Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Turbulence; Large Deviations in Equilibrium Statistical
Mechanics; Lyapunov Exponents and Strange Attractors;
Nonequilibrium Statistical Mechanics: Interaction
Between Theory and Numerical Simulations;
Nonequilibrium Statistical Mechanics (Stationary):
Overview; Phase Transitions in Continuous Systems;
Polygonal Billiards; Regularization for Dynamical Zeta
Functions; Singularity and Bifurcation Theory; von
Neumann Algebras: Introduction, Modular Theory, and
Classification Theory.
Further Reading
Aaronson J (1997) An Introduction to Infinite Ergodic Theory.
Mathematical Surveys and Monographs 50. Providence, RI:
American Mathematical Society.
Bergelson V, Boshernitzan M, and Bourgain J (1994) Some results
on non-linear recurrence. Journal d’analyse mathematique
62: 444–458.
Billingsley P (1965) Ergodic Theory and Information. New York:
Wiley.
Birkhoff G (1931) Proof of t he ergodic theorem. Proceedings
of the National Academy of Sciences of the USA
17: 656–660.
Bowen R (1975) Equilibrium States and the Ergodic Theory of
Anosov Diffeomorphisms. Springer Lecture Notes in Math-
ematics, vol. 470. Berlin: Springer.
Bunimovich LA et al. (2000) Dynamical Systems, Ergodic Theory
and Applications. Encyclopedia of Mathematical Science,
vol. 100. Berlin: Springer.
Cornfeld IP, Fomin SV, and Sinai YaG (1982) Ergodic Theory.
Grundlehren der mathematischen Wissenschaften, vol. 245.
Berlin: Springer.
Darling DA and Kac M (1957) On occupation times for Markov
processes. Transactions of the American Mathematical Society
84: 444–458.
Kolmogorov AN (1958) A new metric invariant of transitive
dynamical systems and Lebesgue space automorphisms. Dok-
lady Akademii Nauk USSR 119: 861–864.
Koopman BO (1931) Hamiltonian systems and transformations in
Hilbert space. Proceedings of the National Academy of
Sciences of the USA 17: 315–318.
Lin M (1971) Mixing for Markov operators. Zeitschrift
fur Wahrscheinlichkei tstheorie und Verwandte Gebiete
19: 231–234.
Ornstein D (1970) Bernoulli shifts with the same entropy are
isomorphic. Advances in Mathematics 4: 337–352.
Pesin YB (1977) Characteristic Lyapunov exponents and
smooth ergodic theory. Russian Mathematic al Surveys
32: 54–114.
Rohlin VI (1964) Exact endomorphism of a Lebesgue space.
American Mathematical Society Translations 39: 1–36.
Ruelle D (1978) An inequality for the entropy of differential
maps. Boletim da Sociedade Brasileira de Mathematica
9: 83–87.
Ruelle D (1997) Entropy production in nonequilibrium statistical
mechanics. Communications in Mathematical Physics
189: 365–371.
Shanonn C (1948) A mathematical theory of communications.
Bell System Technical Journal 27: 379–423, 623–656.
Sinai YaG (1970) Dynamical systems with elastic reflections.
Russian Mathematical Surveys 25: 137–189.
von Neumann J (1932) Proof of the quasi-ergodic hypothesis.
Proceedings of the National Academy of Sciences of the USA
18: 70–82.
Walters P (1981) An Introduction to Ergodic Theory. GTM,
Springer Verlag.
Euclidean Field Theory
F Guerra, Universita
`
di Roma ‘‘La Sapienza,’’
Rome, Italy
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
In this article, we consider Euclidean field theory as
a formulation of quantum field theory which lives in
some Euclidean space, and is expressed in probabil-
istic terms. Methods arising from Euclidean field
theory have been introduced in a very successful
way in the study of concrete models of constructive
quantum field theory.
Euclidean field theory was initiated by Schwinger
(1958) and Nakano (1959), who proposed to study
the vacuum expectation values of field products
analytically continued into the Euclidean region
(Schwinger functions), where the first three (spatial)
coordinates of a world point are real and the last one
(time) is purely imaginary (Schwinger points). The
possibility of introducing Schwinger functions, and
their invariance under the Euclidean group are immedi-
ate consequences of the now classic formulation of
quantum field theory in terms of vacuum expectation
values given by Wightman (Streater and Wightman
1964). The convenience of dealing with the Euclidean
group, with its positive-definite scalar product, instead
of the Lorentz group, is evident, and has been exploited
by several authors, in different contexts.
The next step was made by Symanzik (1966), who
realized that Schwinger functions for boson fields
have a remarkable positivity property, allowing to
introduce Euclidean fields on their own sake.
Symanzik also pointed out an analogy between
Euclidean field theory and classical statistical
mechanics, at least for some interactions (Symanzik
1969).
This analogy was successfully extended, with a
different interpretation, to all boson interactions by
Guerra et al. (1975), with the purpose of using
rigorous results of modern statistical mechanics for
256 Euclidean Field Theory
the study of constructive quantum field theory,
within the program advocated by Wightman (1967),
and further pursued by Glimm and Jaffe (see Glimm
and Jaffe (1981) for an overall presentation).
The most dramatic advance of Euclidean theory
was due to Nelson (1973a, b). He was able to isolate
a crucial property of Euclidean fields (the Markov
property) and gave a set of conditions for these
fields, which allow us to derive all properties of
relativistic quantum fields satisfying Wightman
axioms. The Nelson theory is very deep and rich in
new ideas. Even after so many years since the basic
papers were published, we lack a complete under-
standing of the radical departure from the conven-
tional theory afforded by Nelson’s ideas, especially
about their possible further developments.
By using the Nelson scheme, in particular a very
peculiar symmetry property, it was very easy to prove
(Guerra 1972) the convergence of the ground-state
energy density, and the van Hove phenomenon in the
infinite-volume limit for two-dimensional boson
theories. A subsequent analysis (Guerra et al. 1972)
gave other properties of the infinite-volume limit of
the theory, and allowed a remarkable simplification
in the proof of a very important regularity property
for fields, previously established by Glimm and Jaffe.
Since then, all work on constructive quantum field
theory has exploited in different ways ideas coming
from Euclidean field theory. Moreover, a very
important reconstruction theorem has been estab-
lished by Osterwalder and Schrader (1973), allowing
a reconstruction of relativistic quantum fields from
the Euclidean Schwinger functions, and avoiding the
previously mentioned Nelson reconstruction theorem,
which is technically more difficult to handle.
This article is intended to be an introduction to the
general structure of Euclidean quantum field theory,
and to some of the applications to constructive
quantum field theory. Our purpose is to show that,
50 years after its introduction, the Euclidean theory is
still interesting, both from the point of view of
technical applications and physical interpretation.
The article is organized as follows. In the next
section, by considering simple systems made of a
single spinless relativistic particle, we introduce the
relevant structures in both Euclidean and Minkowski
worlds. In particular, a kind of (pre)Markov property
is introduced already at the one-particle level.
Next we present a description of the procedure of
second quantization on the one-particle structure.
The free Markov field is introduced, and its crucial
Markov property explained. Following Nelson, we
use probabilistic concepts and methods, whose
relevance for constructive quantum field theory
became immediately more and more apparent. The
very structure of classical statistical mechanics for
Euclidean fields is firmly based on these probabil-
istic methods. This is followed by an introduction of
interaction, and we show the connection between
the Markov theory and the Hamiltonian theory, for
two-dimensional space-cutoff interacting scalar
fields. In particular, we present the Feynman–Kac–
Nelson formula that gives an explicit expression of
the semigroup generated by the space-cutoff
Hamiltonian in o space. We also deal with some
applications to constructive quantum field theory.
This is followed by a short discussion about the
physical interpretation of the theory. In particul ar,
we discuss the Osterwalder–Schrader recons truction
theorem on Euclidean Schwinger functions, and the
Nelson reconstruction theorem on Euclidean fields.
For the sake of completeness, we sketch the main
ideas of a proposal, advanced in Guerra and
Ruggiero (1973), accordi ng to which the Euclidean
field theory can be interpreted as a stochastic field
theory in the physical Minkowski sp acetime.
Our treatment will be as simple as possible, by relying
on the basic structural properties, and by describing
methods of presumably very long lasting power. The
emphasis given to probabilistic methods, and to the
statistical mechanics analogy, is a result of the historical
development. Our opinion is that not all possibilities
of Euclidean field theory have been fully exploited
yet, both from technical and physical points of view.
One-Particle Systems
A system made of only one relativistic scalar
particle, of mass m > 0, has a quantum state space
represented by the positive-frequency solutions of
the Klein–Gordon equation. In momentum space,
with points p
, = 0, 1, 2, 3, let us introduce the
upper mass hyperboloid, characterized by the con-
straints p
2
p
2
0
P
3
i = 1
p
2
i
= m
2
, p
0
m, and the
relativistic invariant measure on it, formally given
by d(p) = (p
0
)(p
2
m
2
)dp, where is the step
function (x) = 1ifx 0, and (x) = 0 otherwise,
and dp is the four-dimensional Lebesgue measure.
The Hilbert space of quantum states F is given by
the square-integrable functions on the mass hyper-
boloid equipped with the invariant measure d(p).
Since in some reference frame the mass hyperboloid
is uniquely characterized by the space values of the
momentum p, with the energy given by p
0
!(p) =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2
þ m
2
p
, the Hilbert space F of the states
is, in fact, made of those complex-valued tempered
distributions f in the configuration space R
3
whose
Fourier transforms,
~
f (p), are square-integrable func-
tions in momentum space with respect to the image
of the relativistic invariant measure dp=2!(p), where
Euclidean Field Theory 257
dp is the Lebesgue measure in momentum space.
The scalar product on F is defined by
f ; g
hi
F
¼ð2Þ
3
Z
~
f
ðpÞ
~
gðpÞ
dp
2!ðpÞ
where we have normalized the Fourier transform in
such a way that
f ðxÞ¼
Z
expðip:xÞ
~
f ðpÞdp
~
f ðpÞ¼ð2Þ
3
Z
expðip:xÞ
~
f ðxÞdx
Z
expðip:xÞdp ¼ð2Þ
3
ðxÞ
The scal ar product on F can also be expressed in the
form
f ; g
hi
F
¼
ZZ
f ðx
0
Þ
Wðx
0
xÞgðxÞdx
0
dx
where we have introduced the two-point Wightman
function at fixed time, defined by
Wðx
0
xÞ¼ð2Þ
3
Z
expðip:ðx
0
xÞÞ
dp
2!ðpÞ
A unitary irreducible representation of the Poincare´
group can be defined on F in the obvious way. In
particular, the generators of space translations are
given by mul tiplication by the components of p in
momentum space, and the generator of time transla-
tions (the energy of the particle) is given by !(p).
For the scalar product of time-evolved wave
functions, we can write
expðit
0
Þf ; expðitÞg
hi
F
¼
ZZ
f ðx
0
Þ
Wðt
0
t; x
0
xÞgðxÞdx
0
dx
where we have introduced the two-point Wightman
function, defined by
Wðt
0
t; x
0
xÞ
¼ð2Þ
3
Z
expðiðt t
0
ÞÞexpðip:ðx
0
xÞÞ
dp
2!ðpÞ
To the physical single-particle system living in
Minkowski spacetime, we associate a kind of
mathematical image, living in Euclidean space,
from which all properties of the physical system
can be easily derived. We start from the two-point
Schwinger function
SðxÞ¼
1
ð2Þ
4
Z
expðip xÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2
þ m
2
p
dp
which is the analytic continuation of the previously
given two-point Wightman function into the Schwinger
points. Here x, p 2 R
4
,andp x =
P
4
i = 1
x
i
p
i
.Heredp
and dx are the Lebesgue measures in the R
4
momentum
and configuration spaces, respectively. The function
S(x) is positive and analytic for x 6¼ 0, decreases as
exp (mkxk)asx !1, and satisfies the equation
ð þ m
2
ÞSðxÞ¼ðxÞ
where =
P
4
i = 1
@
2
=@x
2
i
is the Laplacian in four
dimensions.
The mathematical image we are looking for is
described by the Hilbert space N of those tempered
distributions in four-dimensional configuration space
R
4
whose Fourier transforms are square integrable
with respect to the measure dp=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2
þ m
2
p
. The scalar
product on N is defined by
f ; g
hi
N
¼ð2Þ
4
Z
~
f
ðpÞ
~
gðpÞ
dp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2
þ m
2
p
Four-dimensional Fourier transforms are normalized
as follows:
f ðxÞ¼
Z
exp ðip: xÞ
~
f ðpÞ dp
~
f ðpÞ¼ð2Þ
4
Z
exp ðip:xÞ
~
f ðxÞdx
Z
exp ðip:x Þdp ¼ð2Þ
4
ð xÞ
We also write
f ; g
hi
N
¼
ZZ
f
ðxÞSðx yÞgðyÞdx dy
¼ f ; ð þ m
2
Þ
1
g
DE
where ,
hi
is the ordinary Lebesgue product defined
on Fourier transforms and, in momentum space,
( þ m
2
)
1
amounts to a multiplication by
(p
2
þ m
2
)
1
. The Schwinger function S(x y)is
formally the kernel of the operator ( þ m
2
)
1
.
The Hilbert space N is the carrier space of a
unitary (nonirreducible) representation of the four-
dimensional Euclidean group E(4). In fact, let (a, R)
be an element of E(4)
ða; RÞ : R
4
! R
4
x ! Rx þ a
where a 2 R
4
, and R is an orthogonal matrix,
RR
T
= R
T
R = 1
4
. Then the transformation u(a, R)
defined by
uða; RÞ : N ! N
f ðxÞ!ðuða; RÞf ÞðxÞ¼f ðR
1
ðx aÞÞ
provides the representation. In particular, we con-
sider the reflection r
0
with respect to the hyperplane
258 Euclidean Field Theory
x
4
= 0, and the translations u(t) in the x
4
-direction.
Then we have r
0
u(t) r
0
= u(t ), and analogously for
other hyperplanes.
Now we introduce a local structure on N by
considering, for any closed region A of R
4
, the
subspace N
A
of N made by distributions in N with
support on A. We call e
A
the orthogonal projection
on N
A
. It is obvious that if A 2 B then N
A
2 N
B
and
e
A
e
B
= e
B
e
A
= e
A
. A kind of (pre)Markov property
for one-particle systems is introduced as follows.
Consider a closed three-dimensional piecewise
smooth manifold , which divides R
4
in two closed
regions A and B, having in common. Therefore,
2 A, 2 B, A \ B = , A [ B = R
4
. Let N
A
, N
B
, N
,
and e
A
, e
B
, e
be the associated subspaces and
projections, respectively. Then N
N
A
, N
N
B
,
and e
e
A
= e
A
e
= e
, e
e
B
= e
B
e
= e
. It is very
simple to prove the following:
Theorem 1 Let e
A
, e
B
, e
be defined as above, then
e
A
e
B
= e
B
e
A
= e
.
Clearly, it is enough to show that for any f 2 N
we have e
A
e
B
f 2 N
. In that case, e
e
A
e
B
f = e
A
e
B
f ,
from which the theorem easily follows. Since e
A
e
B
f
has support on A, we must show that for any C
1
0
function g with support on A
we have
g, e
A
e
B
fhi= 0. Then e
A
e
B
f has support on , and
the proof is complete. Now we have
g; e
A
e
B
f
hi
¼ð þ m
2
Þg; e
A
e
B
f
N
¼ e
A
ð þ m
2
Þg; e
B
f
N
¼ð þ m
2
Þg; e
B
f
N
¼ g; e
B
f
hi
¼ 0
where we have used the definition of
hi
N
in terms of
hi
, the fact that e
A
( þ m
2
)g = ( þ m
2
)g, since
( þ m
2
)g has support on A
, and the fact that e
B
f
has support on B. This ends the proof of the
(pre)Markov property for one-particle systems.
A very important role in the theory is played by
subspaces of N associated to hyperplanes in R
4
.To
fix ideas, consider the hyperplane x
4
= 0 and the
associated subspace N
0
. A tempered distribution in
N with support on x
4
= 0 has necessarily the form
(f
0
)(x) f (x)(x
4
), with f 2 F. By using the
basic magic formula, for x 0andM > 0,
Z
þ1
1
expðipxÞ
p
2
þ M
2
dp ¼
M
exp ðMxÞ
it is immediate to verify that kf
0
k
N
= kf k
F
.
Therefore, we have an isomorphic and isometric
identification of the two Hilbert spaces F and N
0
.
Obviously, similar considerations hold for any
hyperplane. In particular, we consider the
hyperplanes x
4
= t and the associated subspaces N
t
.
Let us introduce injection operat ors j
t
defined by
j
t
: F ! N
f ! f
t
where f is a generic element of F, with values f (x),
and (f
t
)(x) = f (x)(x
4
t). It is immediate to
verify the following properties for j
t
and its adjoint
j
t
: the range of j
t
is N
t
; moreover, j
t
is an isometry,
so that j
t
j
t
= 1
F
, j
t
j
t
= e
t
, where 1
F
is the identity on
F,ande
t
is the projection on N
t
. Moreover, e
t
j
t
= j
t
and j
t
= j
t
e
t
.
If we introduce translations u(t) along the
x
4
-direction and the reflection r
0
with respect to
x
4
= 0, then we also have the covariance property
u(t) j
s
= j
tþs
, and the reflexivity pro perty r
0
j
0
= j
0
,
j
0
r
0
= j
0
. The reflexivity property is very important.
It tells us that r
0
leaves N
0
pointwise invariant, and
it is an immediate consequence of the fact that
(x
4
) = (x
4
).
Therefore, if we start from N we can obtain F,by
taking the projection j
with respect to some
hyperplane , in particular x
4
= 0. It is also obvious
that we can induce on F a representati on of E(3) by
taking those elements of E(4) that leave invariant.
Let us now see how we can define the Hamiltonian
on F starting from the properties of N. Since we are
considering the simple case of the one-particle system,
we could just perform the following construction
explicitly by hand, through a simple application of
the basic magic formula given earlier. But we prefer
to follow a route that emphasizes Markov property
and can be immediately generalized to more
complicated cases.
Let us introduce the operator p(t)onF defined by
the dilation p(t) = j
0
j
t
= j
0
u(t)j
0
, t 0. Then we
prove the following:
Theorem 2 The operator p(t) is bounded and self-
adjoint. The family {p(t)}, for t 0, is a norm-
continuous semigroup.
Proof Boundedness and continuity are obvious. Self-
adjointness is a consequence of reflexivity. In fact,
p
ðtÞ¼j
0
uðt Þj
0
¼ j
0
r
0
uðtÞr
0
j
0
¼ j
0
uðtÞj
0
¼ pðtÞ
The semigroup property is a consequence of the
Markov property. In fact, let us intro duce
N
þ
, N
0
, N
as sub spaces of N made by distributions
with support in the regions x
4
0, x
4
= 0, x
4
0,
respectively, and call e
þ
, e
0
, e
the respective projec-
tions. By Markov property, we have e
0
= e
e
þ
. Now
write, for s, t 0,
pðtÞpð sÞ¼j
0
uðtÞj
0
j
0
uðsÞj
0
¼ j
0
uðtÞe
0
uðsÞj
0
Euclidean Field Theory 259
If e
0
could be cancelled, then the semigroup
property would follow from the group property of
the translations u(t)u(s) = u( t þ s ) (a miracle of the
dilations!). For this, consider the matrix element
f ; pðtÞpðsÞghi
F
¼ uð tÞj
0
f ; e
0
uðsÞj
0
ghi
N
recall e
0
= e
e
þ
, and use u(s)j
0
g 2 N
þ
and
u(t)j
0
f 2 N
.
Let us call h the generator of p(t), so that
p(t) = exp (th), for t 0. By definition, h is the
Hamiltonian of the physical system. A simple
explicit calculation shows that h is just the energy
! introduced earlier. Starting from the representa-
tion of the Euclidean group E(3) already given and
from the Hamiltonian, we immediately get a
representation of the full Poincare´ group on F.
Therefore, all physical properties of the one-particle
system have been reconstructed from its Euclidean
image on the Hilbert space N.
As a last remark of this section, let us note that we
can consider the real Hilbert spaces N
r
and F
r
,madeof
real elements (in configuration space) in N and F.The
operators u(a, t), u(t), r
0
, j
, j
, e
A
are all reality preser-
ving, that is, they map real spaces into real spaces.
This completes our discussion about the one-
particle system. For more details we refer to Guerra
et al. (1975) and Simon (1974). We have introduced
the Euclidean image, discussed its main properties,
and shown how we can derive all properties of the
physical system from its Euclidean image. In the
next sections, we will show how this kind of
construction carries through the second-quantized
case and the interacting case.
Second Quantization and Free Fields
We begin this section with a short review about the
procedure of second quantization based on prob-
abilistic methods, by following mainly Nelson
(1973b); see also Guerra et al. (1975) and Simon
(1974). Probabilistic methods are particularly useful
in the framework of the Euclidean theory.
Let H be a real Hilbert space with symmetric
scalar product ,
hi
. Let (u) be the elements of a
family of centered Gaussian random variables
indexed by u 2H, uniquely defined by the expecta-
tion values E((u)) = 0, E ((u)(v)) = u, v
hi
. Since
is Gaussian, we also have
EðexpððuÞÞÞ ¼ exp
1
2
2
u; u
hi
and
Eððu
1
Þðu
2
Þðu
n
ÞÞ ¼ ½u
1
u
2
u
n
Here [ ...] is the Hafnian of elements
[u
i
u
j
] = u
i
, u
j
, defined to be zero for odd n, and
for even n given by the recursive formula
½u
1
u
2
u
n
¼
X
n
i¼2
½u
1
u
i
½u
1
u
2
u
n
0
where in [ ...]
0
the terms u
1
and u
i
are suppressed.
Hafnians, from the Latin name of Copenhagen, the
first seat of the theoretical group of CERN, were
introduced in quantum field theory by Caianiello
(1973), as a useful tool when dealing with Bose
statistics.
Let (Q, , ) be the underlying probability space
where are defined as random variables. Here Q is
a compact space, a -algebra of subsets of Q, and
a regul ar, countable additive probability measure
on , normalized to (Q) =
R
Q
d = 1.
The fields (u) are represented by measurable
functions on Q. The probability space is uniquely
defined, but for trivial isomorphisms, if we assume
that is the smallest -algebra with respect to
which all fields (u), with u 2H, are measurable.
Since (u) are Gaussian, they are represented by
L
p
(Q, , ) functions, for any p with 1 p < 1,
and the expectations will be given by
Eððu
1
Þðu
2
Þðu
n
ÞÞ
¼
Z
Q
ðu
1
Þðu
2
Þðu
n
Þd
where, by a mild abuse of notation, (u
i
) on the
right-hand side denote the Q space functions which
represent the random variables (u
i
). We call the
complex Hilbert space F =(H) = L
2
(Q, , ) the
ok space constructed on H, and the function
0
1onQ the ok vacuum.
In order to introduce the concept of second
quantization of operators, we must introduce sub-
spaces of F with a ‘‘fixed number of particles.’’ Call
F
(0)
= {
0
}, wher e is any complex number.
Define F
(n)
as the subspace of F generated by
complex linear combinations of monomials of the
type (u
1
) (u
j
), with u
i
2H, and j n. Then
F
(n1)
is a subspace of F
(n)
. We define F
(n)
, the
n-particle subspace, as the orthogonal complement
of F
(n1)
in F
(n)
, so that
F
ðnÞ
¼F
ðnÞ
F
ðn1Þ
By construction, the F
(n)
are orthogonal, and it is
not difficult to verify that
F¼
M
1
n¼0
F
ðnÞ
260 Euclidean Field Theory
Let us now introduce the Wick normal products
by the definition
:ðu
1
Þðu
2
Þðu
n
Þ:¼ E
ðnÞ
ðu
1
Þðu
2
Þðu
n
Þ
where E
(n)
is the projection on F
(n)
. It is not difficult
to prove the usual Wick theorem (see, e.g., Guerra
et al. (1975), and its inversion given by Caianiello
(1973).
It is inte resting to remark that, in the framework
of the second quantization performed with prob-
abilistic methods, it is not necessary to introduce
creation and destruction operators as in the usual
treatment. However, the two procedures are com-
pletely equivalent, as shown, for example, in Simon
(1974).
Given an operator A from the real Hilbert space
H
1
to the real Hilbert space H
2
, we define its
second-quantized operator (A) throug h the follow-
ing definitions:
ðAÞ
01
¼
02
ðAÞ :
1
ðu
1
Þ
1
ðu
2
Þ...
1
ðu
n
Þ:
¼:
2
ðAu
1
Þ
2
ðAu
2
Þ
2
ðAu
n
Þ:
where we have introduced the probability spaces Q
1
and Q
2
, their vacua
01
and
02
, and the random
variables
1
and
2
, associated to H
1
and H
2
,
respectively. The following remarkable theorem by
Nelson (1973b) gives a full characterization of (A),
very useful in the applications.
Theorem 3 Let A be a contraction from the real
Hilbert space H
1
to the real Hilbert space H
2
. Then
(A) is an operator from L
1
(1)
to L
1
(2)
which is
positivity preserving, (A)u 0 if u 0, and such
that E((A)u) = E(u). Moreover , (A) is a contrac-
tion from L
p
(1)
to L
p
(2)
for any p,1 p < 1. Finally,
(A) is also a contraction from L
p
(1)
to L
q
(2)
, with
q p, if kAk
2
(p 1) = (q 1).
We have indicated with L
p
(1)
, L
p
(2)
the L
p
spaces
associated to H
1
and H
2
, respectively. This is the
celebrated best hypercontractive estimate given by
Nelson. For the proof, we refer to the original paper
of Nelson (1973b); see also Simon (1974).
This completes our short review on the theory of
second quantization based on probabilistic methods.
The usual time-zero quantum field
(u), u 2 F
r
,in
the ok representation, can be obtained through
second quantization starting from F
r
. We call
(
Q,
,
) the underlying probability space, and
F =(F
r
) = L
2
(
Q,
,
) the Hilbert ok space of
the free physical particles.
Now we introduce the free Markov field (f ), f 2
N
r
, by taking N
r
as the starting point. We call
(Q, , ) the associated probability space. We
introduce the Hilbert space N =(N
r
) =
L
2
(Q, , ), and the operators U(a, R) =( u(a, R)),
R
0
=(r
0
), U(t ) =(u(t)), E
A
=(e
A
), and so on, for
which the previous Nelson theorem holds (take
H
1
= H
2
= N
r
).
Since in genera l (AB) =(A)(B), we have
immediately the following expression of the Markov
property E
= E
A
E
B
, where the closed regions
A, B, of the Euclidean space have the same
properties as explained earlier in the proof of the
(pre)Markov property for one-particle systems.
It is obvious that E
A
can also be understood as
conditional expectation with respect to the sub--
algebra
A
generated by the field (f ) with f 2 N
r
and the support of f on A.
The relation, previous ly pointed out, between N
t
subspaces and F are also valid for their real parts
N
rt
and F
r
. Therefore, they carry out through the
second quantization procedure. We introduce
J
t
=(j
t
)andJ
t
=(j
t
); then the following proper-
ties hold. J
t
is an isometric injection of L
p
(
Q,
,
)
into L
p
(Q, , ); the range of J
t
as an operator L
2
!L
2
is obviously N
t
=(N
rt
); moreover, J
t
J
t
= E
t
.Thefree
Hamiltonian H
0
is given for t 0by
J
0
J
t
¼ expðtH
0
Þ¼ðexpðt!ÞÞ
Moreover, we have the covariance property
U(t)J
0
= J
t
, and the reflexivity R
0
J
0
= J
0
, J
0
R
0
= J
0
.
These relations allow a very simple expression for
the matrix elements of the Hamiltonian semigroup
in terms of Markov quantities. In fact, for u, v 2F
we have
u; exp ðtH
0
Þv
hi
¼
Z
Q
ðJ
t
uÞ
J
0
v d
In the next section, we will generalize this
representation to the interacting case.
Finally, let us derive the hypercontractive property
of the free Hamiltonian semigroup.
Since kexp (t!)kexp (tm), where m is the
mass of the particle, we have immediately, by a
simple application of Nelson theorem,
kexpðtH
0
Þk
p;q
1
provided q 1 (p 1) exp (2tm), where k...k
p, q
denotes the norm of an operator from L
p
to L
q
spaces.
Interacting Fields
The discussion of the previous sections was limited
to free fields both in Minkowski and Euc lidean
spaces. Now we must introduce interact ion in order
to get nontrivial theories.
Euclidean Field Theory 261
First, as a general motivation, we will proceed
quite formally and then we will resort to precise
statements.
Let us recall that in standard quantum field theory,
for scalar self-coupled fields, the time-ordered pro-
ducts of quantum fields in Minkowski spacetime can
be expressed formally through the formula
Tððx
1
Þðx
n
Þexpði
R
LdxÞÞ
T expði
R
LdxÞ
where T denotes time ordering, are free fields in
Minkowski spacetime, L is the interaction Lagrangian,
and ...
hi
are vacuum averages. As is well known, this
expression can be put, for example, at the basis of
perturbative expansions, giving rise to terms expressed
through Feynman graphs. The appropriately chosen
normalization provides automatic cancelation of the
vacuum to vacuum graphs.
Now we can introduce a formal analytic continua-
tion to the Schwinger points, as previously done for
the one-particle system, and obtain the following
expression for the analytic continuation of the field
time-ordered products, now called Schwinger
functions,
Sðx
1
; ...; x
n
Þ¼
ðx
1
Þðx
n
Þexp U
hi
exp U
hi
Here x
1
, ..., x
n
denote points in Euclidean space,
are the Euclidean fields introduced earlier. The
chronological time ordering disappears, because the
fields are commutative, and there is no distin-
guished ‘‘time’’ direction in Euclidean space. Here
the symbol ...
hi
denotes the expectation values
represented by
R
...d, as explained earlier, and U
is the Euclidean ‘‘action’’ of the system formally
given by the integral on Euclidean space
U ¼
Z
PððxÞÞdx
if the field self-interaction is produced by the
polynomial P.
Therefore, these formal considerations suggest
that the passage from the free Euclidean theory to
the fully interacting one is obtained through a
change of the free probability measure d to the
interacting measure
exp U d=
Z
Q
exp U d
The analogy with classical statistical mechanics is
evident. The expression exp U acts as Boltzmannfaktor,
and Z =
R
Q
exp U d is the partition function.
Our task will be to make these statements precise
from a mathematical point of view. We will be
obliged to introduce cutoffs, a nd then be involved in
their careful removal.
For the sake of convenience, we make the
substantial simplification of considering only two-
dimensional theories (one space, one time dimension
in the Minkowski region) for which the well-known
ultraviolet problem of quantum field theory gives no
trouble. There is no difficulty in translating the
contents of the previous sections to the two-
dimensional case.
Let P be a real polynomial, bounded below and
normalized to P(0) = 0. We introduce approxima-
tions h to the Dirac function at the origin of the
two-dimensional Euclidean space R
2
, with h 2 N
r
.
Let h
x
be the translate of h by x, with x 2 R
2
. The
introduction of h, equivalent to some ultraviolet
cutoff, is necessary, because local fields, of the
formal type (x), have no rigorous meaning, and
some smearing is necessary.
For some compact region in R
2
, acting as space
cutoff (infrared cutoff), introduce the Q space
function
U
ðhÞ
¼
Z
:Pððh
x
ÞÞ: dx
where dx is the Lebesgue measure in R
2
.Itis
immediate to verify that U
(h)
is well defined,
bounded below and belongs to L
p
(Q, , ), for any
p,1 p < 1. This is the infrared and ultraviolet
cutoff action. Notice the presence of the Wick
normal products in its definition. They provide a
kind of automatic introduction of counterterms, in
the framework of renormalization theory.
The following theorem allows us to remove the
ultraviolet cutoff.
Theorem 4 Let h !, in the sense that the Four ier
transforms
~
h are uniformly bounde d and converge
pointwise in momentum space to the Fourier trans-
form of the -function given by (2)
2
. Then U
(h)
is
L
p
-convergent for any p,1 p < 1, as h !. Call
U
the L
p
-limit, then U
, exp U
2 L
p
(Q,
, ), for
1 p < 1.
The proof uses standard methods of probability
theory, and originates from pioneering work of
Nelson in (1966). It can be found for example in
Guerra et al. (1975), and Simon (1974).
Since U
is defined with normal products, and
the interaction polynomial P is normalized to
P(0) = 0, an elementary application of Jensen
inequality gives
Z
Q
exp U
d exp :
Z
Q
U
d ¼ 1
262 Euclidean Field Theory
Therefore, we can rigorously define the new space
cutoff measure in Q space:
d
¼ exp U
d=
Z
Q
exp U
d
The space-cutoff interacting Euclidean theory is
defined by the same fields on Q space, but with a
change in the measure and, therefore, in the
expectation values. The corre lations for the inter-
acting fields
are the cutoff Schwinger functions
S
ðx
1
; ...; x
n
Þ¼
ðx
1
Þ
ðx
n
Þ
¼ Z
1
ðx
1
Þðx
n
Þexp U
hi
where the partition function is
Z
¼ exp U
hi
We see that the analogy with statistical mechanics
is complete here. Of course, the introduction of the
space cutoff destroys translation invariance. The
full Euclidean covari ant theory must be recovered by
taking the infinite-volume limit !R
2
on field
correlations. For the removal of the space cutoff, all
methods of statistical mechanics are available. In
particular, correlation inequalities of ferromagnetic
type can be easily exploited, as shown, for example,
in Guerra et al. (1975) and Simon (1974).
We would like to conclude this section by giving
the connection between the space-cutoff Euclidean
theory and the space-cutoff Hamiltonian theory in
the physical ok space.
For ‘ 0, t 0, consider the rectangle in R
2
,
ð‘; tÞ¼ ðx
1
; x
2
Þ:
‘
2
x
1
‘
2
; 0 x
2
t
and define the operator in the physical ok space
P
‘
ðtÞ¼J
0
exp U
ð‘; tÞJ
t
where J
0
and J
t
are injections relative to the lines
x
2
= 0 and x
2
= t, respectively. Then the following
theorem, largely due to Nelson, holds.
Theorem 5 The operator P
‘
(t) is bounded and self-
adjoint. The family {P
‘
(t)}, for ‘ fixed and t 0,
is a strongly continuous semigroup. Let H
‘
be
its lower bounded self-adjoint generator, so that
P
‘
(t) = exp (tH
‘
). On the physical ok space, there
is a core D for H
‘
such that on D the equality
H
‘
= H
0
þ V
‘
holds, where H
0
is the free Hamiltonian
introduced earlier and V
‘
is the volume-cutoff
interaction given by
V
‘
¼ lim
Z
‘=2
‘=2
: Pð
ðh
x
1
ÞÞ: dx
1
where h
x
1
are the translates of approximations to
the -function at the origin on the x
1
-space, and the
limit is taken in L
p
, in analogy to what has been
explained for the tw o-dimensional case in the
definition of U
.
While we refer to Guerra et al. (1975) and Simon
(1974) for a full proof, we mention here that
boundedness is related to hypercontractivity of the
free Hamiltonian, self-adjointness is a consequence
of reflexivity, and the semigroup property follows
from Markov property. This theorem is remark-
able, because it expresses the cutoff interacting
Hamiltonian semigroup in an explicit form in the
Euclidean theory through probabilistic expectations.
In fact, we have
u; expð tH
‘
Þv
hi
¼
Z
Q
ðJ
t
uÞ
J
0
v exp U
ð‘; tÞd
We could call this expression as the Feynman–Kac–
Nelson formula, in fact it is nothing but a path
integral expressed in stochastic terms, and adapted to
the Hamiltonian semigroup.
By comparison with the analogous formula given
for the free Hamiltonian semigroup, we see that
the introduction of the interaction inserts the
Boltzmannfaktor under the integral.
As an immediate consequence of the Feynman–
Kac–Nelson formula, together with Euclidean cov-
ariance, we have the following astonishing Nelson
symmetry:
0
; ex pðtH
‘
Þ
0
hi
¼
0
; expð ‘H
t
Þ
0
hi
which was at the basis of Guerra (1972) and Guerra
et al. (1972), and played some role in showing the
effectiveness of Euclidean methods in constructive
quantum field theory.
It is easy to establish, through simple probabilistic
reasoning, that H
‘
has a unique ground state
‘
of
lowest energy E
‘
. For a convenient choice of
normalization and phase factor, one has k
‘
k
2
= 1,
and
‘
> 0 almost everywhere on Q space (for
bosonic systems, ground states have no nodes in
configuration space!). Moreover,
‘
2 L
p
, for any
1 p < 1.If‘>0 and the interaction is not
trivial, then
‘
6¼
0
, E
‘
< 0, and k
‘
k
1
< 1.
Obviously, kexp (tH
‘
)k
2, 2
= exp (tE
‘
).
The general structure of Euclidean field theory, as
explained in this section, has been at the basis of all
applications in constructive quantum field theory.
These applications include the proof of the existence
of the infinite-volume limit, with the establishment of
all Wightman axioms, for two- and three-dimensional
theories. Moreover, the existence of phase transitions
and symmetry breaking has been firmly established.
Euclidean Field Theory 263
Extensions have also been given to theories involving
Fermions, and to gauge field theory. Due to the scope
of this review, limited to a description of the general
structure of Euclidean field theory, we cannot give
a detailed treatment of these applications. Therefore,
we refer to recent general reviews on constructive
quantum field theory for a complete description of
all results (see, e.g., Jaffe (2000)). For recent applica-
tions of Euclidean field theory to quantum fields on
curved spacetime manifolds we refer, for example, to
Schlingemann (1999).
The Physical Interpretation of Euclidean
Field Theory
Euclidean field theory has been considered by most
researchers as a very useful tool for the study of
quantum field theory. In particular, it is quite easy,
for example, to obtain the fully interacting Schw in-
ger functions in the infinite-volume limit in two-
dimensional spacetime. At this point, there arises
the problem of connecting these Schwinger func-
tions with observable physical quantities in Min-
kowski spacetime. A very deep result of
Osterwalder and Schrader (1973) gives a very
natural interpretation of the resulting limiting
theory. In fact, the Euclidean theory, as has been
shown earlier, arises from an analytic continuation
from the physical Minkowski spacetime to the
Schwinger points, through a kind of analytic
continuation in time (also called Wick rotation,
because Wick exploited this trick in the study of
the Bethe–Salpeter equation). Therefore, having
obtained the Schwinger functions for the full
covariant theory, after all cutoff removal, it is
very natural to try to reproduce the inverse analytic
continuation in order to recover the Wightman
functions in Minkowski spacetime. Therefore,
Osterwalder and Schrader have been able to
identify a set of conditions, quite easy to verify,
wich allow us to recover Wight man functions from
Schwinger functions. A key role in this reconstruc-
tion theorem is played by the so-called reflection
positivity for Schwinger functions, a property quite
easy to verify. In this way, a fully satisfactory
solution for the physical interpretation of Euclidean
field theory is achieved.
From a historical point of view, an alternate route
is possible. In fact, at the beginning of the exploita-
tion of Euclidean methods in constructive quantum
field theory, Nelson was able to isolate a set of
axioms for the Euclidean fields (Nelson 1973a),
allowing the reconstruction of the physical theory.
Of course, Nelson axioms are more difficult to
verify, since they also involve properties of the
Euclidean fields and not only of the Schwinger
functions. However, it is still very interesting to
investigate whether the Euclidean fields play only an
auxiliary role in the construction of the physical
content of relativistic theories, or if they have a
more fundamental meaning.
From a physical point of view, the following
considerations could also lead to further developments
along this line. By its very structure, the Euclidean
theory contains the fixed-time quantum correlations in
the vacuum. In elementary quantum mechanics, it is
possible to derive all physical content of the theory
from the simple knowledge of the ground state wave
function, including scattering data. Therefore, at least
in principle, it should be possible to derive all physical
content of the theory directly from the Euclidean
theory, without any analytic continuation.
We conclude this short section on the physical
interpretation of the Euclidean theory with a mention
of a quite surprising result (Guerra and Ruggiero
1973) obtained by submitting classical field theory
to the procedure of stochastic quantization in the sense
of Nelson (1985). The procedure of stochastic
quantization associates a stochastic process to each
quantum state. In this case, in a fixed reference frame,
the procedure of stochastic quantization, applied to
interacting fields, produces, for the ground state, a
process in the physical spacetime that has the same
correlations as Euclidean field theory. This opens the
way to a possible interpretation of Euclidean field
theory directly in Minkowski spacetime. However, a
consistent development along this line requires a new
formulation of representations of the Poincare´group
in the form of measure-preserving transformations in
the probability space where the Euclidean fields are
defined. This difficult task has not been accomplished
as yet.
Acknowledgments
Research connected with this work was supported
in part by MIUR (Italian Minister of Instruction,
University and Research), and by INFN (Italian
National Institute for Nuclear Physics).
See also: Axiomatic Quantum Field Theory; Constructive
Quantum Field Theory; Feynman Path Integrals;
Functional Integration in Quantum Physics; High T
c
Superconductor Theory; Malliavin Calculus; Quantum
Chromodynamics; Quantum Field Theory: A Brief
Introduction; Quantum Fields with Indefinite Metric:
Non-Trivial Models; Relativistic Wave Equations including
Higher Spin Fields; Renormalization: General Theory;
Two-dimensional Models.
264 Euclidean Field Theory
Further Reading
Caianiello ER (1973) Combinatorics and Renormalization in
Quantum Field Theory. Reading, MA: Benjamin.
Glimm J and Jaffe A (1981) Quantum Physics, A Functional
Integral Point of View. Berlin: Springer.
Guerra F (1972) Uniqueness of the vacuum energy density and
van Hove phenomenon in the infinite volume limit for two-
dimensional self-coupled Bose fields. Physical Review Letters
28: 1213.
Guerra F, Rosen L, and Simon B (1972) Nelson’s symmetry and
the infinite volume behavior of the vacuum in P()
2
.
Communication in Mathematical Physics 27: 10.
Guerra F, Rosen L, and Simon B (1975) The P()
2
Euclidean
quantum field theory as classical statistical mechanics.
Annuals of Mathematics 101: 111.
Guerra F and Ruggiero P (1973) New interpretation of the
Euclidean Markov field in the framework of physical
Minkowski space-time. Physical Review Letters 31: 1022.
Jaffe A (2000) Constructive quantum field theory, available on
http://www.arthurjaffe.com.
Nakano T (1959) Quantum field theory in terms of Euclidean
parameters. Progress in Theoretical Physics 21: 241.
Nelson E (1966) A quartic interaction in two dimensions. In:
Goodman R and Segal I (eds.) Conference on the Mathematical
Theory of Elementary Particles. Cambridge, MA: MIT Press.
Nelson E (1973a) Construction of quantum fields from Markoff
fields. Journal of Functional Analysis 12: 97.
Nelson E (1973b) The free Markoff field. Journal of Functional
Analysis 12: 211.
Nelson E (1985) Quantum Fluctuations. Princeton, NJ: Princeton
University Press.
Osterwalder K and Schrader R (1973) Axioms for Euclidean Green’s
functions. Communication in Mathematical Physics 31: 83.
Schlingemann D (1999) Euclidean field theory on a sphere,
available on http://arXiv.org.
Schwinger J (1958) On the Euclidean structure of relativistic field
theory. Proceedings of the National Academy Sciences 44: 956.
Simon B (1974) The P()
2
Euclidean (Quantum) Field Theory.
Princeton, NJ: Princeton University Press.
Streater R and Wightman AS (1964) PCT, Spin and Statistics and
All That. New York: Benjamin.
Symanzik K (1966) Euclidean quantum field theory, I, Equations
for a scalar model. Journal of Mathematical Physics 7: 510.
Symanzik K (1969) Euclidean quantum field theory. In: Jost R
(ed.) Local Quantum Theory. New York: Academic Press.
Wightman AS (1967) An introduction to some aspects of the
relativistic dynamics of quantized fields. In: Levy M (ed.) 1964
Carge´se Summer School Lectures. New York: Gordon and Breach.
Evolution Equations: Linear and Nonlinear
J Escher, Universita
¨
t Hannover, Hannover, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
In this article we present the semigroup approach
to linear and nonlinear evolution equations in
general Banach spaces. In the first part we
introduce the general frame and we explain the
cornerstones of the widely developed theory of
linear evolution equations. Besides the classical
approach to linear evolution equations based on
C
0
-semigroups, we also g ive a brief introduction to
the more recent theory of maximal regularity. The
entire linear theory is not only important on its
own (which we prove by discussing applications to
the heat equation, Schro¨ dinger equation, wave
equation, and Maxwell equations) but it is also
the indispensable basis for the theory of nonlinear
evolution equation, which we present in the second
part.
Linear Evolution Equations
Let E
0
be a Banach space, T > 0, and assume that
A:= {A(t); t 2 [0, T]} is a family of closed linear
operators in E
0
. By this we mean that, given t 2
[0, T], there is a linear subspace D(A(t)) of E
0
and
linear mapping A(t):D(A (t)) E
0
! E
0
such that the
graph {(x, A(t)x); x 2 D(A(t))} of A(t) is a closed
subspace of E
0
E
0
. Given a mapping f : [0, T] !
E
0
and a vector u
0
2 E
0
, we study the following initial-
value problem for (A, f , u
0
): find a function u 2
C
1
((0, T], E
0
)suchthatu(t) 2 D(A(t)) for
t 2 (0, T]and
u
0
ðtÞ¼AðtÞuðtÞþf ðtÞ; t 2ð0; T; uð0Þ¼u
0
½1
Sometimes we call [1] also the Cauchy problem of
the linear evolution equation u
0
(t) = A(t)u(t) þ f (t).
In the following, we will specify differe nt conditions
on (A , f , u
0
) which guarantee the well-posedness of
[1], and we shall discuss several examples of
equations of type [1] which are relevant in mathe-
matical physics.
Autonomous Homogeneous Equations
As in the case of ordinary differential equations in
finite-dimensional spaces, it is convenient to con-
sider first the autonomous version of [1], that is, we
assume that A is trivial in the sense that T = 1 and
that A(0) = A(t) for all t 0. In order to simplify
our notation, we set A := A(0). We consider first the
homogeneous problem
u
0
ðtÞ¼AuðtÞ; t 2ð0; 1Þ; uð0Þ¼u
0
½2
Evolution Equations: Linear and Nonlinear 265