Wavelet based regularization for Euclidean field theory and stochastic quantization

Abstract

It is shown that Euclidean field theory with polynomial interaction, can be regularized using the wavelet representation of the fields. The connections between wavelet based regularization and stochastic quantization are considered with $\phi^3$ field theory taken as an example.

Full text

arXiv:hep-th/0305167v2 30 Oct 2003
Wavelet based regularization for Euclidean
field theory
∗
M V Alta isky
Joint Institute for Nuclear Research, Dubna, 141980, Russia;
and Space Research Institute, Moscow, 117997, Russia;
e-mail: altaisky@mx.iki.rssi.ru
Sep 20, 2002
Abstract
It is shown that Euclidean field theory with polynomial interac-
tion, can be regularized using the wavelet representation of the fields.
The conn ections between wavelet based regularization and stochastic
quantization are considered.
1 Introduction
The connections between quantum field theory and stochastic differential
equations have been calling constant attention for quite a long time [1, 2].
We know that stochastic processes of t en posses self-similarity. The renor-
malization procedure used in quantum field theory is also based on the self-
similarity. So, it is natural to use for the regularization of field theories
the wavelet transform (WT), the decomposition with respect to the repre-
sentation of the affine group. In this paper two ways of regularization are
presented. First, the direct substitution of WT of t he fields into the action
functional leads to a field theory with scale-dependent coupling constants.
∗
Proc. GROUP 24: Physical and Mathematical Aspects of Symmetries, edited by J-P.
Gazeau, R. Kerner, J-P. Antoine, S. Metens, J-Y. Thibon; IOP Publishing, Bristol, 2003
1

Second, the WT, being substituted into the Pa risi-Wu stochastic quantiza-
tion scheme [3], provides a stochastic regularization with no extra vertexes
introduced into the theory.
2 Scalar field theory on affin e group
The Euclidea n field t heory is defined o n R
d
by the generating functional
W
E
[J] = N
Z
Dφ exp

−S[φ(x)] +
Z
d
d
xJ(x)φ(x)

, (1)
where S[φ] is the Euclidean action. In the simplest case of a scala r field with
the fourth power interaction
S[φ] =
Z
d
d
x
1
2
(∂
µ
φ)
2
+
m
2
2
φ
2
+
λ
4!
φ
4
. (2)
The φ
4
theory is of t en referred to as the Ginsburg-Landau model for its
ferromagnetic applications. The φ
3
theory is also a useful model.
The perturbation expansion generated by the functional (1) is usually
evaluated in k-space. The reformulation of the theory from the coordinate
(x) to momentum (k) representation is a particula r case of decomposition of
a function with respect to the representation of a Lie group G. G : x
′
=x+b
for the case of Fourier transform, but o t her groups may be used as well. For
a locally compact Lie gr oup G acting transitively on the Hilbert space H it
is possible to decompose state vectors with respect to the representations of
G [8, 9]
|φi = C
−1
ψ
Z
G
U(g)|ψidµ(g)hψ|U(g)|φi. (3)
The constant C
ψ
is determined by the norm of the action of U(g) on the
fiducial vector ψ ∈ H, C
ψ
= k ψk
−2
R
g∈G
|hψ|U(g)| ψi|
2
dµ(g); dµ(g) is the
left-invariant measure on G.
Using decomposition (3), it is possible to define a field theory on a non-
abelian Lie group. Let us consider the fourth power interaction model
Z
V (x
1
, x
2
, x
3
, x
4
)φ(x
1
)φ(x
2
)φ(x
3
)φ(x
4
)dx
1
dx
2
dx
3
dx
4
.
2

Using the notation U(g)| ψi ≡ |g, ψi, hφ|g, ψi ≡ φ(g), hg
1
, ψ|D|g
2
, ψi ≡
D(g
1
, g
2
), we can rewrite the generating functional (1) in the form
W
G
[J] =
Z
Dφ(g) exp

−
1
2
Z
G
φ(g
1
)D(g
1
, g
2
)φ(g
2
)dµ(g
1
)dµ(g
2
)
−
λ
4!
Z
G
V (g
1
, g
2
, g
3
, g
4
)φ(g
1
)φ(g
2
)φ(g
3
)φ(g
4
)dµ(g
1
)dµ(g
2
)dµ(g
3
)dµ(g
4
)
+
Z
G
J(g)φ(g)dµ(g)

, (4)
where V (g
1
, g
2
, g
3
, g
4
) is the result of the transform φ(g) :=
R
U(g)ψ(x)φ(x)dx
applied to V (x
1
, x
2
, x
3
, x
4
) in all arguments x
1
, x
2
, x
3
, x
4
.
Let us turn to the particular case of the affine group.
x
′
= ax + b, U(g)ψ(x) = a
−d/2
ψ((x − b)/a)), x, x
′
, b ∈ R
d
. (5)
The scalar field φ(x) in the action S[φ] can be written in the form of wavelet
decomposition
φ(x) = C
−1
ψ
R
1
a
d/2
ψ

x−b
a

φ
a
(b)
dadb
a
d+1
,
φ
a
(b) =
R
1
a
d/2
¯
ψ

x−b
a

φ(x)d
d
x.
(6)
In the scale-momentum (a, k) representation the matrix element of the free
field inverse pro pagator ha
1
, b
1
; ψ|D|a
2
, b
2
; ψi =
R
d
d
k
(2π)
d
e
ik(b
1
−b
2
)
D(a
1
, a
2
, k)
has the form
D(a
1
, a
2
, k) = a
d/2
1
ˆ
ψ(a
1
k)(k
2
+ m
2
)a
d/2
2
ˆ
ψ(a
2
k). (7)
The field theory (4) with the propagator D
−1
(a
1
, a
2
, k) gives standard Feyn-
man diagram technique, but with extra wavelet factor a
d/2
ˆ
ψ(ak) on each line
and the integrations over the measure dµ(a, k) =
d
d
k
(2π)
d
da
a
d+1
instead of
d
d
k
(2π)
d
.
Recalling the power law dependence of the coupling constants on the
cutoff momentum resulting from the Wilson expansion, we can define a scalar
field model on the affine group, with the coupling constant dep endent on
scale. Say, the fourth power interaction can be written as
V [φ] =
Z
λ(a)
4!
φ
4
a
(b)dµ(a, b), λ(a) ∼ a
ν
. (8)
3

The one-loop order contribution to the Green function G
2
in the theory
with interaction (8) can be evaluated [11] by integ ration over z = ak:
Z
a
ν
a
d
|
ˆ
ψ(ak)|
2
k
2
+ m
2
d
d
k
(2π)
d
da
a
d+1
=
Z
d
d
k
(2π)
d
C
(ν)
ψ
k
−ν
k
2
+ m
2
, C
(ν)
ψ
=
Z
|
ˆ
ψ(z)|
2
dz
z
1−ν
. (9 )
Therefore, there are no UV divergences for ν > d − 2. This is a kind of
asymptotically free theory which is hardly appropriate, say, to spin systems.
What is required to get a finite theory is an interaction vanishing outside a
given domain of scales. Such model is presented in the next section by means
of the stochastic quantization framework.
3 Stochastic quantization with wave lets
Let us remind the basic ideas of the stochastic quantization [3, 14, 12, 13]. Let
S[φ] be an action of the field φ(x). Instead of calculation of the physical Green
functions, it is possible to introduce the “extra-time” variable τ: φ(x) →
φ(x, τ) and evaluate the moments hφ(x
1
, τ
1
) . . . φ(x
m
, τ
m
)i
η
by averaging over
the random process φ(x, τ, ·) governed by the Langevin equation with the
Gaussian random force
˙
φ(x, τ)+
σ
2
2
δS
δφ(x, τ)
= η(x, τ), hη(x, τ)η(x
′
, τ
′
)i = σ
2
δ(x − x
′
)δ(τ − τ
′
). (1 0)
The physical G reen functions are then obtained by taking the limit τ
1
=
. . . = τ
m
= T → ∞.
The stochastic quantizatio n procedure has been considered as a perspec-
tive candidate fo r the regularization of gauge theories, for it respects local
gauge symmetries in a natural way. However a δ-correlated Gaussian random
noise in the Langevin equation still yields sharp singularities in the pertur-
bation theory. For this reason a number of modifications based on the noise
regularization η(x, τ) →
R
dyR
xy
(∂
2
)η(y, τ) have been proposed [4, 7, 6].
In this paper, following [15], we start with the random processes defined
directly in wavelet space. The use of the wavelet coefficients instead of the
original stochastic processes provides an extra analytical flexibility of the
method: t here exist more than one set of random functions W (a, b, ·) the
images of which have coinciding correlation functions. It is easy to check
that the random process generated by wavelet coefficients with the correlation
4

function h
c
W (a
1
, k
1
)
c
W (a
2
, k
2
)i = C
ψ
(2π)
d
δ
d
(k
1
+k
2
)a
d+1
1
δ(a
1
−a
2
)D
0
has the
same correlation function as the white noise has [17].
As an example, let us consider the Kardar-Parisi-Zhang equation [16]:
˙
Z − ν∆Z =
λ
2
(∇Z)
2
+ η. (11)
Substitution of wavelet transform
Z(x) = C
−1
ψ
Z
exp(ı(kx − k
0
t))a
d
2
ˆ
ψ(ak)
ˆ
Z(a, k)
d
d+1
k
(2π)
d+1
da
a
d+1
into (11), with the random f orce of the form
hbη(a
1
, k
1
)bη(a
2
, k
2
)i = C
ψ
(2π)
d+1
δ
d+1
(k
1
+ k
2
)a
d+1
1
δ(a
1
− a
2
)D(a
2
, k
2
), (12)
leads to the integral equation
(−ıω + νk
2
)
ˆ
Z(a, k) = η(a, k) −
λ
2
a
d
2
ˆ
ψ(ak)C
−2
ψ
R
(a
1
a
2
)
d
2
ˆ
ψ(a
1
k
1
)
ˆ
ψ(a
2
(k − k
1
))
k
1
(k − k
1
)
ˆ
Z(a
1
, k
1
)
ˆ
Z(a
2
, k − k
1
)
d
d+1
k
1
(2π)
d+1
da
1
a
1
d+1
da
2
a
2
d+1
.
From this the one-loop contribution the stochastic Green function follows:
G(k) = G
0
(k) − λ
2
G
2
0
(k)
R
d
d+1
k
1
(2π)
d+1
∆(k
1
)
k
1
(k − k
1
)|G
0
(k
1
)|
2
kk
1
G
0
(k − k
1
) + O(λ
4
),
(13)
where G
−1
0
(k) = −ıω + νk
2
. The difference from the standard approach [16]
is in the appearance of the effective force correlator
∆(k) ≡ C
−1
ψ
Z
da
a
|
ˆ
ψ(ak)|
2
D(a, k), (14)
which has the meaning of the effective force averaged over all scales.
Let us consider a single-band forcing [17] D(a, k) = δ(a − a
0
)D(k) and
the “Mexican hat” wavelet
ˆ
ψ(k) = (2π)
d/2
(−ık)
2
exp(−k
2
/2). In the leading
order in small parameter x = |k|/|k
1
| ≪ 1, the contribution to the stochastic
Green function is:
G(k) = G
0
(k) + λ
2
G
2
0
(k)
S
d
(2π)
d
a
3
0
k
2
ν
2
d − 2
8d
Z
∞
0
D(q)e
−a
2
0
q
2
q
d+1
dq + O(λ
4
).
5

4 Langevin equation for the φ
3
theory with
scale-depend ent noise
Let us apply the same scale-dependent noise (12) to the Langevin equation
for φ
3
theory. The standard procedure of the sto chastic quantization then
comes from the Langevin equation [6]
˙
φ(x, τ) +

−∆φ + m
2
φ +
λ
2!
φ
2

= η(x, τ). (15)
Applying the wavelet transform to this equation, we get:
(−ıω + k
2
+ m
2
)
ˆ
φ(a, k) = ˆη(a, k) −
λ
2
a
d
2
ˆ
ψ(ak)C
−2
ψ
R
(a
1
a
2
)
d
2
ˆ
ψ(a
1
k
1
)
ˆ
ψ(a
2
(k − k
1
))
ˆ
φ(a
1
, k
1
)
ˆ
φ(a
2
, k − k
1
)
d
d+1
k
1
(2π)
d+1
da
1
a
1
d+1
da
2
a
2
d+1
.
(16)
Iterating the integral equation (16), we yield the correction to the stochastic
Green function
G(k) = G
0
(k) + λ
2
G
2
0
(k)
Z
d
d+1
q
(2π)
d+1
∆(q)|G
0
(q)|
2
G
0
(k − q) + O(λ
4
). (17)
The analytical expressions for the stochastic Green functions, can be
obtained in the ω → 0 limit. As an example, we take the D(a, q) =
δ(a − a
0
)D(q) for φ
3
theory and the Mexican hat wavelet. The one loop con-
tribution to the stochastic Green function G(k) = G
0
(k) + G
2
0
λ
2
I
2
3
+ O(λ
4
)
is
lim
ω→0
I
2
3
=
Z
d
d
q
(2π)
d
∆(q)
1
2(q
2
+ m
2
)
·
1
q
2
+ (k − q)
2
+ 2m
2
.
The same procedure can be applied in higher loops. As it can be seen,
for constant o r compactly supported D(q) all integrals are finite due to the
exponential factor coming fro m wavelet ψ.
Acknowlegement
The author is grateful to Profs. H.H¨uffel and V.B.Priezzhev for useful dis-
cussions.
6

References
[1] E. Nelson. Quantum fluctuations. Princeton University Press, New Jer-
sey, 1985.
[2] J. Glim and A. Jaffe. Quantum physics. Springer-Verlag, NY, 1981.
[3] G. Parisi and Y.-S. Wu. Scientica Sinica, 24:483,1981.
[4] J. Breit, S. Gupta, and A. Zaks. Nuclear Physics, B233:61–68, 1984.
[5] Z. Bern, M. Halpern, L. Sadun, and C. Taubes. Physics Letters,
B165:151–156, 1985.
[6] R. Iengo and S. Pugnetti. Nuclear Physics , B300 :1 28–142, 1988.
[7] Z. Bern, M. Halpern, L. Sadun, and C. Taubes. Nuclear Physic s , B284:1–
91, 1987.
[8] A. Carey. Bull. Austr. Math. Soc., 15:1–12, 1976.
[9] M. Duflo and C. Moore. J. Func. Anal., 21:209–243, 1976.
[10] C. Chui. An i ntroduction to wavelets. Academic press, inc, 1991.
[11] M. Altaisky. In B.G. Sidharth and M.V. Altaisky, editors, Frontiers of
fundamental physics 4, pages 124–128, NY, 2001. Kluver Academic /
Plenum Publishers.
[12] P. Damgaard and H. H¨uffel. Physics Reports. 152:227-248, 1987.
[13] M. Namiki. Stochastic quan tiza tion . Springer, Heidelberg, 1992.
[14] J. Zinn-Justin. Nuclear Physics B, 275(FS17):135–159, 1986.
[15] M.V.Altaisky. European Journal of Physics B, 8(4 ) :6 13-617, 1999.
[16] M. Kardar, G. Parisi, and Y.-C. Zhang. Phys. Rev. Lett., 56(9):889–892,
1986.
[17] M. Altaisky. Doklady Physics, 48(9):478-480, 2003.
7