 operators and observables are identified

with self-adjoint operators A acting on H. The

mean value of an observable A at a state  is given

by the following expression:

< A >



¼trðAÞ½1

The time evolution of the isolated system is deter-

mined by the self-adjoint operator H (Hamiltonian)

corresponding to the energy of the system. The

infinitesimal change of state of the isolated system

can be written as

ðt þ dtÞ¼ ðtÞiHdt ðtÞ; or

ðt þ dtÞ¼ðtÞidt½H;

½2

what leads to a reversible purity preserving unitary

dynamics (t) = e

itH

, (t) = e

itH

e

itH

. We use the

notation [A, B]  AB  BA ,{A, B} = AB þ BA and

put h  1. An interaction with environment leads

to irreversible changes of the density matrix trans-

forming, in general, pure states into mixed ones.

Such a process can be modeled phenomenologically

by a transition map V : H7!H leading to

ðt þ dtÞ¼ðtÞþdtVV



 dt



V;g½3

Combining Hamiltonian dynamics with several

irreversible processes governed by a family of

transition operators {V

} we obtain the following

formal evolution equation in the Schro¨ dinger picture

(quantum Markovian master equation)

ðtÞ¼LðtÞ

¼i½H;ðtÞþ

j2I



½V

;ðtÞV



þ½V

ðtÞ ; V







¼DðtÞþðtÞD



þðtÞ½4

with the initial condition (0)= .HereD = iH 

(1=2)



,  =

j2I

V



,andI is a certain

countable set of indices. Assume for the moment

that the Hilbert space H = C

. Then the eqn [4] is

always meaningful and its solution is given in

terms of the exponential (t)=(t) e

.The

linear map  is a general completely positive map

on matrices, which preserves the positivity of 

and  I

preserves positivity of nd nd matrices

for arbitrary d = 1,2, 3, ... A useful Dyson-type

expansion

 ¼WðtÞ þ

k¼1

k1





Wðt  t

ÞWðt

 t

k1

  Wðt

Þ ½5

with W(t)  W(t)W(t)



, W(t) = e

shows that

(t) is also completely positive. It is often conve-

nient to describe quantum evolution in terms of

observables (Heisenberg picture)

< A >

ðtÞ

¼ tr e





¼ tr  e





¼< AðtÞ>



½6

AðtÞ¼L



AðtÞ

¼ i½H; AðtÞ þ

j2I





½AðtÞ; V

þ½V



; AðtÞV





¼ D



ðtÞþðtÞD þ 



ðtÞ½7

with the initial condition A(0) = A, completely

positive 



A =

j2I



and the corresponding

Dyson expansion.

The solutions of eqns [4] and [7] are given in

terms of dynamical semigroups. Their general

mathematical properties and particular examples

will be reviewed in this article. Various methods of

derivation of master equations for open quantum

systems from the underlying Hamiltonian dynamics

of composed systems will also be presented.

Semigroups and Their Generators

For standard quantum-mechanical models it is con-

venient to define quantum dynamical semigr oup in

the Schro¨ dinger picture as a one-parameter family

{(t); t  0} of linear and bounded maps acting on the

Banach space of trace-class operators T (H) equipped

with the norm kk

= tr(



)

1=2

and satisfying the

following conditions:

1. Composition (semigroup) law

ðtÞ  ðsÞ¼ðt þ sÞ; for all t; s  0 ½8

Quantum Dynamical Semigroups 159

2. Complete positivity

ðtÞI

is positive on TðHC

for all d ¼ 1; 2; 3; ... and t  0 ½9

3. Conservativity (trace preservation)

trððtÞÞ¼trðÞ; for all  2 T ðHÞ ½10

4. Continuity (in a weak sense)

lim

t !0

trðAðtÞÞ¼trðAÞ

for all  2 T ðHÞ; A 2 B ðHÞ ½11

From a general theory of one-par ameter semigroups

on Banach spaces it follows that under the condi-

tions (1)–(4) (t) is a one-parameter strongly

continuous semigroup of contra ctions on T (H)

uniquely characterized by a generally unbounded

but densely defined semigroup generator L with the

domain dom(L) T(H) such that for any

 2 dom(L)

ðtÞ¼LðtÞ;ðtÞ¼ðtÞ ½12

One can show that for >0 the resolvent

R() = (I L)

1

can be extended to a bounded

operator satisfying kR()k

1

and, therefore, the

following formula makes sense:

lim

n !1

I 



n

 ¼ ðtÞ; for all  2 T ðHÞ ½13

Under the additional assumption that the generator

L is bounded (and hence everywhere defined) Gorini,

et al. (1976) and Lindblad (1976) proved that eqns

[4] and [7] with bounded H, V

and



provide

the most general form of L. The choice of H and V

not unique and the sum over j can be replaced by an

integral. In the case of n-dimensional Hilbert space

we can always choose the form of eqn [4] with at

most n

 1 V

’s. Sometimes the structure [4] is

hidden as for the following useful example of the

relaxation process to a fixed density matrix 

with

the rate >0:

ðtÞ¼

 ðtÞðÞ ½14

The general structure of an unbounded L is not

known. However, the formal expressions [4] and [7]

with possibly unbounded D and V

are meaningful

under the following conditions:



the operator D generates a strongly continuous

contracting semigroup {e

; t  0} on H;



dom(V

) dom(D), for all j;



<, D > þ <D, > þ

, V

> = 0, for

all , 2 dom(D).

We can solve eqn [4] in terms of a minimal solution.

Defining by Z the generator of the contracting

semigroup  7!e

e



and denoting by J the com-

pletely positive (unbounded) map  7!

j2I

V



one can show that for any >0, J(I  Z)

1

possesses

a unique bounded completely positive extension

denoted by A



with kA



k1. Hence, for any

0 r < 1 there exists a strongly continuous, comple-

tely positive and contracting semigroup 

(r)

(t)withthe

resolvent explicitly given by

ðrÞ

ðÞ¼ðI  ZÞ

1

k¼0



½15

As kR

(r)

()k1 the limit lim

r !1

(r)

() = R(),

where R() is the resolvent of the semigroup (t)

satisfying (1), (2), and (4) and called the minimal

solution of the eqn [4]. The minimal solution need

not be a unique solution or conservative (generally

tr (t) tr (0) and for any other solution



(t)  (t)). There exist useful sufficient conditions

for conservativity, an example of a sufficient and

necessary condition is the following: A



!0 strongly

as n !1 for all >0(Chebotarev and Fagnola

1988).

Examples

Bloch equ ation The simplest two-level system can

be described in terms of spin operators

= (1=2)

, k = 1, 2, 3, wher e 

are Pauli matrices.

The most general master equation of the form [4]

can be written as (Alicki and Fannes 2001, Ingarden

et al. 1997)

d

¼i

k¼1

½S

;þ

k;l¼1

½S

; S



þ½S

;S

g ½16

where h

2 R and [a

]isa3 3 complex,

positively defined matrix. Introducing the magneti-

zation vector M

(t) = tr((t)S

), we obtain the

following Bloch equation used in the magnetic

resonance theory:

MðtÞ¼h ðMðtÞM

ÞFðMðtÞM

Þ½17

where the tensor F (real, symmetric, and positive

3  3 matrix) and the vector M

are functions of

]. In particular, complete positivity implies the

following inequalities for the inverse relaxation

times 

, 

(eigenvalues of F):



 0;

þ 

 



þ 

 

;

þ 

 

½18

160 Quantum Dynamical Semigroups

Damped and pumped harmonic oscillator The

quantum master equation for a linearly damped

and pumped harmonic oscillator with frequency !

and the damping (pumpin g) coeffic ient 

(

) has

form

d

¼i!½a



a;þ



½a; a



þ½a;a



ðÞ



½a



; aþ½a



;aðÞ½19

where a



, a are creation and annihilation operators

satisfying [a , a



] = 1. Taking diagonal elements

= <n, n> in the ‘‘particle number’’ basis



ajn >= njn >, n = 0, 1, 2, ..., which evolve inde-

pendently of the off-diagonal elements, one obtains

the birth and death process,

¼

ðn þ 1Þp

nþ1

þ 

n1

 

n þ 

ðn þ 1Þ



½20

It is convenient to use the Heisenberg picture and

find an explicit solution in terms of Weyl unita ry

operators W(z) = exp[(i=

ﬃﬃﬃ

)(za þ





)],





ðtÞWðzÞ

¼exp 

jzj





1 e

ð



Þt



()

WðzðtÞÞ ½21

where z(t)= exp{(i! þ

(



))t }, t 0. For 

>

the solution of eqn [19] always tends to the stationary

Gibbs state





¼ Z

1

!a



; Z ¼ tre

!a



 ¼

lnð

=

½22

Quasifree semigroups The previous example is the

simplest instance of the dynamical semigroups for

noninteracting bosons and fermions which are

completely determined on the single-p article level.

Such systems are defined by a single-particle Hilbert

space H

and a linear map H

3  7!a



() into

creation operators satisfying canonical commutation

or anticommutation relations (CCRs or CARs,

respectively) for bosons and fermions, respectively

½að Þ; a



ðÞ



¼< ;>

½A; B



 AB ð1ÞBA

½23

In all expressions containing (), sign (þ) refers to

bosons and () to fermions.

Consider a nonhomogeneous evolution equation

on the trace-class operators  2T(H

d

¼i½H

;

fð

 ðÞ

Þ;gþ

½24

with a single-particle Hamiltoni an H

and a damp-

ing (pumping) positive operator 

(

)  0. The

operators H

, 

, and 

need not be bounded

provided i H

 (1=2){(

 ()

) generates a

(contracting in the fermionic case) semigroup

{T(t); t  0} on H

and the formal solution of

eqn [24]

ðtÞ¼TðtÞð0ÞT



ðtÞþQðtÞ

where QðtÞ¼

TðsÞ



ðsÞds ½25

is meaningful. We can now define the quasifree

dynamical semigroup for the many-particle system

described by the Fock space F



)(Alicki and

Lendi 1987, Alicki and Fannes 2001). The simplest

definition involves Heisenberg evolution of the

ordered monomials in a



(

) and a(





ðtÞa



Þa



Það

Það





Det



; QðtÞ



k;l¼1;2;...;r

 a



ðtÞ



ð Þa



ðtÞ



mr

ðÞ

 aT



ðtÞ





aT



ðtÞ



nr



½26

The sum is taken over all partitions {(j

, ..., j

)

(

, ..., 

mr

)}, {(i

, ..., i

)(

, ..., 

nr

} such that

< j

<  < j

, 

<

,  <  <

mr

, i

< i

< 

< r

, 

<

 <

nr

; 

 1, 



is a product of

signatures of the permutations {1, 2, ..., m} 7!

,..., j

,

,..., 

mr

},{1,2,...,n}7!{i

,..., i

,

,...,



nr

}; a permanent Det

is taken for bosons, a

determinant Det



for fermions.

Introducing an orthonormal basis {e

}inH

and

using the notation a



)  a



, we can write a

formal master equation for density matrices on the

Fock space corresponding to eqn [26]:

d

¼i½H

;þ

k;l



½a

; a







þ½a

;a







þ 

½a



; a

þ½a



;a







½27

Again, formally,

k;l

; H

> a



¼< e

; 

>; 

¼< e

; 

½28

Often the formulas [27], [28] are not well-

defined, but replacing the (infinite) mat rices by

(distribution-valued) integral kernels, sums by inte-

grals, and a



, a

by quantum fields, we can obtain

meaningful objects.

Quasifree dynamical semigr oups find applications

in the theory of unstable particles, quantum linear

Quantum Dynamical Semigroups 161