Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2011, Article ID 257916, 14 pages
doi:10.1155/2011/257916
Research Article
Some Comments on Rigorous Finite-Volume
Euclidean Quantum Field Path Integrals in
the Analytical Regularization Scheme
Luiz C. L. Botelho
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal Fluminense,
Rua Mario Santos Braga, 24220-140 Niterói, RJ, Brazil
Correspondence should be addressed to Luiz C. L. Botelho, botelho.luiz@superig.com.br
Received 4 March 2011; Accepted 20 April 2011
Academic Editor: Klaus Kirsten
Copyright q 2011 Luiz C. L. Botelho. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Through the systematic use of the Minlos theorem on the support of cylindrical measures on R∞ ,
we produce several mathematically rigorous finite-volume euclidean path integrals in interacting
euclidean quantum fields with Gaussian free measures defined by generalized powers of finitevolume Laplacian operator.
1. Introduction
Since the result of R.P. Feynman on representing the initial value solution of Schrodinger
Equation by means of an analytically time continued integration on a infinite-dimensional
space of functions, the subject of Euclidean Functional Integrals representations for
Quantum Systems has became the mathematical-operational framework to analyze Quantum
Phenomena and stochastic systems as showed in the previous decades of research on
Theoretical Physics 1–3.
One of the most important open problems in the mathematical theory of Euclidean
Functional Integrals is that related to implementation of sound mathematical approximations
to these Infinite-Dimensional Integrals by means of Finite-Dimensional approximations
outside of the always used computer oriented Space-Time Lattice approximations see 2,
3, chapter 9. As a first step to tackle upon the above cited problem it will be needed
to characterize mathematically the Functional Domain where these Functional Integrals are
defined.
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The purpose of this paper is to present the formulation of Euclidean Quantum Field
theories as Functional Fourier Transforms by means of the Bochner-Martin-Kolmogorov
theorem for Topological Vector Spaces 4, 5, Theorem 4.35 and suitable to define and
analyze rigorously Functional Integrals by means of the well-known Minlos theorem 5,
Theorem 4.312 and 6, part 2 which is presented in full in Appendix A.
We thus present studies on the difficult problem of defining rigorously infinitedimensional quantum field path integrals in general finite volume space times Ω ⊂ Rν ν 2, 4, . . . by means of the analytical regularization scheme 7.
2. Some Rigorous Finite-Volume Quantum Field Path Integral in
the Analytical Regularization Scheme
Let us thus start our analysis by considering the Gaussian measure on L2 R2 defined by
the finite volume, infrared regularized, and α-power Laplacian acting on L2 RN as an
operatorial quadratic form jx ∈ L2 RN see Appendix B
0 Zα,εIR j
−1 1
−α
2
exp − j, χΩ −Δ εIR χΩ j
2
L2 R2 0
≡
dα,εIR μ ϕ exp i j, ϕ L2 R2 .
2.1a
L2 R2 Here χΩ denotes the multiplication operator defined by the characteristic function
χΩ x of the compact region Ω ⊂ R2 and εIR > 0 the associated infrared cutoff.
It is worth calling the reader attention that due to the infrared regularization introduced on 2.1a, the domain of the Gaussian measure 4, 6 is given by the space of square
integrable functions on R2 by the Minlos theorem of Appendix A, since for α > 1, the operator
defines a trace class operator on L2 R2 , namely,
−1 d2 p
α
2
Tr 1 L2 R2 χΩ −Δ εIR χΩ volΩ ×
< ∞.
2α
p ε 2
2.1b
IR
This is the only point of our analysis where it is needed to consider the infrared cut
off. As a consequence of the above remarks, one can analyze the ultraviolet renormalization
program in the following interacting model proposed by us and defined by an interaction
gbare V ϕx, with V x being the Fourier Transformed of an integrable and essentially
bounded measurable real function. It could be as well consider a polynomial interaction
of the form Vn,p x minimum of {ϕxp , n} with p and n positive integers. Note that
L1 < ∞. Note
V k ∈ L1 R ∩ L∞ R by hypothesis | Ω eikϕx · V kdkd2 x| ≤ volΩ||V||
that V x is thus a continuous real function vanishing at the infinite point.
Let us show that by defining an ultraviolet renormalized coupling constant with a
finite volume Ω cutoff built in
gbare α gren
1 − α1/2
2−α π −1/4
2.2
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one can show that the interaction function
d2 xV ϕx
exp −gbare α
2.3
Ω
0
is an integrable function on L1 L2 R2 , dα,εIR μϕ and leads to a well-defined ultraviolet
functional integral in the limit of α → 1.
The proof is based on the following estimates.
Since almost everywhere we have the pointwise limit
exp −gbare α d2 xV ϕx ,
lim
N →∞
N
−1n gbare α n n!
n0
Ω
2.4
×
dk1 · · · dkN V k1 · · · V kN dx1 · · · dxN e
ik1 ϕx1 ···e
ikn ϕxN we have that the upper bound estimate below holds true
n ∞
α −1n gbare α
Zε gbare ≤ dk1 · · · dkn V k1 · · · V kn IR
n0
n!
R
i N k ϕx 0
dx1 · · · dxn dα,εIR μ ϕ e 1
Ω
2.5a
With
ZεαIR gbare
0
dα,εIR μ ϕ exp
2
−gbare α
d xV ϕx
Ω
2.5b
0
we have, thus, the more suitable form after realizing the d2 ki and dα,εIR μϕ integrals, respectively
n n
∞ g
bare α
α ZεIR 0 gbare ≤
V ∞
L R
n!
n0
m −1/2
Gα xi , xj 1≤i≤m .
× dx1 · · · dxm det
Ω
1≤j≤m
2.6
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N
Here Gα xi , xj 1≤i≤N,1≤j≤N denotes the N × N symmetric matrix with the i, j entry
given by the positive Green-function of the α-Laplacian without the infrared cut off here!.
Γ1 − α
Gα xi , xj |xi − xj |2α−1
Γα
e−iπα
4α · π 1/2
2.7
.
At this point, we call the reader attention that we have the formulae on the asymptotic
behavior for α → 1 and α < 1 see 7, Appendix A
⎧
⎨
lim
⎩α → 1
N det−1/2 Gα xi , xj
⎫
⎬
α>1
⎭
∼ eN/2πiα × π N/4 · 2Nα
1
N/2 × 1 − α
.
−1N N − 1 2.8
After substituting 2.8 into 2.6 and taking into account the hypothesis of the
compact support of the nonlinearity V k, one obtains the finite bound for any value gren > 0,
with the finite volume cutoff and producing a proof for the convergence of the perturbative
expansion in terms of the renormalized coupling constant for the model
lim ZεαIR 0 gbare α ≤
α→1
V ∞
n
L∞ R
1/2
n!
n0
≤ exp gren V n
gren
L∞ R
1 − α
volΩ < ∞.
× 1 − αn/2 × volΩn
2.9
Another important rigorously defined functional integral is to consider the following
α-power Klein Gordon operator on Euclidean space-time with f ∈ L2 RN L−1
Ω,f
ν/2
χΩ 2π
α
−Δ fχΩ m
2
−1
×
χΩ 2πν/2
volΩ
2.10
with m2 a positive “mass” parameters.
2
ν
Let us note that L−1
Ω,f is an operator of trace class on L R if and only if the result
below holds true
−1
TrL2 Rν L
dν k
&π
νπ '
1
ν/α−2
cosec
< ∞,
Cνm
×
2α
2α
k2α m2
2.11
namely, if
α>
ν
.
2
2.12
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In this case, let us consider the double functional integral with functional domain
L2 Rν Z j, k 0
dG
βvx ×
0
dLΩ,v μ ϕ
ν × exp i d x jxϕx kxvx ,
2.13
where the Gaussian functional integral on the fields V x has a Gaussian generating functional defined by a 1 -integral operator with a positive-definite kernel g|x − y|, namely,
Z0 k 0
dG βvx exp i dν xkxvx
1
.
dν x
dν y kxg x − y k y
exp −
2 Ω
Ω
2.14
A simple direct application of the Fubbini-Tonelli theorem on the exchange of the
integration order on 2.13 leads us to the effective λϕ4 —Formal functional integral representation
0 Zeff j dL μ ϕ ϕx
2 2
1
× exp −
dν xdν yϕx g x − y ϕ y 2 Ω
× exp i
dν xjxϕx .
2.15
Ω
Note that if one introduces from the beginning a bare mass parameter m2bare depending
on the parameter α, but such that it always satisfies 2.11 one should obtain again 2.15 as
a well-defined measure on L2 Rν . Of course that the usual pure Laplacian limit of α → 1
on 2.10, there will be needed a renormalization of this mass parameters limα → 1 m2bare α ∞! as much as it has been done in previous studies.
Let us continue our examples by showing again the usefulness of the precise
determination of the functional-distributional structure of the domain of the functional
integrals in order to construct rigorously these path integrals without complicated lattice limit
procedures 2.
Let us consider a general Rν Gaussian measure defined by the Generating functional
on SRν defined by the α-power of the Laplacian operator −Δ acting on SRν with a small
infrared regularization mass parameter μ2 as considered in 2.1a
−1 1
α
2
Z0 j exp − j, −Δ μ0 j
2
L2 Rν 0 dα μ ϕ exp iϕ j .
Ealg SRν 2.16
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An explicit expression in momentum space for the Green function of the α-power of
−Δα μ20 given by
α
−Δ
μ20
ν
−1 d k ikx−y
x−y e
2πν
1
2α
k μ20
.
2.17
Here Cν is a ν-dependent finite for ν-values ! normalization factor.
Let us suppose that there is a range of α-power values that can be choosen in such way
that one satisfies the constraint below
2j
0 dα μ ϕ ϕL2j Rν <∞
2.18
Ealg SRν with j 1, 2, . . . , N and for a given fixed integer N, the highest power of our polynomial
field interaction. Or equivalently, after realizing the ϕ-Gaussian functional integration, with
a space-time cuttoff volume Ω on the interaction to be analyzed on 2.16
j −j
ν
d k
dν x −Δα μ2 x, x ≤ volΩ ×
0
2
k2α μ0 Ω
π
νπ j
ν/α−2
Cν μ0 cosec
×
× volΩ < ∞.
2α
2α
2.19
For α > ν/2, one can see by the Minlos theorem that the measure support of the
Gaussian measure 2.16 will be given by the intersection Banach space of measurable
Lebesgue functions on Rν instead of the previous one Ealg SRν 4–6
L2N Rν N
(
L2j Rν .
2.20
j1
In this case, one obtains that the finite-volume pϕ2 interactions
⎧
⎫
N
⎨ j ⎬
ϕ2 x dx ≤ 1
exp − λ2j
⎩ j1
⎭
Ω
2.21
are mathematically well defined as the usual pointwise product of measurable functions and
for positive coupling constant values λ2j ≥ 0. As a consequence, we have a measurable
0
functional on L1 L2N Rν ; dα μϕ since it is bounded by the function 1. Thus, it makes
sense to consider mathematically the well-defined path integral on the full space Rν with
those values of the power α satisfying the constraint 2.17.
Z j ⎧
⎫
N
⎨ ⎬
0
dα μ ϕ exp − λ2j
ϕ2j xdx × exp i
jxϕx .
⎩ j1
⎭
L2N Rν Ω
Rν
2.22
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Finally, let us consider an interacting field theory in a compact space-time Ω ⊂ Rν
defined by an iteger even power 2n of the Laplacian operator with Dirichlet Boundary
conditions as the free Gaussian kinetic action, namely,
)
*
1
Z0 j exp − j, −Δ−2n j 2
L Ω
2
0 d2n μ ϕ exp i j, ϕ L2 Ω
2.23
W2n Ω
here ϕ ∈ W2n Ω—the Sobolev space of order n which is the functional domain of the
cylindrical Fourier Transform measure of the Generating functional Z0 j, a continuous
bilinear positive form on W2−n Ω the topological dual of W2n Ω 4–6.
By a straightforward application of the well-known Sobolev immersion theorem, we
have that for the case of
n−k >
ν
2
2.24
including k as a real number the functional Sobolev space W2n Ω is contained in the
continuously fractional differentiable space of functions Ck Ω. As a consequence, the
domain of the Bosonic functional integral can be further reduced to Ck Ω in the situation
of 2.24
Z0 j Ck Ω
0 d2n μ ϕ exp i j, ϕ L2 Ω .
2.25
That is our new result generalizing the Wiener theorem on Brownian paths in the case
of n 1, k 1/2, and ν 1.
Since the bosonic functional domain on 2.25 is formed by real functions and not
distributions, we can see straightforwardly that any interaction of the form
exp −g
F ϕx dν x
2.26
Ω
with the nonlinearity Fx denoting a lower bounded real function γ > 0
Fx ≥ −γ
2.27
0
is well-defined and is integrable function on the functional space Ck Ω, d2n μϕ by a
direct application of the Lebesgue theorem
ν +
,
exp −g
≤ exp gγ .
d
F
ϕx
x
Ω
2.28
At this point we make a subtle mathematical remark that the infinite volume limit of
2.25-2.26 is very difficult, since one looses the Garding-Poincaré inequality at this limit for
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those elliptic operators and, thus, the very important Sobolev theorem. The probable correct
procedure to consider the thermodynamic limit in our Bosonic path integrals is to consider
solely a volume cutoff on the interaction term Gaussian action as in 2.22 and there search
for volΩ → ∞ 8–11.
As a last remark related to 2.23 one can see that a kind of “fishnet” exponential
generating functional
1
Z j exp − j, exp{−αΔ}j L2 Ω
2
0
2.29
has a Fourier transformed functional integral representation defined on the space of the
infinitely differentiable functions C∞ Ω, which physically means that all field configurations
making the domain of such path integral has a strong behavior like purely nice smooth
classical field configurations.
As a last important point of this paper, we present an important result on the
geometrical characterization of massive free field on an Euclidean Space-Time 11.
Firstly we announce a slightly improved version of the usual Minlos Theorem 4.
Theorem 2.1. Let E be a nuclear space of tests functions and dμ a given σ-measure on its topological
dual with the strong topology. Let , 0 be an inner product in E, inducing a Hilbertian structure on
H0 E, , 0 , after its topological completion.
We suppose the following.
a There is a continuous positive definite functional in H0 , Zj, with an associated
cylindrical measure dμ.
b There is a Hilbert-Schmidt operator T : H0 → H0 ; invertible, such that E ⊂
Range T, T −1 E is dense in H0 and T −1 : H0 → H0 is continuous.
We have thus, that the support of the measure satisfies the relationship
∗
support dμ ⊆ T −1 H0 ⊂ E∗ .
2.30
At this point we give a nontrivial application of ours of the above cited Theorem 2.1.
Let us consider a differential inversible operator L : S RN → SR, together with a
positive inversible self-adjoint elliptic operator P : DP ⊂ L2 RN → L2 RN . Let Hα be the
following Hilbert space:
& '
Hα S RN , P α ϕ, P α ϕ L2 RN , α , for α a real number .
2.31
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We can see that for α > 0, the operators below
P −α :L2 RN −→ H
α
ϕ −→ P −α ϕ ,
P α :H
α −→ L2 RN
2.32
ϕ −→ P α ϕ
are isometries among the following subspaces:
DP −α , , L2 , H
α
2.33
since
P −α ϕ, P −α ϕ
H
α
P α P −α ϕ, P α P −α ϕ L2 RN ϕ, ϕ L2 RN ,
P α f, P α fL2 RN f, fH
α .
2.34
If one considers T a given Hilbert-Schmidt operator on Hα , the composite operator
T0 P α TP −α is an operator with domain being DP −α and its image being the Range P α . T0
is clearly an invertible operator and SRN ⊂ Range T means that the equation TP −α ϕ f has always a nonzero solution in DP −α for any given f ∈ SRN . Note that the condition
that T −1 f be a dense subset on Range P −α means that
*
)
T −1 f, P −α ϕ
L2 RN 0
2.35
has as unique solution the trivial solution f ≡ 0.
Let us suppose too that T −1 : SRN → Hα be a continuous application and the
bilinear term L−1 jj be a continuous application in the Hilbert spaces H
α ⊃ SRN ,
L2
L2
namely, if jn → j, then L−1 : P −α jn → L−1 P −α j, for {jn }n∈ and fn , j ∈ SRN .
By a direct application of the Minlos Theorem, we have the result
1 −1 Z j exp − L j j
dμT exp iT j .
∗
2
T −1 Hα
2.36
Here the topological space support is given by
T −1
∗
Hα P −α T0 P α
−1 ∗ P α SRN ∗ ∗ P α ∗ T0−1
P −α P α S RN
P α T0−1 L2 RN .
2.37
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In the important case of L −Δ m2 : S RN → SRN and T0 T0∗ χΩ 2πN/2 /
−2β
β
N/2
ΓN/2Γ2β −
volΩ−Δ m2 × χΩ ∈ 1 L2 RN since TrT0 T0∗ 1/2m2 m2 /1
N/2/Γβ < ∞ for β > N/4 with the choice P −Δ m2 , we can see that the support of
the measure in the path-integral representation of the Euclidean measure field in RN may be
taken as the measurable subset below
α
β , +
L2 RN
IΩ x −Δ m2
supp d−Δ
m2 μ ϕ −Δ m2
2.38
−1−α
since L−1 P −α −Δ m2 is always a bounded operator in L2 RN for α > −1.
As a consequence each field configuration can be considered as a kind of “fractional
distributional” derivative of a square integrable function as written below of the formal
infinite volume Ω → RN .
ϕx -
−Δ m2
N/4
ε−1 .
f x
2.39
with a function fx ∈ L2 RN and any given ε > 0, even if originally all fields configurations
entering into the path integral were elements of the Schwartz Tempered Distribution
Spaces S RN certainly very “rough” mathematical objects to characterize from a rigorous
geometrical point of view.
We have, thus, made a further reduction of the functional domain of the free massive
Euclidean scalar field of S RN to the measurable subset as given by 2.38 denoted by
WRN -
−1 . 1
−Δ m2 j j
d−Δ
m2 μ ϕ eiϕj
exp −
2
S RN 2 N/4
ε−1
f L2 RN d−Δ
m2 μ f eif,−Δ
m .
2.40
WRN ⊂S RN Appendices
A. Some Comments on the Support of Functional Measures in
Hilbert Space
Let us comment further on the application of the Minlos Theorem in Hilbert Spaces. In this
case one has a very simple proof which holds true in general Banach Spaces E, .
Let us thus, give a cylindrical measures d∞ μx in the algebraic dual Ealg of a given
Banach Space E 4–6.
Let us suppose either that the function x belongs to L1 Ealg , d∞ μx. Then the
support of this cylindrical measures will be the Banach Space E.
The proof is the following.
Let A be a subset of the vectorial space Ealg with the topology of pontual convergence, such that A ⊂ Ec so x ∞ E can always be imbeded as a cylindrical
measurable subset of Ealg —just use a Hammel vectorial basis to see that. Let be the sets
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An {x ∈ Ealg | x ≥ n}. Then we have the set inclusion A ⊂
satisfies the estimates below:
/∞
n0
An , so its measure
μA ≤ lim inf μAn n
&
'
lim inf μ x ∈ Ealg | x ≥ n
n
≤ lim inf
n
lim inf
1
n
Ealg
xd∞ μx
xL1 Ealg ,dμ∞ n
n
A.1
0.
Leading us to the Minlos theorem that the support of the cylindrical measure in Ealg is
reduced to the own Banach Space E.
Note that by the Minkowisky inequality for general integrals, we have that x2 ∈
L1 Ealg , d∞ μx. Now it is elementary evaluation to see that if A−1 ∈ 1 M, when E M, a
given Hilbert Space, we have that
Malg
∞
dA
μx · x2 TrM A−1 < ∞.
A.2
∞
This result produces another criterium for supp dA
μ M the Minlos Theorem, when E M
is a Hilbert Space.
It is easy too to see that if
M
xd∞ μx < ∞
A.3
then the Fourier-Transformed functional
Z j M
eij,xM d∞ μx
A.4
is continuous in the norm topology of M.
Otherwise, if Zj is not continuous in the origin 0 ∈ M without loss of generality,
then there is a sequence {jn } ∈ M and δ > 0, such that jn → 0 with
δ ≤ Z jn − 1 ≤
ijn ,xM
− 1d∞ μx
e
≤
M
M
jn , xd∞ μx
≤ jn a contradiction with δ > 0.
M
xd∞ μx
A.5
−→ 0,
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Finally, let us consider an elliptic operator B with inverse from the Sobelev space
W −2m Ω to W 2m Ω. Then by the criterium given by A.2 if
TrL2 Ω I Δ
m/2 B−1 I Δ
m/2 < ∞,
A.6
we will have that the path integral written below is well defined for x ∈ W 2m Ω and j ∈
W −2m Ω. Namely,
1
−1
dB μx exp i j, x L2 Ω .
exp − j, B j 2
L Ω
2
W 2m Ω
A.7
By the Sobolev theorem which means that the embeded below is continuous with
Ω ⊆ Rν denoting a smooth domain, one can further reduce the measure support to the
Hölder α continuous function in Ω if 2m − ν/2 > α. Namely, we have an easy proof of the
famous Wiener Theorem on sample continuity of certain path integrals in Sobolev Spaces
W 2m Ω ⊂ Cα Ω.
A.8a
The above Wiener Theorem is fundamental in order to construct nontrivial examples
of mathematically rigorous euclidean’s path integrals in spaces Rν of higher dimensionality,
since it is a trivial consequence of the Lebesgue theorem that positive continuous functions
V x generate functionals integrable in {W 2m Ω, dB μϕ} of the form below
V ϕx dx ∈ L1 W 2m Ω, dB μ ϕ .
exp −
Ω
A.8b
As a last important remark on Cylindrical Measures in Separable Hilbert Spaces, let us
point to our reader that the support of such above measures is always a σ-compact set in the
norm topology of M. In order to see such result let us consider a given dense set of M, namely,
{xk }k∈I . Let {δk }k∈I be a given sequence of positive real numbers with δk → 0. Let {εn } be
another sequence of positive real numbers such that ∞
n1 εn < ε. Now it is straightforward
0
0∞
to see that M ⊂ h1 Bxk , δk ⊂ M and thus lim sup μ{ nk1 Bxk , δk } μM 1. As a
0n
consequence, for each n, there is a kn , such that μ kk1
Bxk , δk ≥ 1 − ε.
/∞ 0kn
Now the sets Kμ n1 k1 Bxk , δk are closed and totally bounded, so they are
compact sets in M with μM ≥ 1 − ε. Let us now choose ε 1/n and the associated compact
1 n,μ 0n K,μ. We have that K
1 n,μ ⊆
sets {Kn,μ }. Let us further consider the compact sets K
1
0
∞
1 n,μ 1. So, Supp dμ n1 K
1 n,μ , a σ-compact set of M.
1 n
1,μ , for any n and lim sup μK
K
We consider now a enumerable family of cylindrical measures {dμn } in M satisfying
the chain inclusion relationship for any n ∈ I Supp dμn ⊆ Supp dμn
1 .
n
1 n }, where Supp dμm Now it is straightforward to see that the compact sets {K
0∞ 1 n
is such that Supp {dμm } ⊆ n1 Kn , for any m ∈ I .
A.9
0∞ 1 m
n1 Kn ,
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Let us consider the family of functionals induced by the restriction of this sequence of
1 nn . Namely,
measures in any compact K
μn −→
n Ln f
1 nn
K
fx · dμp x.
A.8c
1 nn . Note that all the above functionals in 0∞
1 n
Here f ∈ Cb K
n1 Cb Kn are bounded by 1.
By the Alaoglu-Bourbaki theorem they form a compact set in the weak star topology of
∗
0
1 n
∞
n1 Cb Kn , so there is a subsequence or better the whole sequence converging to a
unique cylindrical measure μx. Namely,
lim
n→∞
for any f ∈
0∞
n1
M
fxdμn x M
fxdμx
A.8d
n
1 n .
Cb K
B. On the Trace Class Property
A straightforward calculation gives the following expression for the trace of the integral
operator on 2.1a
Tr
1
−1 ∞ ∞ χΩ k − pχΩ p − k
α
2
dp dk,
χΩ −Δ εIR χΩ 2α
2
p εIR
−∞
−∞
B.1a
where
1
χΩ p 2π
∞
−∞
χΩ xeipx d2 x .
B.1b
Since by Parserval’s theorem
∞
−∞
2
dkχΩ k − p d2 k ∞
−∞
χΩ x2 d2 x volΩ
B.1c
one has the validity of the result written in 2.1b.
References
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Academic Press, New York, NY, USA, 1979.
2 B. Simon, The P(φ)2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, NJ, USA,
1974.
3 J. Glimm and A. Jaffe, Quantum Physics, Springer, New York, NY, USA, 2nd edition, 1987.
4 Y. Yamasaki, Measures on Infinite-Dimensional Spaces, vol. 5 of Series in Pure Mathematics, World
Scientific Publishing, Singapore, 1985.
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Advances in Mathematical Physics
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