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

Подождите немного. Документ загружается.

distributions violate the ‘‘Hadamard’’ condition (C4)

and which therefore do not have a well-defined finite

expectation value for the renormalized stress–energy

tensor.

See also: AdS/CFT Correspondence; Algebraic

Approach to Quantum Field Theory; Axiomatic Quantum

Field Theory; Black Hole Mechanics; Bosons and

Fermions in External Fields; Integrability and Quantum

Field Theory; Quantum Fields with Indefinite Metric:

Non-Trivial Models; Quantum Fields with Topological

Defects; Quantum Geometry and Its Applications;

Scattering in Relativistic Quantum Field Theory:

Fundamental Concepts and Tools; Thermal Quantum

Field Theory.

Further Reading

Birrell ND and Davies PCW (1982) Quantum Fields in Curved

Space. Cambridge: Cambridge University Press.

Brunetti R, Fredenhagen K, and Verch R (2003) The generally

covariant locality principle – a new paradigm for local quantum

physics. Communications in Mathematical Physics 237: 31–68.

DeWitt BS (1975) Quantum field theory in curved space-time.

Physics Reports 19(6): 295–357.

Dimock J (1980) Algebras of local observables on a manifold.

Communications in Mathematical Physics 77: 219–228.

Haag R (1996) Local Quantum Physics. Berlin: Springer.

Hartle JB and Hawking SW (1976) Path-integral derivation of

black-hole radiance. Physical Review D 13: 2188–2203.

Hawking SW (1975) Particle creation by black holes. Commu-

nications in Mathematical Physics 43: 199–220.

Hawking SW (1992) The chronology protection conjecture.

Physical Review D 46: 603–611.

Israel W (1976) Thermo-field dynamics of black holes. Physics

Letters A 57: 107–110.

Kay BS (1979) Casimir effect in quantum field theory. (Original

title: The Casimir effect without magic.) Physical Review D

20: 3052–3062.

Kay BS (1993) Sufficient conditions for quasifree states and an

improved uniqueness theorem for quantum fields on space-times

with horizons. Journal of Mathematical Physics 34: 4519–4539.

Kay BS (2000) Application of linear hyperbolic PDE to linear

quantum fields in curved spacetimes: especially black holes,

time machines and a new semi-local vacuum concept. Journe´es

quations aux De´rive´es Partielles, Nantes, 5–9 juin 2000,

GDR 1151 (CNRS): IX1–IX19. (Also available at http://

www.math.sciences.univ-nantes.fr or as gr-qc/0103056.)

Kay BS, Radzikowski MJ, and Wald RM (1997) Quantum field

theory on spacetimes witha compactly generated Cauchy horizon.

Communications in Mathematical Physics 183: 533–556.

Kay BS and Wald RM (1991) Theorems on the uniqueness and

thermal properties of stationary, nonsingular, quasifree states

on spacetimes with a bifurcate Killing horizon. Physics

Reports 207(2): 49–136.

Misner CW, Thorne KS, and Wheeler JA (1973) Gravitation. San

Francisco: W.H. Freeman.

Unruh W (1976) Notes on black hole evaporation. Physical

Review D 14: 870–892.

Visser M (2003) The quantum physics of chronology protection.

In: Gibbons GW, Shellard EPS, and Rankin SJ (eds.) The

Future of Theoretical Physics and Cosmology. Cambridge:

Cambridge University Press.

Wald RM (1978) Trace anomaly of a conformally invariant quantum

field in a curved spacetime. Physical Review D 17: 1477–1484.

Wald RM (1994) Quantum Field Theory in Curved Spacetime

and Black Hole Thermodynamics. Chicago: University of

Chicago Press.

Quantum Field Theory: A Brief Introduction

L H Ryder, University of Kent, Canterbury, UK

By any account quantum field theory occupies a

prominent place in the history of mathematical

physics. This article is, however, not intended to

serve as an overview of this subject, but has the

more modest aim of identifying a few areas which

seem to me interesting and significant.

Historical Remarks; Second Quantization

At the time when quantum field theory was at the

forefront of theoretical physics its raison d’eˆtre was

to complete the quantum description of the sub-

atomic world. Quantum mechanics had been amaz-

ingly successful in solving almost the whole of

atomic physics by making explicit the quantum

(wave) nature of the electron, according to the

formulations of Heisenberg and Schro¨ dinger. The

introduction of the quantum idea into physics,

however, by Planck in 1900 closely followed by

Einstein in 1905 was the proposal of a quantum

(particular) aspect of the electromagnetic field – the

photon. In the mid-1920s the only force in nature to

be considered was the electromagnetic interaction;

this was before the theories of Yukawa and Fermi,

concerning the strong and weak nuclear forces.

Dirac, Heisenberg, Jordan, and others then

addressed themselves to finding a formulation of

quantum electrodynamics (QED) comparable in

mathematical sophistication to the Heisenberg–

Schro¨ dinger formulation of quantum mechanics –

which Planck’s and Einstein’s theories were not.

The idea that was pursued, at least in the early

stages, was that the Schro¨ dinger wave function ,

taken as a wave field, should be ‘‘quantized’’; Dirac

212 Quantum Field Theory: A Brief Introduction

seems to have taken this as a model for photons.

Jordan further proposed that electrons should be

treated as the quanta of an electron field, but

recognized that their fermi onic nature would modify

the quantization procedure. This generic idea

involved what was called ‘‘second quantization’’ –

of a field into a particle.

One of the earliest quantization rules was Bohr’s

condition relating to the periodic orbits of electrons in

atoms, J =

p dq = nh. At the hands of Heisenberg and

Dirac this became upgraded to the commutation

relation

½q; p¼ih

where the operators p and q are ‘‘observables.’’ In

their papers on quantum field theory, Dirac, Jordan

and Wigner, and Heisenberg introduced creation and

annihilation operators which had the function, as

their name implied, of creating and destroying single

particles – quanta of the field. These operators obeyed

the commutation rules (with [A, B] = AB  BA)

½b

; b



¼

; ½b

; b

¼½b



; b



¼0

when the field quanta were bosons, and the anti-

commutation rules

; b



g¼

; fb

; b

g¼fb



; b



g¼0

(with {A, B} = AB þ BA) when the field quanta were

fermions (e.g., electrons). These steps constitute

second quantization, but it may be noted that

the creation and annihilation operators are not

observables, as p and q are in the Heisenberg

commutation relation. In addition, the second

quantization conditions do not involve Planck’s

constant. ‘‘First’’ and ‘‘second’’ quantization are

therefore not so similar as one might like to think.

The question of what exactly is being quantized

was in fact the source of some confusion. In his

paper of 1927, Dirac’s attention is focussed on

electromagnetic radiation, but he nevertheless dis-

cusses the difference between ‘‘a light-wave and the

de Broglie or Schro¨ dinger wave associated with the

light-quanta.’’ As Dirac points out, ‘‘their intensities

are to be interpreted in different ways. The number

of light quanta per unit volume associated with a

monochromatic light-wave equals the energy per

unit volume of the wave divided by the energy

(2h) of a single light quantum. On the other hand

a monochromatic de Broglie wave of amplitude a

(multiplied into the imaginary exponential factor)

must be interpreted as representing a

light quanta

per unit volume for all frequencies.’’ There are at

least two problematic issues here. First, is the

Schro¨ dinger wave function to be considered as a

‘‘real’’ field, whose quanta result in ‘‘real’’ particl es,

or is it a probability field, whose significance lies in

Born’s probabilistic interpretation of quantum

mechanics? Born wrote in 1926, ‘‘[Einstein said

that] the waves are present only to show the

corpuscular light qua nta the way, and he spoke in

the sense of a ‘‘ghost field’’. This determines the

probability that a light quantum, the bearer of

energy and momentum, takes a certain path;

however, the field itself has no energy and no

momentum.’’ This is the first problem. The second

one concerns the nature of the quantization itself. Is

this a quantization of field energy, or a quantization

of the field itself, as a substantial entity? If the field

is real, the second of these does not imply the first.

Ambiguities surrounding the idea of second

quantization survived into the 1960s. Wigner is

recorded as saying, in an interview in 1963, ‘‘just as

we get photons by quantising the electr omagnetic

fields, so we should be able to get material particles

by quantising the Schro¨ dinger field.’’ And Rosenfeld,

also in an interview in 1963, said, ‘‘in some sense or

other, Jordan himself took the wave function, the

probability amplit ude, physically more seriously

than most people [did].’’

It would seem we are justified in concluding that the

idea of second quantization contains flaws, but an even

clearer indication of the need for rethinking is provided

by the story of the Dirac equation. This is a wave

equation for the electron, compatible with special

relativity, and taking explicit account of its spin being

(1/2)h. The equation famously had both positive- and

negative-energy solutions. This potential disaster was

converted by Dirac into a triumph by reinterpreting the

(absence of) negative-energy solutions as (positive-

energy) antiparticles – positrons, particles with positive

charge but the same mass and spin as the electron.

Positrons were eventually discovered by Anderson. It

was later shown that the existence of antiparticles is a

general feature of quantum field theory, not just a

peculiarity of spin-1/2 particles. The significance of this

discovery, however, is that the twin requirements of

relativity and quantum theory are not compatible with

a single-particle state; rather, these requirements result

in a two-particle state. Thus, in some sense the

requirements of relativity and quantum mechanics

already start to take us down the road to a quantum

theory of fields.

Quantum field theory is then constructed on the

following sort of framework: ‘‘classical’’ theories for

fields with any spin may be writt en down and these

are quantized by reinterpreting the field variables as

operators and imposing Heisenberg-type commuta-

tion relations on the field and its corresponding

Quantum Field Theory: A Brief Introduction 213

‘‘momentum’’ variable. So, for example, for spinless

fields we have the equal-time commutation relation

½ðx; tÞ;ðy ; tÞ ¼ ih

ð3Þ

ðx  yÞ

where  = @L=@(@

) and L is the Lagrange density.

The mass and spin of particles are defined with

reference to the Poincare´ group (thereby incorporat-

ing special relativity) and the quantum requirement

is the familiar one that phy sical states are repre-

sented by vectors in Hilbert space. The rest follows:

as Weinberg says, ‘‘quantum field theory is the way

it is because (with certain qualifications) this is the

only way to reconcile quantum mechanics with

special relativity.’’

Renormalization

A notorious problem in quantum field theory is the

occurrence of infinities. In QED, for example, the

electron acquires a self-energy – and therefore a

contribution to its mass – by virtue of the emission

and reabsorption of virtual photons. It turns out

that this self-energy is infinite – it is given by a

divergent integral – even in the lowest order of

perturbation theory. In the early days, this was

recognized as being a serious problem, and in fact it

turns out to be a generic problem in quantum field

theory. It was realized by Dyson, however, that in

some field theories these divergences may be dealt

with by redefining a small number of parameters

(e.g., in QED, the electron mass, charge, and field

amplitude) so that thereafter the theory is finite to

all orders of perturbation theory. Such theories are

called renormalizable, and QED is a renormalizable

field theory.

Some important field theories, however, are not

renormalizable; an example is Fermi’s theory of

weak interactions. To lowest order in perturbation

theory, Fermi’s theory works well (e.g., in account-

ing for the electron spectrum in neutron beta decay),

but to higher orders divergent results are obtained,

which cannot be waved away by redefining a finite

number of parameters; that is to say, as the order of

perturbation increases, so also does the number of

parameters to be redefined. Nonrenormalizable

theories of this type have traditionally been regarded

as highly undesirable, not to say rather nasty.

The modern view of renormalization is, however,

somewhat different. The problem with nonrenormal-

izable theories is that, in order to calculate a physical

process to all orders in perturbation theory, an

infinite number of parameters must be renormalized,

so the theory has no predictive power. In practice,

however, we do not need to calculate to all orders in

perturbation theory, since any physical process (say a

scattering process or a particle decay) will only be

observed at a finite energy and comparison of theory

and experiment therefore only requires calculation up

to a finite order of perturbation theory. So even

nonrenormalizable theories are perfectly acceptable

as low-energy theories. This amounts to a philosophy

of effective field theories; an effective field theory is a

model which holds good up to a particular energy

scale, or equivalently down to a particular length

scale.

An important addition to the theoretical armoury

is the renormalization group. Renormalization is

implemented first of all by a scheme of regulariza-

tion, which enables the divergences to be exhibited

explicitly. The simplest type of regularization is the

introduction of a cutoff in the momentum integrals,

but in modern particle physics the favored scheme is

dimensional regularization. The dimensionality of

the integrals in momentum space is taken to be

d = 4  " and the divergent quantities have an

explicit depend ence on " (which, of course, as the

‘‘real’’ world is approached, approaches zero). At

the same time, a mass parameter  is introduced in

order to define dimensionless quantities, for exam-

ple, a dimensionless coupling constant. The renor-

malized quantities then depend on the ‘‘bare’’

(unrenormalized) quantities an d on  and ". The

arbitrariness of  enables a differential equation, for

scattering amplitudes, for example, to be written

down. While at first sight this renormalization

group equation might seem to have no physical

importance, in fact it gives a powerful way of

studying scattering behavior at large momenta.

Most interestingly, the concept of the renormali-

zation group also arises in condensed matter physics.

Here, rather than, for exampl e, a cutoff in momen-

tum space, the relevant parameter is a distance scale.

In the Ising model in statistical mechanics, for

example, in which spins are located on a lattice,

the parameter is the lattice spacing. To construct a

theory that describes the physics on the macroscopic

scale involves integrating out the details on the

microscopic scale and one way to do this is via the

‘‘block spin’’ transformation originally introduced

by Kadanoff. In this way the renormalization group

has had a large impact in condensed matter physics,

for example, in the study of critical phenomena.

Particle Physics and Cosmology

Probably the most spectacular success of quantum

field theory in the twentieth century has been in

particle physics. The ‘‘standard model’’ accounts for

the strong, electromagnetic, and weak interactions

214 Quantum Field Theory: A Brief Introduction

between elementary particles with outstanding

success. The interactions are generalizatio ns of Max-

well’s electrodynamics, which is invariant under a

symmetry group U(1) of gauge transformations. An

enlargement of this group to SU(2)  U(1) accounts

for the unified electroweak interaction (the unifica-

tion resulting from the fact that the two U(1)’s above

are not exactly the same; there is some on-diagonal

mixing), and the strong interactions between quarks,

which binds them into hadrons, are invariant under an

SU(3) group of gauge transformatio ns. The gauge

fields are the photon , the W and Z bosons (both

heavy; of the order of 100 times the proton mass), and

the (massless) gluons mediating the force between

quarks (quantum chromodynamics, QCD). An

important feature of the standard model is sponta-

neous symmetry breaking, which is the mechanism by

which the W and Z particles acquire a mass (but the

photon does not, and neither do the gluons). This goes

by the name of the Higgs mechanism.

The quantization of the standard model is most

successfully carried out using the path-integral

formalism, rather than canonical quantization, and

the proof of the renormalizability of the model (of

nonabelian gauge theories with spontaneous sym-

metry breaking) was given by ’t Hooft. Details of

these topics are now available in many textbooks.

Confidence that this is a realistic model of elemen-

tary particles – that is to say, of quarks and leptons –

depends, of course, on particular experiments and

their interpretation and an important milestone on this

journey was Feynman’s quark–parton model of deep

inelastic electron–proton scattering. The interpretation

of the data required a picture of an electron scattering

from an individual quark in the proton, and this in

turn required a negligible interaction between quarks;

in other words, that at small distances (inside the

proton) the quarks are (almost) free – despite the fact

that at large distances they most certainly are not! The

proof, by Gross, Politzer, and Wilczek, that nonabe-

lian gauge are indeed asymptotically free (asymptotic

in momentum space, that is) was therefore an

important event in helping to establish the credibility

of the standard model.

A characteristic contribution of quantum field theory

to our view of the physical world is its picture of the

vacuum, as being populated with virtual particle–

antiparticle pairs. A consequence of this is the phenom-

enon of vacuum polarization – that the presence of an

electric charge in free space polarizes these virtual pairs.

This in turns leads to the phenomenon of screening in

QED, and antiscreening in QCD, SU(3) having a more

complicated structure than U(1). It also leads to a

nonzero (in fact, quadratically divergent!) value for the

energy of the vacuum. This is in effect the contribution

of the zero-point energies of all the oscillators in the

Fourier expansion of the scalar field operator. In any

other interaction than gravity, this zero-point energy

may be ignored, but in gravity it may be expected to

have observable consequences, and indeed it turns out

that it plays the same role as a cosmological constant ,

and therefore acts as an agent of acceleration, rather

than deceleration, of the universe.

A final topic worth noting is one whose existence

would have been inconceivable in the early days of this

subject. The nonlinearity of the (nonabelian) gauge

field equations and the existence of a nontrivial group

space allows new types of topologically nontrivial

solutions to these equations: solitons, bounces, instan-

tons, sphalerons, and so on. Effects such as fractional

spin and nonconservation of fermion number also

appear, and, on the cosmological scale, domain walls

and cosmic strings. There is something here for

theoretical physicists of many differing interests.

See also: Algebraic Approach to Quantum Field Theory;

Axiomatic Quantum Field Theory; BRST Quantization;

Constrained Systems; Constructive Quantum Field

Theory; Deformation Quantization; Electroweak Theory;

Euclidean Field Theory; Exact Renormalization Group;

Integrability and Quantum Field Theory; Nonperturbative

and Topological Aspects of Gauge Theory; Perturbative

Renormalization Theory and BRST; Quantum

Chromodynamics; Quantum Electrodynamics and Its

Precision Tests; Quantum Fields with Indefinite Metric:

Non-Trivial Models; Quantum Fields with Topological

Defects; Renormalization: General Theory; Standard

Model of Particle Physics; Symmetries and Conservation

Laws; Symmetries in Quantum Field Theory of Lower

Spacetime Dimensions; Topological Defects and Their

Homotopy Classification; Topological Quantum Field

Theory: Overview; Twistors.

Further Reading

Cao TY (1997) Conceptual Developments of 20th Century Field

Theories. Cambridge: Cambridge University Press.

Davies P (ed.) (1989) The New Physics. Cambridge: Cambridge

University Press.

Gross F (1993) Relativistic Quantum Mechanics and Field

Theory. New York: Wiley.

Huang K (1998) Quantum Field Theory. New York: Wiley.

Itzykson C and Zuber J-B (1980) Quantum Field Theory.

New York: McGraw-Hill.

Maggiore M (2005) A Modern Introduction to Quantum Field

Theory. Oxford: Oxford University Press.

Rubakov V (2002) Classical Theory of Gauge Fields. Princeton:

Princeton University Press.

Schweber SS (1994) QED and the Men Who Made It. Princeton:

Princeton University Press.

Schwinger J (ed.) (1958) Quantum Electrodynamics. New York:

Dover.

’t Hooft G (1997) In Search of the Ultimate Building Blocks.

Cambridge: Cambridge University Press.

Quantum Field Theory: A Brief Introduction 215

Weinberg S (1995, 1996) The Quantum Theory of Fields, vol. 1

and 2. Cambridge: Cambridge University Press.

Wentzel G (1960) Quantum theory of fields (until 1947). In: Fierz M

and Weisskopf VF (eds.) Theoretical Physics in the Twentieth

Century,(ReprintedinMehraJ(1973)The Physicist’s Conception

of Nature. Dordrecht: Reidel.) New York: Interscience.

Zee A (2003) Quantum Field Theory in a Nutshell. Princeton:

Princeton University Press.

Quantum Fields with Indefinite Metric: Non-Trivial Models

S Albeverio and H Gottschalk, Rheinische

Friedrich-Wilhelms-Universita

t Bonn, Bonn, Germany

Introduction

The nonperturbative construction of quantum field

models with nontrivial scattering in arbitrary dimen-

sion d of the underlying Minkowski spacetime is

much simpler in the framework of quantum field

theory with indefinite metric than in the positive-

metric case. In particular, there exist a number of

solutions in the physical dimension d = 4, where up

to now no positive-metric solutions are known. The

reasons why this is so are reviewed in this article,

and some examples obtained by analytic continua-

tion from the solutions of Euclidean covariant

stochastic partial differential equations (SPDEs)

driven by non-Gaussian white noise are discussed.

The Hilbert Space Structure Condition

It has been proved by F Strocchi that a quantum

gauge field in a local, covariant gauge cannot act on

a Hilbert space with a positive-definite inner

product. But it is possible to overcome this obstacle

by passing from a Hilbert space representation of

the algebra of the quantum field to Krein space

representations in order to preserve locality and

covariance under the Poincare´ group.

A Krein space K is an inner-product space which

also is a Hilbert space with respect to some auxiliary

scalar product. The relation between the inner

product h.,.i and the auxiliary scalar product (. , .)

is given by a self-adjoint linear operator J : K!K

with J

= 1

and h.,.i = (. , J.). J is called the metric

operator. A quantum field acting on such a space is

called a quantum field with indefinite metric. The

formal definition is as follows.

Let DKbe a dense linear space and  2Da

distinguished vector (henceforth called the vacuum).

Let S = S(R

, C

) be the space of Schwartz test

functions with values in C

. A quantum field  by

definition is a linear mapping from S to the linear

operators on D. One usually assumes that D is

generated as the linear span of vectors generated by

repeated application of field operators to the

vacuum. The following properties should hold for

the quantum field :

1. Temperedness: f

! f in S)h, (f

)i!

h, (f )i8,  2S.

2. Covariance: There exists a weakly continuous

representation U of the covering of the

orthochronous, proper Poincare´group

linear operators on D which is J-unitary, that is,

[]

= U

1

with U

[]

= JU



and leaves 

invariant.  is said to be covariant with respect

to U and a representation  of the covering of the

orthochronous, proper Lorentz group

U(g)(f )U(g)

1

= (f

), where f

(x) = ()f ( 

1

(x  a)), g = f, ag,  2

, a 2 R

3. Spectrality: Let U(a), a 2 R

, be the representa-

tion of the translation group and let

 = [

,2D

suppF(h, U(.)i) with F the Fourier

transform (in the sense of tempered distribu-

tions). Formally,  is the joint spectrum of the

generators of spacetime translations U(a). The

spectral condition then demands that  



the closed forward light cone in energy–momentum

space.

4. Locality: There is a decomposition C

= 



such that for each f , h 2Staking values in a V



and having sp acelike separated supports one has

either [(f ), (h)] = 0orf(f ), (h)g= 0, where

[.,.] is the commutator and f.,.g the

anticommutator.

5. Hermiticity: There is an involution  on S such

that (f )

[]

= (f



The quan tum-mechanical interpretation of the

inner product of two vectors in K as a probability

amplitude, however, gets lost. It has to be restored

by the construction of a physical subspace of K

where the restriction of the inner product is non-

negative. This is called the Gupter–Bleuler gauge

procedure. Typically, one first considers the problem

of constructing quantum fields with indefinite

metric, that is, the dynamical pro blem is addressed.

This is often followed by the construction of the

physical states, which involves implementation of

quantum constraints.

216 Quantum Fields with Indefinite Metric: Non-Trivial Models

The vacuum expectation values (VEVs), also called

Wightman functions, of the quantum field theory

with indefinite metric (IMQFT) are defined as

ðf

f

Þ = h;ðf

Þðf

Þi

; ...; f

2S ½1

An axiomatic framework for (unconstrained)

IMQFT has been suggested by G Morchio and

F Strocchi in terms of the Wightman functions

, n 2 N

. Previous work on the topic had

been done by J Yngvason. These generalized Wight-

man axioms of Morchio and Strocchi replace the

positivity condition on the Wightman functions by a

so-called Hilbert space structure condition (HSSC):

for n 2 N

there exist p

a Hilbert seminorm on S

n

such that

nþm

ðf  hÞj  p

ðf Þp

ðhÞ8n; m 2 N

f 2S

n

; h 2S

m

½2

This condition makes sure that a field algebra on a

Krein space with VEVs equal to the given set of

Wightman functions can be constructed. The

remaining axioms of the Wightman framework –

temperedness, covariance, spectral condition, local-

ity, and Hermiticity – remain the same. Clustering of

Wightman functions is assumed at least for massive

theories:

lim

t!1

nþm

ðf  h

Þ = W

ðf ÞW

ðhÞ8n; m 2 N

f 2S

n

; h 2S

m

½3

for spacelike a 2 R

. It fails to hold in certain

physical contexts where multiple vacua (also called

-vacua) accompanied with massless Goldstone

bosons occur due to spontaneous symmetry

breaking.

In the original Wightman axioms, there are

essentially two nonlinear axioms: positivity and

clustering. Here nonlinear means that checking that

condition involves more than one VEV with a given

number of field operators. The cluster condition can

be linearized by an operation on the Wightman

functions called ‘‘truncation.’’ The equations

ðf

f

I2P

ðnÞ

;...;j

g2I

<<j

ðf

f

½4

recursively define the truncated Wightman functions

for n 2 N. Here P

(n)

stands for the set of all

partitions of f1, ..., ng into disjoint, nonempty sets.

Unfortunately, the positivity cond ition (at least

when combined with nontrivial scattering) becomes

highly nonlinear for truncated Wightman functions.

This can be seen as one explanation why it is so

difficult to find nontrivial (i.e., corresponding to

nontrivial interactions) solutions to the Wightman

axioms.

But it turns out that, in co ntrast to positivity, the

HSSC is essentially linear for truncated Wightman

functions.

Theorem 1 If there exists a Schw artz norm jj  jj on

S such that W

is continuous with respect to jj  jj

n

for n 2 N then the associated sequence of Wightman

functions fW

g fulfills the HSSC [2].

Note that jj  jj

n

is well defined a s S is a nuclear

space. This theorem makes it much easier to

construct IMQFTs. In particular, all known solu-

tions of the linear program for truncated

Wightman functions lead to an abundance of

mathematical solutions to the a xioms of IMQFT,

as long as the singularities of truncated Wightman

functions in position and energy–momentum space

do not become increasingly stronger with growing n.

For example, the perturbative solutions to Wight-

man functions of Ostendorf and Steinmann provide

solutions when the perturbation series is truncated at

agivenorder.

Relativistic Fields from Euclidean

Stochastic Equations

In the classical work on constructive quantum field

theory, relativistic fields in spacetime dimensions

d = 2 and 3 have been constructed by analytic

continuation from Euclidean random fields. This, in

particular, has led to firm connections between

quantum field theory and equilibrium statistical

mechanics. Let us discuss one specific class of

solutions of the axioms of IMQFT for arbitrary d

which also stem from random fields related to an

ensemble of statistical mechanics of classical, con-

tinuous particles. Mathematically, this is connected

with using random fields with Poisson distribution.

As in constructive QFT, the moments, also called

Schwinger functions, of the random field can be

analytically continued from Euclidean imaginary

time to relativistic real time. That this is possible

results from an explicit calculation. Axiomatic results

cannot be used, as they depend on positivity or

reflection positivity in the Euclidean spacetime,

respectively.

By definition, a mixing Euclidean covariant

random field ’ is an almost surely linear mapping

from S

= S(R

, R

) to the space of real-valued

Quantum Fields with Indefinite Metric: Non-Trivial Models 217

measurable functions (random variables) on some

probability space that fulfills the following

properties:

1. Temperedness: f

! f in S

)’(f

) !

’(f ).

2. Covariance: ’(f )

’(f

) 8f 2S

, g = f, ag,

 2SO(d), a 2R

, f

(x) = ()f (

1

(x  a)) for

some continuous representation  : SO(d) !GL(N).

3. Mixing: lim

t!1

E[AB

] = E[A]E[B] for all

square-integrable random variables A = A(’),

B = B(’), and B

= B(’

), ’

(f ) = ’(f

) 8f 2S

a 2R

nf0g.

The mixing condition in the Euclidean spacetime

plays the same role as the cluster property in the

generalized Wightman axioms.

In particular, we consider random fields ’

obtained as solutions of the SPDE D’ = . In this

equation,  is a noise field, that is,  is -covariant

for some representation of SO(d), (f ) has infinitely

divisible probability law and (f ), (h) are indepen-

dent 8f , h 2S

with supp f \ supp h = ;. D is a

-covariant (i.e., ()D()

1

= D 8 2 SO(d))

partial differential operator with constant coeffi-

cients (also pseudodifferential operators D could be

considered). From the classification of infinitely

divisible probability laws, it is known that 

essentially consists of Gaussian white noise and

Poisson fields and derivati ves thereof. Such a Gauss–

Poisson noise field by the Bochner–Minlos theorem

is characterized by its Fourier transform. Direct

relations with QFT arise if one chooses

E½e

iðf Þ

 = exp

ðf Þf  

pðÞf dx



f 2S

½5

where : R

! C is a Le´vy function,

ðt Þ = ia  t 

t  

þ z

nf0g

ðe

its

 1ÞdrðsÞ

t 2 R

½6

Here the centered dot represents a -invariant scalar

product on R

,  a positive-semidefinite -invariant

N  N matrix, z  0 a real number and r is a

-invariant probability measure on R

nf0g with all

moments. Further, 

,

= (@

(t)=@t





t = 0

and p : [0, 1) ! [0, 1) is a polynomial depending

on D.If

1

, the Fourier-transformed inverse of D,

exists, it can be represented by

1

ðkÞ =

ðkÞ

l = 1

ðjkj

þ m



½7

Here Q

(k) is a complex N  N matrix with

polynomial entries being -covariant, () Q

(

1

k)()

1

= Q

(k) 8 2 SO(d), k 2 R

. 

N and m

2 Cn(1, 0) are parameters with the

interpretation of the mass spectrum (m

, ..., m

)

and (

, ..., 

) the dipole degrees of the related

masses. We restrict ourselves to the case of positive

mass spectrum where m

> 0, and in this case

pðtÞ = pðt; DÞ =

l = 1

ðt þ m



l = 1

2

; t > 0 ½8

One can show that ’ obtained as the unique

solution of the SPDE D ’ =  is a Euclidean covariant,

mixing random field. The Schwinger functions

(moments) of ’ are given by

ðf

f

= E½’ðf

Þ’ðf

Þ; f

; ...; f

½9

Now the Schwinger functions can be calculated

explicitly. They are determined by the truncated

Schwinger functions, cf. [4], as follows: for n = 2,

2;

;

ðx

; x

2;

;

ði r

l ¼1

2

l ¼1

ð þ m



ðx

 x

Þ½10

and for n  3

n;



ðx

; ...; x

¼ Q

n;



ði r



j ¼1

l ¼1

ð þ m



ðx

 xÞdx ½11

where

n;



ði r

Þ¼C





l ¼1

E;

;



i



½12

with





= ðiÞ

ðt Þ



@t





t = 0

½13

and the Einstein convention of summation and raising/

lowering of indices on R

with respect to the invariant

inner product  is applied. The Schwinger functions

fulfill the requirements of -covariance, symmetry,

clustering, and Hermiticity from the Osterwalder–

Schrader axioms of Euclidean QFT.

While there is no known general reason why a

relativistic QFT should exist for a given set of

Schwinger functions, one can take advantage of the

explicit formulas [10]–[13] in order to calculate the

analytic continuation from Euclidean to relativistic

times explicitly.

218 Quantum Fields with Indefinite Metric: Non-Trivial Models

It simplifies the considerations to exclude dipole

fields, that is, one assumes that 

= 1 for

l = 1, ..., n. In physical terms, the no-dipole condi-

tion guarantees that the asymptotic fields in Min-

kowski spacetime fulfill the Klein–Gordon equation

and thus generate particles in the usual sense if

applied to the vacuum. If this condition is not

imposed, asymptotic fields might only fulfill a dipole

equation (

þm

)



in=out

= 0 or a related hyper-

bolic equation of even higher order, and the particle

states generated by application of such fields to the

vacuum require a gauge fixing (constraints) in order

to obtain a physical interpretation. Given the no-

dipole condition, one obtains by expansion into

partial fractions

l = 1

ðjkj

þ m

l = 1

ðjkj

þ m

½14

with b

2 (0, 1) uniquely determined and b

6¼ 0.

For the truncated Schwinger functions, this implies

(n  3) that

n;



ðx

; ...; x

¼ Q

n;



ði r

;...; l

¼1



r ¼1

j ¼1

ð þ m

1

ðx  x

Þ dx ½15

At this point, a lengthy calculation yields a repres-

entation of the functions

j = 1

( þ m

)

1

(x  x

)dx as the Fourier–Laplace transform of a

distribution

n,m

,...,m

that fulfills the spectral

condition. This is equivalent to the statement that

the analytic continuation of such functions to

relativistic times yields W

n,m

,...,m

, where the latter

distribution is the inverse Fourier transform of

n , m

,..., m

. This distribution up to a constant that

can be integrated into Q

is given by

j = 1

j1

l = 1





ðk

ð1Þ

 m

l = jþ1



ðk

;





l = 1



½16

Here 



(k) = ( k

)(k

 m

), wher e  is the

Heaviside step function and k

= k

jkj

. On the

other ha nd, the partial differential ope rator Q

can

be analytically continued in momentum space:

ððk

; k

Þ; ...; ðk

; k

ÞÞ

= Q

ððik

; k

Þ; ...; ðik

; k

ÞÞ

½17

, ..., k

2 R

. With the definition

2;



ðk

; k

Þ¼ð2Þ

ðdþ1Þ

2;



ðk

; k

l ¼1



l ¼1





ðk

Þðk

þ k

Þ½18

and

n;



ðk

; ...; k

¼ Q

n;



ðk

; ...; k



;...; l

¼1

j ¼1

n;m

;...; m

ðk

; ...; k

Þ½19

the analytic continuation of Schw inger functions can

be summarized as follows:

Theorem 2 The truncated Schwinger functions

have a Fourier–Laplace representation with

defined in eqns [18] and [19]. Equivalently, S

is the

analytic continuation of W

from purely real

relativistic time to purely imaginary Euclidean

time. The truncated Wight man functions W

fulfill

the requirements of temperedness, relativistic covar-

iance with respect to the representation of the

orthochronous, prop er Lorentz group

 : L

(d) !

Gl(L), locality, spectral property, and cluster prop-

erty. Her e

 is obtained by analytic continuation of 

to a representation of the proper complex Lorentz

group over C

(which contains SO(d) as a real

submanifold) and restriction of this representation

to the real orthochronous proper Lor entz group.

Again making use of the explicit formula in

Theorem 2, the condition of Theorem 1 can be verified.

This proves the existence of IMQFT models associated

with the class of random fields under discussion.

Theorem 3 The Wightman functions defined in

Theorem 2 fulfill the HSSC [2]. In particular, there

exists a QFT with indefinite metric such that the

Wightman functions are given as the VEVs of that

IMQFT.

Nontrivial Scattering

Theories as described in Theorem 2 obviously have

trivial scattering behavior if the noise field  is

Gaussian, that is, if, in [7], z = 0. In the case where

there is also a Poisson component in , that is, z > 0,

higher-order truncated Wightman functions do not

vanish and such relativistic theories have nontrivial

scattering.

Before the scattering of the models can be

discussed, some comments about scattering in

IMQFT in general are in order. The scattering

Quantum Fields with Indefinite Metric: Non-Trivial Models 219

theory in axiomatic QFT, Haag–Ruelle theory, relies

on positivity. In fact, one can show that in the class

of models under discussion, the LSZ asymptotic

condition is violated if dipole degrees of freedom are

admitted. In that case more complicated asymptotic

conditions have to be used. In any case, the Haag–

Ruelle theory cannot be adapted to IMQFT.

Nevertheless, asymptotic fields and states can be

constructed in IMQFT if one imposes a no-dipole

condition in a mathematically precise way. Then the

LSZ asymptotic condition leads to the construction of

mixed VEVs of asymptotic in- and out-fields with local

fields. The collection of such VEVs is called the form-

factor functional. After constructing this collection of

mixed VEVs, one can try to check the HSSC for this

functional and obtains a Krein space representation for

the algebra generated by in- local and out-fields.

Following this line, asymptotic in- and out-particle

states can be constructed for the given mass spectrum

, ..., m

). If a

in=outy

, l

(k), l = 1, ..., P,denotesthe

creation operator for an incoming/outgoing particle

with mass m

, spin component , and energy–momen-

tum k, the following scattering amplitude can be derived

for r incoming particles with masses m

, ..., m

and

n  r outgoing particles with masses m

rþ1

, ..., m

iny



ðk

Þa

iny



ðk

Þ; a

outy



rþ1

ðk

rþ1

Þa

outy



ðk

Þ

¼ð2ÞiQ



;...;

ðk

; ...; k

; k

rþ1

; ...; k



j ¼1



ðk

ÞðK

 K

out

Þ½20

in=out

stand for the total energy–momentum of

in- and out-particles, that is, K

j = 1

and

out

j = rþ1

Two immediate consequences can be drawn from

[20]. First, choosing a model with nonvanishing

Poisson part such that C



6¼ 0 and a differential

operator D containing in its mass spectrum the

masses m and  with m > 2, one gets a nonvanish-

ing scattering amplitude for the pro cess

½21

even though in- and out-particle states consist of

particles with well-defined sharp masses. Thus, for the

incoming particle, the energy uncertainty, which for a

particle at rest is proportional to the mass uncertainty,

vanishes but still the particle undergoes a nontrivial

decay and must have a finite decay time. This appears

to be a contradiction to the energy–time uncertainty

relation, which therefore seems to have an unclear

status in IMQFT (i.e., in QFT including gauge fields).

The origin of this inequality, which of course is

experimentally very well tested, apparently has to be

located in the constraints, that is, in the procedure of

implementing a gauge, of the theory and not in the

unconstrained IMQFT.

Second, one can replace somewhat artificially the

polynomials Q

in [17] by any other symmetric and

relativistically covariant polynomial. If the sequence of

the ‘‘new’’ Q

is of uniformly bounded degree in any

of the arguments k

, ..., k

, the redefined Wightman

functions in [17] still fulfill the requirements of

Theorem 1 and thus define a new relativistic, local

IMQFT. The scattering amplitudes of such a theory

are again well defined and given by [20]. For example,

in the case of only one scalar particle with mass m ,one

can show that arbitrary Lorentz-invariant scattering

behavior of bosonic particles can be reproduced by

such theories for energies below an arbitrary maximal

energy up to arbitrary precision. This kind of

interpolation theorem shows that the outcome of an

arbitrary scattering experiment can be reproduced

within the formalism of (unconstrained) IMQFT as

long as it is in agreement with the general requirements

of Poincare´ invariance and statistics.

List of Symbols

converges to

convergence in law

set of natural numbers

set of natural numbers and zero

set of real numbers

set of complex numbers

identity mapping

restricted to D

and x

time and spatial part of

x = (x

, x) 2 R  R

d1

gradient operator on R

See also: Algebraic Approach to Quantum Field Theory;

Euclidean Field Theory; Indefinite Metric; Perturbative

Renormalization Theory and BRST; Quantum Field

Theory in Curved Spacetime; Quantum Field Theory: A

Brief Introduction; Stochastic Differential Equations.