On scaling limits in Euclidean quantum field theory

0.22 MB·

by  Svetoslav Zahariev

ON SCALING LIMITS IN EUCLIDEAN QUANTUM FIELD
THEORY
7
1
SVETOSLAVZAHARIEV
0
2

r
a LaGuardia Community College of The City University of New York,
M
MEC Department, 31-10 Thomson Ave.
Long Island City, NY 11101, U.S.A.
3
[email protected]
1

] Abstract. We present a general scaling limit construction of proba-
h
p bility measures obeying the Glimm-Jaffe axioms of Euclidean quantum
- field theory in arbitrary space-time dimension. Inparticular, weobtain
h
measures that may be interpreted as corresponding to scalar quantum
t
a fields with arbitrary continuous self-interaction. It remains however an
m open problem whether this general construction contains non-Gaussian
measures.
[

2
Contents
v
9
1. Introduction 2
6
5 2. Preliminaries 3
5 2.1. Characteristic functionals 3
0
2.2. Constructing measures from functionals 4
.
1 2.3. Equicontinuity and analyticity 4
0
3. Finite volume measures 5
7
1 3.1. Isometry invariant mollifiers 5
: 3.2. Isometry invariant measures 5
v
i 4. Reflection positivity 8
X
5. The infinite volume limit 10
r
a 5.1. Conformal maps, Laplacians, and the stereographic projection 10
5.2. The scaling limit 11
5.3. Euclidean invariance 13
5.4. Verification of the Glimm-Jaffe axioms 16
References 18
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2 SVETOSLAVZAHARIEV
1. Introduction
EstablishingtheexistenceofinteractingfieldmodelssatisfyingtheWight-
man axioms [14] of relativistic quantum field theory (QFT) in four dimen-
sionalspace-time has remainedan importantopenprobleminmathematical
physics since the 1960s. Many models in two and three space-time dimen-
sions[13]havebeenconstructedsofaremployingtheframeworkofEuclidean
QFTinwhichMinkowskispaceisreplacedwithEuclideanspace. Inthemid
1970s Osterwalder and Schrader [11] discovered a set of axioms formulated
on Euclidean space that is equivalent to the axioms of Wightman.
In this paper we present a general construction of probability measures
satisfyingtheGlimm-JaffeaxiomsofEuclideanQFTasstatedin[7,Chapter
6] (except possibly ergodicity) in arbitrary space-time dimension. More
explicitly, we obtain Euclidean invariant reflection-positive measures on the
spaceofdistributionsonRd possessingcharacteristic functionalsthatsatisfy
appropriate analyticity and regularity conditions.
Ourmethodmay besummarizedas follows. Given asequenceof densities
satisfying certain growth conditions, we construct a sequence of measures
on the space of distributions on the standard d-dimensional sphere Sd. We
observe that one can obtain such a sequence of densities from an arbitrary
real continuous function (representing a self-interaction) and an appropriate
sequence of coupling constants. We then transfer these measures to Rd
via a scaling procedure analogous to that developed in [12, Section 3] in the
context of algebraic EuclideanQFT,andshow thatthetransferredsequence
contains a weakly convergent subsequence. Heuristically, this scaling limit
may be envisaged as a process in which the radius of the sphere tends to
infinity, the subgroup of the isometry group of Sd preserving a given point
becomes the rotation group of Rd, and the remaining rotations of Sd are
identified with translations in Rd.
While one would expect that in many special cases the limit measures
so obtained coincide with the free scalar field measure, we believe that our
construction is sufficiently general to possibly contain non-Gaussian exam-
ples as well, including in four space-time dimensions. We also conjecture
that feeding the scaling limit construction with the densities corresponding
to the P(φ) model on S2 (cf. [1]) produces the well-known P(φ) model on
2 2
R2.
The paper is organized as follows. In Section 2 we briefly review for
reader’s convenience the results regarding measures on infinite dimensional
spaces that will be needed in the sequel. Section 3 is dedicated to a general
construction of limits of isometry invariant measures on distributions on a
closed Riemannian manifold. Sequences of measures are produced from a
given Gaussian referencemeasureandacontinuous interaction function; the
existence of a (weak) limit point is established via a simple Gaussian upper
bound. The reflection positivity of these limit measures is established in
Section 4.
ON SCALING LIMITS 3
In Section 5 the scaling limit passage from Sd to Rd from is performed. A
sequence of measures on distributions on Sd is transferred to distributions
on Rd employing the natural unitary equivalences induced by the scaling
transformations on Rd and the stereographic projection. The existence of
a (weak) limit point is shown via the same Gaussian upper bound as in
Section 3 and the Glimm-Jaffe axioms are verified.
Acknowledgment. TheauthorisgratefultoWojciechDybalski,Leonard
Gross, Nikolay M. Nikolov and Yoh Tanimoto for many helpful suggestions
and stimulating discussions at various stages of this project.
2. Preliminaries
In this section we present some well-known results from measure theory
on locally convex spaces; more detailed review may be found in [2, Section
7.13]. For background on topological vector spaces the reader is referred to
[16].
2.1. Characteristic functionals. In this article by a measure on a topo-
logical space we shall always mean a finite Radon measure. Let F be a real
locally convex vector space and let F′ be its dual space considered with the
weak* topology. The characteristic functional (or Fourier transform) of a
measure µ on F′ is a functional on F defined by
S (f)= eiφ(f)dµ(φ), f ∈F. (2.1)
µ
F′
Z
Recall that a sequence of measures µ on a topological space X converges
n
weakly to a measure µ if
fdµ → fdµ
n
ZX ZX
for every bounded continuous function f on X. Clearly the weak conver-
gence of sequenceof measures implies the convergence of their characteristic
functionals. It turns out that the converse is also true for finite dimensional
spaces (L´evy’s theorem) and also for a wide class of locally convex spaces if
equicontinuity is assumed.
Given a smooth manifold M, we denote by D(M) the space of the real-
valuedsmoothcompactlysupportedfunctionsonM equippedwiththeusual
C∞-topology, and by D′(M) its topological dual equipped with the weak*
topology. Recall that D(M) is a barrelled nuclear locally convex space.
Theorem 2.1. (cf. [6, Chapter III, Corollary 2.6 and Example 2.3]) A se-
quence of measures on D′(M) converges weakly if their characteristic func-
tionals are equicontinuous at zero and converge pointwise.
4 SVETOSLAVZAHARIEV
2.2. Constructing measures from functionals. The following result is
known as the Bochner-Minlos theorem.
Theorem 2.2. (see [6,ChapterIII,Theorem1.3]) Let F be a nuclear locally
convex space. For a functional on F to be the characteristic functional of
a measure on F′ it is sufficient, and necessary if F is barrelled, that it be
positive definite and continuous at zero.
We recall that a measure µ on a locally convex space F is called Gaussian
if for every φ ∈ F′ the image measure φ•µ on R is Gaussian.
Now assume that the nuclear space F is continuously embedded into a
real Hilbert space H. Then by Theorem 2.2 every nonnegative selfadjoint
boundedoperatorC onHdefinesameanzeroGaussianprobabilitymeasure
µ whose characteristic functional is given by
C
SµC(f) =e−12hCf,fi, f ∈ F. (2.2)
The operator C is called the covariance of the measure µ .
C
2.3. Equicontinuity and analyticity. Thefollowinggeneralization ofthe
Arzel`a-Ascoli theoremisaspecialcaseof[3,ChapterX,§2, No. 5, Corollary
1].
Theorem 2.3. Let {Φn}n∈N be an equicontinuous sequence of complex-
valued functions on a topological space X and assume that Φ (x) is bounded
n
for every x ∈ X. Then {Φ } contains a subsequence converging uniformly
n
on compact subsets to a continuous function on X.
Recall that a functional on a complex locally convex space is called
(Fr´echet) analytic if itis continuous andits restrictions tofinitedimensional
subspaces are analytic (see e.g. [9, Chapter 3])
Theorem 2.4. (cf. [9, Theorem 3.1.5(c)]) A uniformly locally bounded se-
quence of analytic functionals on a complex locally convex space is equicon-
tinuous.
Inwhatfollows, wedenotethecomplexification ofarealvectorspaceF by
FC. If µ is a measure on the dual of a real locally convex space F we denote
the extension of its characteristic functional to FC by SµC. This extension
(which may or may not exist) is defined by setting
φ(f +if )= φ(f )+iφ(f ), f ,f ∈ F (2.3)
1 2 1 2 1 2
in the integral in (2.1).
Given a Riemannian manifold M, we denote by L2(M) the Hilbert space
of real-valued functions on M which are square integrable with respect to
the measure induced by the metric. If µ is a Gaussian measure on D′(M)
C
C
with covariance operator C, the extension S is analytic and moreover one
µC
has
eφ(f)dµC(φ) ≤ eKCkfk2L2, f ∈ D(M), (2.4)
ZD′(M)
where K is a positive number depending only on the L2-norm of C.
C
ON SCALING LIMITS 5
3. Finite volume measures
3.1. Isometry invariant mollifiers. Let M be a closed connected d -
dimensional Riemannian manifold with group of isometries G. We observe
that G acts naturally on D(M) and hence on D′(M). Further, there is a
natural unitary representation of G on L2(M).
Given a neighborhood U of the diagonal in M ×M, we denote by d(U)
the largest distance between a point in U and the diagonal.
Lemma 3.1. There exists a sequence of smoothing continuous operators
A :D′(M) → D(M) satisfying:
k
(1) The restriction of A to L2(M) is a trace class operator and one has
k
TrA = A (x,x)dx,
k k
ZM
where A (x,y) stands for the smooth integral kernel of A ;
k k
(2) The operators A are uniformly bounded on L2(M);
k
(3) A → 1 strongly on L2(M) as k → ∞;
k
(4) d(supp(A (x,y))) → 0;
k
(5) A commutes with the action of G on L2(M).
k
Proof. The existence of mollifiers B satisfying (1)-(4) is a standard result
k
(see e.g. [16, Chapter II.7]). To obtain isometry invariant mollifiers, we
average over G as follows. We set
A (x,y) = B (g−1x,g−1y)dg,
k k
ZG
where dg denotes the normalized Haar measure on the compact group G.
Then an application of Fubini’s theorem shows that the operators A with
k
kernels A (x,y) satisfy (1) and (2). Similarly, (3) holds by Lebesque’s dom-
k
inated convergence theorem. Finally, (4) is true since G is a group of isome-
tries. (cid:3)
3.2. Isometry invariant measures. Given a measure µ on D′(M), we
denote by Lp(µ) the space of the complex-valued Lp functions with respect
to µ. Let µ be a centered Gaussian probability measure on D′(M) with
C
covariance operator C and let ρ ∈ L1(µ ) be a sequence of nonnegative
k C
functions. We define a sequence of probability measures µ on D′(M) by
k
setting
µ = N ρ µ , (3.1)
k k k C
where
N−1 = ρ dµ .
k k C
ZD′(M)
Proposition 3.2. Assume that the densities ρ satisfy
k
ρ dµ ≥ K , (3.2)
k C 1
ZD′(M)
6 SVETOSLAVZAHARIEV
ρ2dµ ≤ K (3.3)
k C 2
ZD′(M)
for some positive constants K and K .
1 2
(a)ThenthesequenceS ofthecharacteristic functionalsofµ isequicon-
µk k
tinuous.
(b) There exists a subsequence of S converging to the characteristic
µk
functional S of a probability measure µ on D′(M).
µ0 0
(c) There exists a constant K > 0 such that
kΨk ≤ KkΨk (3.4)
L1(µ0) L2(µC)
for every bounded continuous Ψ.
Proof. (a) It follows from the Cauchy-Schwartz inequality that
N kΨρ k ≤ N kΨk kρ k
k k L1(µC) k L2(µC) k L2(µC)
for every bounded continuous function Ψ on D′(M). Hence, using the esti-
mates (3.2) and (3.3), one obtains
kΨk ≤ K−1K kΨk . (3.5)
L1(µk) 1 2 L2(µC)
Now a short computation employing (2.2) shows that
eiφ(f)−eiφ(g) 2dµC(φ) = 2(1−e−kC1/2(f−g)k2L2/2), (3.6)
Z
(cid:12) (cid:12)
where f,g ∈ (cid:12)D(M) and φ((cid:12)f),φ(g) are the corresponding functionals on
D′(M). Combining (3.5) and (3.6), one finds that
−kC1/2(f−g)k2 /2
|Sµk(f)−Sµk(g)| ≤ const·(1−e L2 )
from which the equicontinuity of S follows since convergence in D(M)
µk
implies convergence in L2(M).
(b) The existence of a converging subsequence of S follows from part
µk
(a) and Theorem 2.3. The limit of such a subsequence is the characteristic
functional of a probability measure µ by Theorem 2.2.
0
(c) Sinceby Theorem 2.1 µ converges weakly to µ , taking the limit k →
k 0
∞ in (3.5), one finds that (3.4) holds for all bounded continuous functions
on D′(M). (cid:3)
We now fix a subgroup G of the isometry group G and a bounded non-
0
negative selfadjoint operator C on L2(M) that commutes with the action
of G on L2(M). For example, one can take C = (m2 + △)−1, where △
0
stands for the unique selfadjoint extension of the Laplacian on L2(M) and
m > 0 (we use the sign convention in which △ is a nonnegative operator.)
By Theorem 2.2 the operator C is the covariance of a unique mean zero
Gaussian probability measure µ on D′(M) whose characteristic functional
C
is given by (2.2).
ON SCALING LIMITS 7
Example 3.3. (Bounded interaction) We fix a bounded continuous function
F : R → R and, using the mollifiers A from Lemma 3.1, define a sequence
k
of densities
ρ = exp F(A φ)(x)dx , φ∈ D′(M), (3.7)
F,k k
(cid:16)ZM (cid:17)
and denote the corresponding probability measures given by (3.1) by µ .
F,k
Using the boundedness of F, Proposition 3.2(b) and Theorem 2.1 imply
that the sequence µ contains a weakly convergent subsequence. We fix
F,k
such a subsequence and denote its limit by µ . Observe that the measures
F
µ and hence µ are G -invariant since so are the densities in (3.7) and
F,k F 0
µ .
C
Example 3.4. (Regularized unbounded interaction) Let C be as above and
let F : R → R be a possibly unbounded continuous function. We set V =
k
{x : |F(x)| ≤ k} and define bounded continuous functions
F(x), x ∈ V
k
F (x) =
k (k, x ∈ R\Vk
for every k ∈ N. We define a sequence of densities
ρ = exp −λ F (A φ)(x)dx , φ ∈ D′(M), (3.8)
F,k k k k
(cid:16) ZM (cid:17)
where λ is a sequence of positive numbers chosen so that the assumptions
k e
of Proposition 3.2 are satisfied. For example, one can take
λ = 1/(sup{|F(x)| : x ∈ V }), (3.9)
k k
obtaining uniformly bounded densities. We consider the corresponding se-
quence of probability measures given by (3.1) and, as above, obtain a G -
0
invariant limit measure that we shall denote by µ .
F
Remark 3.5. The sequence λ in the example above may be interpreted as
k
e
a sequence of renormalized coupling constants, while the convergence of λ
k
to 0 may be interpreted as an asymptotic freedom property of the theory.
Example 3.6. (P(φ) models) Let C be as above and set C = A∗CA . We
2 k k k
define the n-th Wick power of f ∈ D(M) with respect to C by setting (cf.
k
[7, Eq. (8.5.5)])
[n/2]
(−1)jn!
:f(x)n: = C (x,x)jf(x)n−2j,
Ck (n−2j)!j!2j k
j=0
X
where C (x,y) is the integral kernel of the smoothing operator C and [·]
k k
denotes the integer part of a number.
More generally, for every bounded from below polynomial P, one can
consider the corresponding polynomial in the Wick powers which we denote
8 SVETOSLAVZAHARIEV
by :P(f): . Further, we define a sequence of functionals via
Ck
:P(φ): = :P(A φ): , φ ∈D′(M)
Ck k Ck
ZM
and probability measures on D′(M) via
µP,k = Nke−:P(φ):CkdµC(φ),
where N are normalization constants.
k
Now assume that M is the standard two-dimensional sphere S2 and take
C = (m2 +△)−1. Then proceeding exactly as in [7, Chapter 8], one can
prove that that :P(φ): converge in L2(µ ) to a limit :P(φ): as n → ∞
Ck C C
and that e−:P(φ):C belongs to L1(µC) (see also [1, Section 11.1]). Using this,
one can show as in the proof of Eq. (0.8) in [5] that
e−:P(φ):Ck → e−:P(φ):C
in Lp(µ ) for 1 ≤ p < ∞. It follows that the measures µ satisfy the
C P,k
assumptions of Proposition 3.2, hence there exists a subsequence of µ
P,k
converging weakly to a measure µ .
P
4. Reflection positivity
In this section the Riemannian manifold M is notassumed to becompact
unless otherwise specified. Supposethat we are given a pair M ,M of dis-
+ −
jointopensubmanifoldsof M and anisometry Θmappingdiffeomorphically
M onto M . We denote the induced unitary operator on L2(M) by Θ as
± ∓
well and for every open U ⊂ M consider the following algebra of functionals
on D′(M):
A = Ψ : Ψ(φ) = z eiφ(fi), f ∈ D(U), z ∈ C (4.1)
U i i i
n Xi o
Definition 4.1. (a) We say that a bounded positive operator A on L2(M)
is reflection positive with respect to (M ,Θ) if AΘ = ΘA and hAf,Θfi≥ 0
±
for every f ∈ L2(M) that is supported in M .
+
(b) A functional S on D(M) is reflection positive with respect to (M ,Θ)
±
if the matrix S = S(f − Θf ) is positive semi-definite for every finite
ij i j
sequence f ,...,f ∈ D(M) supported in M .
1 k +
(c) A probability measure µ on D′(M) is reflection positive with respect
to (M ,Θ) if
±
hΨ,ΘΨi ≥ 0 (4.2)
L2(µ)
for all Ψ ∈ A (We denote the operator on L2(µ) induced by Θ again by
M+
Θ.)
Lemma 4.2. (a) If an operator A on L2(M) is reflection positive, the func-
tional S(f) = e−hAf,fi on D(M) is reflection positive as well.
(b) A probability measure µ on D′(M) is reflection positive if and only if
its characteristic functional S is reflection positive.
µ
ON SCALING LIMITS 9
Proof. (a) See [7, Theorem 6.2.2].
(b) Suppose first that µ is reflection positive. We fix f ,...,f ∈ D(M )
1 k +
and observe that
S = S (f −Θf )= eφ(fi−Θfj)dµ(φ).
ij µ i j
ZD′(M)
Hence for every sequence z ,...,z ∈ C one has
1 k
ziSijz¯j = ( zieiφ(fi))(Θ zieiφ(fi))dµ(φ) ≥ 0 (4.3)
i,j ZD′(M) i i
X X X
which implies that S reflection positive.
µ
Conversely, suppose that S is reflection positive. Then Eq. (4.3) implies
µ
that µ is reflection positive. (cid:3)
Remark 4.3. Denote by r : D′(M) → D′(M ) the natural restriction map.
+
Given aprobability measure µ on D′(M), we defineR (M ) to bethe space
µ +
consisting of those functionals Ψ ∈ L2(µ) for which there exists a functional
Ψ ∈L2(D′(M ),r•µ) such that Ψ= Ψr (here r•µ the denotes the image of
+
µ under r.)
Thenaprobability measureµon D′(M)is reflection positive withrespect
to (M ,Θ) if and only if (4.2) holds for every Ψ ∈ R (M ). This follows
± µ +
from the fact that A is dense in L2(D′(M ),r•µ) (see e.g. [2, Corollary
M+ +
7.12.2]).
We now imposeadditional restrictions onthedata (M ,Θ)as in [10]. We
±
assume that M = M ∪M ∪M where M ,M and M are disjoint, M
− 0 + 0 − + 0
is a closed submanifold of codimension 1 and M are open submanifolds.
±
Further, we suppose that Θ fixes the points in M and induces hyperplane
0
reflections on the tangent spaces at points in M .
0
Consider a sequence of densities ρ as in Section 3.2 and probability mea-
k
sures µ defined via (3.1). Assume that ρ have the form
k k
ρ (φ) = exp η (φ)(x)dx , φ ∈ D′(M), (4.4)
k k
(cid:16)ZM (cid:17)
with
η (φ) = g F (A φ),
k k,i k,i k
i
X
whereg and F are continuous real-valued functions on M and R respec-
k,i k,i
tively, F (0) = 0, and A are given by Lemma 3.1.
k,i k
Proposition 4.4. Assume that the covariance operator C used in the con-
struction of the measures µ is reflection positive with respect to (M ,Θ)
k ±
and that the characteristic functionals of µ converge pointwise to the char-
k
acteristic functional of a measure µ . Then µ is reflection positive with
0 0
respect to (M ,Θ).
±
10 SVETOSLAVZAHARIEV
Proof. For every τ > 0 we denote by Mτ the set of all points in M whose
± ±
distancefromM isgreater thanτ. We shallshowthatforevery τ > 0there
0
exists sufficiently large k such that themeasure µ is reflection positive with
k
respect to (Mτ,Θ).
±
For every open U ⊂ M we denote by Ψ (U) the functional given by
k
D′(M) ∋ φ7→ exp η (φ)(x)dx . (4.5)
k
(cid:16)ZU (cid:17)
Given τ > 0, by Lemma 3.1(4) one can find N such that A φ is supported
k
inM forallφ ∈ D′(M) supportedin Mτ andall k > N. Itfollows thatthe
+ +
functionals Ψ (M ) belong to R (Mτ) defined in Remark 4.3. Indeed, one
k + µ +
can define continuous functionals on compactly supported distributions in
Mτ via (4.5) and extend them by continuity to functionals Ψ on D′(Mτ)
+ k +
that satisfy Ψ (M ) = Ψ r.
k + k
Next we note that
Ψ (M) =Ψ (M )Ψ (M )= Ψ (M )ΘΨ (M ),
k k + k − k + k +
which,usingRemark4.3andthefactthatµ isreflectionpositivebyLemma
C
4.2 (a), easily implies that µ are reflection positive with respect to (Mτ,Θ)
k ±
for all k > N.
ByLemma4.2(b)thecharacteristic functionalsS arereflectionpositive
µk
with respect to (Mτ,Θ) for all k > N, i.e. the matrices with entries
±
Sk = S (f −Θf )
ij µk i j
arepositive semi-definite for every finite sequencef ,...,f ∈ D(Mτ). Now
1 k +
given a sequence f ,...,f supported in M , one can find τ >0 sufficiently
1 k +
small so that f ,...,f are supported in Mτ and it follows that µ is reflec-
1 k + 0
tion positive. (cid:3)
Example 4.5. ThesequencesofdensitiesdefinedinExamples3.3,3.4,assum-
ing F(0) = 0, and in 3.6 are all of the form (4.4), hence the corresponding
limit measures are reflective positive provided that C is reflection positive.
A natural choice for C is (m2 +△)−1, the covariance of the free Euclidean
scalar quantum field on M with mass m. This operator is reflection positive
by [10, Theorem 1].
5. The infinite volume limit
5.1. Conformal maps, Laplacians, and the stereographic projec-
tion. Let(M ,g )and(M ,g )beapairofd-dimensionalRiemannianman-
1 1 2 2
ifolds and let η : M → M be a conformal diffeomorphism with conformal
1 2
factor Λ ∈ C∞(M ), i.e. η∗g = Λ2g (Here and below upper ∗ stands for
η 1 2 η 1
the pull-back map acting on functions and tensors.) Then the operator U
η
given by
U f = Λd/2η∗f, f ∈ D(M ) (5.1)
η η 2
extendstoaunitaryoperatorfromL2(M2)toL2(M1)suchthatUη−1 = Uη−1.